Loop Quantum Gravity vs. M-Theory.
Mark William Hopkins
them into a unified theory?
Demian Cho
What is M theory? :-)
Look at some recent papers by Lee Smolin.
Demian Cho
"Mark William Hopkins" <whop...@alpha2.csd.uwm.edu> wrote in message
news:9ehndj$jb2$1...@uwm.edu...
John Baez
Mark William Hopkins <whop...@alpha2.csd.uwm.edu> wrote:
>Is there a way to reconcile M-theory with loop quantum gravity, combining
>them into a unified theory?
If we knew, we would not be wasting time reading usenet newsgroups.
However, we're working on it! Try recent papers by Lee Smolin and
Yi Ling available here:
http://xxx.lanl.gov/form/hep-th
such as these:
hep-th/0104050
Title: The exceptional Jordan algebra and the matrix string
Authors: Lee Smolin
Comments: LaTex 15 pages, no figures
Subj-class: High Energy Physics - Theory; Quantum Algebra
hep-th/0009018
Title: Holographic Formulation of Quantum Supergravity
Authors: Yi Ling, Lee Smolin
Comments: 30 pages, no figure
Journal-ref: Phys.Rev. D63 (2001) 064010
hep-th/0006137
Title: The cubic matrix model and a duality between strings and loops
Authors: Lee Smolin
Comments: Latex, 32 pages, 7 figures
Subj-class: High Energy Physics - Theory; Quantum Algebra
hep-th/0003285
Title: Eleven dimensional supergravity as a constrained
topological field theory
Authors: Yi Ling, Lee Smolin
Comments: 15 pages+7, Appendix added
Journal-ref: Nucl.Phys. B601 (2001) 191-208
hep-th/9904016
Title: Supersymmetric Spin Networks and Quantum Supergravity
Authors: Yi Ling, Lee Smolin
Comments: 21 pages, LaTex, 22 figures, typos corrected and
references completed
Journal-ref: Phys.Rev. D61 (2000) 044008
hep-th/9903166
Title: A candidate for a background independent formulation of M theory
Authors: Lee Smolin
Comments: Latex 46 pages, 21 figures, new results included which
lead to a modification of the statement of the basic conjecture.
Presentation improved.
Journal-ref: Phys.Rev. D62 (2000) 086001
Also see my not-yet-written papers, which will make use of the math
described here:
http://math.ucr.edu/home/baez/oct.ps
Lubos Motl
> them into a unified theory?
Some people have enumerated few papers of Lee Smolin and I want to point
out that on June 5th, Lee Smolin's book
"Three Roads to Quantum Gravity"
will appear. Search for "Lee Smolin" at amazon.com.
As you see the title "Three Roads to Quantum Gravity", you might ask "Who
is the third?" Well, you will find the answer in the book. :-) Maybe it's
some Roger Penrose's picture of twistors, pretending few mathematical
observations on commuting projective spinors to be a physical theory? :-)
But I want to suggest my answer, too.
Actually, my answer to the question "can they be reconciled" would be
"most likely not". There are many reasons. Loop quantum gravity works in 4
dimensions and requires a lot of changes to be put in higher dimensions
(10 or 11 is certainly not better). String theory, on the contrary,
predicts that we live in 10 or 11 dimensions.
Loop Quantum Gravity predicts quantized areas etc., string theory quite
certainly predicts exact moduli spaces where the shapes of dimensions etc.
can be changed continuously.
String/M-theory predicts unification of all forces (and
electromagnetism-like U(1) can appear via Kaluza-Klein mechanism etc. and
therefore various interactions are interconnected a lot) while Loop
Quantum Gravity is supposed to be a theory of gravitation only and assumes
that forces can be separated from each other.
There is however an easier answer to the question. String/M-theory cannot
be reconciled with Loop Quantum Gravity simply because String/M-theory is
correct (although not understood completely) while Loop Quantum Gravity is
probably wrong. ;-) The moderators can include an educational comment, but
I believe that they will not delete this message because it contains the
previous sentence which expresses the opinion of most high-energy
theorists as well as mine.
Loop Quantum Gravity is based on the assumption that the holy Einstein's
questions are more beatiful than anything else and therefore they must
hold even and Planckian and transPlanckian distances (where we have no
experimental data at all) without any corrections whatsoever - and that
the apparent nonrenormalizability of the Einstein's equations is just a
perturbative illusion. As a result, the Loop Quantum Gravity physicists
derived an amusing theory with SU(2) spin networks, quantized areas and
similar interesting stuff predicted at ultrashort distances - but they
cannot predict whether the theory really resembles usual general
relativity at long distances (which was in fact the only physical
motivation).
In fact, they need to say that the usual physics is not reproduced at long
distances: if one computes the entropy of a black hole (which should be
A/4G, a quarter of the area of horizons over Newton's constant - setting
c=hbar=1) from Loop Quantum Gravity, one gets a wrong result by a
universal factor of ln(2)/sqrt(3). So they need to claim that the Newton's
constant is redefined between low energies and high energies by this
factor. But unfortunately renormalization group is not allowed in LCQ
(also because they want to eliminate any divergences etc. from the
structure - and because several workers in the field do not understand RG)
so it is hard to justify such a change. I think that the only fair answer
is that LCG predicts a wrong entropy of the black hole.
I would be happy if LCG worked and could be unified with M-theory - just
like Supergravity community was included into String community after
the discovery of M-theory etc. I am just afraid that in this case the
situation is not so optimistic - even the founder of the new variables
(Ashtekar) has more or less abandoned the project...
Best wishes
Lubos
______________________________________________________________________________
E-mail: lu...@matfyz.cz Web: http://www.matfyz.cz/lumo tel.+1-805/893-5025
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Superstring/M-theory is the language in which God wrote the world.
Steve Carlip
> Loop Quantum Gravity predicts quantized areas etc., string
> theory quite certainly predicts exact moduli spaces where
> the shapes of dimensions etc. can be changed continuously.
How certain is this? It's true, of course, in perturbative string
theory. But we know that the perturbation series doesn't
converge, and that perturbative string theory gives only an
asymptotic series. In the nonperturbative approaches I know
(e.g., the AdS/CFT correspondence), it's not clear, at least
to me, how to even define an area operator.
In loop quantum gravity, the quantization of area is a non-
perturbative effect. If a corresponding result exists in string
theory, I would expect it to be nonperturbative as well. But
without something like an area operator (or a ``shapes of
dimensions'' operator, for that matter), how can you tell?
Is it clear that areas and shapes of dimensions even exist
as fundamental quantities in string theory, and are not
just effective descriptions in some ranges of energy and
coupling?
> the Loop Quantum Gravity physicists derived an amusing
> theory with SU(2) spin networks, quantized areas and similar
> interesting stuff predicted at ultrashort distances - but they
> cannot predict whether the theory really resembles usual
> general relativity at long distances
That's true. To be fair, though, string theory has the opposite
problem---while it's easy enough to get something like the
usual general relativity at long distances (modulo questions
of the dilaton), it's not yet clear what nonperturbative short
distance theory these results come from. To be sure, there is
something known about nonperturbativbe string theory. But
from the point of view of someone working in quantum gravity,
the string theorists are just barely beginning to get to the hard
problems: the required nonlocality of diffeomorphism-invariant
observables, the ``problem of time,'' the question of what can
replace a smooth classical spacetime at short distances, etc.
This isn't really a criticism of string theory, of course. A lot
of people have asked these questions from a lot of different
perspectives, and no one seems especially close to an answer.
String theory has some very nice features, like (probably)
holography, that might be important. But it seems premature
to criticize loop quantum gravity, which is after all still a rather
young field, if you're not applying the same standards to its
alternatives.
Steve Carlip
John Baez
Lubos Motl <mo...@physics.rutgers.edu> wrote:
>Loop Quantum Gravity is based on the assumption that the holy Einstein's
>questions are more beautiful than anything else and therefore they must
>hold even and Planckian and transPlanckian distances (where we have no
>experimental data at all) without any corrections whatsoever - and that
>the apparent nonrenormalizability of the Einstein's equations is just a
>perturbative illusion.
No, it's not based on this assumption. The goal of loop quantum gravity
is to try to quantize Einstein's equations of general relativity without
using a background metric, and SEE WHAT HAPPENS. If it works, this will
be very interesting. If it does not work, this too will be very interesting.
One does not need to believe that Einstein's equations are "holy" to consider
this to be a worthwhile project. They are interesting and important
equations, so we should find out whether or not they can be quantized in
a sensible way. Perturbative quantization was not sensible, at least
not for understanding the behavior at short length scales.
Another correction: Most people I know who work on loop quantum
gravity expect a running coupling constant, just as one gets from
perturbation theory or numerical simulations using the Regge calculus
or dynamical triangulations. If loop quantum gravity works, there will
be a *finite* renormalization of the Newton constant, because there will
be a shortest length scale roughly around the Planck scale, so that the
renormalization group flow carries the observed Newton constant to a
*finite* bare value.
>As a result, the Loop Quantum Gravity physicists
>derived an amusing theory with SU(2) spin networks, quantized areas and
>similar interesting stuff predicted at ultrashort distances - but they
>cannot predict whether the theory really resembles usual general
>relativity at long distances (which was in fact the only physical
>motivation).
We cannot do this yet, but we are working on it. Starting with
a background-free theory, one must work hard to get a perturbative
theory that reduces to Newtonian gravity in a suitable limit. The
problem is converse to that in string theory, which was initially
formulated perturbatively: there folks are having to work very hard
to get a background-free formulation.
Every approach to quantum gravity has its Achilles heel. This is the
Achilles heel of loop quantum gravity. String theory has its own
Achilles heels. I will not list them here, since 1) we've already
discussed them many times here on sci.physics.research and 2) I'm not
interested in yet another "battle of the theories". I'd be glad to
list them if you like! But for now, I'll just say this: both string
theory and loop quantum gravity require a certain optimism that one
will eventually solve problems that have plagued the theory from the
very beginning.
>In fact, they need to say that the usual physics is not reproduced at long
>distances: if one computes the entropy of a black hole (which should be
>A/4G, a quarter of the area of horizons over Newton's constant - setting
>c=hbar=1) from Loop Quantum Gravity, one gets a wrong result by a
>universal factor of ln(2)/sqrt(3). So they need to claim that the Newton's
>constant is redefined between low energies and high energies by this
>factor.
As one of the authors of the paper you're referring to, I'm aware of
many problems with it... but they are not the problems you mention!
Loop quantum gravity contains a parameter called the Immirzi parameter.
This is an unavoidable artifact of the quantization procedure, and
there is presently no known way to determine it by calculations done
purely within loop quantum gravity.
The Immirzi paramter shows up in the formula for black hole entropy.
By an appropriate choice of this parameter one gets the right formula
for the black hole entropy. One might not like this - it certain
suggests that we don't understand some things here! - but that's how
it goes.
>But unfortunately renormalization group is not allowed in LCQ
>(also because they want to eliminate any divergences etc. from the
>structure - and because several workers in the field do not understand RG)
>so it is hard to justify such a change.
I don't know anybody working in loop quantum gravity who says that
the renormalization group is "not allowed". We don't use it much
yet, but only because nobody understands the renormalization group
very well in the background-free context. (You *do* see it showing
up in numerical calculations based on theories closely related to
loop quantum gravity, like the Regge calculus and dynamical triangulations.)
>I think that the only fair answer
>is that LCG predicts a wrong entropy of the black hole.
No: we don't get a "wrong" result. We see that the area is
asymptotically proportional to entropy; the constant of
proportionality depends on the Immirzi parameter, and we
can use this to determine the Immirzi parameter as a function
of the bare Newton constant.
I will be glad to remind you of the deficiencies of the string
theory calculation of black hole entropy, too, if you like....
>I would be happy if LCG worked and could be unified with M-theory - just
>like Supergravity community was included into String community after
>the discovery of M-theory etc. I am just afraid that in this case the
>situation is not so optimistic - even the founder of the new variables
>(Ashtekar) has more or less abandoned the project...
Huh? Where did you hear that? Ashtekar is *very* active in loop quantum
gravity. I just visited him a while ago at the CGPG, and he is very
enthusiastic about Bojowald's new work on loop quantum cosmology. He
and I will both be giving talks on the subject in Stony Brook the week
after next at a conference in honor of Dennis Sullivan's 60th birthday:
http://www.math.sunysb.edu/events/dennisfest/index.html
Come on over and see! I'll give a 4-part introduction to the math of
loop quantum gravity. Ashteker's talk is titled "Quantum Geometry in
Action: Black Holes and Big Bang" - about using loop quantum gravity
to study black hole and cosmology.
Gordon D. Pusch
> Another correction: Most people I know who work on loop quantum
> gravity expect a running coupling constant, just as one gets from
> perturbation theory or numerical simulations using the Regge calculus
> or dynamical triangulations. If loop quantum gravity works, there
> will be a *finite* renormalization of the Newton constant, because
> there will be a shortest length scale roughly around the Planck scale,
> so that the renormalization group flow carries the observed Newton
> constant to a *finite* bare value.
John, would you willing to expand on the above a bit more ???
In particular, why should one expect a _finite_ renormalization
of 'G', and a _finite_ ``bare'' value of 'G', as opposed to zero
or infinite values? Why can't the renormalization constant
run to zero or infinity over a finite range of scales?
Also, if there is a ``shortest'' length-scale, would that
also mean the _other_ ``bare'' constants that ``run''
(e.g., the charge on the electron) might be finite,
by the same logic ???
-- Gordon D. Pusch
perl -e '$_ = "gdpusch\@NO.xnet.SPAM.com\n"; s/NO\.//; s/SPAM\.//; print;'
Aaron Bergman
>
>Another correction: Most people I know who work on loop quantum
>gravity expect a running coupling constant, just as one gets from
>perturbation theory or numerical simulations using the Regge calculus
>or dynamical triangulations. If loop quantum gravity works, there will
>be a *finite* renormalization of the Newton constant, because there will
>be a shortest length scale roughly around the Planck scale, so that the
>renormalization group flow carries the observed Newton constant to a
>*finite* bare value.
This brings up an interesting point, though. Can you accomodate
higher R terms in lqg? Might you expect them to be generated
dynamically?
Aaron
--
Aaron Bergman
<http://www.princeton.edu/~abergman/>
John Baez
Gordon D. Pusch <gdp...@NO.xnet.SPAM.com> wrote:
>ba...@galaxy.ucr.edu (John Baez) writes:
>> Another correction: Most people I know who work on loop quantum
>> gravity expect a running coupling constant, just as one gets from
>> perturbation theory or numerical simulations using the Regge calculus
>> or dynamical triangulations. If loop quantum gravity works, there
>> will be a *finite* renormalization of the Newton constant, because
>> there will be a shortest length scale roughly around the Planck scale,
>> so that the renormalization group flow carries the observed Newton
>> constant to a *finite* bare value.
>John, would you willing to expand on the above a bit more???
If you insist. Note the crucial clause in the above paragraph:
"IF LOOP QUANTUM GRAVITY WORKS". That means I'm talking about
guesses and hopes. Please bear that in mind... and don't sue
me if I'm wrong.
>In particular, why should one expect a _finite_ renormalization
>of 'G', and a _finite_ ``bare'' value of 'G', as opposed to zero
>or infinite values?
The first answer is: otherwise we're screwed, and loop quantum
gravity probably won't work!
The second answer is... well, it's related to this other question
of yours:
>Why can't the renormaliz[ed] constant
>run to zero or infinity over a finite range of scales?
This can happen, and it's called a Landau pole when it goes
to infinity. QED seems to have a Landau pole, for example.
However, a simple one-loop calculation suggests that this pole
occurs at a ridiculously short distance scale - MUCH smaller
than the Planck length.
In fact, this is probably the smallest distance that any
sane physicist has ever written down with a straight face:
it's about exp(-3 pi/ alpha 2) times the Compton wavelength
of the electron, where alpha is the fine structure constant,
i.e. about 1/137. That's a ridiculously small distance!
But what about quantum gravity? This is a nonrenormalizable
theory, so it's tricky to get detailed information about the
running of the coupling constant. However, people have done
clever hand-wavy calculations and also numerical simulations,
and the consensus is that:
1) the coupling constant increases at short distances, but
2) there is no Landau pole, and in fact
3) the coupling constant may even approach a fixed value at
very short distances - a so-called "ultraviolet fixed point".
I should warn you, conclusion 3) is less solid than the other
two. For details, see below.
>Also, if there is a ``shortest'' length-scale, would that
>also mean the _other_ ``bare'' constants that ``run''
>(e.g., the charge on the electron) might be finite,
>by the same logic ???
Yes indeed: if we had quantum gravity coupled to QED, and
our theory of quantum gravity imposed a shortest distance
scale larger than that ridiculously small distance I mentioned
above, it's possible that quantum gravity would save QED from
its Landau pole!
It's ironic: while quantum gravity has a reputation of
being the nastiest of quantum field theories, thanks to
ultraviolet divergences, if it turned out to involve a shortest
distance scale it might be able to HELP us with the ultraviolet
divergences of all other quantum field theories.
Of course, most string theorists don't like this idea, because
they are looking for a more or less unique "theory of everything",
not "quantum gravity as a universal cure for quantum field theories
with nasty ultraviolet problems". Indeed, Witten once told me
that he hopes loop quantum gravity will fail, because otherwise
we will have a lot of trouble determining the correct "theory of
everything". While I understand this hope, it doesn't strike me
as a very convincing argument that loop quantum gravity WILL fail.
So for now, I will keep marching on trying to get it to work....
....................................................................
Also available at http://math.ucr.edu/home/baez/week139.html
September 19, 1999
This Week's Finds in Mathematical Physics (Week 139)
John Baez
[stuff deleted]
Suppose we have any old quantum field theory with a coupling constant
G in it. In fact, G will depend on the length scale at which we
measure it. But using Planck's constant and the speed of light we
can translate length into 1/momentum. This allows us to think of G
as a function of momentum. Roughly speaking, when you shoot particles
at each other at higher momenta, they come closer together before
bouncing off, so measuring a coupling constant at a higher momentum
amounts to measuring at a shorter distance scale.
The equation describing how G depends on the momentum p is called
the "Callan-Symanzik equation". In general it looks like this:
dG
------- = beta(G)
d(ln p)
But all the fun starts when we use our quantum field theory to calculate
the right hand side, which is called - surprise! - the "beta function"
of our theory. Typically we get something like this:
dG
------- = (n - d)G + aG^2 + bG^3 + ....
d(ln p)
Here n is the dimension of spacetime and d is a number called the
"upper critical dimension". You see, it's fun when possible to think
of our quantum field theory as defined in a spacetime of arbitrary
dimension, and then specialize to the case at hand. I'll show you
how work out d in a minute. It's harder to work out the numbers
a, b, and so on - for this, you need to do some computations using the
quantum field theory in question.
What does the Callan-Symanzik equation really mean? Well, for starters
let's neglect the higher-order terms and suppose that
dG(p)
------- = (n - d)G
d(ln p)
This says G is proportional to p^{n-d}. There are 3 cases:
A) When n < d, our coupling constant gets *smaller* at higher momentum
scales, and we say our theory is "superrenormalizable". Roughly, this
means that at larger and larger momentum scales, our theory looks more
and more like a "free field theory" - one where particles don't interact
at all. This makes superrenormalizable theories easy to study by
treating them as a free field theory plus a small perturbation.
B) When n > d, our coupling constant gets *larger* at higher momentum
scales, and we say our theory is "nonrenormalizable". Such theories
are hard to study using perturbative calculations in free field theory.
C) When n = d, we are right on the brink between the two cases above.
We say our theory is "renormalizable", but we really have to work out
the next term in the beta function to see if the coupling constant
grows or shrinks with increasing momentum.
Consider the example of general relativity. We can figure out
the upper critical dimension using a bit of dimensional analysis
and handwaving. Let's work in units where Planck's constant and the
speed of light are 1. The Lagrangian is the Ricci scalar curvature
divided by 8 pi G, where G is Newton's gravitational constant. We
need to get something dimensionless when we integrate the Lagrangian
over spacetime to get the action, since we exponentiate the action
when doing path integrals in quantum field theory. Curvature has
dimensions of 1/length^2, so when spacetime has dimension n, G must
have dimensions of length^{n-2}.
This means that if you are a tiny little person with a ruler X
times smaller than mine, Newton's constant will seem X^{n-2} times
bigger to you. But measuring Newton's constant at a length scale
that's X times smaller is the same as measuring it at a momentum scale
that's X times bigger. We already solved the Callan-Symanzik equation
and saw that when we measure G at the momentum scale p, we get an
answer proportional to p^{n-d}. We thus conclude that d = 2.
(If you're a physicist, you might enjoy finding the holes in the
above argument, and then plugging them.)
This means that quantum gravity is nonrenormalizable in 4 dimensions.
Apparently gravity just keeps looking stronger and stronger at
shorter and shorter distance scales. That's why quantum gravity has
traditionally been regarded as hard - verging on hopeless.
However, there is a subtlety. We've been ignoring the higher-order
terms in the beta function, and we really shouldn't!
This is obvious for renormalizable theories, since when n = d, the
beta function looks like
dG
------- = aG^2 + bG^3 + ....
d(ln p)
so if we ignore the higher-order terms, we are ignoring the whole
right-hand side! To see the effect of these higher-order terms let's
just consider the simple case where
dG
------- = aG^2
d(ln p)
If you solve this you get
c
G = -------------
1 - ac ln p
where c is a positive constant. What does this mean? Well, if a < 0,
it's obvious even before solving the equation that G slowly *decreases*
with increasing momentum. In this case we say our theory is
"asymptotically free". For example, this is true for the strong
force in the Standard Model, so in collisions at high momentum quarks
and gluons act a lot like free particles. (For more on this, try "week94".)
On the other hand, if a > 0, the coupling constant G *increases* with
increasing momentum. To make matters worse, it becomes INFINITE
at a sufficiently high momentum! In this case we say our theory has
a "Landau pole", and we cluck our tongues disapprovingly, because it's
not a good thing. For example, this is what happens in quantum
electrodynamics when we don't include the weak force. Of course,
one should really consider the effect of even higher-order terms in
the beta function before jumping to conclusions. However, particle
physicists generally feel that among renormalizable field theories,
the ones with a < 0 are good, and the ones with a > 0 are bad.
Okay, now for the really fun part. Perturbative quantum gravity
in 2 dimensions is not only renormalizable (because this is the
upper critical dimension), it's also asympotically free! Thus
in n dimensions, we have
dG
------- = (n - 2)G + aG^2 + ....
d(ln p)
where a < 0. If we ignore the higher-order terms which I have
written as "....", this implies something very interesting for
quantum gravity in 4 dimensions. Suppose that at low momenta
G is small. Then the right-hand side is dominated by the first
term, which is positive. This means that as we crank up the
momentum scale, G keeps getting bigger. This is what we already
saw about nonrenormalizable theories. But after a while, when G
gets big, the second term starts mattering more - and it's negative.
So the growth of G starts slowing!
In fact, it's easy to see that as we keep cranking up the momentum,
G will approach the value for which
dG
------- = 0
d(ln p)
We call this value an "ultraviolet stable fixed point" for the
gravitational constant. Mathematically, what we've got is a flow
in the space of coupling constants, and an ultraviolet stable fixed
point is one that attracts nearby points as we flow in the direction
of higher momenta. This particular kind of ultraviolet stable fixed
point - coming from an asymptotically free theory in dimensions above
its upper critical dimension - is called a "Wilson-Fisher fixed point".
So: perhaps quantum gravity is saved from an ever-growing Newton's
constant at small distance scales by a Wilson-Fisher fixed point!
But before we break out the champagne, note that we neglected the
higher-order terms in the beta function in our last bit of reasoning.
They can still screw things up. For example, if
dG
------- = (n - 2)G + aG^2 + bG^3
d(ln p)
and b is positive, there will not be a Wilson-Fisher fixed point
when the dimension n gets too far above 2. Is 4 too far above 2?
Nobody knows for sure. We can't really work out the beta function
exactly. So, as usual in quantum gravity, things are a bit iffy.
John Baez
Aaron Bergman <aber...@princeton.edu> wrote:
We're having enough fun (or trouble) already quantizing plain old gravity,
so there haven't been attempts to do loop quantum gravity starting
with a Lagrangian that includes higher R terms. However, I personally
expect them to be generated dynamically.
It's easiest for me to understand this in terms of spin foam models
like the Barrett-Crane model. The Barrett-Crane model gives a
formula for the "amplitude" of a 4-simplex with triangles labelled
by areas. In the limit of large areas, this should be asymptotic to
something like exp(iS) where S is the Regge action... a discretized
version of the Einstein-Hilbert action. There is pretty good evidence
for this - it's just a matter of doing the stationary phase approximation
of a certain integral.
For smaller 4-simplexes there will be noticeable deviations from the
Regge action, and I expect these to be interpretable as higher R terms.
After all, what else could they be? It would be nice to calculate
these deviations - it's just a matter of messing with integrals - but
it's not my cup of tea.
However, fundamentally, at the Planck scale, there is no Lorentzian
manifold, no Riemann curvature, and no Lagrangian in the Barrett-Crane
model: just a formula for a 4-simplex with triangles labelled by areas!
For more on the phrase "something like", see below! This stuff
is about the Riemannian Barrett-Crane model, but the Lorentzian
one should work similarly, now that Barrett and I have shown that
the integrals involved actually converge. It'll be technically
more difficult, but morally similar.
........................................................................
Also available at http://math.ucr.edu/home/baez/week128.html
January 4, 1999
This Week's Finds in Mathematical Physics (Week 128)
John Baez
[stuff deleted]
Barrett and Crane have a theory of quantum gravity, which I've also
worked on; I discussed it last in "week113" and "week120". Before I
describe it I should warn the experts that this theory deals with
Riemannian rather than Lorentzian quantum gravity (though Barrett and
Crane are working on a Lorentzian version, and I hear Friedel and
Krasnov are also working on this). Also, it only deals with vacuum
quantum gravity - empty spacetime, no matter.
In this theory, spacetime is chopped up into 4-simplices. A 4-simplex
is the 4-dimensional analog of a tetrahedron. To understand what I'm
going to say next, you really need to understand 4-simplices, so let's
start with them.
It's easy to draw a 4-simplex. Just draw 5 dots in a kind of circle and
connect them all to each other! You get a pentagon with a pentagram
inscribed in it. This is a perspective picture of a 4-simplex
projected down onto your 2-dimensional paper. If you stare at this
picture you will see the 4-simplex has 5 tetrahedra, 10 triangles,
10 edges and 5 vertices in it.
The shape of a 4-simplex is determined by 10 numbers. You can take
these numbers to be the lengths of its edges, but if you want to be
sneaky you can also use the areas of its triangles. Of course, there
are some constraints on what areas you can choose for there to *exist* a
4-simplex having triangles with those areas. Also, there are some
choices of areas that fail to make the shape *unique*: for one of these
bad choices, the 4-simplex can flop around while keeping the areas of
all its triangles fixed. But generically, this non-uniqueness doesn't
happen.
In Barrett and Crane's theory, we chop spacetime into 4-simplices and
describe the geometry of spacetime by specifying the area of each
triangle. But the geometry is "quantized", meaning that the area
takes a discrete spectrum of possible values, given by
sqrt(j(j+1))
where the "spin" j is a number of the form 0, 1/2, 1, 3/2, etc. This
formula will be familiar to you if you've studied the quantum mechanics
of angular momentum. And that's no coincidence! The cool thing about
this theory of quantum gravity is that you can discover it just by
thinking a long time about general relativity and the quantum mechanics
of angular momentum, as long as you also make the assumption that
spacetime is chopped into 4-simplices.
So: in Barrett and Crane's theory the geometry of spacetime is described
by chopping spacetime into 4-simplices and labelling each triangle with
a spin. Let's call such a labelling a "quantum 4-geometry". Similarly,
the geometry of space is described by chopping space up into tetrahedra
and labelling each triangle with a spin. Let's call this a "quantum
3-geometry".
The meat of the theory is a formula for computing a complex number
called an "amplitude" for any quantum 4-geometry. This number plays the
usual role that amplitudes do in quantum theory. In quantum theory, if
you want to compute the probability that the world starts in some state
psi and ends up in some state psi', you just look at all the ways the
world can get from psi to psi', compute an amplitude for each way, add
them all up, and take the square of the absolute value of the result.
In the special case of quantum gravity, the states are quantum 3-geometries,
and the ways to get from one state to another are quantum 4-geometries.
So, what's the formula for the amplitude of a quantum 4-geometry? It
takes a bit of work to explain this, so I'll just vaguely sketch how it
goes. First we compute amplitudes for each 4-simplex and multiply all
these together. Then we compute amplitudes for each triangle and
multiply all these together. Then we multiply these two numbers.
(This is analogous to how we compute amplitudes for Feynman diagrams
in ordinary quantum field theory. A Feynman diagram is a graph whose
edges have certain labellings. To compute its amplitude, first we
compute amplitudes for each edge and multiply them all together. Then
we compute amplitudes for each vertex and multiply them all together.
Then we multiply these two numbers. One goal of work on "spin
foam models" is to more deeply understand this analogy with Feynman
diagrams.)
Anyway, to convince oneself that this formula is "good", one would like
to relate it to other approaches to quantum gravity that also involve
4-simplices. For example, there is the Regge calculus, which is a
discretized version of *classical* general relativity. In this approach
you chop spacetime into 4-simplices and describe the shape of each
4-simplex by specifying the lengths of its edges. Regge invented a
formula for the "action" of such a geometry which approaches the usual
action for classical general relativity in the continuum limit. I
explained the formula for this "Regge action" in "week120".
Now if everything were working perfectly, the amplitude for a 4-simplex
in the Barrett-Crane model would be close to exp(iS), where S is the
Regge action of that 4-simplex. This would mean that the Barrett-Crane
model was really a lot like a path integral in quantum gravity. Of
course, in the Barrett-Crane model all we know is the areas of the triangles
in each 4-simplex, while in the Regge calculus we know the lengths of
its edges. But we can translate between the two, at least generically,
so this is no big deal.
Recently, Barrett came up with a nice argument saying that in the limit
where the triangles have large areas, the amplitude for a 4-simplex in
the Barrett-Crane theory is proportional, not to exp(iS), but to cos(S):
5) John W. Barrett, The asymptotics of an amplitude for the 4-simplex,
preprint available as gr-qc/9809032.
This argument is not rigorous - it uses a stationary phase approximation
that requires further justification. But Regge and Ponzano used a
similar argument to show the same sort of thing for quantum gravity in 3
dimensions, and their argument was recently made rigorous by Justin
Roberts, with a lot of help from Barrett:
6) Justin Roberts, Classical 6j-symbols and the tetrahedron, preprint
available as math-ph/9812013.
So one expects that with work, one can make Barrett's argument rigorous.
But what does it mean? Why does he get cos(S) instead of exp(iS)?
Well, as I said, the same thing happens one dimension down in the
so-called Ponzano-Regge model of 3-dimensional Riemannian quantum
gravity, and people have been scratching their heads for decades trying
to figure out why. And by now they know the answer, and the same
answer applies to the Barrett-Crane model.
The problem is that if you describe 4-simplex using the areas of its
triangles, you don't *completely* know its shape. (See, I lied to you
before - that's why you gotta read the whole thing.) You only know it
*up to reflection*. You can't tell the difference between a 4-simplex
and its mirror-image twin using only the areas of its triangles! When
one of these has Regge action S, the other has action -S. The Barrett-
Crane model, not knowing any better, simply averages over both of them,
getting
(1/2)(exp(iS) + exp(-iS)) = cos(S)
So it's not really all that bad; it's doing the best it can under
the circumstances. Whether this is good enough remains to be seen.
(Actually I didn't really *lie* to you before; I just didn't tell you
my definition of "shape", so you couldn't tell whether mirror-image
4-simplices should count as having the same shape. Expository prose
darts between the Scylla of overwhelming detail and the Charybdis of
vagueness.)
Lubos Motl
Quantum Gravity so that we enjoy plurality here. :-) The purpose of the
text below (as well as the previous mail) is not to ban Loop Quantum
Gravity :-) but rather explain why I personally (and probably other
people, too) do not currently find LQG neither as the correct direction of
research in QG, nor as a possible dual description of M-theory, despite
the high standards of intelligence and originality of many people involved
in LQG. Of course, I might be wrong - but in this case there must exist
some rational arguments showing why my statements are incorrect (like in
every case when people were wrong in the past) - and not just statements
like "LQG is still pretty young, you will see one day!". ;-)
I appologize that in my previous mail abbreviations such as LCG, LCQ were
used. These stand for Light Cone Gauge and Light Cone Quantization :-) and
are very similar to LQG, therefore the confusion. :-) Sorry.
On Wed, 6 Jun 2001, Steve Carlip wrote:
>> LM: Areas are not always quantized in string theory but they are in LQG.
> SC: How certain is this? It's true, of course, in perturbative string
> theory. But we know that the perturbation series doesn't ...
LM: I am convinved that virtually everyone in the field would agree that
the existence of exact moduli spaces which allow certain geometric
parameters (areas of 2-cycles etc.) to change continuously has nothing to
do with perturbative expansions. It is rather a direct consequence of
supersymmetry. BTW while the perturbative expansion of string theory is
roughly as divergent as in field theory (the most exact result is gotten
by summing the power series up to the minimal term c_k.g^k where k is of
order 1/g - then the error of this calculation is of the same order as
first nonperturbative corrections exp(-c/g) related to D-branes), we know
in many cases that there exists a finite answer for any value of the
coupling constant.
Supersymmetry is an important feature of theories connected with string
theory. It is seen perturbatively but holds also exactly and most of the
successes of string theory in the 90s are connected with the fact that
people finally understood a lot about the behavior of string theories at
strong coupling - i.e. completely nonperturbative features of string
theory.
We can find BPS objects (that preserve some supersymmetries) which are
completely stable and whose properties such as their mass can be
calculated precisely - not only perturbatively - from the supersymmetry
algebra. Their properties precisely agreed with dualities, i.e. with
conjectures that various theories are related (and their BPS objects -
strings, branes, KK monopoles, KK modes etc. are exchanged by dualities).
Today we can calculate much more than the statements apparent from the
SUSY algebra.
If we have at least 8 supercharges i.e. N=2 in 4 dimensions, there
generically exist exact moduli spaces (the scalar fields must remain
precisely massless because massive particles transform in bigger
representations of SUSY). This can be proved from supersymmetry, too, and
has nothing to do with perturbative expansions. The most beautiful
(mathematical) objects in M-theory involve a lot of supersymmetry (the
real world corresponds to a sector of M-theory that has at most 4 broken
supercharges) and these high-SUSY theories (in the language of M-theory,
high-SUSY vacua) are reasonably well understood. The less SUSY we have,
the less control we have over the system.
One possible loophole [of the "quantization" argument why M-theory and LQG
are different] is the (co)dimension of the areas. I claim that areas of
some (supersymmetric, i.e. very important) 2-cycles, 3-cycles etc. (up to
7-cycles at least) in your geometry can change smoothly in vacua with at
least 8 supercharges. But it is much harder to say something about
8-cycles or 9-cycles (codimension 1 objects or so) because one would have
to deal with stringy cosmology which is poorly understood. But my feeling
was that Loop Quantum Gravity predicts quantization of areas of arbitrary
dimensions (volumes, lengths etc.) and this certainly does not hold in
M-theory.
Of course, I agree that today people cannot define operators such as "the
area of the star" in AdS-CFT (and also in Matrix theory such geometric
notions sound very obscure). But I want to emphasize that string
theorists understand a lot of nonperturbative physics of M-theory even
without explicit and complete formulations of the theory around some
backgrounds (such as AdS-CFT or Matrix theory). Most of these insights
rely on supersymmetry (holomorphy of various superpotentials etc. together
with the knowledge of poles and zeroes determines them often uniquely).
To repeat, a minimalistic conclusion is that we definitely know that the
areas of supersymmetric 2-cycles or 3-cycles or 4-cycles in string theory
with at least N=2 SUSY (and these are the vacua that string theorists
really like because they can calculate many things precisely there) can be
continuously changed (we know very well what the "area" means in these
cases although one can always redefine the variables - the choice of
coordinates in spacetime nor in the configuration space is physical) and
this is an exact nonperturbative fact.
> Is it clear that areas and shapes of dimensions even exist
> as fundamental quantities in string theory, and are not
> just effective descriptions in some ranges of energy and
> coupling?
If a manifold (say, the Calabi-Yau manifold of compact dimensions) is
large enough, the notions of classical geometry make sense. One can define
scalar fields (such as "the volume of this S^3 inside your Calabi-Yau
manifold") in the effective theory that precisely agree with the geometric
intuition when the manifolds are large - but we can also study them in the
"nongeometric" region. Those scalar fields are continuous and often
parameterize exact moduli spaces. In various limits of the moduli space
one can interpret these parameters as sizes of various manifolds, Wilson
lines etc. - but in various limits of the moduli space different effective
geometric description can appear (they are related by dualities); those
numbers have therefore different interpretations in different limits.
The existence of the continuous parameters (scalar fields) is more general
than their geometric interpretation; the geometric interpretation becomes
meaningful in the asymptotic limits only and depends on which limit we
take.
> That's true. To be fair, though, string theory has the opposite
> problem---while it's easy enough to get something like the
> usual general relativity at long distances (modulo questions
I certainly disagree with the word "easy". The task to construct a quantum
theory that behaves as general relativity at long distances
(semiclassically) was probably the biggest problem that theoretical
physics ever faced. The central problem of physics of the 20th century,
some people say. This is/was much harder task than to reconcile QM with
special relativity. To include gravity was much more difficult than to
explain weak interactions (whose Fermi's theory was nonrenormalizable just
like the naive Einstein's action and this sickness was finally identified
with new physics at short distances - we think the same about the
gravitational divergences, too).
And we still do not understand many consequences of the solution that some
of us ;-) think is the only correct solution of this problem.
And this problem [to derive semiclassical quantum gravity from a
consistent theory] is also the physical motivation to study both string
theory and LQG, I think. We know that there exists gravity above 70
microns :-) but there is no evidence about standard general relativity at
Planckian and shorter scales. On the contrary, I would emphasize that it
is very easy to construct some "nonperturbative physics" such as Loop
Quantum Gravity that looks covariant to some extent etc. The difficult
part of the task is to derive the low-energy gravity from such a quantum
theory. If someone believes that the correct low-energy physics is
guaranteed just because we have put the most beautiful (Einstein's)
equations in, he/she is a victim of a completely unjustified religion.
> problems: the required nonlocality of diffeomorphism-invariant
> observables, the ``problem of time,'' the question of what can
> replace a smooth classical spacetime at short distances, etc.
I agree with identification of some of the problems. What string theory
has not explained at all so far, is what happens when one falls to the
black hole, how the horizon encodes the information about the interior
(nature of the holographic code in general), whether and why the degrees
of freedom inside are just different representation of the Hawking
radiation, in what sense the theory is nonlocal i.e. why the Hawking's
proof of loss of information is wrong (or is it correct? Then how should
we generalize QM?), how do we define the theory at cosmological
backgrounds, which time coordinates we can choose and what is the solution
of the problem of time, why does the universal formula for entropy of
horizons work etc. Note that LQG has solved any of these problems either.
I think that string theory has already said a lot about "what replaces a
smooth classical spacetime at short distances". T-duality, noncommutative
geometry, topology transitions etc. are pretty radical insights in this
direction. We expect many more. Some of them might have some features
similar to LQG. But what I have problem with is the relatively simple
machinery of Loop Quantum Gravity, with Einstein-Hilbert action postulated
at all distance scales with no corrections at all, written in some simple
Yang-Mills variables - and with people who claim that this is the ultimate
solution of Quantum Gravity.
What also disturbs me is that as far as I know, no surprising consistency
checks in LQG appeared (in deep contrast with string theory). In fact I do
not see too many surprises at all. One of the nice things of LQG is the
quantization of areas etc. but of course it is not too surprising that
operators such as J^2 (angular momentum) in a SU(2) spin-network theory of
some sort have quantized eigenvalues of some sort - in fact physicists put
this information in. The rest seems to me as a religion based on the
"exceptional" nature of classical Einstein's equations - and people in the
field of LQG seem to keep our view of the world essentially identical to
the opinion in the 1920s.
String theory is different. Some people complain that it does not predict
- but it in fact predicts a lot. To include gravity consistently to a
quantum theory, we were forced to go to 10 dimensions, to discover
supersymmetry, to allow topology change etc. Some people criticize string
theory that it predicts nothing new, some people complain the string
theory involves/predicts too much new stuff (excited strings, higher
dimensions, SUSY etc.). Some critics happily belong to both categories and
they do not realize how inconsistent their position is... ;-)
Toby Bartels
>But what I have problem with is the relatively simple
>machinery of Loop Quantum Gravity, with Einstein-Hilbert action postulated
>at all distance scales with no corrections at all, written in some simple
>Yang-Mills variables - and with people who claim that this is the ultimate
>solution of Quantum Gravity.
>What also disturbs me is that as far as I know, no surprising consistency
>checks in LQG appeared (in deep contrast with string theory). In fact I do
>not see too many surprises at all.
Your argument seems to be simply <LQG is too simple to be true.>.
>String theory is different. Some people complain that it does not predict
>- but it in fact predicts a lot. To include gravity consistently to a
>quantum theory, we were forced to go to 10 dimensions, to discover
>supersymmetry, to allow topology change etc. Some people criticize string
>theory that it predicts nothing new, some people complain the string
>theory involves/predicts too much new stuff (excited strings, higher
>dimensions, SUSY etc.). Some critics happily belong to both categories and
>they do not realize how inconsistent their position is... ;-)
String theory predicts nothing that can be tested,
and a whole lot of things that can't be tested.
LQG, OTOH, predicts only a few things that can't be tested ^_^.
I think that you've described the fundamental difference well:
String theory is complicated, and LQG is simple.
You conclude that string theory is more likely to be correct;
but my sympathies here lie with those of William of Ockham.
-- Toby
to...@math.ucr.edu
Demian H.J. Cho
> As a gravitational physicist who is very symphathetic to string theory I
> don't have anything to add on the top of what everyone already said. But,
> if you allow me I want to express my personal view of the whole issue
> which is nothing to do with science.
First of all. I personally believe so called "theory of everything", if
it exists, will be very, very surprising one, and non of the current
theories can claim to be one. Don't ask me why. It's just a personal
belief.
Second, which is my bigger concern is that there has been always some
too early claims, and arrogances in science, and I don't think they are
good for the development of science in general. It's like "I am a number
theorist, the King of mathematics, and you are just a mere analyst" -
Sorry John, kind of attitude. When we attack a problem like quantum
gravity, or unification we need all the possible direction and effort. I
believe that Nature (or the Lord) is very subtle.
In fact, my impression from many people I met, some of them string
theorists, some of them loop gravity - most of them quite well known, is
that they are very careful talking about what they've acheived so
far. some of them even worrying about the fact that some of their
collegues alienate the other areas of physics simply by making
"outrageous" claims. Let me remind you that the success of gauge theory
is largely came from the spontaneous broken gauge symmetry - which is
developed by solid state physicists. Who guessed "chauvists" solid state
physicists has a key to our holygrail?
So, let's quit arguing. We have far way to go. We better encourage each other.
Bless all,
--
Demian H.J. Cho
Center for Gravitation and Cosmology
University of Wisconsin-Milwaukee
John Baez
In article <Pine.SOL.4.10.101060...@strings.rutgers.edu>,
Lubos Motl <mo...@physics.rutgers.edu> wrote:
>Supersymmetry is an important feature of theories connected with string
>theory. It is seen perturbatively but holds also exactly [...]
Except, of course, in the real world. This is what makes me
nervous. Somehow supersymmetry must be broken to obtain a
theory that's consistent with experiment. I haven't heard
any good explanation of how this is supposed to work. What
is your attitude towards this issue?
>On the contrary, I would emphasize that it
>is very easy to construct some "nonperturbative physics" such as Loop
>Quantum Gravity that looks covariant to some extent etc. The difficult
>part of the task is to derive the low-energy gravity from such a quantum
>theory.
Yup. I've spent a lot of time talking to Ashtekar, Rovelli,
Smolin, and other people who work on loop quantum gravity.
They all agree that this is the big task. Most of them are
working on it in one way or another. (Smolin is probably
spending more time trying to forge connections with string
theory.)
>If someone believes that the correct low-energy physics is
>guaranteed just because we have put the most beautiful (Einstein's)
>equations in, he/she is a victim of a completely unjustified religion.
I agree. I don't know any such people - do you?
>... what I have problem with is the relatively simple
>machinery of Loop Quantum Gravity, with Einstein-Hilbert action postulated
>at all distance scales with no corrections at all, written in some simple
>Yang-Mills variables - and with people who claim that this is the ultimate
>solution of Quantum Gravity.
Do you know anyone who has made such a claim? I don't. It's possible
you've just talked to people who say that some *other* people make this
claim.
As for "relatively simple machinery", I think that's good rather than
bad.
As for "the Einstein-Hilbert action postulated at all distance scales
with no corrections at all", I've explained in a reply to Aaron Bergman
that this is almost surely NOT how things will work.
>String theory is different. Some people complain that it does not predict
>- but it in fact predicts a lot. To include gravity consistently to a
>quantum theory, we were forced to go to 10 dimensions, to discover
>supersymmetry, to allow topology change etc.
The remaining big task is to explain why the world looks 4-dimensional,
why it doesn't look supersymmetric, why we don't see topology change,
etc.. That is, why the low-energy effective limit matches the world
we see around us. Note that this task is very much like the remaining
big task of loop quantum gravity!
A.J. Tolland
I'd like to critique some of the statements you made about quantum
gravity. My basic problem is that you are making the assertion that
quantum gravity and M theory are the same thing. This assertion is
completely unjustified.
Any theory which gives rise to Einstein's gravity in the classical
limit is a theory of quantum gravity... As has been said before on this
newsgroup, any theory which has diffeomorphism invariance and a dynamical
metric below some energy scale implies Einstein's gravity in the classical
limit. Thanks to the enormity of the Planck mass, the metric's classical
low-energy behavior is essentially governed by the leading terms in the
effective action. These leading terms are just the Einstein-Hilbert
action with cosmological constant.
My point here is that there won't necessarily be a unique quantum
theory of gravitation; there may be many quantum theories which imply
Einstein's equations. This is essentially a mathematical question, and
since we don't yet have an algebraic characterization of M-theory, there
is no theorem showing that all quantum theories of gravity are string
theories of some kind. Indeed, the LQG people think they are hot on the
trail of a counterexample to this statement. No surprise, I guess; many
people believe that some effective theories can be made into fully
rigorous quantum theories without reincorporating the higher energy
degrees of freedom. In all honesty, I don't think LQG will correspond to
the real world -- my prejudice at the moment is that the metric variables
are effective, valid only at sub-Planck energies -- but I won't be at all
surprised if the loop quantum gravity people construct their theory long
before we string theory types construct ours. In this sense, LQG is every
bit as valid a direction for quantum gravity as string theory.
Furthermore, string theorists might do well to pay some attention
to LQG in the future. It is the simplest possible approach to making a
quantum theory with 4D gravity, and although it does depend on certain
features of the 4D Lorentz group, it may well teach us a few lessons about
background free quantum theories.
Now, let me ask you a question. Do you believe that M-theory is
(a) the _only_ quantum theory of gravity in the mathematical sense used
above, or were you claiming that string theory is (b) the only theory
correctly describing the gravity of our physical universe? I would love
to hear your reasons for believing in the truth of either statement. As
you probably guess, I don't consider the usual litany of string theory's
many interesting aspects to be a good reason to believe that it is either
(a) or (b).
--A.J.
Steve Carlip
Gravity vs. M-Theory II.'' There are two different issues there: a
general discussion of loop quantum gravity and string theory, and
a much more specific question about quantization of volumes in
string theory. I'm splitting things up, and focusing on the latter
here.
Lubos Motl <mo...@physics.rutgers.edu> wrote:
>>> LM: Areas are not always quantized in string theory but they
>>> are in LQG.
>> SC: How certain is this? It's true, of course, in perturbative string
>> theory. But we know that the perturbation series doesn't ...
> LM: I am convinved that virtually everyone in the field would agree that
> the existence of exact moduli spaces which allow certain geometric
> parameters (areas of 2-cycles etc.) to change continuously has nothing to
> do with perturbative expansions. It is rather a direct consequence of
> supersymmetry. [...]
> To repeat, a minimalistic conclusion is that we definitely know that
> the areas of supersymmetric 2-cycles or 3-cycles or 4-cycles in string
> theory with at least N=2 SUSY (and these are the vacua that string
> theorists really like because they can calculate many things precisely
> there) can be continuously changed (we know very well what the "area"
> means in these cases although one can always redefine the variables
> - the choice of coordinates in spacetime nor in the configuration
> space is physical) and this is an exact nonperturbative fact.
Let me show off my ignorance in an attempt to clarify the issue. In old
fashioned perturbative string theory, you start with a Polyakov-like
action for a string in a background. Imposing Weyl invariance gives
you a set of equations for the background, from which you obtain things
like Calabi-Yau manifolds with cycles that have computable areas. I
understand that supersymmetry then lets you make statements that
go beyond the perturbation theory you began with, but that's not really
my question. Rather, my question is this:
In the perturbation theory, the background metric is not a quantum
field in its own right. It is, rather, a collective excitation of more
fundamental degrees of freedom, and as such has more in common
with an expectation value than with an eigenstate of a ``geometry''
operator of some sort. Indeed, at least for nearly flat metrics, it is
an expectation value of a coherent state obtained by exponentiating
a graviton vertex operator.
Now, even if eigenvalues of area are quantized, expectation values
certainly need not be. In particular, expectation values in coherent
states can vary continuously---just think about the expectation
value of the Hamiltonian in a harmonic operator coherent state,
which can vary continuously even though energy is quantized.
So the question is whether the continuously varying area of a two-
cycle, say, is really a continuous eigenvalue of an area operator, or
whether it merely indicates the existence of a continuous family of
coherent states. (And, to make things fun, if it's an eigenvalue
of an area operator, what Hilbert space does that operator act on?)
> If a manifold (say, the Calabi-Yau manifold of compact dimensions)
> is large enough, the notions of classical geometry make sense. One
> can define scalar fields (such as "the volume of this S^3 inside your
> Calabi-Yau manifold") in the effective theory that precisely agree
> with the geometric intuition when the manifolds are large - but we
> can also study them in the "nongeometric" region. Those scalar
> fields are continuous and often parameterize exact moduli spaces.
Same question. When you talk about the ``value'' of one of these
fields (in an effective theory!), what do you mean? The eigenvalue
of an operator on the true Hilbert space of the theory? Or an
expectation value?
Note, to connect with the earlier discussion of loop quantum gravity,
that in loop quantum gravity it's the eigenvalues of the area operator
that are quantized, not he expectation value, which can certainly vary
continuously as the state changes.
Steve Carlip
Steve Carlip
quantum gravity and string theory, and a much more specific
question about quantization of volumes in string theory. I'm
here.
Lubos Motl <mo...@physics.rutgers.edu> wrote:
> Good that someone - even someone famous - (tries to) defend(s)
> Loop Quantum Gravity so that we enjoy plurality here. :-)
I'm not actually ``defending loop quantum gravity'' in the sense of
arguing that it's right. It's more that I have a stronger feeling of
uncertainty and ignorance than you do: I don't know the right way
to quantize gravity, and I suspect that no one else does either.
This comes in part from work in (2+1)-dimensional quantum
gravity---certainly a hugely oversimplified model, but one that
might tell us something. Our experience there is that
(1) There are lots of ways to quantize gravity, and they're not
all equivalent;
(2) It's not necessary to make the leap to string theory in order
to get a quantum theory; and
(3) The big problems are the ``conceptual'' ones that come from
trying to quantize a theory of the structure of spacetime---the
``problem of time,'' the problem of reconstructing a classical
limit from nonlocal diffeomorphism-invariant observables,
the problem of finding good observables to begin with, etc.
[Note that I'm *not* saying that the way to solve ``conceptual
issues'' is to think vague philosophical thoughts about the
concepts!]
Now, it may well be that (3+1)-dimensional quantum gravity is
fundamentally different, and that there are a whole new set of
issues that don't show up in the lower-dimensional model. But,
first, I think this is an open question, and, second, even if it's
true, it's not going to let you escape the problems that are already
there in 2+1 dimensions.
>The task to construct a quantum theory that behaves as general
> relativity at long distances (semiclassically) was probably the
> biggest problem that theoretical physics ever faced. The central
> problem of physics of the 20th century, some people say.
> And we still do not understand many consequences of the
> solution that some of us ;-) think is the only correct solution
> of this problem. [...]
> I would emphasize that it is very easy to construct some
> "nonperturbative physics" such as Loop Quantum Gravity
> that looks covariant to some extent etc. The difficult part of
> the task is to derive the low-energy gravity from such a quantum
> theory.
Well, I'm not at all sure that it's so easy to construct a covariant
nonperturbative theory that looks anything like general relativity.
But the more important point, I think, is that I believe it's too early
to call string theory ``the correct solution of this problem'': it's
more a promising program for finding an as yet unknown solution.
Let me explain by means of an analogy. We've actually known
for more than 30 years how to construct a quantum theory that
behaves like general relativity at long distances. Start with the
Einstein-Hilbert action and add a curvature squared term with
a small coefficient. This makes propagators go as 1/k^4 at high
momenta, and Stelle showed in 1977 that the resulting theory is
perturbatively renormalizable, and has general relativity as its
low energy limit.
There's a catch, of course: perturbatively, the theory has negative
norm states. For simpler theories, though, there's a known
mechanism (the Lee-Wick mechanism) for getting rid of such
states, and it's not unreasonable to hope for such a mechanism
in gravity.
Does this mean we've solved the problem? I, and I assume you,
would say, ``No.'' The answer is ``no'' because the ``solution''
depends on a postulated nonperturbative mechanism that hasn't
been shown to work for the case of physical interest. So R+R^2
gravity isn't a quantum theory of gravity; it's at best a program
for finding one by working out the existence of a nonperturbative
method of getting rid of negative norm states.
I would argue that as far as quantum gravity is concerned, the status
of string theory isn't so different. Perturbative string theory has
general relativity as a long distance limit, but the perturbation series
is not Borel summable, so the theory only really makes sense as an
asymptotic expansion of some nonperturbative theory. And even
though we have a bunch of hints about features of nonperturbative
string theory, we don't really know what it is, or even if it is.
Now, I happen to think that string theory is a very good guess for a
program that will eventually give us a quantum theory of gravity.
But I also know that there have been a lot of other guesses that
looked good at the time. So I think a bit of caution is in order.
> I think that string theory has already said a lot about "what
> replaces a smooth classical spacetime at short distances". T-duality,
> noncommutative geometry, topology transitions etc. are pretty
> radical insights in this direction.
T-duality certainly seems to provide a radical insight, though I'm
not quite sure what that insight is. The traditional interpretation
that the string scale provides a minimum length seems to be wrong,
since D0 branes can probe smaller distances. But there's certainly
*something* there.
Noncommutative geometry, on the other hand, arises even in
condensed matter physics (electrons restricted to their lowest
Landau level), but no one thinks that says something deep about
the nature of space and time. Maybe noncommutative geometry
in string theory is deeper, and I haven't read enough; the stuff
I've seen mostly refers to particular background configurations.
As for topology change, the idea's been around since the '50s.
It's certainly very nice that string theory can implement some
particular examples, but again, I don't think anything can be
said systematically yet. These are interesting directions; I just
don't think they're yet interesting *answers*.
> But what I have problem with is the relatively simple machinery
> of Loop Quantum Gravity, with Einstein-Hilbert action postulated
> at all distance scales with no corrections at all, written in some
> simple Yang-Mills variables - and with people who claim that
> this is the ultimate solution of Quantum Gravity.
Who claims that? This is the same kind of stereotype that some
loop quantum gravity people have about string theorists---that
they postulate a bunch of little loops running around on a fixed
background and claim that this is the ultimate solution of Quantum
Gravity. If you want to criticize loop quantum gravity, that's
great. But make the criticisms real: setting up straw men to knock
down doesn't do anyone any good.
> What also disturbs me is that as far as I know, no surprising
> consistency checks in LQG appeared (in deep contrast with
> string theory). In fact I do not see too many surprises at all.
Well, there's the fact that spin network states are eigenstates of
the area operator---something not at all obvious, and not really
put in by hand---and there's the equivalence of the analytic
``generalized connection'' approach and the combinatoric spin
network approach, again something that certainly wasn't
obvious a priori.
On the other hand, I agree that string theory has some amazing
consistency checks, of the sort that loop quantum gravity doesn't.
On the other hand, QED doesn't, either. In some sense, this
makes loop quantum gravity much more like an ordinary quantum
theory, and not, as you charge, ``a religion based on the "exceptional"
nature of classical Einstein's equations.
Steve Carlip
Squark
<9g3btk$b8o$1...@news.state.mn.us>):
>
>A few random comments:
>
>In article <Pine.SOL.4.10.101060...@strings.rutgers.edu>,
>Lubos Motl <mo...@physics.rutgers.edu> wrote:
>>On the contrary, I would emphasize that it
>>is very easy to construct some "nonperturbative physics" such as Loop
>>Quantum Gravity that looks covariant to some extent etc. The difficult
>>part of the task is to derive the low-energy gravity from such a quantum
>>theory.
>
>Yup. I've spent a lot of time talking to Ashtekar, Rovelli,
>Smolin, and other people who work on loop quantum gravity.
>They all agree that this is the big task. Most of them are
>working on it in one way or another.
What surprises me is why so much effort is put into proving quantum
gravity has the correct classical limit, in contrast to, say, that QED has
the correct classical limit? Is there any special reason for it, other
than the general believe in better understand of QED? Or maybe the later
fact is somehow fundumentally much more obvious?
Best regards,
Squark.
--------------------------------------------------------------------------------
Write to me at:
[Note: the fourth letter of the English alphabet is used in the later
exclusively as anti-spam]
dSdqudarkd_...@excite.com
Lubos Motl
the question whether the areas are quantized in string theory so that
string theory can be compatible with Loop Quantum Gravity. SC says that
the areas might be quantized nonperturbatively while LM says that the
areas can be changed continuously in supersymmetric string vacua even if
one takes nonperturbative physics into the account.
It seems that SC now agrees that the question has nothing to do with
perturbative expansions (the background areas of the cycles are continuous
even perturbatively) but he raised quite an interesting issue (I think)
of the difference between eigenvalues and expectation values...
> SC: In the perturbation theory, the background metric is not a quantum
> field in its own right.
LM: This sentence sounds like a tautology to me. Background is defined as
the classical configuration (solution of the classical equations of
motion, in fact) of vacuum expectation values of various fields (in this
case we talk about the metric) around which we expand. So, of course, the
background itself is not a quantum field, by definition. ;-)
> It is, rather, a collective excitation of more fundamental degrees of
> freedom, and as such has more in common with an expectation value than
> with an eigenstate of a ``geometry'' operator of some sort.
Yes and no. The background is always a collective excitation of the
fundamental degrees of freedom. The state where Higgs has a nonzero vev
can be also thought of as a coherent state made of the Higgses. The
difference in string theory is only in the nature of the fundamental
building blocks: the particles are really strings. However one can make a
coherent state of the stringy graviton modes and the effect of this
coherent state is absolutely equivalent to the change of the background
metric, just like we expect from the field theory intuition; therefore we
know that physics of string theory is background independent (although we
would like to find also a *formulation* that is manifestly background
independent).
However I understand your point that the eigenvalue is not the same as the
expectation value (the latter can change continuously, of course). Good
point. My description of the Calabi-Yau spaces was meant to represent a
clear example: of course, all the conclusions must be valid in this case,
too. While I think that your points are interesting, they will probably
not change my opinion so far.
Well, in LQG the situation looks as follows: the eigenvalues of the area
operators are quantized, however the expectation values in a state do not
have to be quantized. Because LQG is completely discrete, in a sense, I am
not sure whether you can create anything like the coherent states (are
there any observables in LQG that have continuous eigenvalues at all?).
This is related to the problems with the low energy limit of LQG.
In string theory, imagine a pure state, a second quantized vacuum of type
IIA strings on a Calabi-Yau with a fixed geometry etc. This geometry is
just a background, a vacuum expectation value of some fields (that
determine the geometry etc.). The fields themselves are the vevs PLUS the
quantum fluctuations. Yes, one cannot claim so easily that the vacuum is
an eigenstate of the (total) field. In fact, string theory does not allow
you to consider the value of the fields as a rigorous notion at all,
simply because it is not a field theory. It is holographic. The only
perturbative observables you can compute are on-shell S-matrix elements,
not the full off-shell Green's functions. String theory is not a field
theory in this sense - but this should not be understood as a handicap; it
is a feature of string theory. Perturbatively, only S-matrix elements are
physical and rigorously defined. (Of course, the other notions can be
given an approximate definition,)
Nevertheless I still think - although no clear proof is available - that
the condition that the vev in coherent states is continuous is equivalent
to the condition that the eigenvalues of the given operator are
continuous. You cannot create much of a coherent state from a discrete
operator, I think.
> states can vary continuously---just think about the expectation
> value of the Hamiltonian in a harmonic operator coherent state,
Harmonic oscillator, right? ;-) Note that you mixed different operators
in this example. You are talking about the eigenvalues of the Hamiltonian
but the coherent states are not parametrized by the energy! They are
parameterized by z=x+ip - and x, p as well as the annihilation operator
x+ip have a continuous spectrum. Nevertheless I think that your point is
serious. You can probably construct the "coherent" states exp(iJ_+)|m=-l>
in which the expectation value of j_z is continuous - although the
eigenvalue must be quantized. That's true.
> So the question is whether the continuously varying area of a two-
> cycle, say, is really a continuous eigenvalue of an area operator, or
> whether it merely indicates the existence of a continuous family of
> coherent states.
Because I cannot prove that there is no loophole in the interesting point
you have raised - I will try to think about it in both ways - let me tell
you at least one more point. If you average the operator of the area of a
2-cycle (in a Calabi-Yau) over a large region of spacetime, you might say
that the vacuum *is* an eigenstate of this operator with an arbitrary
(continuous) eigenvalue. Well, I know that this does not prove much - by
averaging you can get a continuous spectrum in LQG, too, it seems to me.
> (And, to make things fun, if it's an eigenvalue of an area operator,
> what Hilbert space does that operator act on?)
In perturbative string theory you know very well what the Hilbert space
is. The Hilbert space is the Fock space of all possible particles
corresponding to string harmonics, with the appropriate statistics. Beyond
the free theory approximation, the Hilbert space contains all the possible
scattering states with their correct masses etc. (one must calculate a
bit). It is however much harder to define the operator "area of the given
cycle of the Calabi-Yau at the point (t,x,y,z) in the large four
dimensions". It is because string theory really does not contain local
fields - that behave as well defined operators as (t,x,y,z) -, due to
holography. (The fact that perturbative string theory gives on-shell
result, is in fact an old appearance of "holography" in it.)
OK, let me summarize. It is true that we know that string theory has a
continuous spectrum of the areas understood as the expectation values -
but it is not clear about the eigenvalues of the area operators
corresponding to a given 2-surface in the coordinate space. As I have
mentioned, it is likely that such operators will never have a rigorous
definition in string theory simply because it is not a field theory living
in on a classical spacetime of fixed topology where coordinates make an
accurate sense (like LQG): i.e. because of holography and the quantum foam
that forces us to consider not only graviton, metric and geometry at
substringy distances, but also all the "massive counterparts" associated
with massive stringy harmonics.
One can also imagine that such "area operators" can have some definition
in string theory - and they might even match LQG-like rules. If this was
proved, it would be really exciting. But don't be too optimistic. There
are potentially many limits where the geometric interpretation is
different - the moduli can even change from one phase to another through
spacetime. It is hard for me to believe that one could define an overall
system of geometric operators in a stringy spacetime. You know, string
theory modifies the notion of geometry at Planckian distances. The usual
geometric concepts do not have a meaning at ultrashort distances - just
like the value of the Fermi's interaction term in weak interactions does
not make sense if you compute it at a too short distance scales.
Lubos Motl
I tried to figure out your first name, but I failed: A.J. Tolland
is the maximum I can see anywhere. :-) Thanks for your critique.
> I'd like to critique some of the statements you made about quantum
> gravity. My basic problem is that you are making the assertion that
> quantum gravity and M theory are the same thing. This assertion is
> completely unjustified.
It is justified in the sense that I can refer to about 10,000 scientific
papers that justify it and bring pieces of evidence for this assertion. If
this is not enough, I give up - I cannot justify the statement more than I
did. :-) The assertion has not been accepted to the Bible yet, for
example.
> Any theory which gives rise to Einstein's gravity in the classical
> limit is a theory of quantum gravity...
Yes, I agree. Any consistent theory reducing to Einstein's gravity (maybe
plus other fields) at low energies is a theory of quantum gravity.
Therefore I say that M-theory in the broad sense is the only theory of
quantum gravity. If you prove me wrong, it will be extremely exciting!
> low-energy behavior is essentially governed by the leading terms in the
> effective action. These leading terms are just the Einstein-Hilbert
> action with cosmological constant.
Maybe you should pay your attention to the adjective "consistent" little
bit more. It is of course easy to construct an inconsistent theory of
quantum gravity. ;-)
> My point here is that there won't necessarily be a unique quantum
> theory of gravitation; there may be many quantum theories which imply
> Einstein's equations.
Yes, this conjecture was plausible a priori but over 50 years of research
have shown that it is very unlikely.
> This is essentially a mathematical question, and since we don't yet
> have an algebraic characterization of M-theory, there is no theorem
> showing that all quantum theories of gravity are string theories of
> some kind.
Yes, I agree, the question about the uniqueness of quantum gravity
theories is essentially a mathematical question. The answer also depends
on how general definition of "what we still call M-theory" we accept.
> Indeed, the LQG people think they are hot on the trail of a
> counterexample to this statement. No surprise, I guess; many people
> believe that some effective theories can be made into fully rigorous
> quantum theories without reincorporating the higher energy degrees of
> freedom.
OK, some people also believe that the Universe was created 6,000 years
ago. The history of science has shown that the nonrenormalizable theories
were always nonrenormalizable because they neglected important physics at
very short distances. This is a general lesson given to us by the
renormalization group ideas. Nonrenormalizable theories become infinitely
strongly coupled at short distances - something else must always take
over.
> before we string theory types construct ours. In this sense, LQG is every
> bit as valid a direction for quantum gravity as string theory.
The only difference is that the airplanes don't land. ;-)
> Now, let me ask you a question. Do you believe that M-theory is
> (a) the _only_ quantum theory of gravity in the mathematical sense used
> above, or were you claiming that string theory is (b) the only theory
> correctly describing the gravity of our physical universe? I would love
> to hear your reasons for believing in the truth of either statement.
Once again: yes, I believe that M-theory (in the broad sense) is the only
mathematically consistent quantum theory of gravity above 3 dimensions. It
is hard to tell you the reason. I have personally tried a lot to construct
a different consistent quantum theory that would reduce to Einstein's
equations at long distances. And I also know that thousands of people have
been trying to do the same for 50 years or so and they failed so far. If
you try really hard, you learn a lot of things, e.g. some general deffects
of broad classes of theories you might try. Quantum gravity is an
extremely difficult task. One should appreciate that the consistence of a
quantum field theory in 4D is a very subtle issue. There are very few
consistent quantum field theories, in a sense.
Yes, (b) is more about beliefs, but I believe that the universe, including
gravity, is described by a (single) consistent mathematical theory. And
because the only mathematically consistent quantum theory with gravity is
M-theory in the broad sense, I also believe that M-theory is the only
theory describing our physical universe, including gravity. (I do not
agree that at the fundamental level you can divide the Universe into
"gravity" and the "rest".) I think that while the previous paragraph is
justified by decades of unsuccessful attempts of many people, this
paragraph is more about my belief.
> As you probably guess, I don't consider the usual litany of string
> theory's many interesting aspects to be a good reason to believe that
> it is either (a) or (b).
Yes, I guessed so and I also guess that my mail will change nothing about
it. ;-)
Daniel Doro Ferrante
[trimmed previous text]
>
> What surprises me is why so much effort is put into proving quantum
> gravity has the correct classical limit, in contrast to, say, that QED has
> the correct classical limit? Is there any special reason for it, other
> than the general believe in better understand of QED? Or maybe the later
> fact is somehow fundumentally much more obvious?
>
Also, why couldn't this "Quantum Gravity" predict some "Classical"
effect that we haven't seen yet?! Also, once we're all accepting that some kind
of _new_ physics should emerge, why shouldn't we expect some "changes" (in one
way or another ;) in the "classical limit"?!
My understanting is that, when Quantum Mechanics appeared last century,
all of the above questions ended up with a "Yes!" kind of answer...
I guess that this could be considered "off topic", but, ... Once we're
all looking for something *new*, we should expect something new... ;) Pretty
obvious statement, but some people seem to have difficulty with changes...
still!
--
Daniel
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> | www.het.brown.edu www.cecm.usp.br <
> Daniel Doro Ferrante | <
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Aaron Bergman
>
>What surprises me is why so much effort is put into proving quantum
>gravity has the correct classical limit, in contrast to, say, that QED has
>the correct classical limit? Is there any special reason for it, other
>than the general believe in better understand of QED? Or maybe the later
>fact is somehow fundumentally much more obvious?
Just scatter electrons in QED and you can see that the formulae
you get have the correct limit.
If someone could scatter gravitons in lqg, then we could see if
the result agrees with the expected result from the cutoff
Einstein-Hilbert action.
Lubos Motl
moderators can put it in a new thread? ;-)
> John Baez (LQG vs. M-theory discussion): Except, of course, in the
> real world. This is what makes me nervous. Somehow supersymmetry must
> be broken to obtain a theory that's consistent with experiment. I
> haven't heard any good explanation of how this is supposed to work.
> What is your attitude towards this issue?
Well :-), the problem you mention is called "supersymmetry breaking".
There are hundreds of papers dealing with this problem. Joe's book has a
separate chapter focusing on it. Many things are known but we are very far
from the complete understanding.
Let me give you a course on SUSY breaking by a SUSY breaking beginner.
First of all, if you deal with MSSM - minimal SUSY standard model - you
account for the SUSY breaking by adding the so-called "soft SUSY breaking
terms". These are terms with coefficients having a positive power of mass
- such as squark masses (greeting to squark) and other masses etc.
Because they have a positive power of mass, they do not change anything
substantial about the ultraviolet behavior of the amplitudes. For
instance, the cancellation of the quadratic divergences to the Higgs mass
is still guaranteed to hold, even after you add the soft SUSY terms.
While SM has 19 parameters or so, MSSM with all those soft terms has about
105 parameters.
But there is in principle dynamics that predicts the values of all those
soft SUSY breaking terms. Usually those terms can be computed as one loop
diagrams where something breaking SUSY is included in the loop. More
precisely, to break SUSY - and it is not *qualitatively* different from a
spontaneous breaking of the electroweak symmetry, for example - you need
two things: a source of the SUSY breaking and a messenger that
communicates it.
The source of breaking are F-terms in SUSY (as far as we know, D-terms do
not break it) or e.g. the gaugino condensation - a nonzero vev
<0|gaugino.gaugino|0> (one can also imagine a global, "topological"
constraints breaking SUSY). The gaugino condensate can be understood as an
F-term for a composite field. This breaking should be thought of as a
complete analogue of the Higgs breaking of the electroweak symmetry.
Imagine that an effective action contains gaugino^4 terms that cause the
minimum of the energy to be at a nonzero value of <0|gaugino.gaugino|0>.
Then you need to communicate this breaking to the standard model to
generate the soft terms for the standard model fields. According to the
messenger, you can distinguish gravity-mediated, gauge-mediated,
anomaly-mediated, moduli-driven, dilaton-drive and other types of SUSY
breaking.
Let me tell you a particular example. The real world was understood as the
E8 x E8 heterotic string on a Calabi-Yau in the 80s. In 1995 Horava and
Witten realized that at strong coupling, E8 x E8 string looks like
M-theory on a line interval; each 10D boundary carries a single E8 gauge
supermultiplet. Our standard model lives at one boundary - a GUT group is
embedded into the left E8 - and the other E8 has a gaugino condensation on
it, breaking SUSY. Membranes stretched between the domain walls can be
charged under both E8's and they can mediate (as well as simply gravitons)
the SUSY breaking on our domain wall because they couple to both sectors.
The resulting superpartner masses will be much smaller than the SUSY
breaking scale on the other brane.
One can say - again - that people understand many possible mechanisms of
the SUSY breaking. The problem is that there are too many possibilities
and no clear formulation of the stringy theories that can pick up the
correct model. So far. But certainly, SUSY breaking is allowed by the
theory. The problem then is to keep the cosmological constant very small
after the supersymmetry breaking - and not of the expected order m^4 where
m is the SUSY breaking scale. But this is, maybe, a different question.
If someone thinks that I have omitted something essential, please write it
down!
Lubos Motl
> So, let's quit arguing. We have far way to go. We better encourage each other.
>
> First of all. I personally believe so called "theory of everything", if
> it exists, will be very, very surprising one, and non of the current
> theories can claim to be one. Don't ask me why. It's just a personal
> belief.
Well, thank you for encouraging all of us! :-) More seriously, I also
believe that the ultimate theory of everything will involve notions that
would look unfamiliar to us today. But I also believe that it will predict
the existence of all the supersymmetric vacua etc. that string theorists
study. If I had serious enough doubts and I thought that we might be on a
completely wrong track - or that we might be working on random one of ten
completely different theories (and only one is correct - so that the
chances are less than 10%), I would probably leave theoretical physics.
This is a completely psychological question but I believe that string
theory (in the broad sense of the word) is the only correct description of
the real world (including quantum gravity) with probability greater than
50%.
> Second, which is my bigger concern is that there has been always some
> too early claims, and arrogances in science, and I don't think they are
> good for the development of science in general. It's like "I am a number
My opinion is closer to the opposite one. My goal here is not to defend
the arrogance in general ;-), but I certainly think that the people who
made the most important contributions to physics usually believed that the
structure that they studied was unique, exceptional and far-reaching. The
other, less influential scientists did not believe those principles so
seriously and they were more "tollerant", "open-minded" etc. One can be
more pleasant for others. But taking your theories seriously is almost
certainly (positively) correlated with your chances to succeed. I think
that if you analyze the dependence between physicists' belief that they
work on the right thing - and between their productivity, number of
citations etc., there will be a significant correlation.
Albert Einstein wanted to find nothing less than the theory of everything.
This is why he dedicated his life to physics. Among many string theorists,
Edward Witten is not only the most productive and influential one, but he
also takes string theory most seriously, I think. And he often shocks
participants of a mathematics conference by claims such as "mathematics of
the 3rd millenium will be dominated by string theory" or "string theory
has the remarkable property of predicting gravity", "learning about the
way how string theory incorporates gravity was the strongest intellectual
thrill of my life", "string theory is a science of the 21nd century that
fell to the 20th century by an accident" etc.
Such a "belief" is a motivation to work more intensely - and vice versa,
if you work intensely and things work, it is more likely that you will
take the subject under ivestigation more seriously.
> Sorry John, kind of attitude. When we attack a problem like quantum
> gravity, or unification we need all the possible direction and effort. I
I would agree that we must be able and ready to deal with ideas of very
many different types. But it is not true that all the directions of
research we may imagine will play a role in the ultimate theory of
everything, I think. We must be also ready to abandon completely wrong
ideas as soon as they are proved wrong. We must simply use the standard
rules of thinking, estimating the correct directions of research,
abandoning the conjectures that have been proved wrong (theoretically or
experimentally) etc.
> is largely came from the spontaneous broken gauge symmetry - which is
> developed by solid state physicists. Who guessed "chauvists" solid state
> physicists has a key to our holygrail?
Yes, I think that none wants to claim that solid state physics is less
important or that it cannot give us new interesting insights. But I do not
think that anyone is justified to think or to claim that he or she is as
important physicist as Peter Higgs (or even Ed Witten) just because he or
she is not working on string theory and is "different". Difference does
not imply quality!
Bless all
Lubos Motl
Some poor soul whom Lubos Motl forgot to cite wrote:
> John Baez wrote:
> >Yup. I've spent a lot of time talking to Ashtekar, Rovelli,
> >Smolin, and other people who work on loop quantum gravity.
> >They all agree that this is the big task. Most of them are
> >working on it in one way or another.
> What surprises me is why so much effort is put into proving quantum
> gravity has the correct classical limit, in contrast to, say, that QED has
> the correct classical limit? Is there any special reason for it, other
> than the general believe in better understand of QED? Or maybe the later
> fact is somehow fundamentally much more obvious?
QED is a standard quantum field theory that starts with a classical field
and then quantizes it in the canonical fashion, usual in particle physics.
The parameters are then renormalized using the standard rules of QFT and
the renormalization group. The classical limit (say the Coulomb's law) can
be calculated as the zero momentum-exchange limit of the scattering
amplitudes etc. This is a straightforward task to do.
LQG looks naively similar. You start with Einstein's fields, write them in
some noncanonical variables and then quantize them in a "new way". So why
cannot you just say that at long distances the expectation values of the
fields in LQG just satisfy the same equations as the classical Einstein
metric? I think that the main problem is that this simple prescription is
known to give the wrong results. More precisely, if one calculates the
entropy of the black hole, one does not get the correct A/4G where G was
the "classically" substituted Newton's constant and A is the horizon area.
One gets a result wrong by the factor of ln(2)/sqrt(3). In my
interpretation, this factor is called the Immerzi parameter gamma and it
measures how much wrong Loop Quantum Gravity calculations are. This gamma
should equal one but it differs by ln(2)/sqrt(3). The only way how can one
argue that LQG is not ruled out is to say that Gnewton is allowed to be
renormalized, so that this Gnewton inserted to the formulae using the
classical intuition differs from the Gnewton measured at long distances by
the factor gamma. Correct me if I am wrong, but I think that gamma really
equals the renormalization factor of the Newton's constant between
the Planckian distances and low energies.
Because LQG has nothing to do with the standard ways to quantize local
field theories, the usual machinery of the renormalization is not
applicable. Maybe there is a way to show that at low energies, LQG behaves
again as general relativity, although with a redefined value of the
coupling constant. But maybe this discrepancy between the correct and the
calculated black hole entropy should be understood as a proof that LQG is
inconsistent. It is fine that Ashtekar, Smolin, Rovelli and others realize
that the ignorance about the low energy dynamics is the most obvious flaw
of LQG (at least of LQG as understood today) but we will see whether the
laws of mathematics will allow them to resolve the paradox.
Note that string theory predicts the correct entropy of all the extremal
black holes (as well as near extremal and a couple of Schwarzschild
ones...) and does not suffer from any problems of this sort.
> I agree. I don't know any such people - do you?
Yes, I do. But I would prefer not to mention their names. It is not you,
John, however! :-)
>>... what I have problem with is the relatively simple
>>machinery of Loop Quantum Gravity, with Einstein-Hilbert action postulated
>>at all distance scales with no corrections at all, written in some simple
>>Yang-Mills variables - and with people who claim that this is the ultimate
>>solution of Quantum Gravity.
> Do you know anyone who has made such a claim? I don't. It's possible
> you've just talked to people who say that some *other* people make this
> claim.
Well, I am not completely sure which claim of mine you find so
controversial, but the claims can be found e.g. in Rovelli's review of LQG
http://arXiv.org/abs/gr-qc/9710008
> As for "relatively simple machinery", I think that's good rather than
> bad.
If I compare two working theories, I prefer the theory with the simple
machinery, too. ;-) But note the word "working". I do not think that you
can take Newton's equations, for example, and claim that they solve the
problem of Quantum Gravity. They are simple, they are fine. But the
problem is that they do not solve it. Quantum Gravity is one of the
greatest problems in physics - and it requires very strong weapons!
> As for "the Einstein-Hilbert action postulated at all distance scales
> with no corrections at all", I've explained in a reply to Aaron Bergman
> that this is almost surely NOT how things will work.
Good! :-)
>>String theory is different. Some people complain that it does not predict
>>- but it in fact predicts a lot. To include gravity consistently to a
>>quantum theory, we were forced to go to 10 dimensions, to discover
>>supersymmetry, to allow topology change etc.
> The remaining big task is to explain why the world looks 4-dimensional,
Let me remind you that LQG is much further from explaining the dimension
of our real world than string theory. You put the dimension "4" in
(already at the beginning) and LQG people can, of course, construct
similar theories in 3, 11 (as we discussed recently) or 2001 dimensions.
String theory is different: it is a unique structure that transmutes even
the question of the number of large dimensions into a dynamical question
that can be in principle determined. But it is not sure that the theory
will determine it. On the contrary, I think that the SUSY vacua in many
dimensions, for example, will remain exact solutions of the ultimate
formulation of string theory. We will be just forced to accept that there
are also other vacua, just like there are other continents than America.
:-) Nevertheless, with a collection of anthropic arguments, you could
finally single out the correct vacuum of string theory (which we live in)
uniquely and you should be able to calculate all its parameters. Maybe,
our vacuum is the only sufficiently stable state with 4 large dimensions
and broken SUSY.
> why it doesn't look supersymmetric, why we don't see topology change,
There is a better solution than just to try to explain SUSY breaking etc.:
wait till 2006 and observe at LHC that the world *is* supersymmetric; at
the energies available at LHC, SUSY is slowly becoming restored. String
theory is not a vague theory that must hide its predictions. On the
contrary, it (more or less) boldly asserts that SUSY exists and I have bet
$1000 that it will be observed. ;-) There are many bold predictions of
particular stringy scenarios and it is likely that one of them will be
confirmed soon experimentally.
The question "why we do not see topology change" was answered already in
the first papers that proved that topology change (in the form of flops of
Calabi-Yau) is possible. In fact, this is precisely the way how they
proved that topology change is possible. :-)
> etc..
Not sure what this "etc." stands for, but I believe that you can get a
fair answer for any question of this sort. The question is whether you
want to learn more about the correct answers or whether the questions are
meant just to make doubts about something.
> That is, why the low-energy effective limit matches the world we see
> around us. Note that this task is very much like the remaining big
> task of loop quantum gravity!
The difference is that we have very specific stringy models that
qualitatively predict the whole spectrum of particles of the Standard
Model, together with gravity, in a coherent fashion. ;-) There is of
course still a long way to go, but it is not fair to say that the state of
both fields is equal. Among many things, such a claim implies that you think
that any paper on LQG is 50 times (or so) more important than an average
stringy paper. Sorry, I definitely disagree and any number above 1
(instead of 50) is completely unacceptable for me. ;-)
Lubos Motl
moderators can put it in a new thread? ;-)
[Moderator's note: This post has the new Subject: heading you gave it.
But in a threaded newsreader, it will still appear to be part of the
original thread, because the References: header is intact. For
future reference, if you want to make sure a post starts an entirely
new thread, delete the References: header before submitting it. (I
could have done that for you, but changing the Subject: header while
maintaining the threading is a reasonable and quite common way to
organize big threads whose subject drifts, so I decided to leave it
the way it is.) -TB]
> John Baez (LQG vs. M-theory discussion): Except, of course, in the
> real world. This is what makes me nervous. Somehow supersymmetry must
> be broken to obtain a theory that's consistent with experiment. I
> haven't heard any good explanation of how this is supposed to work.
> What is your attitude towards this issue?
Well :-), the problem you mention is called "supersymmetry breaking".
Best wishes
Lubos Motl
> >Yup. I've spent a lot of time talking to Ashtekar, Rovelli,
> >Smolin, and other people who work on loop quantum gravity.
> >They all agree that this is the big task. Most of them are
> >working on it in one way or another.
>
> What surprises me is why so much effort is put into proving quantum
> gravity has the correct classical limit, in contrast to, say, that QED has
> the correct classical limit? Is there any special reason for it, other
> than the general believe in better understand of QED? Or maybe the later
> fact is somehow fundumentally much more obvious?
QED is a standard quantum field theory that starts with a classical field
> I agree. I don't know any such people - do you?
Yes, I do. But I would prefer not to mention their names. It is not you,
John, however! :-)
>>... what I have problem with is the relatively simple
>>machinery of Loop Quantum Gravity, with Einstein-Hilbert action postulated
>>at all distance scales with no corrections at all, written in some simple
>>Yang-Mills variables - and with people who claim that this is the ultimate
>>solution of Quantum Gravity.
> Do you know anyone who has made such a claim? I don't. It's possible
> you've just talked to people who say that some *other* people make this
> claim.
Well, I am not completely sure which claim of mine you find so
controversial, but the claims can be found e.g. in Rovelli's review of LQG
http://arXiv.org/abs/gr-qc/9710008
> As for "relatively simple machinery", I think that's good rather than
> bad.
If I compare two working theories, I prefer the theory with the simple
machinery, too. ;-) But note the word "working". I do not think that you
can take Newton's equations, for example, and claim that they solve the
problem of Quantum Gravity. They are simple, they are fine. But the
problem is that they do not solve it. Quantum Gravity is one of the
greatest problems in physics - and it requires very strong weapons!
> As for "the Einstein-Hilbert action postulated at all distance scales
> with no corrections at all", I've explained in a reply to Aaron Bergman
> that this is almost surely NOT how things will work.
Good! :-)
>>String theory is different. Some people complain that it does not predict
>>- but it in fact predicts a lot. To include gravity consistently to a
>>quantum theory, we were forced to go to 10 dimensions, to discover
>>supersymmetry, to allow topology change etc.
>
> The remaining big task is to explain why the world looks 4-dimensional,
Let me remind you that LQG is much further from explaining the dimension
of our real world than string theory. You put the dimension "4" in
(already at the beginning) and LQG people can, of course, construct
similar theories in 3, 11 (as we discussed recently) or 2001 dimensions.
String theory is different: it is a unique structure that transmutes even
the question of the number of large dimensions into a dynamical question
that can be in principle determined. But it is not sure that the theory
will determine it. On the contrary, I think that the SUSY vacua in many
dimensions, for example, will remain exact solutions of the ultimate
formulation of string theory. We will be just forced to accept that there
are also other vacua, just like there are other continents than America.
:-) Nevertheless, with a collection of anthropic arguments, you could
finally single out the correct vacuum of string theory (which we live in)
uniquely and you should be able to calculate all its parameters. Maybe,
our vacuum is the only sufficiently stable state with 4 large dimensions
and broken SUSY.
> why it doesn't look supersymmetric, why we don't see topology change,
There is a better solution than just to try to explain SUSY breaking etc.:
wait till 2006 and observe at LHC that the world *is* supersymmetric; at
the energies available at LHC, SUSY is slowly becoming restored. String
theory is not a vague theory that must hide its predictions. On the
contrary, it (more or less) boldly asserts that SUSY exists and I have bet
$1000 that it will be observed. ;-) There are many bold predictions of
particular stringy scenarios and it is likely that one of them will be
confirmed soon experimentally.
The question "why we do not see topology change" was answered already in
the first papers that proved that topology change (in the form of flops of
Calabi-Yau) is possible. In fact, this is precisely the way how they
proved that topology change is possible. :-)
> etc..
Not sure what this "etc." stands for, but I believe that you can get a
fair answer for any question of this sort. The question is whether you
want to learn more about the correct answers or whether the questions are
meant just to make doubts about something.
> That is, why the low-energy effective limit matches the world we see
> around us. Note that this task is very much like the remaining big
> task of loop quantum gravity!
The difference is that we have very specific stringy models that
qualitatively predict the whole spectrum of particles of the Standard
Model, together with gravity, in a coherent fashion. ;-) There is of
course still a long way to go, but it is not fair to say that the state of
both fields is equal. Among many things, such a claim imply that you think
that any paper on LQG is 50 times (or so) more important than an average
stringy paper. Sorry, I definitely disagree and any number above 1
(instead of 50) is completely unacceptable for me. ;-)
A.J. Tolland
> One can say - again - that people understand many possible mechanisms of
> the SUSY breaking. The problem is that there are too many possibilities
> and no clear formulation of the stringy theories that can pick up the
> correct model.
This seems to me to be more than a problem of formulation.
You've skipped over the deepest and nastiest part of the problem: We may
know a number of mechanisms for breaking SUSY, but we have no clue how
M-theory chooses a vacuum! Maybe you're happy with the anthropic
principle. I'm not. Too many questions in M-theory -- indeed most of its
explanatory power -- hang on vacuum selection. I want a mechanism, and
I'm not willing to call M-theory complete until we know of one.
--A.J.
A.J. Tolland
> Hi A.J. Tolland,
>
> I tried to figure out your first name, but I failed: A.J. Tolland
> is the maximum I can see anywhere. :-) Thanks for your critique.
Hi Lubos,
I've used the nickname "A.J." for more than a decade now, long
before I had any presence on USENET or the WWW.
> [The claim that "Quantum Gravity = M-theory"] is justified in the sense
> that I can refer to about 10,000 scientific papers that justify it and
> bring pieces of evidence for this assertion. If this is not enough, I
> give up - I cannot justify the statement more than I did. :-) The
> assertion has not been accepted to the Bible yet, for example.
It is definitely not enough. Those 10,000 scientific papers
support the claim that M-theory is _a_ theory of quantum gravity, not
_the_ theory of quantum gravity. This distinction is crucial. In any
case, sheer volume of writing justifies nothing. Just look at how much
has been written about the truths in the Bible!
In perfect honesty, I don't really think that it's very important
that M-theory be the unique quantum theory of gravity. Gravity may be an
excellent clue as to the nature of ultra-high energy physics, since it
seems to be a massively suppressed effect stemming from from Planck scale
physics.. But I find M-theory interesting as physics, not so much because
it reconciles Einstein and Planck -- like I said, I won't be surprised if
many theories can do this -- but because
(a) it reduces many deep and seemingly unrelated physical
questions -- Why D=4, why such a bizarre mass hierarchy, why 3
generations, why spontaneously broken symmetries, why these values for our
physical constants, etc? -- to a single problem of vacuum selection, and
(b) it could teach us a great deal about the non-perturbative
structure of quantum physics.
> Yes, I agree. Any consistent theory reducing to Einstein's gravity (maybe
> plus other fields) at low energies is a theory of quantum gravity.
> Therefore I say that M-theory in the broad sense is the only theory of
> quantum gravity. If you prove me wrong, it will be extremely exciting!
I think you are wrong, but in a very un-exciting sense. I do not
believe we have any working quantum theories of gravity yet, i.e. I think
the "M" stands for "Missing". You're from Rutgers; I suppose you think it
stands for "Matrix"? :)
> > My point here is that there won't necessarily be a unique quantum
> > theory of gravitation; there may be many quantum theories which imply
> > Einstein's equations.
>
> Yes, this conjecture was plausible a priori but over 50 years of research
> have shown that it is very unlikely.
50 years of research have shown that it is exceedingly difficult
to construct such a theory. In my opinion, the difficulties mankind has
encountered are more a testament to our poor abilities than a proof that
M-theory is the mathematically unique quantum theory of gravity.
This is actually related to an interesting, but more concrete
physics question ("interesting" in the sense that Nobel prizes might hang
on it): How secure is our claim that stringy/braney theories require
SUSY? Is there a solid physical argument that any self-consistent
non-perturbative theory of brane-like objects must be supersymmetric, or
do we merely think that this is the case because we don't know how to
construct anything else?
> The history of science has shown that the non-renormalizable theories
> were always nonrenormalizable because they neglected important physics
> at very short distances.
The history of science has shown that nonrenormalizable
_effective_ QFTs were non-renormalizable because they neglected important
short distance physics. I do not know of any proof that the fundamental
theory must be renormalizable.
To be honest, I have yet to hear of any way of realizing
renormalization theory within the context of, say, C*-algebraic QFT.
Does anyone know if such a thing exists, or if renormalization should only
show up when one starts thinking about approximate field coordinates on
the net of algebras?
> Once again: yes, I believe that M-theory (in the broad sense) is the only
> mathematically consistent quantum theory of gravity above 3 dimensions.
As Jacques Distler has occasionally said in other contexts:
"Prove it."
> There are very few consistent quantum field theories, in a sense.
In what sense? There are very few consistent renormalizable field
theories with 4D Poincare symmetry and finitely many particles species,
yes, but that can be explained by the fact that -- in these theories --
you can only write down renormalizable interactions for the particles with
very low spins. There are also very few consistent 4D QFTs in the
axiomatic sense: None, so far as I know. :)
Would you please make your statement more precise?
--A.J.
Robert C. Helling
On 12 Jun 2001 21:51:07 GMT, Lubos Motl <mo...@physics.rutgers.edu> wrote:
>Nevertheless I still think - although no clear proof is available - that
>the condition that the vev in coherent states is continuous is equivalent
>to the condition that the eigenvalues of the given operator are
>continuous. You cannot create much of a coherent state from a discrete
>operator, I think.
C'mon, from the home work excercises to my quantum mechanics 1 course:
The Hamiltonian for the harmonic oscillator H = w (a^\dagger a +1/2) =w (N+1/2)
clearly has discrete eigenvalues. Now, consider the state
|r> = exp(-r^2/2) \sum_n r^n/sqrt(n!) exp( -iw(n+1/2)t ) |n>
Check
1) |r> solves the time dependant Schroedinger equation
2) <r|r> = 1
3) <r|H|r> = w (r^2 + 1/2) i.e. continuous!
Robert
--
.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oOo.oO
Robert C. Helling Institut fuer Physik
Humboldt-Universitaet zu Berlin
print "Just another Fon +49 30 2093 7964
stupid .sig\n"; http://www.aei-potsdam.mpg.de/~helling
Lubos Motl
> A.J.: It is definitely not enough. Those 10,000 scientific papers
> support the claim that M-theory is _a_ theory of quantum gravity, not
> _the_ theory of quantum gravity. This distinction is crucial. In any
> case, sheer volume of writing justifies nothing. Just look at how much
> has been written about the truths in the Bible!
Well, the Bible contains 30,000 verses which equals a small fraction of
the stringy literature. Should not there exist more people who take
strings seriously - at least equal to the number of Christians? :-) Well,
I am kidding, of course, and I agree that the volume does not imply
everything. BTW while those 10,000 papers suggest that M-theory is *a*
theory of quantum gravity, the other zero or 100 reasonable papers (and
other papers that do not exist) about a definition of quantum gravity
suggest that M-theory is *the* theory. :-)
> physics.. But I find M-theory interesting as physics, not so much because
> it reconciles Einstein and Planck -- like I said, I won't be surprised if
> many theories can do this -- but because
I would be extremely surprised if you found *one* new theory that
reconciles them and showed that it gives you a consistent S-matrix for
graviton scattering. You would certainly become very famous.
> generations, why spontaneously broken symmetries, why these values for our
> physical constants, etc? -- to a single problem of vacuum selection, and
OK, sure, the choice of vacuum is responsible more or less for all the
physical properties.
> (b) it could teach us a great deal about the non-perturbative
> structure of quantum physics.
I do not understand what you mean by the general term "nonperturbative
structure of quantum physics". Different theories (with different
"Hamiltonians") have generally completely different nonperturbative
physics. Even the same theory - such as string theory - can admit many
perturbative descriptions and in each of them the definition of
"nonperturbative physics" is different. Before you say what is the
Hamiltonian (dynamics) and what is the coupling constant that defines the
perturbative expansion, the term "nonperturbative structure of quantum
physics" is empty. Furthermore from a more general viewpoint, dividing
theory into perturbative and nonperturbative physics is purely a technical
question. There is one theory only - for example QCD - and perturbative or
nonperturbative approaches are merely different technical tools to
calculate different things in the same theory. Many quantum theories are
understood precisely and they have no magic nonperturbative physics
waiting for us to analyze. Quantum field theories were first understood
perturbatively (Feynman diagrams) but using lattices etc. you can
calculate anything exactly in principle. It was only string theory that
was defined perturbatively only for about 20 years, but this is not the
case anymore.
> I think you are wrong, but in a very un-exciting sense. I do not
> believe we have any working quantum theories of gravity yet, i.e. I think
> the "M" stands for "Missing". You're from Rutgers; I suppose you think it
> stands for "Matrix"? :)
Some time ago, Lenny Susskind told me that I should change my first name
because then I would be known to the world as Matrix Motl. Of course, I
mostly disagree that "M" stands for "Missing" and e.g. the matrix model is
a complete definition of M-theory on certain backgrounds. Type IIA string
theory can be defined exactly using MST (= My String Theory or Motl String
Theory or also Matrix String Theory), later promoted and improved by DVV.
;-) A lot of things is known so M-theory can be missing only partially. :-)
> 50 years of research have shown that it is exceedingly difficult
> to construct such a theory. In my opinion, the difficulties mankind has
> encountered are more a testament to our poor abilities than a proof that
> M-theory is the mathematically unique quantum theory of gravity.
Maybe. And maybe we cannot move by superluminal speeds only because we are
too lazy. ;-) Well, I do not think so. The world seems comprehensible
(although this property is the most imcomprehensible thing about the
world) and it seems that the human brain has the capacity to figure out
essentially everything, after a sufficient effort.
> This is actually related to an interesting, but more concrete
> physics question ("interesting" in the sense that Nobel prizes might hang
> on it): How secure is our claim that stringy/braney theories require
> SUSY? Is there a solid physical argument that any self-consistent
It is likely but it is not completely sure. Indeed, there are stringy
models that break SUSY already at the string scale etc. But supersymmetry
is certainly a close friend of string theory (SUSY was also discovered in
stringy context) and there are also other reasons to believe that there is
a low energy SUSY in our world.
> The history of science has shown that nonrenormalizable
> _effective_ QFTs were non-renormalizable because they neglected important
> short distance physics. I do not know of any proof that the fundamental
> theory must be renormalizable.
I do not understand this paragraph at all. First of all, the fundamental
theory cannot be a quantum field theory, because it contains gravity, so
it is not clear why you ask whether it is renormalizable. Probably because
you assumed that the fundamental theory can be a quantum field theory. OK,
let us accept this (incorrect) assumption. Then you think that the
fundamental theory could be nonrenormalizable in the sense that we should
believe that the probabilities are infinite? I am not sure what you mean.
The fundamental theory should first of all make sense and therefore there
must be ways how to extract finite results out of it. Concerning the
adjective "effective", today we understand all quantum field theories as
effective field theories of some sort, and thus I do not know why you
underlined this (trivial) adjective.
> To be honest, I have yet to hear of any way of realizing
> renormalization theory within the context of, say, C*-algebraic QFT.
> Does anyone know if such a thing exists, or if renormalization should only
> show up when one starts thinking about approximate field coordinates on
> the net of algebras?
My answer can hardly satisfy you. But I think that you should not expect
that you can hide the renormalization group when you talk about quantum
field theories properly. Any C*-algebraic QFTs or anything must be
expressed with respect to an energy scale, accepting the rules of the
renormalization group, otherwise it is a physically irrelevant game with
some mathematical objects.
> > Once again: yes, I believe that M-theory (in the broad sense) is the only
> > mathematically consistent quantum theory of gravity above 3 dimensions.
>
> As Jacques Distler has occasionally said in other contexts:
> "Prove it."
Well, maybe next time. ;-) I could prove it, of course, by definition.
M-theory is defined to be the unified collection of all the good and
working ideas about the fundamental physics, containing quantum mechanics
and general relativity at low energies.
> > There are very few consistent quantum field theories, in a sense.
>
> Would you please make your statement more precise?
I think that you did a good job. ;-) I meant something similar: 4D quantum
field theories do not allow you spin greater than 1, even the spin 1
fields must be associated with gauge symmetries whose anomalies must be
cancelled. A truly consistent QFT should be either asymptotically free or
at least have a UV fixed point. In other words, QFT does not allow you to
add fields of spin 2 for example (necessary for gravity) or many other
features that one might consider "easy" a priori.
Toby Bartels
>But I do not
>think that anyone is justified to think or to claim that he or she is as
>important physicist as Peter Higgs (or even Ed Witten) just because he or
>she is not working on string theory and is "different". Difference does
>not imply quality!
I don't know of anybody who believes that they are a good physicist
*because* they are working on something different from string theory.
But I do know people who think that another person is a *bad* physicist
because the other is working on something different from string theory.
Difference certainly does not imply quality --
but it doesn't imply the lack of quality either!
It is my personal opinion that string theory is wrong;
nevertheless, I want people to continue to work on it,
in case *I* am wrong. This is tolerance, if you like,
but it in no way compromises the strength of my opinion.
-- Toby
to...@math.ucr.edu
A.J. Tolland
> What surprises me is why so much effort is put into proving quantum
> gravity has the correct classical limit, in contrast to, say, that QED has
> the correct classical limit? Is there any special reason for it, other
> than the general believe in better understand of QED? Or maybe the later
> fact is somehow fundumentally much more obvious?
QED, or rather the Standard Model, is already formulated as an
effective field theory, thanks to the use of path integrals. (Asymptotic
freedom lets us pretend that the particle degrees of freedom are
fundamental rather than effective, if we like.) String theory and LQG on
the other hand aren't really formulated in this way, so you have to check
to make sure that the effective description is correct.
--A.J.
Lubos Motl
> The first answer is: otherwise we're screwed, and loop quantum
> gravity probably won't work!
Yes, I think there are reasons to believe that the renormalization can be
finite - although I have heard of lattice numerical calculations involving
spin networks and showing that G diverges at finite distances... Good to
hear more optimistic comments from John.
> Yes indeed: if we had quantum gravity coupled to QED, and
> our theory of quantum gravity imposed a shortest distance...
To make quantum gravity predictions finite, one is forced to make the
geometry at Planckian distances fuzzy. String theory is an example (and
most likely the only working example) how to smooth out the Planckian
deadly undulations. And string theory also automatically implies the UV
finiteness of all the interactions. In this sense, curing gravitational UV
problems also cures other UV problems. In string theory this can be
explicitly computed but I have not heard of evidence why LQG should be
able to do the same with other forces. In fact I have not even heard of a
well-defined theory coupling LQG to other fields and thus the claims "LQG
might solve something" sound unjustified to me. Furthermore it seems that
in the LQG context, one can add the other interactions "by hand" only as
field theories and therefore they should finally satisfy the standard
rules of Quantum Field Theories, including divergences and
renormalization.
> Of course, most string theorists don't like this idea, because
> they are looking for a more or less unique "theory of everything",
> not "quantum gravity as a universal cure for quantum field theories
> with nasty ultraviolet problems". Indeed, Witten once told me
> that he hopes loop quantum gravity will fail, because otherwise
I do not want to make anyone too frustrated, but I agree with Witten's
position. The constraints for a theory of everything must be tough -
otherwise there could be hundreds of completely different candidates and
we had no chance to figure out the correct one. String theory is an
extremely rigid theoretical structure that allows no fluctuations from its
"stringiness" - any modification would make the theory inconsistent. But
even string theory has a large collection of allowed "discrete" choices.
Their number might be reduced in the future when we understand the
selection mechanisms better.
> we will have a lot of trouble determining the correct "theory of
> everything". While I understand this hope, it doesn't strike me
> as a very convincing argument that loop quantum gravity WILL fail.
> So for now, I will keep marching on trying to get it to work....
Good luck. Yes, this line of Witten's thinking is partly a religion, but
it is a well justified religion. The experience in physics showed us that
anytime we had to solve big problems, we were forced to make dramatic
revolutions in our thinking. And a good theory finally turned out to be
essentially unique and gave us a lot of new predictions that we did not
put in. At the current state, the Standard Model (plus general relativity)
allows us to predict anything we have observed so far from 19 parameters.
I see no reasons to believe that at the (almost) end this successful and
beautiful strategy would break down; on the contrary, things seem
increasingly constraining as we approach more far-reaching theories. Based
on my knowledge of history of physics, I do not believe that it is
possible to formulate a viable candidate for the theory of everything in
many ways which are not dual to each other. Even if the people "glued"
together some naive theory similar to LQG that would seem to be able to
match the observations of gravity in a quantum framework, people could not
be satisfied. We would still face the same questions: why there are just 3
generations of particles with the given quantum numbers? Why the
parameters take those values? Even if some people felt that they succeeded
with LQG, they would have to return to (something like) string theory at
the end.
I personally do not find useful the attempts to quantize gravity
"separately". The reason why we want to quantize gravity, in my
understanding, is that gravity is not the only force and we also observe
many phenomena based on other forces and particles, satisfying the rules
of Quantum Mechanics. We are working on Quantum Gravity because we need to
unify gravity with the other forces and matter that require a quantum
framework. Attempts that cannot formulate a theory containing both gravity
and other forces, are therefore - in my opinion - a failure to construct a
theory of Quantum Gravity.
An important criterion whether we are making progress, is also measured by
our ability to explain data that we could not explain before. Today, in
the Standard Model, we cannot explain about 19 ultimate parameters only
(plus about 10 because of the neutrino masses that are slowly revealed to
be nonzero). If we gave up the task to explain all those numbers at the
end, I think that it would mean that we gave up the challenge to make
further progress in theoretical physics. And to explain those numbers
etc., one must find a mechanism underlying all the forces, explaining
their properties from a more fundamental and unified starting point: just
like we understand spectra of hydrogen and helium from dynamics between
protons and electrons. String theory is able to do this. But I do not
understand the motivation of an effort that gives up the task to
calculate numbers which we do not understand yet.
So hopefully I have explained why I feel that LQG is (even if it gets more
successful than so far) pretty orthogonal to what I would consider
"progress in fundamental physics". For me, it is mostly an attempt of the
people to defend the old ways of thinking (to say that there is nothing
wrong about them, in this case about the local field theories applied to
all the forces including gravity) - much like some people tried to avoid
the revolutions of relativity and/or quantum mechanics. But furthermore I
believe that a non-stringy quantization of gravity above 3 dimensions is
necessarily inconsistent.
> a "Landau pole", and we cluck our tongues disapprovingly, because it's
> not a good thing. For example, this is what happens in quantum
> electrodynamics when we don't include the weak force. Of course,
It happens even if you include the weak force, the hypercharge U(1) -
close to the electromagnetic one - still gets (infinitely) stronger at
short distance scales. You must embed the Standard Model into a bigger
theory (Grand Unified Theory etc.) without U(1)'s to make it
asymptotically free.
> Okay, now for the really fun part. Perturbative quantum gravity
> in 2 dimensions is not only renormalizable (because this is the
> upper critical dimension), it's also asympotically free!
Not sure how you meant this. Quantum gravity in 2 dimensions has minus one
physical polarizations - i.e. it is enough to kill one scalar. Quantum
gravity in 2 dimensions is well-defined as a conformal (Weyl rescaling
invariant) field theory - an interaction whose strength is
scale-independent (G in 2 dimensions is dimensionless.) If you thought
that this is a counterexample of a consistent quantum theory of gravity
which is not string theory, I would like to warn you: this exceptional
consistent quantum theory of gravity is called "perturbative string theory
formulated on the worldsheet". ;-) The Einstein Hilbert action is a
topological invariant in 2D and is responsible for the correct power of
the string coupling, depending on the topology of the Riemann surface
(its genus counts the number of loops, the Riemann surface is a stringy
Feynman diagram, representing a history of splitting and joining strings).
> In fact, it's easy to see that as we keep cranking up the momentum,
> G will approach the value for which
>
> dG
> ------- = 0
> d(ln p)
>
> We call this value an "ultraviolet stable fixed point" for the
> gravitational constant. Mathematically, what we've got is a flow
This is completely unfair. You forgot to say that the claim that 4D
gravity has a 4D UV fixed point is just a conjecture - and most likely an
incorrect conjecture. Certainly there is no evidence.
> of higher momenta. This particular kind of ultraviolet stable fixed
> point - coming from an asymptotically free theory in dimensions above
> its upper critical dimension - is called a "Wilson-Fisher fixed point".
Well, you kind of confused it. Gravity is not asymptotically free (even in
2D it's not) so one also does not expect a Wilson-Fisher fixed point.
Lubos Motl
>> One can say - again - that people understand many possible mechanisms of
>> the SUSY breaking. The problem is that there are too many possibilities
>> and no clear formulation of the stringy theories that can pick up the
>> correct model.
A.J.Tolland:
> This seems to me to be more than a problem of formulation.
> You've skipped over the deepest and nastiest part of the problem: We may
> know a number of mechanisms for breaking SUSY, but we have no clue how
> M-theory chooses a vacuum! Maybe you're happy with the anthropic...
Yes, I tend to agree. I am the last one who would be happy with the
anthropic explanations. For instance our paper
http://arXiv.org/abs/hep-th/0007206
can be understood as a general criticism of the anthropic ideas. But we
cannot rule out that the ultimate theory has many "vacua" and "our" vacuum
is chosen by accident or because it is among the most viable or potent
vacua that allow an intelligent life. I believe that the Universe is
described by a unique theory - and that our experience can serve as
evidence. But this theory can have many or very many solutions
(spacetimes). While I would be happier if even the solution was unique, I
do not see evidence for this belief (as opposed to the belief that there
is a unique theory of everything).
The reason why I say that it is the problem of the formulation is the
following: today we cannot predict whether a string vacuum with one
Calabi-Yau manifold can (and tends to) decay into another vacuum and where
string theory wants to stabilize its scalar fields/moduli; consider a
"vacuum" to be a minimum of some function (a potential) defined on some
complicated configuration space. It is not only because we are poor
mathematicians: we even do not know the exact conditions that create a
potential at the generic points of the configuration space etc. While we
understand many things qualitatively and most things perturbatively (and
we have exact quantitative definitions of some vacua - usually those
irrelevant for the real world), we must learn the correct rules that
govern the vacuum selection, SUSY breaking etc. We need to find out the
formulation. But if we find it one day, the problem will reduce to a
well-defined mathematical question. And it may happen that the
mathematical conditions will have several (or many) solutions: five, one
hundred or 10^{10^{120}}. But if this is the case, you will not be
justified to protest against it. The existence of many vacua will become a
true property of the real world. We could be still able to pick the
correct one - but there will not be a truly scientific explanation why we
live in this vacuum, just like there is no truly scientific explanation
why you live in the USA.
> principle. I'm not. Too many questions in M-theory -- indeed most of its
> explanatory power -- hang on vacuum selection. I want a mechanism, and
> I'm not willing to call M-theory complete until we know of one.
Yes, I agree. I certainly also do not call M-theory as understood today a
"complete" theory. :-) This is why people should try to complete it.
Steve Carlip
> Well, in LQG the situation looks as follows: the eigenvalues of the area
> operators are quantized, however the expectation values in a state do not
> have to be quantized. Because LQG is completely discrete, in a sense,
> I am not sure whether you can create anything like the coherent states
You almost certainly can: see Sahlmann, Thiemann, and Winkler,
gr-qc/0102038, for example, or Corichi and Reyes, gr-qc/0006067.
It's not clear that these are the ``right'' coherent states, but they're
coherent (so to speak) attempts to understand the classical limit.
> OK, let me summarize. It is true that we know that string theory has a
> continuous spectrum of the areas understood as the expectation values
> - but it is not clear about the eigenvalues of the area operators
> corresponding to a given 2-surface in the coordinate space. As I have
> mentioned, it is likely that such operators will never have a rigorous
> definition in string theory simply because it is not a field theory living
> in on a classical spacetime of fixed topology
OK. I think we agree.
> One can also imagine that such "area operators" can have some definition
> in string theory - and they might even match LQG-like rules. If this was
> proved, it would be really exciting. But don't be too optimistic.
I'm not especially optimistic that this will happen. Actually, I think
the situation is worse: even if it did happen, it might not mean much.
The problem is that when you're talking about quantum gravity, you
don't have a whole lot of experimental tests of things like spectra of
areas. So ``area operator'' doesn't really mean much more than ``some
operator that looks like area in the classical limit.'' Put this way, it's
clear that the ``area operator'' is not unique---you can add any term
you want that's proportional to Planck's constant and still have the
same classical limit.
(Imagine doing quantum mechanics without the hydrogen atom
spectrum. There are lots of orderings you can choose for the
Hamiltonian, and they'll give you different spectra. Nowadays we
cleverly set up the problem so that one ordering seems ``natural,''
but that's because we already know the answer.)
So when loop quantum gravity folks say that ``area'' is quantized with
a particular spectrum, they're really only talking about ``some operator
(that looks nice in loop variables) whose classical limit is area.'' If
you invented an operator in string theory that had the same spectrum,
that wouldn't inherently mean too much, since there's so much
arbitrariness to begin with in what you mean by ``the'' area operator.
It would take more---something like a string theory area operator
that was ``obviously'' right in a natural set of string variables. If
*that* happened, it would be exciting, not so much because of what
it said about area, but because it would suggest a natural connection
between string and loop variables.
I'm not holding my breath.
Steve Carlip
Josh Willis
In article <Pine.SOL.4.10.101060...@strings.rutgers.edu>,
Lubos Motl <mo...@physics.rutgers.edu> writes (in part):
> Steve Carlip wrote:
>> That's true. To be fair, though, string theory has the opposite
>> problem---while it's easy enough to get something like the
>> usual general relativity at long distances (modulo questions
>I certainly disagree with the word "easy".
You are right to disagree with it, but a little careless in your own
use of it, in my opinion...
[stuff deleted]
>And this problem [to derive semiclassical quantum gravity from a
>consistent theory] is also the physical motivation to study both string
>theory and LQG, I think. We know that there exists gravity above 70
>microns :-) but there is no evidence about standard general relativity at
>Planckian and shorter scales. On the contrary, I would emphasize that it
>is very easy to construct some "nonperturbative physics" such as Loop
>Quantum Gravity that looks covariant to some extent etc.
Very easy? Name *one* such theory. Aside from loop quantum gravity.
This is a point I have heard Ashtekar and others make repeatedly: we
are not saying that loop quantum gravity is "the" correct way to
quanize gravity---the jury is still out on that. But after many
people working for many years there was, prior to LQG, no
nonperturbative quantization of gravity *at all*. Thus, even should
LQG turn out to be incorrect, we feel we should learn quite a lot
just from the effort.
As Steve Carlip has pointed out on this thread, things are known to be
complicated in 2+1 dimensional gravity, where there are several
inequivalent quantizations. It could be the case that 3+1 dimensional
gravity, unlike its lower dimensional counterpart, for some reason has
an essentially unique quantization. If so, it's not clear that we
have it yet, in either loop quantum gravity or string/M theory. But
it also could be the case that 3+1 dimensional gravity also has many
inequivalent quantizations, but we haven't yet been clever enough to
figure some/any of them out. We have one advantage over 2+1
dimensional gravity in this respect: we can (at least hope to)
experimentally test 3+1 dimensional quantum gravity theories.
However, we are not yet to the point that we can take advantage of
this advantage, if we ever shall be. But certainly people are working
on this.
>The difficult
>part of the task is to derive the low-energy gravity from such a quantum
>theory. If someone believes that the correct low-energy physics is
>guaranteed just because we have put the most beautiful (Einstein's)
>equations in, he/she is a victim of a completely unjustified religion.
I think you should either back up your claims about the beliefs of
workers in LQG with some hard evidence, or be a little more cautious
in those claims. No one I know in the loop quantum gravity community
argues that LQG "necessarily" has general relativity as a
semiclassical limit, simply because it uses the Einstein-Hilbert
action as a starting point. (Actually, to be technically
correct, it doesn't even do that, at least according to the way the
terminology is normally used. Einstein-Hilbert usually refers to the
action written in terms of metric, rather than connection variables.)
Right now it is a central problem in LQG to try to see whether or not
it has GR as a semiclassical limit, and many people are working on
this. Thiemann and collaboraters are working on a coherent state
approach to this using constructions by Brian Hall; Madhavan
Varadarajan is working on another (C* algebraic) approach for U(1)
Maxwell theory which he and Ashtekar and collaboraters are seeking to
better understand and then generalize to SU(2). The reason people are
only now working on this is that other parts of the theory had to be
in place first: constructing the Hilbert space and operators on it.
Even these issues are by no means fully resolved, but I don't see how
progress on the semiclassical limit of the theory could have been made
without at least some understanding of the structure of the full
quantum theory. One can't do everything at once.
[more snipping with abandon]
>> problems: the required nonlocality of diffeomorphism-invariant
>> observables, the ``problem of time,'' the question of what can
>> replace a smooth classical spacetime at short distances, etc.
>I agree with identification of some of the problems. What string theory
>has not explained at all so far, is what happens when one falls to the
>black hole, how the horizon encodes the information about the interior
>(nature of the holographic code in general), whether and why the degrees
>of freedom inside are just different representation of the Hawking
>radiation, in what sense the theory is nonlocal i.e. why the Hawking's
>proof of loss of information is wrong (or is it correct? Then how should
>we generalize QM?), how do we define the theory at cosmological
>backgrounds, which time coordinates we can choose and what is the solution
>of the problem of time, why does the universal formula for entropy of
>horizons work etc. Note that LQG has solved any of these problems either.
I think more could certainly be said here about problems with string/M
theory; for example, how does one calculate entropy for non-extremal
black holes? Both LQG and String/M-theory (which I shall henceforth
abbreviate as "SMT" since I am a slow and lazy typist) have both
conceptual issues to address, as you indicate above, but also a dearth
of hard numbers to verify or disprove. Indeed, I don't know that's
accurate to say that SMT has classical GR as a semiclassical limit:
yes you can expand around a given classical solution of Einstein's
equation, but has it been shown, perturbatively or otherwise, that
these solutions or an approximation to them are also a part of the
solutions of SMT? Michael Duff indicated something about recovering
Einstein's equation from SMT when he gave a talk here last semester
(or the one before, I forget), but I did not sufficiently understand
what he was saying is true of SMT, and whether it was a perturbative
or nonperturbative statement.
>I think that string theory has already said a lot about "what replaces a
>smooth classical spacetime at short distances". T-duality, noncommutative
>geometry, topology transitions etc. are pretty radical insights in this
>direction. We expect many more. Some of them might have some features
>similar to LQG. But what I have problem with is the relatively simple
>machinery of Loop Quantum Gravity, with Einstein-Hilbert action postulated
>at all distance scales with no corrections at all, written in some simple
>Yang-Mills variables - and with people who claim that this is the ultimate
>solution of Quantum Gravity.
>
>What also disturbs me is that as far as I know, no surprising consistency
>checks in LQG appeared (in deep contrast with string theory).
I don't know why you would say this. Certainly there are consistency
checks performed within LQG: the closing of the constraint algebra,
and indeed the first-class nature of the constraints come immediately
to mind. Of course the checks are not the *same* as in M-theory,
because the structure of the theories is different.
I am beginning to feel that perhaps you are not very familiar with all
of the details of what goes on in LQG. That's fine; no one can know
everything about everything. I know very little detailed stuff about
string theory (and there's a lot I don't know about LQG too, for that
matter). But it is wise to be aware of the limitations of our
knowledge of a field when criticizing it (and, for that matter, when
supporting it). I think a lack of awareness of this type contributes
to the "religious fervor" that too frequently attaches to discussions
like this; I say more about this below.
>In fact I do
>not see too many surprises at all. One of the nice things of LQG is the
>quantization of areas etc. but of course it is not too surprising that
>operators such as J^2 (angular momentum) in a SU(2) spin-network theory of
>some sort have quantized eigenvalues of some sort - in fact physicists put
>this information in. The rest seems to me as a religion based on the
>"exceptional" nature of classical Einstein's equations - and people in the
>field of LQG seem to keep our view of the world essentially identical to
>the opinion in the 1920s.
Part of what you write here is just plain wrong. Though it is true
that Penrose originally studied spin networks as a postulated
structure of spacetime, LQG does not *postulate* this; it is
*derived*. What is assumed is the action, written in terms of the new
connection variables. Originally, the canonical program sought to
describe the "state" of three-geometry at a given instant in time as a
functional of connections on space (not spacetime) in much the same
way that ordinary QM describes a state as a wavefunction on space.
Then Rovelli and Smolin noticed that, at least formally, you could
make a transform to functions on loops, a kind of nonlinear Fourier
transform. Pullin, Gambini and others noticed that in this framework
solutions to the diffeomorphism constraint could be given by knot
invariants. Ashtekar, Isham, and Lewandowski made rigourous the loop
transform that Rovelli and Smolin proposed, and thereby rigorously
constructed a space of "connections modulo gauge transformations",
usually called A/G (actually with a bar over it, but I'm not going to
attempt that in ASCII. Cognoscenti should assume it is there in what
I write below), on which measure theory could be done. Baez and others
found still other measures on this space. It was then shown that spin
networks form an orthonormal basis of certain Hilbert spaces on
(A/G,d\mu) for various different measures \mu, and that's why spin
networks are so important in loop quantum gravity. Logically, this
all depends ultimately on your starting assumptions, both in choice of
action, variables for that action, and indeed the choice to do
canonical quantization at all. But it is inaccurate to say that spin
networks were put in by hand--they weren't, and Ashtekar et al. were
surprised when they turned up. I have heard him say so.
>String theory is different. Some people complain that it does not predict
>- but it in fact predicts a lot. To include gravity consistently to a
>quantum theory, we were forced to go to 10 dimensions, to discover
>supersymmetry, to allow topology change etc. Some people criticize string
>theory that it predicts nothing new, some people complain the string
>theory involves/predicts too much new stuff (excited strings, higher
>dimensions, SUSY etc.). Some critics happily belong to both categories and
>they do not realize how inconsistent their position is... ;-)
I think what people criticize is predictions that have not yet been
experimentally verified, which includes all of the predictions you
list above. And all of the predictions of loop quantum gravity. More
to the point, what is a valid criticism is any claim that a particular
theory is "right" when it has not yet withstood experimental test.
That by no means argues that such theories must be abandoned---only
when they make predictions that are in disagreement with experiment.
And even then they may be resurrected, with modifications (though I
think Dirac's quote on this issue does fall under the category of
"easy for him to say").
More to the point, the reason loop quantum gravity is an active
program is not because it is in any way guaranteed to be right, nor
because its adherents necessarily "believe" that it is "the" theory.
Rather, one has to start somewhere, and loop quantum gravity and
string theory, both historically and philosophically, start from
different places. It should not be too surprising, then, if they end
up in different places; what would be more surprising---and
interesting---are areas where they might overlap.
Though this has been expounded many times in this newsgroup, perhaps
it bears repeating nonetheless (if this is cut by the moderators, I
guess not!). Loop quantum gravity is worked on mainly by people who
come from the relativity community; i.e., they had gravity, now they
want to quantize it. They therefore bring with them a certain mode
of thinking, and certain beliefs. For example, that gravity is really
geometry, that the diffeomorphism invariance of Einstein's theory is
telling us something deep, and we should try to hold onto that when we
quantize.
SMT, on the other hand, come from particle physics; they had quantum
field theory, and now they wanted to add gravity. This community also
has a certain mode of thinking---but it is different from that of LQG.
In SMT, people feel that the lesson from the Standard Model and its
successes was that we should pay attention to renormalizablility; it
has so far been a very reliable guide in selecting just those
interactions that nature prefers. It also emphasizes unification with
the other forces, since this was a dominant theme a notable success of
high energy theory in the latter half of this century.
Both sets of assumptions could be criticized; simply on the grounds
that they are assumptions, but also for deeper reasons as well. On
the SMT side, why should we believe that renormalizablility is an
infallible guide? In the only dimensions (i.e. less than four) where
we can rigorously do interacting quantum field theory, we have field
theories that can be shown to exist but are non-renormalizable. Also,
one could argue that with the modern understanding a la Wilson of
renormalization, we understand why we see only renormalizable
theories: they are simply effective theories described by a theory
that may be very different at higher energy scales. Certainly it is
if either LQG or M-theory is correct. On the LQG side, one has the
usual problems with time and diffeomorphism invariant operators, as
well as reliance on an action that has been experimentally probed only
at very low energy scales. The choice of canonical quantization in
the first place, and even the choice of the particular flavor of
canonical quantization that is made in LQG are subject to criticism;
if there is one thing we have learned it is that "quantization" is
anything but unique or easy.
Indeed, one could argue that there are unjustified assumptions common
to both LQG and SMT, and that both approaches draw too much assurance
from their successes. In fact, many have argued precisely this. For
example, both LQG and String/M-theory have drawn great comfort from
their derivation of black hole entropy. But Carlip and others have
presented arguments (so far only in the 2+1 dimensional case, as far
as I know) that (*very* roughly state) this entropy may be a generic
feature of *any* quantum gravity theory. Moreover, Ted Jacobson has
shown further that the entropy/area law is equivalent to GR---and
therefore argued that it may make no more sense to quantize
Einstein's equation than to quantize the equation of state of an ideal
gas.
One can go further still. LQG people criticize string theorists for
using a background metric, but what gives LQG the right to start from
a differentiable manifold, especially if we don't end up with one when
all is said and done? Chris Isham I know has made this point and
investigated other possibilities; possibly others have as well that I
{don't know}/{can't remember}. There is the causal net approach that
Sorkin and others have worked on; one could certainly argue that it is
highly suggestive that the causal structure of a 4-manifold determines
the metric up to conformal structure. So perhaps this too is a clue to
be followed.
My answer to all of these criticisms: more power to them!
Particularly in the absence of experiment to guide us, we must make
some assumptions to guide us. We will not all make the same
assumptions. From a psychological point of view, it is natural that
people are more attracted to one approach over another, particularly
if they have made important contributions to one approach. But
psychology is not the ultimate judge of these matters. However, it
should be noted that great progress in the past---quantum theory
springs immediately to mind---has been made by starting from some
wrong and even inconsistent assumptions, which later provided the hint
needed to put things on a firmer footing. Whether the finite
perturbation terms of string theory or the general covariance of loop
quantum gravity will turn out to be such hints, or simply red
herrings, I do not know. But most of the workers I know in LQG (and
one runs into a fair number at Penn State) do not have LQG as a
"religion", but rather as a starting set of assumptions with something
natural to commend them, but by no means without possible problems.
But the problems in either SMT or LQG can only be uncovered by further
work, not religious enlightenment. Your signature file notwithstanding.
Now some comments on later statements in this thread:
>> I'd like to critique some of the statements you made about quantum
>> gravity. My basic problem is that you are making the assertion that
>> quantum gravity and M theory are the same thing. This assertion is
>> completely unjustified.
>Yes, I agree. Any consistent theory reducing to Einstein's gravity (maybe
>plus other fields) at low energies is a theory of quantum gravity.
>Therefore I say that M-theory in the broad sense is the only theory of
>quantum gravity. If you prove me wrong, it will be extremely exciting!
No, no, no! It is *your* burden to prove what you say is right, not
anyone else's to prove that it is wrong! And you have not come close to
doing so. For starters: *define* M-theory. No waving of hands, no,
"well, we expect it looks like 11-dimensional supergravity in a low
energy limit"; give a complete definition of the theory you are
claiming is the one and only quantum theory of gravity.
Next, show that it has GR as its low energy limit. Not that you can
do perturbation theory about any solution, but that if I hand you an
arbitrary classical solution of Einstein's equation, you can show that
in some suitable sense it is well approximated by some solution in
M-theory, and evolves approximately according to Einstein's equation.
This is what people in the LQG field are now trying to do for that
theory; if you claim that M-theory is superior on the basis of its
classical limit, then you should do the same.
>> My point here is that there won't necessarily be a unique quantum
>> theory of gravitation; there may be many quantum theories which imply
>> Einstein's equations.
>Yes, this conjecture was plausible a priori but over 50 years of research
>have shown that it is very unlikely.
It has done no such thing. Were you not paying attention to Steve
Carlip's illustrations from 2+1 dimensional gravity? Even if you are
correct in that ultimately M-theory provides a tenable quantum theory
of gravity, that does *nothing* to prove that it is the *only* such
theory. Nature is under no obligations to simplify itself in
accordance with our ingenuity. To take a historical example, there
was a big difference in being unable to find a solution by radicals of
the quintic, and proving that no such solution existed. Most
physicists don't even take the approach of trying to show that their
theory is a unique one; they just try to come up with any theory that
works, and then compare against ever more experiments.
From yet another followup to this thread:
>LQG looks naively similar. You start with Einstein's fields, write them in
>some noncanonical variables and then quantize them in a "new way". So why
>cannot you just say that at long distances the expectation values of the
>fields in LQG just satisfy the same equations as the classical Einstein
>metric? I think that the main problem is that this simple prescription is
>known to give the wrong results. More precisely, if one calculates the
>entropy of the black hole, one does not get the correct A/4G where G was
>the "classically" substituted Newton's constant and A is the horizon area.
You keep saying this, but it does not become true just through your
repetition. As John has already explained, there is a quantization
ambiguity present in the LQG scheme; the presence of a parameter known
as the Imirizi parameter. For different values of this parameter the
classical theories are all the same; it is only in the quantum
theories that they are inequivalent. This kind of thing happens in
other quantum theories as well; quantization is by now well known to
be an ambigious process. And that's really not all that surprising;
it's not as if Nature starts with a classical theory and then has to
try and cook up a quantum theory; it is quantum all along.
Your claim that LQG predicts the "wrong" value of black hole entropy
is based on a particular choice of that parameter, and you have no
justification for making that choice. If it were to turn out that a
different value of the parameter were needed in order to make some
other calculation give the correct result, then *that* would be bad
news indeed for LQG. But that hasn't happened yet, and until it does
your claim that LQG gives the wrong value for black hole entropy is
just false.
>Note that string theory predicts the correct entropy of all the extremal
>black holes (as well as near extremal and a couple of Schwarzschild
>ones...) and does not suffer from any problems of this sort.
And extremal black holes are to date of no physical relevance for our
universe. If you want to criticize LQG, make sure you are applying the
same standard to SMT as well.
>There is a better solution than just to try to explain SUSY breaking etc.:
>wait till 2006 and observe at LHC that the world *is* supersymmetric; at
>the energies available at LHC, SUSY is slowly becoming restored. String
>theory is not a vague theory that must hide its predictions. On the
>contrary, it (more or less) boldly asserts that SUSY exists and I have bet
>$1000 that it will be observed. ;-) There are many bold predictions of
>particular stringy scenarios and it is likely that one of them will be
>confirmed soon experimentally.
And what will the masses of those supersymmetric particles be? One of
the big objections that many of us in LQG have to SMT is that it is
far from being just one theory: depending on how you compactify and
how you break supersymmetry you can get it to predict a great variety
of things, though it is still not clear that our universe is among
them. And M-theory suffers from the more serious drawback of not
being, well, formulated, as I have already mentioned.
I'm not discouraging people from working on SMT (nor under the
illusion that my discouragement, were I to offer it, would have any
effect :), I am simply emphasizing that when you compare LQG to SMT
you must be fair in your comparisons. And you have not been.
Intuition is a useful and as far as I can see necessary guide for each
individual scientist to base his or her research upon, but that should
not confuse the issue of how to compare competing theories. In that
comparison, it is experiment and not intuition that must be the
arbiter. And as I have said already, neither LQG or SMT has met any
experimental test, and neither is even complete enough to have, yet, a
claim to be a quantum theory of gravity.
Josh
John Baez
Steve Carlip <sjca...@ucdavis.edu> wrote:
>Lubos Motl wrote:
>>Because LQG is completely discrete, in a sense, I am not sure
>>whether you can create anything like the coherent states [...]
>You almost certainly can: see Sahlmann, Thiemann, and Winkler,
>gr-qc/0102038, for example, or Corichi and Reyes, gr-qc/0006067.
There are even older papers on coherent states in loop quantum
gravity, e.g.
gr-qc/9412014
Coherent State Transforms for Spaces of Connections
Abhay Ashtekar, Jerzy Lewandowski, Donald Marolf, Jose
Mourao, Thomas Thiemann
but now the topic is heating up. Ashtekar & Co. are busy writing
a paper on this subject, which modifies the Sahlmann-Thiemann-Winkler
proposal a bit. That other proposal fixed a graph in space
and uses that to construct coherent states that approximate
any given metric/extrinsic curvature pair in the classical theory.
Ashtekar & Co. do an integral over graphs, which seems more
natural. The idea is to randomly spinkle points on your
Riemannian manifold, form a graph by taking the Voronoi
diagram of these points, then get a spin network state, and
finally form a superposition of these spin network states by
integrate over all ways of sprinkling the points.
Do I hear someone muttering that they don't know what a Voronoi
diagram is? Tut tut! Go here:
http://www.cs.cornell.edu/Info/People/chew/Delaunay.html
click on "Voronoi Diagram", and use your mouse to sprinkle
points on the rectangle shown. The program will automatically
construct the corresponding Voronoi diagram. You can either
figure out for yourself what it's doing, or cheat and read
the accompanying text.
You'll note that in this 2d example the Voronoi diagrams
are generically 3-valent graphs. In 3d they are generically
4-valent graphs. That's nice, because that's the case where
our understanding of the quantum tetrahedron applies. 4-valent
graphs are also basic to spin foam models like the Barrett-Crane
model. So, the continuum approaches and the combinatorial
approaches to loop quantum gravity are beginning to merge a
bit more!
More precisely, this technology for building coherent states will
pretty soon allow us to define states approximating classical
geometries in purely combinatorial approaches to loop quantum
gravity, like the Barrett-Crane model.
Josh Willis
some sort of usenet black hole, so I will try again with a modified
version to in part take into account what others have since said on
this thread.
[Moderator's note: The first version in fact got posted-- I
just saw it here, at least-- but I'll post this since it's
been modified. Note that there's some delay associated with
newsgroup moderation. My apologies to people who see this
twice. -MM]
In article <Pine.SOL.4.10.101060...@strings.rutgers.edu>,
Lubos Motl <mo...@physics.rutgers.edu> writes (in part):
>
>> That's true. To be fair, though, string theory has the opposite
>> problem---while it's easy enough to get something like the
>> usual general relativity at long distances (modulo questions
>
>I certainly disagree with the word "easy". The task to construct a quantum
[stuff deleted]
[more snipping with abandon]
because the structure of the theories is different. And there are
criticisms levelled by proponents of LQG at parts of the construction
precisely because of consistency issues; for example, some of the
criticisms of Thiemann's Hamiltonian constraint could perhaps be seen
in this light.
attempt that in ASCII. Cognosgenti should assume it is there in what
Paul D. Shocklee
> Do I hear someone muttering that they don't know what a Voronoi
> diagram is? Tut tut! Go here:
> http://www.cs.cornell.edu/Info/People/chew/Delaunay.html
> click on "Voronoi Diagram", and use your mouse to sprinkle
> points on the rectangle shown. The program will automatically
> construct the corresponding Voronoi diagram. You can either
> figure out for yourself what it's doing, or cheat and read
> the accompanying text.
Cool!
These actually seem rather reminiscent of grid diagrams for
string networks in Type IIB, which are basically dual descriptions
of the networks.
We discuss them in our recent paper:
hep-th/0101080
Authors: Paul Shocklee, Larus Thorlacius
Zero-Mode Dynamics of String Webs
http://arXiv.org/abs/hep-th/?0101080
I had always kind of hoped that string networks and spin networks
would turn out to be related in some way, which is why I got
interested in string networks in the first place.
--
Paul Shocklee
Graduate Student, Department of Physics, Princeton University
Researcher, Science Institute, Dunhaga 3, 107 Reykjavik, Iceland
Phone: +354-525-4429
A.J. Tolland
> the arrogance in general ;-), but I certainly think that the people who
> made the most important contributions to physics usually believed that the
> structure that they studied was unique, exceptional and far-reaching. The
> other, less influential scientists did not believe those principles so
> more pleasant for others. But taking your theories seriously is almost
> certainly (positively) correlated with your chances to succeed.
What you say may be true, but remember, you're posting to USENET.
Around here, when you make claims which you are unable to prove, you run
the risk of being labeled a crackpot. :)
More seriously: I agree that it is important to take your work
seriously. I wouldn't be studying string theory if I didn't find it so
damned fascinating. But I also think it's tremendously important to be
careful when doing science about what you believe or conjecture and about
what you can honestly claim is true. In particular, I believe that you
either need experimental confirmation or, failing that, absolute
mathematical rigor before you can claim that a theory is correct. So far
as I am aware, neither M-theory nor loop gravity meets either of these
criteria.
I'll admit that this attitude is probably a result of my
intellectual training: I spent most of the last 4 years in experimental
particle physics, and I learned my math in Chicago's math department,
which tends to be rather traditional about things like rigorous proof. :)
--A.J.
Lubos Motl
> C'mon, from the home work excercises to my quantum mechanics 1 course:
>
> The Hamiltonian for the harmonic oscillator H = w (a^\dagger a +1/2) =w (N+1/2)
> clearly has discrete eigenvalues. Now, consider the state
>
> |r> = exp(-r^2/2) \sum_n r^n/sqrt(n!) exp( -iw(n+1/2)t ) |n>
Well, I know that the expectation value of an operator in a state can be
continuous even in 2-dimensional Hilbert spaces (where a Hermitean
operator has 2 eigenvalues only). The problem of your example is - as we
have already said - that "r" is really naturally a complex variable x+ip
and therefore parameterizes a two-dimensional space and "x,p" have a
continuous spectrum indeed. I do not consider the coherent state to be
"coherent with respect to H" (in the sense in which stringy vacua are
coherent with respect to the moduli) if you understand me. Because your
simple example does not say anything we did not know, it does not change
my opinion that the areas in those stringy vacua must be continuous i.e.
that those moduli behave locally just like "x" or "p" and not like "H" if
you find any reasonable definition for them at all.
A.J. Tolland
FYI, I've quoted (and responded to) your comments out of order. I
hope you'll read the 2nd half; that's where most of the fun is.
> everything. BTW while those 10,000 papers suggest that M-theory is *a*
> theory of quantum gravity, the other zero or 100 reasonable papers (and
> other papers that do not exist) about a definition of quantum gravity
> suggest that M-theory is *the* theory. :-)
You might as well argue that the fact that 3,5, and 7 are prime
suggests that all odd numbers are prime. Your remarks don't prove
anything, and I'm not willing to take such grand claims on faith.
> I would be extremely surprised if you found *one* new theory that
> reconciles them and showed that it gives you a consistent S-matrix for
> graviton scattering. You would certainly become very famous.
Yeah, well, I'll try my best when I'm less busy trying to
understand my advisor's papers. :) Go easy on me. I only started
graduate school this week. Besides, we were talking about the fact that
we can't prove that there aren't any such theories, not about the fact
that we don't know how to construct them.
Incidentally, have you ever written anything about your
explorations of defective candidates for quantum gravity? I'd be
interested in seeing them (although I wouldn't encourage you to write them
up if you have real work to do :).
> Some time ago, Lenny Susskind told me that I should change my first name
> because then I would be known to the world as Matrix Motl. Of course, I
> mostly disagree that "M" stands for "Missing" and e.g. the matrix model is
> a complete definition of M-theory on certain backgrounds. Type IIA string
> theory can be defined exactly using MST (= My String Theory or Motl String
> Theory or also Matrix String Theory), later promoted and improved by DVV.
> ;-) A lot of things is known so M-theory can be missing only partially. :-)
Hmm... We still have explored only a small set (of "measure 0"?)
of M-theory's vacua. I suggest we amend its name to MM-theory, for
"Mostly Missing." Or "Matrix Motl's", if you like. :) Again, I think
this is important. We have descriptions of M-theory at certain, rather
special vacua. (Indeed a lot of our detailed knowledge of M-theory comes
from the study of rather special situations: BPS states and the like.)
I don't know that it's safe to draw general conclusions from this
information.
The situation seems analagous to the one with GR singularities in
the 60s. People knew that certain, very special GR solutions had a
certain property (i.e. singularities), but it took Hawking and Penrose to
prove that this property was a feature of generic solutions. Hopefully
string theory will meet with similar success in dealing with its vacua,
but I don't think it has yet.
> > (b) it could teach us a great deal about the non-perturbative
> > structure of quantum physics.
> I do not understand what you mean by the general term "nonperturbative
> structure of quantum physics".
Ah... I owe you an apology for not clarifying my terminology here.
I'm using my own weird private language here. I probably should have said
something like "non-effective" to indicate that I don't like the
separation of physics in path integral quantization into free objects and
interactions. This notion of perturbative differs from the notion of
perturbative that shows up when we are "Taylor expanding" quantities we
want to calculate. I remember being very excited when I saw for the first
time that perturbative string theory is perturbative in the second sense,
but not in the first.
Instead of responding point by point to your other comments, let
me concentrate on what I see as a crucial difference in our perspectives:
You seem to believe that all QFTs are effective theories; you claim, for
instance, that the adjective "effective" is trivial when applied to QFT.
I do not agree. I think "effective" should be taken literally; effective
field theory is an approximation to a more complete theory.
I prefer to think of quantum theories (QFT and otherwise) in
terms of observable algebras and state functionals; this is the only
perspective I know of which is intrinsically quantum physical. Effective
field theory is a useful tool for calculating expectation values in
certain regimes. It's really just a quantization technique, a method for
introducing a rough coordinatization of the observable algebra. I don't
think that it's true that a consistent QFT must have a fixed point at some
high (possibly infinite) energy scale; this is merely a sufficient
condition for path integral quantization to function as a definition of
our theory below some energy.
You can dismiss all of this, if you like, as just a mathematical
game. But let me remind you that mathematicians have tried very hard for
_50 years_ to find a rigorous formulation of Feynman's weighted
sum-over-states. What they have discovered is that the usual DX measure
on path space is pure fantasy; the only thing which exists is the measure
DU "=" DX*exp(iS_free), which is only good for doing perturbative (in the
1st sense) quantization. By your own "50 years of research" standard,
their discoveries suggest that the path integral is nothing more than a
quantization tool, and should not be considered as a definition of QFT
except in certain special circumstances.
Now for a word about renormalization. To do path integral
quantization, we have to appeal to "Wilson's Principle", which says that
the low energy physics doesn't depend on the details of the high energy
physics. This principle guarantees (a) that our Hamiltonian can be
written as a sum of terms which are products of the effective variables
and their derivatives, and (b) that our correlation functions at low
energy are independent of our cutoff scale M. In exchange for keeping our
effective degrees of freedom fixed when we change M, we have to accept the
fact that the coupling constants (which govern the relative strength of
individual Hamiltonian terms) are functions of M. Some constants grow
with M, some are marginal, others shrink. The latter kind are called
renormalizable couplings; they dominate in the low energy effective
theory. And they have a number of nice features, including sometimes one
which has historically led to much confusion: If we set all
non-renormalizable couplings to zero at one mass scale M, and then shift
to another mass scale M', we generate a whole bunch of non-zero
non-renormalizable couplings. Remarkably, however, we can redefine our
fields and coupling constants to set all the non-renormalizable constants
back to zero. This wacky process is the usual notion of renormalization.
(I want to note here that one of my criteria for a final theory is
that Wilson's Principle, the observed organization of physics by mass
scale, should be derivable from the axioms.)
The point of all this is that our notions of renormalizable are
tied deeply with effective theory; they are apparently a consequence of
two things: (a) the fact that we are keeping our effective degrees of
freedom fixed when we change M, and (b) the miraculous properties of the
couplings which dominate low energy physics. It's worth noting that we
know of consistent theories which are not renormalizable in this sense.
Now, as to the criticisms you leveled at my claim that gravity
need not be fundamentally renormalizable. You assumed that I was claiming
that we could have a fundamental theory which was (a) effective, a QFT in
your sense, and (b) included the gravitational interaction as a
non-renormalizable coupling. I agree with you; this is probably nonsense.
But it is not what I was claiming. I don't think that the fundamental
theory will necessarily be formulatable as an effective QFT; it may be a
QFT in the more general sense of local quantum physics, or (more likely)
it may be something else entirely. In either case, I see no need at all
for the interaction which gives rise to gravity at low energies to be
renormalizable in the rather limited sense offered by effective field
theory.
--A.J.
Greg Kuperberg
Lubos Motl <mo...@physics.rutgers.edu> wrote:
>[Witten] often shocks participants of a mathematics conference by claims
>such as "mathematics of the 3rd millenium will be dominated by string
>theory" or "string theory has the remarkable property of predicting
>gravity", "learning about the way how string theory incorporates
>gravity was the strongest intellectual thrill of my life", "string
>theory is a science of the 21nd century that fell to the 20th century
>by an accident" etc.
Is he shocking us or merely confusing us? :-)
I can believe for strictly external reasons that string theorists are
on the right track. (The reasons have to be external since I don't know
string theory.) I also understand that believing a theory is very useful
for studying it. (E.g. it is hard to do evolutionary biology if you don't
believe evolution.) But some of these statements by Witten do strike me
as strange salesmanship. Statements 2 and 3 are perfectly reasonable.
Statement 4 could be technically correct, but it is also a conceit.
Alan Guth could have said the same thing about inflation, but he didn't
as far as I know. One could also say the same thing about quantum
computation, which is a topic that I do understand some, but it would
not be an encouraging comment. And I refuse to accept statement 1.
It implies either that mathematicians of the 3rd millenium won't have
much imagination, or that string theory will devolve into an axiomatic
foundation of mathematics.
Maybe Witten actually gave a more conservative version of statement 1,
like "much of mathematics in the 3rd millenium will be motivated by
string theory." That I could believe.
--
/\ Greg Kuperberg (UC Davis)
/ \
\ / Visit the Math ArXiv Front at http://front.math.ucdavis.edu/
\/ * All the math that's fit to e-print *
Aaron Bergman
>I think more could certainly be said here about problems with string/M
>theory; for example, how does one calculate entropy for non-extremal
>black holes?
It's been done for large classes of near-extremal black holes.
The proportionality to the area has been derived for general
Schwarzchild black holes and probably some others through Matrix
theory.
> Both LQG and String/M-theory (which I shall henceforth
>abbreviate as "SMT" since I am a slow and lazy typist) have both
>conceptual issues to address, as you indicate above, but also a dearth
>of hard numbers to verify or disprove. Indeed, I don't know that's
>accurate to say that SMT has classical GR as a semiclassical limit:
>yes you can expand around a given classical solution of Einstein's
>equation, but has it been shown, perturbatively or otherwise, that
>these solutions or an approximation to them are also a part of the
>solutions of SMT?
You can show that in order to have a conformal worldsheet theory,
the background has to satisfy Einstein's equations (plus higher
order corrections is alpha'). This comes from requiring the beta
functions to vanish.
[snip much stuff coming to the comparison between where
loopy and string people come from]
>SMT, on the other hand, come from particle physics; they had quantum
>field theory, and now they wanted to add gravity. This community also
>has a certain mode of thinking---but it is different from that of LQG.
>In SMT, people feel that the lesson from the Standard Model and its
>successes was that we should pay attention to renormalizablility; it
>has so far been a very reliable guide in selecting just those
>interactions that nature prefers.
This just isn't a postulate, you know. There are good reasons why
the interactions we observe are renormalizably. The "particle
physics community" as you call has long since abandoned the idea
that non-renormalizability == bad. In the last twenty years, the
understanding of quantum field theory, especially in regards to
the renormalization group, has increased significantly. In fact,
I would go so far as to say that any theory of general quantum
fields that does not refer to the renormalization group is going
to simply be wrong. RG flow is essential to understanding QFTs.
>It also emphasizes unification with
>the other forces, since this was a dominant theme a notable success of
>high energy theory in the latter half of this century.
Actually, you can write down string theories that have the
standard model at the string scale.
>Both sets of assumptions could be criticized; simply on the grounds
>that they are assumptions, but also for deeper reasons as well. On
>the SMT side, why should we believe that renormalizablility is an
>infallible guide?
See above. This question is couched in so many incorrect
assumptions that it is difficult to respond to.
>Indeed, one could argue that there are unjustified assumptions common
>to both LQG and SMT, and that both approaches draw too much assurance
>from their successes. In fact, many have argued precisely this. For
>example, both LQG and String/M-theory have drawn great comfort from
>their derivation of black hole entropy. But Carlip and others have
>presented arguments (so far only in the 2+1 dimensional case, as far
>as I know) that (*very* roughly state) this entropy may be a generic
>feature of *any* quantum gravity theory.
String theory gets more than just the proportionality correct. It
gets the constant correct and it gets the grey body factors
correct. This shouldn't be discounted -- many string theorists
will tell you that the calculation of Strominger and Vafa is why
they believe in string theory.
Ralph E. Frost
Lubos Motl <mo...@physics.rutgers.edu> wrote in message
news:Pine.SOL.4.10.101061...@strings.rutgers.edu...
> We need to find out the
> formulation. But if we find it one day, the problem will reduce to a
> well-defined mathematical question. And it may happen that the
> mathematical conditions will have several (or many) solutions: five, one
> hundred or 10^{10^{120}}. But if this is the case, you will not be
> justified to protest against it. The existence of many vacua will become a
> true property of the real world. We could be still able to pick the
> correct one - but there will not be a truly scientific explanation why we
> live in this vacuum, just like there is no truly scientific explanation
> why you live in the USA.
I'd think you'd want to embrace the anthropic principle since that IS what
crosslinks the current trial theory into coherence -- like you sort of say.
Then, at least you wouldn't be beginning with the prejudice that you don't
like it and are working to erase it (even though, well, it is in
operation).
If you embrace it and it works maybe it will lead you to a place where you
can grok a new trial theory that really falsifies the anthropic principle.
But, until then, I think denial or avoidance is being invoked first and
that leads to a fairly irrational set of expressions.
IMO
zir...@my-deja.com
>> or Corichi and Reyes, gr-qc/0006067.
I took a look at this paper and was hoping that a LQG expert (like JB) could
elaborate on what the following sentences mean (quoted from the top paragraph on
page 10) -
"...not every spin-network state is an eigenstate of the area operator in the
case when the vertex has valence 4 or more... This particular feature leads to
the conclusion that the area operators of two surfaces that intersect along a
line fail to commute for states which have 4 or higher valent vertex in the
intersection. Thus, the quantum Reimannian geometry that arises from loop
gravity is intrinsically noncommutative."
What is this quantum Riemann geometry like and is it similar to the quantum
Riemann manifolds in Shahn Majid's general noncommutative Riemann geometry? See
page 58 of
Lubos Motl
paper is mostly about the three-dimensional deSitter space. This paper is
an example what I mean when I say that 3 people in the world who
understand gravity in 3 dimensions best, are string theorists.
http://xxx.lanl.gov/abs/hep-th/0106113
On Fri, 15 Jun 2001, Josh Willis wrote:
> Very easy? Name *one* such theory. Aside from loop quantum gravity.
Roger Penrose's theory of twistors is claimed to be another example
(although none of us is sure why he thinks that it is a physical theory at
all). Lee Smolin's new book "Three Roads to Quantum Gravity" presents yet
another example (as is clear already from the title), some information
networks or something. ;-)
I am ready not only name such a theory but I can construct it in real
time. Let's call it Membrane Quantum Gravity (MQG, also stands for Motl's
Quantum Gravity). :-) The Hilbert space is defined to contain all the
2-dimensional surfaces with arbitrary junctions of 3 surfaces (and no
boundaries); spin networks are replaced by spin networks of membranes. At
every minimal 2D area, there is a single "J" operator transforming as an
SU(2) vector. The length of a line is defined as the sum of sqrt[j(j+1)]
over all the intersections of the line with the 2-dimensional surfaces and
j(j+1) is the eigenvalue of J^2 living at this portion of the
2-dimensional surface, to make the analogy with LQG more obvious.
Well, this construction obviously looks "dual" to LQG in some sense but it
is probably not equivalent. But it is equally covariant. Why do the people
study LQG and not MQG, for example? Only because LQG is related to the
gauge formulation of the initial problem in GR? The low-energy limit has
not been found for either of them. I would be happy to hear why MQG is
really a worse direction of research than LQG. ;-) Unlike string theory
where all the questions about the number of dimensions etc. are uniquely
fixed, there are almost no constraints in LQG-like attempts and all LQG,
MQG or similar theories are comparably arbitrary and unjustified. Anything
goes.
> This is a point I have heard Ashtekar and others make repeatedly: we
> are not saying that loop quantum gravity is "the" correct way to
> quanize gravity---the jury is still out on that. But after many
> people working for many years there was, prior to LQG, no
> nonperturbative quantization of gravity *at all*. Thus, even should
> LQG turn out to be incorrect, we feel we should learn quite a lot
> just from the effort.
Yes, today we already have a nonperturbative quantization of quantum
gravity (at least in some backgrounds) but it is not LQG whom should be
grateful for it. ;-)
> As Steve Carlip has pointed out on this thread, things are known to be
> complicated in 2+1 dimensional gravity, where there are several
> inequivalent quantizations. It could be the case that 3+1 dimensional
> gravity, unlike its lower dimensional counterpart, for some reason has
> an essentially unique quantization. If so, it's not clear that we
This is what I call "religion in physics". There is no reason known why
"canonical" quantum gravity in 3+1 dimensions should be better than in
other dimensions. Lee Smolin has in fact generalized the tools of LQG to
11 dimensions, for example. String theory on the contrary tells you
completely rigorously that gravity fundamentally lives in 10 or 11
(nonperturbatively) dimensions. Why 10? The bc ghosts have central charge
1-3.3^2=-26. They are cancelled by 26 dimensions in the bosonic string,
but in the case of superstring you add beta-gamma superconformal ghosts
(to have worldsheet supersymmetry) with central charge 3.2^2-1=11.
-26+11=-15. So we need to cancel remaining central charge of 15. This is
the central charge of 10 bosons and their 10 fermionic superpartners
(10/2=5). This is why perturbative quantum gravity lives in 10 dimensions.
String theorists also believe that the compactification to 4 large
dimensions could turned out to be "special" in some way, but apart from
this religion, they also have rigorous technical results.
> I think you should either back up your claims about the beliefs of
> workers in LQG with some hard evidence, or be a little more cautious...
I prefer to back up my claims. Take as an example Rovelli's review of LQG
http://arXiv.org/abs/gr-qc/9710008
At page 8 you will find a section "One additional assumption" where he
handwaves that the argument that Wilson loops have really UV-divergent
correlators is hopefully incorrect, because of special properties of the
coordinate invariance etc. A well-established mathematical formalism from
gauge theories suggests that you cannot define UV-finite Wilson lines.
There is no well-established formalism that shows the opposite - but LQG
must believe it because of "special properties" of the coordinate
invariance.
Or take page 3, "What is the problem, view of a relativist". He argues
that gravity is special, it is not just one of the forces etc., deserves a
special treatment, can be studied separately etc. The arrogance of LQG is
reflected by the claims that you do not need to unify gravity with other
interactions and study gravity "exceptionally". Today we know that the
weak force required us to unify the weak and electromagnetic forces, for
example. Why do they assume the opposite? And even if LQG succeeded with
its "modest" task, how it could explain the other forces etc.?
> in those claims. No one I know in the loop quantum gravity community
> argues that LQG "necessarily" has general relativity as a
> semiclassical limit, simply because it uses the Einstein-Hilbert
> action as a starting point. (Actually, to be technically
> correct, it doesn't even do that, at least according to the way the
> terminology is normally used. Einstein-Hilbert usually refers to the
> action written in terms of metric, rather than connection variables.)
I do not understand why the same action should be given different names if
you use different variables.
> Even these issues are by no means fully resolved, but I don't see how
> progress on the semiclassical limit of the theory could have been made
> without at least some understanding of the structure of the full
> quantum theory. One can't do everything at once.
Maybe I am naive but for me, LQG is a simple discrete model in 3
dimensions and one could be able to derive the limits at least by
numerical calculations (in fact, the degrees of freedom living at the
links can be understood as discrete degrees of freedom, unlike the case of
the lattice QCD). It should be comparably difficult to lattice QCD.
Green+Schwarz's anomaly cancellation in 10D SUGRA coupled to SO(32) Super
Yang-Mills was an incomparably more difficult task to compute, I think.
> I think more could certainly be said here about problems with string/M
> theory; for example, how does one calculate entropy for non-extremal
> black holes?
I have written a lot about this and enumerated many references to the
papers on the entropy of Schwarzschild black holes etc. This is an example
how extremely badly informed people from LQG are. The entropy of black
holes in string theory is a well-undestood subject. After the
Strominger+Vafa's fundamental paper, I think that the most relevant
reference is the Maldacena+Callan's paper
http://arXiv.org/abs/hep-th/9602043
and 335 papers that cite it. This paper calculates the entropy of both
extremal and (many) non-extremal black holes in string theory, as well as
their temperatures. The entropy of black holes (including non-extremal,
certainly near-extremal) is a solved problem in string theory. This is
definitely not among the big holes of our knowledge that we shold fill in.
> Both LQG and String/M-theory (which I shall henceforth
> abbreviate as "SMT" since I am a slow and lazy typist) have both
> conceptual issues to address, as you indicate above, but also a dearth
> of hard numbers to verify or disprove. Indeed, I don't know that's
> accurate to say that SMT has classical GR as a semiclassical limit:
That's definitely an accurate and proved statement.
> yes you can expand around a given classical solution of Einstein's
> equation, but has it been shown, perturbatively or otherwise, that
> these solutions or an approximation to them are also a part of the
> solutions of SMT?
Sure, it is a completely basic material covered by chapters 3 both of
Polchinski's book as well as the book by Green+Schwarz+Witten. The
conformal invariance requires that the background solves the (generalized)
Einstein's equations, any solution of the Einstein's equations satisfying
some non-singularity and causal constraints is also a solution of string
theory (if the curvatures are much smaller than Planckian/stringy
curvatures) - and the action that can be deduced from the scattering
amplitudes of gravitons around a background is the same action the
describes the consistent backgrounds. Physics of string theory is easily
showed to be background independent.
> Michael Duff indicated something about recovering Einstein's equation
> from SMT when he gave a talk here last semester (or the one before, I
> forget), but I did not sufficiently understand what he was saying is
> true of SMT, and whether it was a perturbative or nonperturbative
> statement.
Yes. You can recover the Einstein's equations by perturbative calculations
in string theory and you can also derive the Einstein's - or more
precisely supergravity limits - in M-theory etc. in all the perturbative
and nonperturbative definitions we have.
> I don't know why you would say this. Certainly there are consistency
> checks performed within LQG: the closing of the constraint algebra,
> and indeed the first-class nature of the constraints come immediately
> to mind. Of course the checks are not the *same* as in M-theory,
I mean, the closure is mostly a check that you wrote your formulae
properly; of course, the canonical quantization of general relativity (in
any variables, as long as they are equivalent) must pass these trivial
classical checks (because of the consistency of string theory from which
you can derive some "classical" relicts). But I meant some surprising
numerical coincidences where you do not have reasons to expect that it
should work - but it does.
String theory is full of such examples, therefore it smells like God. But
LQG smells like Man just because its checks are really not of "the same"
sort as you say very well.
> Part of what you write here is just plain wrong. Though it is true
> that Penrose originally studied spin networks as a postulated
> structure of spacetime, LQG does not *postulate* this; it is
> *derived*.
I know how to derive the spin network basis of the Hilbert space from the
loop variables, this relation is quite straightforward. Maybe it looks
nontrivial to you but it does not look too nontrivial to me.
> networks are so important in loop quantum gravity. Logically, this
> all depends ultimately on your starting assumptions, both in choice of
> action, variables for that action, and indeed the choice to do
> canonical quantization at all.
Yes, it depends on all those initial assumptions, therefore LQG smells
like Man. String theory does not depend on any initial conditions that you
mentioned. It was first seen in the form of some amplitudes that were
supposed to describe the strong interactions (but they were not correct).
But there were many different ways how the full structure of string theory
could be revealed. People could have found supersymmetry first, then
construct the maximal eleven-dimensions SUGRA and try do define its UV
completion consistently. If they compactified SUGRA on a circle, they
would have found that the M2-branes become very light strings; an
investigation of their properties would also lead to string theory. String
theory does not depend on your choice of the action. It does not depend on
your choice of variables. There is only one consistent way how to quantize
it and understand it. This makes a difference!
> But it is inaccurate to say that spin networks were put in by
> hand--they weren't, and Ashtekar et al. were surprised when they
> turned up. I have heard him say so.
I believe you that Ashtekar was surprised. I just do not think that the
appearance of spin networks in a covariant type of a lattice-QCD-like
theory *should* be surprising. Or did you mean that whatever surprises Mr.
Ashtekar Himself must be nontrivial? ;-)
> I think what people criticize is predictions that have not yet been
> experimentally verified, which includes all of the predictions you
No predictions of any theory going beyond the Standard Model (or general
relativity, in the astronomic context) have been verified - simply because
everything that has been seen in this world agreed with the Standard Model
(with neutrino masses added) so far. I do not understand how this simple
fact can be transmuted into a criticism of a particular theory going
beyond SM. If this criticism has any value at all, it is also a criticism
of all the other theories including LQG.
> list above. And all of the predictions of loop quantum gravity. More
> to the point, what is a valid criticism is any claim that a particular
> theory is "right" when it has not yet withstood experimental test.
String theory is the only known consistent theory that can predict a world
containing all the forces and particles we see. In this limited sense it
passes experimental tests. Yes, more or less no *new* predictions of
string theory (that are not contained in the simpler structures of the
Standard Model and General Relativity) have been experimentally verified
yet. But what one means by the word "right" has to do with mathematical
consistency. And this is a very tough constraint for a theory of quantum
gravity.
> More to the point, the reason loop quantum gravity is an active
> program is not because it is in any way guaranteed to be right, nor
> because its adherents necessarily "believe" that it is "the" theory.
Loop Quantum Gravity is an active program? Can you elaborate on this
statement?
> Rather, one has to start somewhere, and loop quantum gravity and
> string theory, both historically and philosophically, start from
> different places. It should not be too surprising, then, if they end
> up in different places; what would be more surprising---and
> interesting---are areas where they might overlap.
There is one world only and it should be described by one theory. Either
LQG and string theory are equivalent in some subtle sense (or more
precisely LQG is included in string theory, because LQG is less general),
or at least one of them must be wrong. I do not understand your attempt to
divide the world to different places that just overlap somewhere. Physics
is not politics.
> Though this has been expounded many times in this newsgroup, perhaps
> it bears repeating nonetheless (if this is cut by the moderators, I
> guess not!). Loop quantum gravity is worked on mainly by people who
> come from the relativity community; i.e., they had gravity, now they
> want to quantize it. They therefore bring with them a certain mode
Yes, this is a sociological problem of LQG - that the people are
*relativists* only. To quantize something like field theory, you should
know quantum field theory.
> of thinking, and certain beliefs. For example, that gravity is really
> geometry, that the diffeomorphism invariance of Einstein's theory is
> telling us something deep, and we should try to hold onto that when we
> quantize.
Yes, this is why I say that this enterprise is mostly a religion. General
covariance is the most subtle local symmetry principle (the most difficult
one to quantize etc.), but otherwise it is just one of many gauge
invariances. In fact, Kaluza-Klein theory explains the electromagnetic
U(1) gauge symmetry (or nonabelian ones) as a subgroup of general
covariance (in higher dimension). It does not seem reasonable (and
democratic) to try to single out one of the gauge invariances. Furthermore
as we know very well today, geometric concepts - such as space, time *and*
general covariance - are not meaningful at ultrashort distances (in
perturbative string theory, for example, general covariance becomes just
the lowest level of an infinitely bigger system of symmetry principles,
associated with a string, if we study the dynamics at stringy distances).
The full theory has a Hilbert space, some dynamics but the geometric
interpretation of the states in the Hilbert space exists only if the size
of the geometry is much greater than the Planck scale.
Trying to defend that coordinates, space, time, geometry and general
covariance has the same meaning at all distance scales, is very similar to
trying to argue that Newtonian equations of motion are valid for any
speed, that arbitrary light particles have well defined momentum and
position - or that Fermi's four-fermion interaction is valid at arbitrary
short distances. One of the obvious lessons that (QG-related) theoretical
physics of last 30 years has taught us is that the notions of space and
time - as well as their reparametrization - are derived concepts that
agree with our intuition (from General Relativity) at long distances only.
> SMT, on the other hand, come from particle physics; they had quantum
> field theory, and now they wanted to add gravity. This community also
> has a certain mode of thinking---but it is different from that of LQG.
The historical details do not matter. Today string theory contains at
least as good bunch of relativists as LQG.
> In SMT, people feel that the lesson from the Standard Model and its
> successes was that we should pay attention to renormalizablility; it
String theory is not a field theory and therefore it does not match the
usual discussions about the renormalizability either. Both general
covariance and renormalizability are derived concepts valid at long
distances (in the effective field theory description) where both of them
must be taken seriously - and of course, I take seriously both general
covariance and renormalizability (and who does not, ignores about one half
of theoretical physics). At Planckian distances, neither of these two
field-theoretical notions is justified and indeed, string theory shows
that neither of them is an exact notion in the full theory.
> Both sets of assumptions could be criticized; simply on the grounds
> that they are assumptions, but also for deeper reasons as well. On
> the SMT side, why should we believe that renormalizablility is an
> infallible guide? In the only dimensions (i.e. less than four) where
As I said, for the full string theory, renormalizability is not directly
applicable. The stringy amplitudes are always automatically UV-finite, for
example, they have no ultraviolet divergences. Nevertheless at long
distances a good theory *must* be reduced to the very well checked
effective field theories such as the Standard Model. String theory does.
And in those theories the renormalizability is essential, otherwise they
do not make sense: probabilities are really not infinite. But we know
experimentally that they *do* make sense.
> we can rigorously do interacting quantum field theory, we have field
> theories that can be shown to exist but are non-renormalizable. Also,
I am not sure which theories you are talking about. You can define a
theory using a RG flow from a specific starting point (the irrelevant
terms then do not matter at all) but what you mean by a theory that
"exists but is non-renormalizable"? You can view renormalizability as
consistency combined with our ability to compute something in a theory
with the action having some parameters (that "renormalize" with scale).
> one could argue that with the modern understanding a la Wilson of
> renormalization, we understand why we see only renormalizable
> theories: they are simply effective theories described by a theory
> that may be very different at higher energy scales.
Good.
> Indeed, one could argue that there are unjustified assumptions common
> to both LQG and SMT, and that both approaches draw too much assurance
> from their successes. In fact, many have argued precisely this. For
> example, both LQG and String/M-theory have drawn great comfort from
> their derivation of black hole entropy. But Carlip and others have
I am not sure which successes you talk about on the LQG side, especially
if it gives a numerical coefficient for the entropy that seems to be
wrong.
> presented arguments (so far only in the 2+1 dimensional case, as far
> as I know) that (*very* roughly state) this entropy may be a generic
> feature of *any* quantum gravity theory.
Sure, every consistent theory of quantum gravity must predict the entropy
of the black hole equal to A/4G - because this value can be calculated
semiclassically, following Hawking. String theory does; LQG with the
natural choice of its parameters is off by a factor of ln(2)/sqrt(3). This
is not just an example but a general rule: string theory is the only
consistent of quantum gravity.
> One can go further still. LQG people criticize string theorists for
> using a background metric, but what gives LQG the right to start from
This "criticism" is just a reflection of the fact that LQG people are not
well-informed. Physics of string theory *is* background independent and a
choice of background is *necessary* to compute the S-matrix in any theory;
therefore the S-matrix formulation *must* pick a background metric (and
other fields). If some physicists are not ready to do so, their theory
will never be able to give the correct S-matrix. And S-matrix is almost
the only one exact observable we know of so far that makes sense in a
generally covariant theory.
> a differentiable manifold, especially if we don't end up with one when
> all is said and done? Chris Isham I know has made this point and
The conclusion at the end-point is correct: quantum gravity makes the
usual notions of geometry incorrect at Planckian distances. This, of
course, shows that the starting point of LQG (fields in a fixed topology
etc.) is not well-justified. The full physics does not have reasons to be
describable by fields localized in some geometry - and most likely, it
cannot be described in this way.
> No, no, no! It is *your* burden to prove what you say is right, not
> anyone else's to prove that it is wrong! And you have not come close to
Not at all, it is very unlikely that you can show rigorously that
something is unique or something else does not exist. So far string theory
is a unique consistent quantum theory giving us semiclassical general
relativity - and it is your task to find a counterexample, if you think
that we are wrong. Judges would probably agree with me. If someone says
that a $10 bill is the only thing that he has ever stolen - and some LQG
people claim that he stole millions - it is *them* who must show that this
$10 bill was not unique. Right?
> doing so. For starters: *define* M-theory. No waving of hands, no,
No problem. For more details read http://arXiv.org/abs/hep-th/9610043 or a
recent review http://arXiv.org/abs/hep-th/0101126
M-theory in 11 dimensions, for example, is the maximally supersymmetric
matrix quantum mechanics with gauge group U(N) and large N, for example. I
do not need to wave my hands. ;-)
> Next, show that it has GR as its low energy limit.
No problem either. You must be kidding, everything like that is quite
simple and understood very well. You must know that such objections of
yours end up very painfully. There are hundreds of papers that check the
claim, let us choose randomly
http://arXiv.org/abs/hep-th/9809070
> Not that you can do perturbation theory about any solution, but that
> if I hand you an arbitrary classical solution of Einstein's equation,
> you can show that in some suitable sense it is well approximated by
> some solution in M-theory, and evolves approximately according to
> Einstein's equation.
Yes, you say "not the proof you had before" - just to show that you can
pretend that you do not see the proofs. Unfortunately for you, there is a
plenty of new proofs.
> This is what people in the LQG field are now trying to do for that
> theory; if you claim that M-theory is superior on the basis of its
> classical limit, then you should do the same.
Sure - and we did. I think you must be joking. The gravitational
low-energy limit of string theory was proved already in the middle 70s. If
we had any doubts whether or not M/string theory reproduces gravity where
it should, I would certainly never be able to argue that it is the theory
of quantum gravity.
> It has done no such thing. Were you not paying attention to Steve
> Carlip's illustrations from 2+1 dimensional gravity? Even if you are
> correct in that ultimately M-theory provides a tenable quantum theory
> of gravity, that does *nothing* to prove that it is the *only* such
> theory. Nature is under no obligations to simplify itself in
OK, you can say the same with any theory, even with those that have been
checked beyond reasonable doubts. Relativity is not necessarily the only
explanation of the Morley-Michelson experiment, right? The evolution
theory is not necessarily the only explanation of the origin of species,
right? The Standard Model is not the only effective theory that describes
the particles at the accelerators, is it? Your arguments are really
unclear to me. If you say that you won't accept string theory as the
correct description of reality and quantum gravity even after it will be
checked and confirmed, is it still science? I do not think so.
> accordance with our ingenuity. To take a historical example, there
> was a big difference in being unable to find a solution by radicals of
> the quintic, and proving that no such solution existed. Most
Yes, this example shows that it is plausible that one day in the future,
people will find a rigorous argument why string theory is the only theory
of quantum gravity. But this is not what motivates me. I do not care much
about such proofs. I do not care about inconsitent theories and theories
that do not exist. What I care about is the correct description of
reality.
> You keep saying this, but it does not become true just through your
> repetition. As John has already explained, there is a quantization
> ambiguity present in the LQG scheme; the presence of a parameter known
> as the Imirizi parameter. For different values of this parameter the
Well, I am not sure whether you have really something to say about the
Immirzi parameter if you cannot even spell it properly. I have explained
the nature of Immirzi parameter and I expect that John Baez will write
something extra along these lines. You can also read Immirzi's paper
http://xxx.lanl.gov/abs/gr-qc/9612030
You can see that he does not hide that the value "1" or "i" looks natural
to him. This choice gives a wrong black hole entropy. I think it is
correct to say that a different choice than "1" or "i" has not been
justified at all. Of course, by fine-tuning a parameter you can get one
result to be anything you wish.
> And extremal black holes are to date of no physical relevance for our
> universe. If you want to criticize LQG, make sure you are applying the
> same standard to SMT as well.
Sure that I do. The entropy of non-extremal black holes has been
calculated in string theory, too, as I analyzed above.
> And what will the masses of those supersymmetric particles be? One of
> the big objections that many of us in LQG have to SMT is that it is
> far from being just one theory: depending on how you compactify and
String theory is one theory definitely. 10 years ago we did not know it,
now we do.
> how you break supersymmetry you can get it to predict a great variety
> of things, though it is still not clear that our universe is among
> them. And M-theory suffers from the more serious drawback of not
> being, well, formulated, as I have already mentioned.
Well, you should address this criticism to God, not to string theorists.
If the ultimate theory has many solutions how to compactify the dimensions
or break SUSY, there really exist many worlds. It is nice if someone
wishes that there is only one solution but it is not unsimilar to the
dreams of the people in the past that our planet is the only planet in the
Universe, in fact just in the middle.
It is still likely that there are very few solutions - minima of a generic
superpotential. But even if there are many, it does not mean that string
theory will not be able to calculate all the parameters to an arbitrary
number of digits.
> effect :), I am simply emphasizing that when you compare LQG to SMT
> you must be fair in your comparisons. And you have not been.
Well, this is just your very private opinion. If you measure how
interesting string theory vs. LQG is, you should choose some objective
criteria. If you measure, for instance, the number of papers written, you
get something like 10:1? Yes, I am sure that such criteria are also not
good enough or you because they give a "wrong" result. ;-)
LQG people should realize that they are a part of the theoretical physics
community, but a minor part. But they certainly should not isolate
themselves from their colleagues. Imagine that John Schwarz in the 70s
ignored Quantum Chromodynamics and insisted that the "dual model" was a
correct description of the strong force. Impossible. In this case he could
hardly make his later important contributions.
A.J. Tolland
> If we set all non-renormalizable couplings to zero at one mass scale
> M, and then shift to another mass scale M', we generate a whole bunch
> of non-zero non-renormalizable couplings. Remarkably, however, we can
> redefine our fields and coupling constants to set all the
> non-renormalizable constants back to zero. This wacky process is the
> usual notion of renormalization.
I should add here that these redefinitions (and their accompanying
violations of naive scaling laws) are not really necessary. We can
redefine our fields and couplings in any way we please, but there'd be no
point if it didn't vastly simplify the form of our Hamiltonian.
> The point of all this is that our notions of renormalizable are
> tied deeply with effective theory; they are apparently a consequence of
> two things: (a) the fact that we are keeping our effective degrees of
> freedom fixed when we change M, and (b) the miraculous properties of the
> couplings which dominate low energy physics. It's worth noting that we
> know of consistent theories which are not renormalizable in this sense.
It's also worth noting that we know of examples from string theory
where renormalizable and non-renormalizable interactions are interchanged:
anything with a UV/IR duality. This is one of the reasons that I suspect
that renormalization theory is relevant only to effective field theory,
and not to more complete theories.
--A.J.
zir...@my-deja.com
says...
>>> or Corichi and Reyes, gr-qc/0006067.
>"...not every spin-network state is an eigenstate of the area operator in the
>case when the vertex has valence 4 or more... This particular feature leads to
>the conclusion that the area operators of two surfaces that intersect along a
>line fail to commute for states which have 4 or higher valent vertex in the
>intersection. Thus, the quantum Reimannian geometry that arises from loop
>gravity is intrinsically noncommutative."
I forgot to mention that here the authors refer to a paper (which I have not yet
looked at):
Quantum Theory of Geometry III: Non-commutativity of Riemannian Structures,
A. Ashtekar, A. Corichi, J.A. Zapata gr-qc/9806041
Ralph Hartley
If I understand correctly, black hole entropy is not a test of
consistency, nor is it an experimental test. It is a test of the
semiclassical limit of a theory. In general it may be hard to tell
what the predictions of a theory are at normal scales. Black hole
entropy is a property that is sometimes relatively easy to calculate
and for which we know the "right" value (In the sense of being
compatible with like GR, QFT etc at large scales). So I guess it is OK
to use it as a test, because theories that don't pass may be
consistent, but it isn't really right to call them "quantum gravity".
String theory (using the term broadly) passes the test, at least for
some cases (and doesn't fail for any we know about), and it doesn't
need to be fine tuned at all to make it pass (and couldn't be fine
tuned anyway). This is a Good Thing.
LQG only passes the test for a particular value of a parameter. That
doesn't mean that LQG isn't a consistent theory of quantum gravity. It
does mean that LQG with the "obvious" parameters isn't. It also means
that LQG can't claim to predict the constant factor. I'm not sure how
much predicting the form of the relationship is worth. (It all depends
on how may potential theories get the wrong answer. Theories should be
graded on a curve. A wrong answer is just wrong, but a right answer is
worth more if other theories get it wrong.)
The result is that the number of free parameters in LQG is reduced by
1. How good that is depends on how obvious the "obvious" value of the
parameter is. If it is so obvious that you wouldn't even consider
other values if you didn't have to, then there is no net gain; you
added a new parameter to the theory and then fixed its value, and you
"used up" the black hole entropy so it can never be used to test the
theory. If, on the other hand, the value was so arbitrary that there
was (or would have been) serious consideration of other values even
without the black hole result, then the theory is better constrained
than before, which is good.
After all, string theory isn't consistent at all in the obvious number
of dimensions (4). It is a point in it's favor that the number of
dimensions is constrained, but I'm pretty sure that if it had turned
out that string theory was only consistent in 4D, you would be saying
"string theory predicts the dimensionality of space-time". Since that
is not the case, it is fair to say "string theory does not predict the
dimensionality of space-time", (Note: I don't know of any theory that
does). A less fair (but valid if your argument above is) statement
would be "string theory makes a prediction for the dimensionality of
space-time that is off by a factor of 11/4". What that means depends
on how "obvious" 4 dimensions is to you.
Ralph Hartley
Alfred Einstead
> Is there a way to reconcile M-theory with loop quantum gravity, combining
> them into a unified theory?
From Lubos Motl <mo...@physics.rutgers.edu>:
> Actually, my answer to the question "can they be reconciled" would be
> "most likely not". There are many reasons. Loop quantum gravity works in 4
> dimensions and requires a lot of changes to be put in higher dimensions
> (10 or 11 is certainly not better). String theory, on the contrary,
> predicts that we live in 10 or 11 dimensions.
Perhaps another perspective will offer some insights.
It's a little known fact that a cone is flat. It only looks curved,
but the curvature is entirely concentrated at one point -- the vertex.
The vertex can be characterized, precisely, by a certain number.
It you move a vector, via parallel transport, in a circle anywhere
on the cone, it will come back N degrees out of alignment if the
circle encloses the vertex, otherwise 0 degrees. The N is the
property of the vertex: its deficit angle.
Thus, a cube contains 8 curvature points each with 90 degrees
deficit, for a total of 720 degrees -- as is true for any
closed surface that looks like a sphere to a topologist.
I actually tried out that construction one, just arbitrarily
picking out a bunch of points, cutting out wedges and carefully
making sure the angles added up to 720, and sure enough, the
sheet closes up into a ball.
So, the sphere can be thought of as a surface with a uniform
distribution of vertices of density (1/R^2) angles/unit area.
The 1/R^2 is, of course, the measure of the sphere's curvature.
You can take a sheet of paper and cut out a bunch of vertices.
If the angles are small and the vertices are closely spaced and
arranged correctly, you'll approximate a general curved surface.
In effect, the curvature of that surface measures the density of
vertices on the sheet.
So, a long time ago, I got this really neat idea: let's create
a particle-based theory of gravity by pretending that all of
space-time curvature actually resides at points and that
these vertices are actually the particles of gravity.
Of course, as you know, this isn't possible. Once you move up
an extra dimension, a circle no longer objectively encloses
a point, so there are no longer any characteristic measures
for point vertices.
That's because the key items of interest are NOT points, but
subspaces of (n-2) dimensions. Thus, in 3-D, one has to
conceive of the curvature as being concentrated on lines,
not points. And the lines either have to close up as
circles or otherwise go off to infinity.
But, by analogy, the same thing holds true. If you parallel
transport a x-y-z coordinate frame in a circle along the (i-j) plane
(i,j=1,2,3) then the frame will come back with a fixed
rotation ONLY IF the loop links one of these singular
lines or circles. And, like before, one finds that associated
with each singularity is a characteristic value which
indicates by how much a general frame will be transformed
for each cycle it takes around one of these singularities.
This will be a function mapping (i-j)'s into rotations, or:
Characteristic Property: Bivectors -> Rotations
This, of course, is just the Riemannian tensor, concentrated
on lines.
But now with that minor correction we could actually go back and
reconsider the idea once again. But this time, in 3+1 space, the
singularities will be 2-dimensional; e.g. 1-dimensional loops or
infinite curves propagating in time or instantaneous 2-dimensional
surfaces that pop in and out at instants.
The curvature in space-time can then be thought of as actually
measuring the density of these objects, each of which is
characterized by a set of general winding numbers relating to
frame rotations.
A "particle-based" theory of gravity would then actually be a
string theory for these curvature singularities. And the
manifold, itself, would be thought of as being nothing more
than a flat space populated by a dense thicket of these
singularities. But the underlying space is purely secondary.
The manifold really reduces to NOTHING BUT a these singularities
and the "underlying space" is really just us projecting a geometry
on them by pretending they reside on a singular locally-flat
manifold.
Toby Bartels
>>(Actually, to be technically
>>correct, it doesn't even do that, at least according to the way the
>>terminology is normally used. Einstein-Hilbert usually refers to the
>>action written in terms of metric, rather than connection variables.)
>I do not understand why the same action should be given different names if
>you use different variables.
Because they give different quantum theories, I assume.
>>I think more could certainly be said here about problems with string/M
>>theory; for example, how does one calculate entropy for non-extremal
>>black holes?
>I have written a lot about this and enumerated many references to the
>papers on the entropy of Schwarzschild black holes etc. This is an example
>how extremely badly informed people from LQG are. The entropy of black
>holes in string theory is a well-undestood subject. After the
>Strominger+Vafa's fundamental paper, I think that the most relevant
>reference is the Maldacena+Callan's paper
> http://arXiv.org/abs/hep-th/9602043
(I see the string theorists' arrogance right up there on page 1.)
I also see that it deals only with "near-extremal" black holes,
the only type that you have ever committed yourself too.
Often you say "non-extremal", but usually you qualify with "near-extremal".
The Ashtekar et al calculation works regardless of extremality.
(OTOH, it only works for nonrotating holes. So we all have our flaws.)
And don't repeat the part about the "wrong answer" again;
I remark on that below.
>>But it is inaccurate to say that spin networks were put in by
>>hand--they weren't, and Ashtekar et al. were surprised when they
>>turned up. I have heard him say so.
>I believe you that Ashtekar was surprised. I just do not think that the
>appearance of spin networks in a covariant type of a lattice-QCD-like
>theory *should* be surprising. Or did you mean that whatever surprises Mr.
>Ashtekar Himself must be nontrivial? ;-)
I doubt it; we don't idolise Ashtekar and treat his word as gospel.
It just proves that he didn't put them in by hand, which is what was required.
>>More to the point, the reason loop quantum gravity is an active
>>program is not because it is in any way guaranteed to be right, nor
>>because its adherents necessarily "believe" that it is "the" theory.
>Loop Quantum Gravity is an active program? Can you elaborate on this
>statement?
I know a guy who works on it. See the last paragraph of this post.
>>No, no, no! It is *your* burden to prove what you say is right, not
>>anyone else's to prove that it is wrong!
>Not at all, it is very unlikely that you can show rigorously that
>something is unique or something else does not exist. So far string theory
>is a unique consistent quantum theory giving us semiclassical general
>relativity - and it is your task to find a counterexample, if you think
>that we are wrong. Judges would probably agree with me. If someone says
>that a $10 bill is the only thing that he has ever stolen - and some LQG
>people claim that he stole millions - it is *them* who must show that this
>$10 bill was not unique. Right?
The burden of proof there depends on who can be punished.
Switch prosecutor and defendant in the example and it goes the other way.
>Well, I am not sure whether you have really something to say about the
>Immirzi parameter if you cannot even spell it properly.
The moderators must have been in a good mood the day your post came by.
(Since you like religion, see Matthew 7:5.)
>Of course, by fine-tuning a parameter you can get one
>result to be anything you wish.
Not this result:
All large nonrotating black holes have the same value of S/A.
This value is unpredicted, but that they're the same is not unpredicted.
Of course, if we were string theorists (some string theorists),
then we'd say that there is a unique theory of quantum black holes,
it just doesn't predict where in the (1D!) moduli space we are.
>>And extremal black holes are to date of no physical relevance for our
>>universe. If you want to criticize LQG, make sure you are applying the
>>same standard to SMT as well.
>Sure that I do. The entropy of non-extremal black holes has been
>calculated in string theory, too, as I analyzed above.
Physically relevant ones?
>Well, this is just your very private opinion. If you measure how
>interesting string theory vs. LQG is, you should choose some objective
>criteria. If you measure, for instance, the number of papers written, you
>get something like 10:1? Yes, I am sure that such criteria are also not
>good enough or you because they give a "wrong" result. ;-)
I didn't realise that "how interesting" was the question.
But what your criterion measures is <how popular> instead.
>LQG people should realize that they are a part of the theoretical physics
>community, but a minor part.
I just wish that SMT people would realise that
LQG people are part of the theoretical physics community.
(Of course, many do.)
-- Toby
to...@math.ucr.edu
Kevin A. Scaldeferri
Lubos Motl <mo...@physics.rutgers.edu> wrote:
>completely rigorously that gravity fundamentally lives in 10 or 11
>(nonperturbatively) dimensions.
This statement is, of course, completely false.
Beyond the fact that it is false on its surface, that little "or" is a
good indicator of why one should be skeptical of such pronouncements.
>I do not understand why the same action should be given different names if
>you use different variables.
This will undoubtedly make Polyakov very sad.
>> I don't know why you would say this. Certainly there are consistency
>> checks performed within LQG: the closing of the constraint algebra,
>> and indeed the first-class nature of the constraints come immediately
>> to mind. Of course the checks are not the *same* as in M-theory,
>I mean, the closure is mostly a check that you wrote your formulae
>properly; of course, the canonical quantization of general relativity (in
>any variables, as long as they are equivalent) must pass these trivial
>classical checks (because of the consistency of string theory from which
>you can derive some "classical" relicts). But I meant some surprising
>numerical coincidences where you do not have reasons to expect that it
>should work - but it does.
>
>String theory is full of such examples, therefore it smells like God. But
>LQG smells like Man just because its checks are really not of "the same"
>sort as you say very well.
Facetiously, I will point out that people frequently do not notice
their own BO. Also that while there is a long history of people
finding evidence of God in numerical coincidence, most scientists are
not impressed by such activities.
More seriously, what he said was that the checks are not the same, not
that they are not of the same sort, by which you seem to insinuate
that they are not of the same caliber.
>> networks are so important in loop quantum gravity. Logically, this
>> all depends ultimately on your starting assumptions, both in choice of
>> action, variables for that action, and indeed the choice to do
>> canonical quantization at all.
>Yes, it depends on all those initial assumptions, therefore LQG smells
>like Man. String theory does not depend on any initial conditions that you
>mentioned.
String theory depends on a different set of initial assumptions. If
the initial assumptions weren't different, they would be the same
theory, now wouldn't they?
Moreover, I'm not aware that anyone has actually explored to what
extent string theory is independent of the choices made in quantizing
the theory. (However, it is possible that I am just ignorant here.)
>String theory does not depend on your choice of the action.
Really? So, string theory follows from, say, the Dirac or
Klein-Gordon action?
>There is only one consistent way how to quantize
>it and understand it.
I don't think you mean this in the same way that LQG people mean
this. In particular, I am quite certain you can't prove this.
>> But it is inaccurate to say that spin networks were put in by
>> hand--they weren't, and Ashtekar et al. were surprised when they
>> turned up. I have heard him say so.
>I believe you that Ashtekar was surprised. I just do not think that the
>appearance of spin networks in a covariant type of a lattice-QCD-like
>theory *should* be surprising. Or did you mean that whatever surprises Mr.
>Ashtekar Himself must be nontrivial? ;-)
But, string theorists are not arrogant.
At any rate, I know many people who think that anything that surprises
Witten is nontrivial. Of course, opinions of triviality differ and
I can give you examples of things that I find trivial that Witten
finds surprising, but it would take us afield of physics.
>String theory is the only known consistent theory that can predict a world
>containing all the forces and particles we see.
The last time I asked the question, does anyone know of a
compactification of string theory that yields the Standard Model (or,
even, the MSSM) the answer I got was no. Are you claiming that this
is no longer the case?
>> Though this has been expounded many times in this newsgroup, perhaps
>> it bears repeating nonetheless (if this is cut by the moderators, I
>> guess not!). Loop quantum gravity is worked on mainly by people who
>> come from the relativity community; i.e., they had gravity, now they
>> want to quantize it. They therefore bring with them a certain mode
>Yes, this is a sociological problem of LQG - that the people are
>*relativists* only. To quantize something like field theory, you should
>know quantum field theory.
Are you actually aware of JB's background?
>Well, this is just your very private opinion. If you measure how
>interesting string theory vs. LQG is, you should choose some objective
>criteria. If you measure, for instance, the number of papers written, you
>get something like 10:1? Yes, I am sure that such criteria are also not
>good enough or you because they give a "wrong" result. ;-)
Well, since you like this metric, we should all quit doing physics and
head to chemistry or biology, where the volume of papers is _vastly_
higher than physics.
--
======================================================================
Kevin Scaldeferri Calif. Institute of Technology
The INTJ's Prayer:
Lord keep me open to others' ideas, WRONG though they may be.
Lubos Motl
> I'd think you'd want to embrace the anthropic principle since that IS what
> crosslinks the current trial theory into coherence -- like you sort of say.
> Then, at least you wouldn't be beginning with the prejudice that you don't
> like it and are working to erase it (even though, well, it is in
> operation).
Physicists do not like the anthropic principle and I think that it is a
completely scientific and justified position, not a prejudice. While it is
true that some drastic changes of the laws of physics would kill life of
any sort remotely similar to ours, we should not go too far because we
would become a kind of "racists"; we should not assume that our life is
too "special" unless we have some evidence.
Just like there exist other species, there may exist other forms of life
with different compounds replacing the aminoacids or completely different
science replacing biochemistry etc. Life of some sort could possibly
appear in the worlds where the fine structure constant is 1/136.5, too. An
anthropic explanation should be the last thing we try after we give up
everything else. And an anthropic explanation is fine only if one has a
working theoretical framework that offers a huge number of possibilites
(or "vacua").
Science could have been stopped many times in the past if people gave up
and accepted an anthropic explanation. In fact, the anthropic principle is
comparably anti-scientific as the old religious ideas about the principles
underlying the Universe. People were also thinking that all the things in
the world had to be arranged in this way because our Lord is the greatest
being in the world and beyond: therefore He knew why he should have
created exactly these elements, the animals we observe etc. Asking what is
the rational reason behind the specific data was something that
transcended human abilities.
Science has changed this approach completely. People were suddenly allowed
to ask "WHY" and they did. And often, they found the answers. Science is
not just searching for new phenomena: trying to unify and explain the
known phenomena from a unified perspective is equally important, at least.
Few centuries ago, you could also think that the properties of carbon,
hydrogen and oxygen must be exactly what they are, otherwise we would die.
From many universes where the parameters describing these elements can
take arbitrary (and independent) values, only the Universe with the
correct properties (our Universe) allows life. Fortunately people
discovered electrons and Rutherford made his experiments. And at the end
we know that the properties of all the elements can be derived from a
single starting point by completely well-defined calculations. Millions of
data concerning frequencies and other properties of spectral lines can be
completely calculated from three parameters or so. This is what I call
"science" and if people just agreed that the choices had to be this way,
otherwise we could not be here, I would think that it was not science. I
expect that this line of success should continue.
Weinberg wrote that a physicist talking about the anthropic principle runs
the same risk as a cleric talking about pornography: no matter how much
you say you're against it, some people will think you'are a little too
interested. ;-)
> If you embrace it and it works maybe it will lead you to a place where you
> can grok a new trial theory that really falsifies the anthropic principle.
Well, different ideas can lead you to different places, history of science
is mostly unpredictable. But I think that the anthropic principle is
little bit like religion: once you accept it, you are not allowed (or
driven) to search for a rational explanation of the "remaining" things
that you do not understand.
> But, until then, I think denial or avoidance is being invoked first and
> that leads to a fairly irrational set of expressions.
I think that the expressions are rational and it is correct that we try to
avoid the anthropic arguments as much as possible. People should try to
solve their problems in a reasonable way first - and only if they become
really desperate, they can think about drugs etc. :-)
Aaron Bergman
>This will undoubtedly make Polyakov very sad.
Actually, I don't think Polyakov wants the thing named after him.
ba...@galaxy.ucr.edu
Lubos Motl <mo...@physics.rutgers.edu> wrote:
>So hopefully I have explained why I feel that LQG is (even if it gets more
>successful than so far) pretty orthogonal to what I would consider
>"progress in fundamental physics".
You've explained to me quite clearly why you feel this way. Of course
I don't agree that you are *right* to feel this way. But I can live
with this disagreement, which only time and experiment will settle.
I doubt further debate will change anyone's mind, so it's probably best
if we each work on our own favorite approach, while doing our best to
learn about other approaches.
>For me, it is mostly an attempt of the
>people to defend the old ways of thinking (to say that there is nothing
>wrong about them, in this case about the local field theories applied to
>all the forces including gravity) - much like some people tried to avoid
>the revolutions of relativity and/or quantum mechanics.
To me it seems odd that you would say this, given that most people
working on loop quantum gravity have abandoned the assumption that
spacetime is a manifold, and are instead working on an approach
based on quantum field theory on a Lie group, with spacetime itself
arising as a superposition of lots of Feynman diagrams that show up
in this quantum field theory.
I'm talking, of course, about the Rovelli-Perez spin foam model,
which has now been proved finite order by order in perturbation theory.
Unfortunately I missed the conference in Warsaw where everybody was
all excited about this, but I got a nice description of it from Ashtekar
when he flew over here to Stony Brook. By the way, he was rather
amused to hear you had claimed he no longer works on loop quantum
gravity! It turns out he has only been avoiding work on spin foams
in order to let Rovelli have this territory. Ashtekar now wants to tackle
the Immirzi parameter problem by looking at how it relates to Lorentzian
spin foams. Interestingly, people have recently figured out a formulation
of 2+1 quantum gravity as an SU(2) gauge theory. When you set it up
this way, the Immirzi parameter shows up there, very much as in the 3+1
case! Since we also know ways to do 2+1 quantum gravity without
an Immirzi parameter, this should shed some interesting light on this
puzzle.
>But furthermore I
>believe that a non-stringy quantization of gravity above 3 dimensions is
>necessarily inconsistent.
Of course people also thought this for 3-dimensional quantum gravity...
until people figured out the right sort of mathematics to tackle the problem.
So: you are free to believe this, but there is certainly no proof.
On another note... I had written:
>> In fact, it's easy to see that as we keep cranking up the momentum,
>> G will approach the value for which
>>
>> dG
>> ------- = 0
>> d(ln p)
>>
>> We call this value an "ultraviolet stable fixed point" for the
>> gravitational constant. Mathematically, what we've got is a flow
and you replied:
>This is completely unfair. You forgot to say that the claim that 4D
>gravity has a 4D UV fixed point is just a conjecture [...]
I did not forget to say this: perhaps I should remind you what I
actually said. Briefly, I started by writing this equation:
dG
------- = (n - 2)G + aG^2 + ....
d(ln p)
where a < 0. Then I wrote: "If we ignore the higher-order terms [...]
this implies something very interesting for quantum gravity in 4 dimensions."
Then I talked about this ultraviolet fixed point idea.
Then, however, I added: "But before we break out the champagne, note that
we neglected the higher-order terms [....] They can still screw things up."
And I concluded by emphasizing that the question of the existence of
an ultraviolet fixed point in 4d quantum gravity was still open.
>- and most likely an
>incorrect conjecture. Certainly there is no evidence.
The conjecture may not be correct, but the material you quoted
came from an issue of This Week's Finds where I discussed numerical
evidence in favor of it, so you are wrong to say there is NO evidence.
See
http://math.ucr.edu/home/baez/week139.html
for details.
But personally I am NOT staking my hopes on the existence of an
ultraviolet fixed point. Instead, I'm hoping that spacetime
has a subtle kind of discreteness at the Planck scale. As
discussed earlier on this newsgroup, a perfectly sensible theory
on a discrete spacetime can look like a nonrenormalizable field
theory on the continuum at large distance scales.
John Baez
Paul D. Shocklee <shoc...@phoenix.Princeton.EDU> wrote:
>John Baez (ba...@galaxy.ucr.edu) wrote:
>> http://www.cs.cornell.edu/Info/People/chew/Delaunay.html
>Cool!
>
>These actually seem rather reminiscent of grid diagrams for
>string networks in Type IIB, which are basically dual descriptions
>of the networks.
Could you say a bit about what string networks are? A long time
ago I posted a request for you to do this, but I'm afraid it got
eaten by a black hole. That's one of the dangers of working on
quantum gravity....
I could of course read your paper, but being lazy, I'm hoping
you can start by giving me the gist of the idea. I'm always looking
for ways to relate string theory and loop quantum gravity.
Btw, on the actual subject of this thread, here's the latest
news from Warsaw:
The Thiemann-et-al coherent states of loop quantum gravity
describe the "Higgs phase", while the Ashtekar-et-al coherent
states describe the "Coulomb phase" - to use 't Hooft's classification
of gauge theories according to whether the E and/or B fields are
confined. Perturbative quantum gravity lies in the Coulomb phase
(neither E nor B confined), so I guess Ashtekar et al's coherent
states, based on Voronoi diagrams, are more relevant for finding
a sector of loop quantum gravity which acts like perturbative
quantum gravity at large distance scales.
Greg Kuperberg
Lubos Motl <mo...@physics.rutgers.edu> wrote:
>completely scientific and justified position, not a prejudice.
Okay, if you don't like the anthropic principle, how come we live on a
large celestial object which, unlike the vast majority of those in the
universe, just HAPPENS to be short on hydrogen and helium and long on
higher elements? What an AMAZING coincidence. :-)
Lubos Motl
>> String theory on the contrary tells you
>> completely rigorously that gravity fundamentally lives in 10 or 11
>> (nonperturbatively) dimensions.
> This statement is, of course, completely false.
I do not think that one must be a physicist to know *what* is the
dimension of spacetime according to string theory. It is enough to read
the press and popular books. A theoretical physicist should be at least
able to reproduce the calculation that shows that perturbative string
theory requires total of 10 dimensions, i.e. to answer *why*. I can do it.
If you are interested, just ask! ;-)
> Beyond the fact that it is false on its surface, that little "or" is a
> good indicator of why one should be skeptical of such pronouncements.
The total number of dimensions is not a completely well-defined notion. If
a dimension is smaller than the fundamental length, you should be careful
when you want to make a geometric interpretation of such a claim. However
what makes sense, is the question what is the maximal number of dimensions
that you can decompactify. In M-theory, it is simply equal to eleven.
M-theory compactified on a circle of radius R (in 11-dimensional Planck
units) is equivalent to type IIA string theory in 10 dimensions with
coupling constant g=R^{3/2}. If R is small, g is small and the
10-dimensional type IIA description is more relevant. If R is large, g is
also large and physics is better understood in 11-dimensional terms.
Geometrical notions, including the number of dimensions, are well-defined
in the long distance limit only.
The discovery of a hidden, 11-th dimension of string theory was a dramatic
breakthrough in string theory (Witten, 1995, based on works of other
people). But it does not change the perturbative insights; it just leads
us to look at them from a more general perspective.
> >I do not understand why the same action should be given different names if
> >you use different variables.
> This will undoubtedly make Polyakov very sad.
Polyakov action is however not the same action as the Nambu-Goto action,
for example. It contains much more stuff, namely b,c ghosts. Even
classically you cannot say that these actions are completely equivalent.
The Fadeev-Popov ghosts are not just redundant degrees of freedom (or
Langrange multipliers) in the Polyakov approach!
Regardless of those technical details, the most important thing in this
respect still holds: all the (consistent) ways how to quantize a string
are completely physically equivalent (apart from the few discrete choices
of boundary conditions, that lead to type I,IIA,IIB,HE,HO theories etc.,
that can be shown to be equivalent only after 10 years of research). And
it would be very bad if the reality was different; the uniqueness of the
theory would be seriously damaged.
> >String theory is full of such examples, therefore it smells like God. But
> >LQG smells like Man just because its checks are really not of "the same"
> >sort as you say very well.
> Facetiously, I will point out that people frequently do not notice
> their own BO. Also that while there is a long history of people
> finding evidence of God in numerical coincidence, most scientists are
> not impressed by such activities.
Well, then one should note that the "miracles" that string theory shows
are slightly too huge and unbelievable phenomena to be called
"coincidences". Most fascinating things in physics have the character of
such "coincidences". For example, the experiment does not tell you
"directly" that you should compute loop diagrams. Feynman (and people
before him) had to figure out the rules theoretically. And then the
higher-order calculation of the electron's magnetic moment agrees to 12
decimals or so with the experiment. Is this also just a "coincidence that
most scientists are not impressed by"? ;-) The situation in string theory
is analogous but the "experiments" have a theoretical character, too. The
technical situation of the current civilization simply does not allow to
do the relevant experiments for any theory going beyond the Standard
Model.
> More seriously, what he said was that the checks are not the same, not
> that they are not of the same sort, by which you seem to insinuate
> that they are not of the same caliber.
I assumed that they are not of the same caliber because I know (roughly,
at least) the checks in both fields and one can see that the checks are
not of the same caliber as in string theory.
> >Yes, it depends on all those initial assumptions, therefore LQG smells
> >like Man. String theory does not depend on any initial conditions that you
> >mentioned.
> String theory depends on a different set of initial assumptions. If
> the initial assumptions weren't different, they would be the same
> theory, now wouldn't they?
You would have to specify which "initial assumptions" you mean. Because
the only initial assumptions I know of are that the fundamental objects
are 1-dimensional and that the theory is quantum-mechanically consistent,
I cannot comment on your points much.
> Moreover, I'm not aware that anyone has actually explored to what
> extent string theory is independent of the choices made in quantizing
> the theory. (However, it is possible that I am just ignorant here.)
Every individual description of string theory that we know has one way to
quantize it only. For example, BFSS matrix models have a well-defined
Lagrangian of the sort "P^2+V(X)" and therefore all the quantizations are
equivalent, just like in ordinary QM of this sort. RNS string is a
conformal field theory. Once you fix which theory you work with, all the
OPEs etc. have a unique interpretation etc. Furthermore all these
apparently different approaches give the same results. String theory is
unique, beyond any reasonable doubt.
> Really? So, string theory follows from, say, the Dirac or
> Klein-Gordon action?
No, on the contrary. String theory is more fundamental than the Dirac or
Klein-Gordon action. You can derive both as effective descriptions of
parts of your physics from string theory.
If you wanted to construct something based exactly on Dirac or
Klein-Gordon action and call it string theory, well, this is not the
correct path. Klein-Gordon field and Dirac field do not respect principles
of string theory and therefore they are not, of course, the ultimate model
nor string theory. Consequently, they also do not describe gravity. ;-)
String theory does not allow on any modifications that you would like to
change in your action etc. But of course, you must still work with string
theory if you want to claim that your results are relevant for string
theory or quantum gravity. Dirac field in 4 dimensions is *not* string
theory. ;-)
> At any rate, I know many people who think that anything that surprises
> Witten is nontrivial. Of course, opinions of triviality differ and
> I can give you examples of things that I find trivial that Witten
> finds surprising, but it would take us afield of physics.
Well, there is certainly no law that Witten is surprised by nontrivial
results only. I know a few very unclear counterexamples. There are
probably not too many. Witten is the most productive (and arguably, the
brightest) living physicist but he is also a human being.
> >String theory is the only known consistent theory that can predict a world
> >containing all the forces and particles we see.
> The last time I asked the question, does anyone know of a
> compactification of string theory that yields the Standard Model (or,
> even, the MSSM) the answer I got was no. Are you claiming that this
> is no longer the case?
Are you kidding? There are thousands of choices. In fact, too many, I
would be personally happier if the number was smaller. Initially people in
the 80s studied HE heterotic strings on Calabi-Yau 3-folds. Even the
simplest, orbifold models can give GUT theories (or directly MSSM, if
broken in the proper way) plus gravity at low energies (sometimes plus a
couple of exotics). Then I personally liked the free fermionic heterotic
models - there are about 100 papers like
http://arXiv.org/abs/hep-ph/9405357
Recently, David Berenstein et al. also constructed the first "Standard
Model on D-branes"
http://arXiv.org/abs/hep-ph/0105042
The way how they get the correct quantum numbers, without anything
stemming from the Grand Unification, is pretty amusing. There are
different descriptions (B. Acharya is working on M-theory on singular
7-manifolds with G2 holonomy, there are examples of F-theory on Calabi-Yau
4-folds etc.). Finally all of them are likely to be shown dual to each
other and a universal potential should pick a few choices of 4D vacua with
broken N=1 SUSY (or maybe one - but maybe many).
String theory is certainly not lacking realistic vacua. On the contrary,
string theorists need to find a sharp way (accurate enough calculational
tools) that can rule out all of those pretty well-looking vacua except for
the truly correct one.
> >Yes, this is a sociological problem of LQG - that the people are
> >*relativists* only. To quantize something like field theory, you should
> >know quantum field theory.
> Are you actually aware of JB's background?
Yes, I apologize. There should have been another sentence "And this is
not the case with the exception of John Baez and one is too little". ;-)
> Well, since you like this metric, we should all quit doing physics and
> head to chemistry or biology, where the volume of papers is _vastly_
> higher than physics.
Thanks. But let us not compare apples with oranges. Maybe one could then
send you to work with Madonna or the Pope or anyone else. ;-) From another
point of view, chemistry is not as fundamental as physics but it is
probably true that chemistry and biology is currently more active than
theoretical and particle physics, I think.
John Baez
<zir...@my-deja.com> wrote:
>I took a look at this paper and was hoping that a LQG expert (like JB) could
>elaborate on what the following sentences mean (quoted from the top paragraph
>on page 10) -
>"...not every spin-network state is an eigenstate of the area operator in the
>case when the vertex has valence 4 or more... This particular feature leads to
>the conclusion that the area operators of two surfaces that intersect along a
>line fail to commute for states which have 4 or higher valent vertex in the
>intersection. Thus, the quantum Riemannian geometry that arises from loop
>gravity is intrinsically noncommutative."
The last sentence is just a summary of the preceding ones. Given
a 3-dimensional manifold S, there is a Hilbert space of "quantum
geometries", which has a basis given by spin networks embedded in
S. Any unit vector in this Hilbert space describes the quantum
analogue of a Riemannian metric on S together with the extrinsic
curvature of S as it sits in 4d spacetime. There are lots of
interesting operators on this Hilbert space. The simplest are
the "area operators". For each surface X embedded in S, there is
an operator A(X) which measures the area of this surface. These
operators have discrete spectra of eigenvalues, so area is discrete
rather than continuous in this theory. Moreover, if two surfaces
intersect, their corresponding area operators do not commute.
>What is this quantum Riemann geometry like [...?]
For a nice easy introduction, try this:
Abhay Ashtekar, Quantum Mechanics of Geometry, available
as http://xxx.lanl.gov/abs/gr-qc/9901023
For more details, try these:
Abhay Ashtekar and Jerzy Lewandowski, Quantum theory of geometry I:
area operators, available as
http://xxx.lanl.gov/abs/gr-qc/9602046">gr-qc/9602046
Abhay Ashtekar and Jerzy Lewandowski, Quantum theory of geometry II:
volume operators, available as
http://xxx.lanl.gov/abs/gr-qc/9711031
Abhay Ashtekar, Alejandro Corichi and Jose A. Zapata,
Quantum theory of geometry III: Non-commutativity of Riemannian
structures, available as http://xxx.lanl.gov/abs/gr-qc/9806041
>[...] and is it similar to the quantum
>Riemann manifolds in Shahn Majid's general noncommutative Riemann
>geometry?
Not really. Majid and others replace the commutative algebra
of functions on space by a noncommutative algebra. Ashtekar and
company take the kinematical phase space of general relativity
and quantize that. This is more directly motivated by physics.
Toby Bartels
>Lubos Motl wrote:
>>Physicists do not like the anthropic principle and I think that it is a
>>completely scientific and justified position, not a prejudice.
>Okay, if you don't like the anthropic principle, how come we live on a
>large celestial object which, unlike the vast majority of those in the
>universe, just HAPPENS to be short on hydrogen and helium and long on
>higher elements? What an AMAZING coincidence. :-)
And what are the chances that we'd live on any celestial object at all,
rather than in the much more prevalant cold vacuum of intergalactic space?
Perhaps what Greg means to point out is the difference between
the *weak* and *strong* anthropic principles.
-- Toby
to...@math.ucr.edu
Squark
<9h08ev$ag$1...@glue.ucr.edu>):
>The Thiemann-et-al coherent states of loop quantum gravity
>describe the "Higgs phase", while the Ashtekar-et-al coherent
>states describe the "Coulomb phase" - to use 't Hooft's classification
>of gauge theories according to whether the E and/or B fields are
>confined.
A bit off subject, but still interesting: what models are there that have
both E and B confined? It is possible, I suppose?
Best regards,
Squark.
--------------------------------------------------------------------------------
Write to me at:
[Note: the fourth letter of the English alphabet is used in the later
exclusively as anti-spam]
dSdqudarkd_...@excite.com
A.J. Tolland
> An anthropic explanation should be the last thing we try after we give
> up everything else. And an anthropic explanation is fine only if one
> has a working theoretical framework that offers a huge number of
> possibilites (or "vacua").
I'm curious about the second of these two statements: Do you mean
that we can use the "anthropic technique" to guess which region of the
vacuum moduli space corresponds to our world by eliminating from
consideration vacua which clearly do not correspond to reality? I hope
you're not claiming that -- as a last resort -- we should try to use the
anthropic principle to figure out which vacuum is ours. The anthropic
principle is fairly lousy as a physical principle, because it can't make
exact predictions; it can only eliminate wrong answers. The problem is
that no one will ever notice if Newton's constant changes by a factor of
1.0000000000000000000000000000000000002.
Then again maybe the final theory will get around this by coming
up with some form of "sufficient discreteness" in the moduli space, so
that anthropic elimination of obvious wrong answers will actually lead us
to a unique correct answer.
> People should try to solve their problems in a reasonable way first -
> and only if they become really desperate, they can think about drugs
> etc. :-)
From the Unix fortune program, source and repository of all
wisdom: "Reality is a crutch for those who can not cope with drugs."
I believe the quote is one of Mae West's zingers.
--A.J.
Lubos Motl
> If I understand correctly, black hole entropy is not a test of
> consistency, nor is it an experimental test. It is a test of the
> semiclassical limit of a theory.
Yes, I agree in a sense. But I count the semiclassical limit, compatible
with the other phenomena in the semiclassical (and classical) limit, to be
a part of the consistency of a given theory. Your other comments look fine
to me, too.
> After all, string theory isn't consistent at all in the obvious number
> of dimensions (4). It is a point in it's favor that the number of
> dimensions is constrained, but I'm pretty sure that if it had turned
> out that string theory was only consistent in 4D, you would be saying
> "string theory predicts the dimensionality of space-time". Since that
> is not the case, it is fair to say "string theory does not predict the
> dimensionality of space-time",
String theory - at least as we know it today - does not predict the number
of large dimensions (other theories don't predict it either, as you noted
very well). But in the stringy sense the theory predicts the total number
of dimensions (including the compact ones): perturbatively it is ten,
nonperturbatively it is eleven in the context of M-theory.
> (Note: I don't know of any theory that does). A less fair (but valid
> if your argument above is) statement would be "string theory makes a
> prediction for the dimensionality of space-time that is off by a
> factor of 11/4". What that means depends on how "obvious" 4 dimensions
> is to you.
:-) This is based on your incorrect belief that the number 10 or 11 of the
total number of dimensions is wrong. But this number is probably correct.
The extra six or seven dimensions at every point must be taken seriously,
we think. It is hard to get used to this pretty radical change but this is
certainly a general change of paradigm that string theory brought us, just
like special relativity implies a general speed limit, for example. And
these extra dimensions are a very good thing, not bad. The detailed
structure and properties of the non-gravitational forces (and particles)
are connected with the precise shape of the extra dimensions.
Paul D. Shocklee
> Could you say a bit about what string networks are? A long time
> ago I posted a request for you to do this, but I'm afraid it got
> eaten by a black hole. That's one of the dangers of working on
> quantum gravity....
Okay.
In Type IIB string theory, you have two kinds of one-dimensional
objects - fundamental strings (call them F-strings) and one-dimensional
D-branes (call them D-strings). (There are also lots of higher-dimensional
D-branes, but let's just consider these for the moment.) You can also
consider bound states of p F-strings and q D-strings, called (p,q) strings.
These are bound whenever p and q are relatively prime (and marginally
unstable otherwise). In this notation, then, F-strings are (1,0) strings,
and D-strings are (0,1) strings.
(Another way of thinking about this is that the low-energy effective theory
of Type IIB string theory, namely Type IIB supergravity, has two kinds of
2-form fields, B and B', and you can have solitonic solutions with
charge p under B and charge q under B'. Furthermore, Type IIB supergravity
has an SL(2,R) symmetry, under which you can take a state of charge (p,q)
and map it to a state
/a b\ /p\ ,
\c d/ \q/
where ad-bc=1. In the quantum string theory, this SL(2,R) is broken by
Dirac-type quantization conditions to SL(2,Z).)
Now, D-strings are objects on which fundamental strings can end, so you
can make junctions of F- and D-strings, which look like this:
(1,1)
/
/
/
F -----o
|
|
|
D
The third leg of the junction is a (1,1) string. In fact, you can make
a whole web of these guys. The only rules are
1. The charges must sum to zero with respect to the vertex (taking
into account the fact that a (-p,-q) string is just a (p,q) string
with the opposite orientation).
and
2. The tensions must balance (for a stable junction).
The tension for a (p,q) string is
T_(p,q) = T_(1,0) |p + q tau|,
where tau is a complex number called the axion-dilaton modulus of
Type IIB. For simplicity, let me set tau equal to i, which corresponds
to looking at string coupling of order 1. Then,
T_(p,q) = T_(1,0) |p + i q| = T_(1,0) sqrt(p^2 + q^2),
and we can satisfy rule 2 quite easily by the following prescription.
Let's orient each of our (p,q) strings so that they lie along
the vector (p+iq) in the complex plane. Then (1,0) strings are
all horizontal, (0,1) strings are all vertical, (1,1) strings
are at a 45 degree angle, etc.
And of course, they're all in a single plane.
It then turns out that this configuration preserves 1/4 of the
supersymmetry of Type IIB, no matter how many strings you have
in the web (as was shown by Sen in hep-th/9711130).
So, what we have is a big network of 1-dimensional objects, with
SL(2,Z) representations on the links, and some constraints at
the vertices.
You can now play with these things, or, in other words, work out
their dynamics. At low energies, this basically boils down to
oscillation modes (worked out by Callan and Thorlacius in
hep-th/9803097) and rigid string modes (worked out in our recent
paper).
The rigid modes are things like this:
(1,2)
/
/
/
o
/|
/ |
/ |
/ |
/ |
o-----o
/ \
/ \
/ \
/ \
/ \
(-2,-1) (1,-1)
My ascii art sucks, but I hope you see what I mean. You just blow up
a three-string vertex into a triangle.
You can also have "flop" transitions, where you go from
something like this:
|
|
o---
/
/
---o
|
|
to something like this:
|
|
---o
\
\
o---
|
|
(I seem to recall that both of these show up in the evolution rules for
spin networks.)
Now, the number of rigid modes is fixed once you've specified the charges
of the external strings, and to calculate it, you construct something called
the "grid diagram", which is basically a dual picture of the web constructed
on an integer lattice. The number of internal points in the grid diagram
gives you the number of rigid modes.
The same seems to be true for the Voronoi diagrams, in that the number of
internal faces in the web is equal to the number of internal points in
the diagram. (I don't know if there is any relation to the Delaunay
Triangulation, though.)
So there do seem to be some things in common between string networks and
spin networks:
1. Both are networks.
2. Both have group representations on the links.
3. Both have some constraint on the representations at the vertex.
4. Their evolution rules have some similarities.
And then there are the differences:
1. String networks are embedded in space, while spin networks are
supposed to represent states of space.
2. SL(2,Z) vs SU(2).
3. The role of the intertwiner for the spin network. I'm not sure
if the charge condition is quite analogous.
4. String networks evolve smoothly, while (I think) spin networks
evolve discontinuously. Although, now that I think about it, from
the spin foam perspective, they look about the same in spacetime...
Also, the smooth evolution of the string networks is just at the
classical level. These are nonperturbative objects, so we don't
know quite how to quantize them. (This ties into the other recent
thread about whether area and volume are quantized in string theory.)
5. Spin networks represent rules for calculating amplitudes. Nothing
similar seems to exist for string networks.
6. Supersymmetry.
7. The string networks I've described are planar, while spin networks
are three-dimensional. Actually, you can construct 3-dimensional
string networks if you compactify some directions. You then get
(p,q,r) strings of SL(3,Z), and you preserve 1/8 of supersymmetry.
(Since you start out with 32 in Type II, that leaves you with 4.)
You can also make stable string networks that satisfy the tension
condition, but are not supersymmetric.
8. String networks are coupled to all of the other fields of Type IIB,
while spin networks seem to only involve gravity.
> Btw, on the actual subject of this thread, here's the latest
> news from Warsaw:
> The Thiemann-et-al coherent states of loop quantum gravity
> describe the "Higgs phase", while the Ashtekar-et-al coherent
> states describe the "Coulomb phase" - to use 't Hooft's classification
> of gauge theories according to whether the E and/or B fields are
> confined. Perturbative quantum gravity lies in the Coulomb phase
> (neither E nor B confined), so I guess Ashtekar et al's coherent
> states, based on Voronoi diagrams, are more relevant for finding
> a sector of loop quantum gravity which acts like perturbative
> quantum gravity at large distance scales.
Interesting. String networks (and brane networks in general) also
play important roles in the Higgs and Coulomb phases of supersymmetric
Yang-Mills. Basically, if you take N D3-branes and look at the physics
of open strings ending on them, you find an SU(N) n=4 SYM theory in
3+1 dimensions. From the perspective of the brane, a (p,q) string
endpoint looks like a (p,q) dyon, and a network of strings looks
like a bound state of dyons. Deformations of the string network
correspond to moving about in the moduli space of the Coulomb phase,
and other brane networks correspond to the Higgs phase.
--
Paul Shocklee
Graduate Student, Department of Physics, Princeton University
Researcher, Science Institute, Dunhaga 3, 107 Reykjavik, Iceland
Phone: +354-525-4429
Lubos Motl
personally like a lot. I could not resist many times and showed them to
many people (independently of AE). The icosahedron, dodekahedron, cube,
tetrahedron, octahedron - or anything with the same topology - has the
total deficit angle 4.pi. This deficit angle is equally distributed over
the sphere.
If one computes the total deficit angle for a different topology, he
always gets 2.pi.chi where chi is the Euler character. This topological
invariant is equal to 2-2g for closed oriented surfaces; g is the genus,
the number of handles. (Every hole (boundary) subtracts 1 from chi just
like one crosscap but it is not relevant here.) A flat model of torus,
made of paper, has the total deficit angle equal to zero, for example.
There are various places in string theory where one can see this counting
in action. For example, F-theory is a 12-dimensional theory that makes
sense only when we compactify it on something that is locally a 2-torus.
Then we get type IIB string theory.
However we can also take this 12-dimensional F-theory and compactify it on
a eliptically fibered K3 surface. A K3 surface is 4-dimensional, has
holonomy SU(2) and if it is eliptically fibered, it looks like a sphere
S^2 with a small 2-torus (the fiber) attached to every point. The fibers
can become singular at isolated points.
The sphere has deficit angle 4.pi. This determines the total amount of
singular fibers that we must find on the sphere. This number is equal 24
in some rough way (24 is the Euler character of K3 surfaces, too); each of
the singularities has a deficit angle pi/6, so to say. Therefore Cumrun
Vafa had to come up with a desciption using algebraic geometry where the
S^2, understood as a complex plane, has a polynomial of the 24th degree
defined on it. The roots of this polynomial define the singular fibers
(7-branes).
Those things can sound arbitrary but in fact, this is the only solution
how can we get a theory with 16 supercharges in 8 dimensions in such a
geometric way. F-theory on K3 is dual to heterotic strings on T^2 etc. and
one can do a lot of checks of this duality.
On 19 Jun 2001, Alfred Einstead wrote:
> So, a long time ago, I got this really neat idea: let's create
> a particle-based theory of gravity by pretending that all of
> space-time curvature actually resides at points and that
> these vertices are actually the particles of gravity.
Well, maybe someone should warn you that the sick problems of quantized
general relativity arise exactly because the curvature is "squeezed" to
one point, in a sense. The electromagnetic force has origins in the
exchange of photons. A particle can suddenly send a virtual photon and the
other can absorb it. The gravitational force has gravitons instead, but a
corresponding calculation leads to nasty divergences. And they appear
exactly because the curvature, caused by a graviton that was suddenly
emitted, is squeezed into a too small region.
Otherwise I understand that your idea is slightly different and
interesting. String theory - and probably every consistent quantum gravity
- cures the divergences in such a way that it allows the curvature to be
dissolved over a region. The distances smaller than the Planck length do
not have a geometric interpretation, therefore the curvature cannot be
squeezed to that small regions and the divergences go away if you are
lucky and you choose a good theory. ;-)
Lubos Motl
> Really? So, string theory follows from, say, the Dirac or
> Klein-Gordon action?
Maybe, last time I did not understand precisely your point. I thought that
you meant 4-dimensional Klein-Gordon and Dirac fields. These are, of
course, just a very small part of the structure of string theory.
Then I realized that you might have meant two-dimensional Klein-Gordon and
Dirac fields.
If this was the case, the answer is: yes, string theory can be derived
from the 2D Klein-Gordon fields as well as 2D Dirac fields (if one knows
how to convert them into string theory with all of its special tools), and
furthermore the results are equivalent and lead to the same string theory.
The proof is called "bosonization" or "fermionization", depending on your
starting point. Two components of a (massless) Dirac field - let us denote
them c,b - are equivalent to a single Klein-Gordon (massless) Phi. The map
is very nonlinear, c=exp(i.Phi), b=exp(-i.Phi), partial Phi = cb. However
the results are identical. If we postulate the correct boundary
conditions, bosons and fermions make the same job.
In fact, there are examples where the boundary conditions are easy on both
sides. Let us construct the heterotic string. (C) Rohm, Martinec, Gross,
Harvey 1985. It is "heterotic" because it is a hybrid ("heterosis" in
Greek) of the 26-dimensional bosonic string, used for left-moving
excitations of the heterotic string, and the 10-dimensional type I/II
superstring, used for the right-movers.
Because there are 16 extra coordinates on the left, they must be
compactified. In fact, they must be compactified on a 16-torus R^16/Gamma,
where Gamma is an even self-dual lattice in 16 dimensions. Both conditions
are required by consistency (modular invariance - the torus partition
function must not depend on whether we rotate the torus by 90 degrees).
The lattice must be even because of the so-called level-matching (so that
we find some states in each sector). And it must be self-dual because the
momenta and the allowed windings form lattices dual to each other, and
momentum must equal to winding if we want the scalar to be really
left-moving, (partial_tau+partial_sigma) X = 0. (The integral of
partial_sigma(X) is the winding while the integral of partial_tau(X) is
the total momentum.)
A well-known mathematical fact is that there are 2 even self-dual lattices
in 16 dimensions, Gamma16 and Gamma8+Gamma8. They are the root lattices of
the groups spin(32)/Z2 and E8 x E8. The Cartan algebra U(1)^16 of the
gauge groups is encoded in the 16 bosons via the standard Kaluza-Klein
mechanism, used for the left-moving (chiral) bosons. The remaining
generators have gauge bosons which are strings with winding equal to the
corresponding root.
In any case, there are 2 consistent heterotic string theories in 10
dimensions. They have gauge groups E8 x E8 and spin(32)/Z2. The anomaly
cancellation (of the low-energy approximation of these string theories,
which is a type I supergravity coupled to a Super Yang-Mills) was proved
by Green+Schwarz in 1984. It works for these two groups only (and also for
apparently physically uninteresting cases U(1)^496 and U(1)^248 x E8).
These calculations work like miracles but they are not real miracles: the
explanation is the powerful structure of string theory underlying them.
This bosonic construction of the heterotic string was close to the
bosonic string etc. But there is another, fermionic construction.
Let us fermionize the 16 left-moving bosons, i.e. replace them by 32 real
fermions lambda_i (sigma). The modular invariance now requires that there
are two possible choices for their boundary conditions. The first choice
is that all 32 are required to be *all* periodic (P) or *all* antiperiodic
(A). This leads to the spin(32)/Z2 theory with the gauge bosons in the A
sector.
Or we can divide them to two equal groups of 16 real fermions and allow
independent perioridicities for these two groups. Therefore we deal with 4
sectors, AA, AP, PA, PP. In the AA sector we find the gauge bosons for
spin(16) x spin(16) and the AP and PA sectors add the remaining generators
of the gauge group, transforming as a spinor 128 under one of these two
spin(16)'s. (This spinor comes because of quantizing the zero modes of the
periodic fermions and it is chiral because of a GSO-like projection).
Therefore, the gauge group is happily enhanced to E8 x E8.
Today we know that the two heterotic string theories are related by
various dualities. For example, in 17+1 dimension, the lattices Gamma16
and Gamma8+Gamma8, with an added Lorentzian Gamma_{1,1}, become
isometric. There is a single even self-dual lattice in 17+1 dimensions,
Gamma_{17,1}. This is the reason why two heterotic string theories are
T-dual to each other. The compactification on a circle adds two extra
U(1)s (from Kaluza-Klein graviphoton and the B-field), and with
appropriate Wilson lines, a compactification of one heterotic string
theory on radius R is equivalent to the other on radius 1/R, using coms
correct units.
The heterotic E8 x E8 theory at strong coupling becomes M-theory in 11
dimensions living on a "layer" of 11D spacetime with two 9+1-dimensional
boundaries (Horava+Witten). Two factors E8 of the gauge group live at the
boundaries of the line interval (11th dimension). A boundary of M-theory
must carry exactly a single E8 multiplet to cancel the anomalies. And
there are two end points of a line interval.
The strong coupling limit of the spin(32)/Z2 heterotic string theory is
the type I theory with the gauge group SO(32) - in fact, nonperturbatively
it is again precisely spin(32)/Z2. The 32 fermions in the fermionic
construction of the heterotic string arise from open strings stretched
between a D1-brane and one of 32 D9-branes. The D1-brane of type I theory
is the same object as the heterotic SO(32) string.
What I wanted to explain is that string theory does not depend on any
initial assumptions or choices. Whatever is our starting point to study
string theory, we are always forced to discover the same structure which
we are discovering today. String theory is a unique structure that does
not admit any consistent departures from its stringiness. For example, we
can start from Klein-Gordon boson field or fermionic Dirac fields in 2D.
The results will be identical. Therefore it smells like God. Therefore
(among other reasons) people take it so seriously. Other theories that do
not enjoy this sort of rigidity smell like Men.
John Baez
Squark <dSdqudarkd_...@excite.com> wrote:
>On Fri, 22 Jun 2001 20:05:51 +0000 (UTC), John Baez wrote (in
><9h08ev$ag$1...@glue.ucr.edu>):
>>The Thiemann-et-al coherent states of loop quantum gravity
>>describe the "Higgs phase", while the Ashtekar-et-al coherent
>>states describe the "Coulomb phase" - to use 't Hooft's classification
>>of gauge theories according to whether the E and/or B fields are
>>confined.
>A bit off subject, but still interesting: what models are there that have
>both E and B confined? It is possible, I suppose?
I don't know such models, but I'm really not an expert on this
stuff, and it's been a long time since I read 't Hooft's original
paper on order/disorder operators and confinement of electric/
magnetic fields in gauge theory. Maybe someone else knows some
models like this? It would be interesting.
John Baez
Alfred Einstead <whop...@csd.uwm.edu> wrote:
>It's a little known fact that a cone is flat.
Mathematicians have a sneaky habit of redefining words
so they can go around saying stuff like this. :-)
>So, a long time ago, I got this really neat idea: let's create
>a particle-based theory of gravity by pretending that all of
>space-time curvature actually resides at points and that
>these vertices are actually the particles of gravity.
This idea works really well in 2+1-dimensional gravity,
since in this dimension, the Einstein equations say that
spacetime is flat in the vacuum, and point particles give
conical singularities in the metric.
If you want a truckload of references on how people have
used this to study quantum gravity coupled to point particles,
let me know. The results are very cool, but there is much
more left to do. My pal Kirill Krasnov is working on this
topic - he's a postdoc at U. C. Santa Barbara.
>Of course, as you know, this isn't possible. Once you move up
>an extra dimension, a circle no longer objectively encloses
>a point, so there are no longer any characteristic measures
>for point vertices.
Right. A somewhat related sad fact is that in 3+1 dimensions,
the Einstein equations no longer say that spacetime is flat in
the vacuum.
Your further remarks about bivectors and curvature were also
dear to my heart. This connection is why "quantum bivectors"
play a fundamental role in loop quantum gravity, and especially
spin foam models. The fact that bivectors are also handy for
describing angular momentum is what puts the "spin" in "spin
network".
Ralph E. Frost
news:9h68pe$pht$1...@glue.ucr.edu...
Can you expand on what you mean when you refer to the *weak* and *strong*
varieties of anthropic principles, please? I looked for "anthropic" in
the FAQ and didn't find anything.
Ralph Hartley
>
> On Fri, 22 Jun 2001, Ralph Hartley wrote:
> > After all, string theory isn't consistent at all in the obvious number
> > of dimensions (4). It is a point in it's favor that the number of
> > dimensions is constrained, but I'm pretty sure that if it had turned
> > out that string theory was only consistent in 4D, you would be saying
> > "string theory predicts the dimensionality of space-time". Since that
> > is not the case, it is fair to say "string theory does not predict the
> > dimensionality of space-time",
>
> String theory - at least as we know it today - does not predict the number
> of large dimensions (other theories don't predict it either, as you noted
> very well). But in the stringy sense the theory predicts the total number
> of dimensions (including the compact ones): perturbatively it is ten,
> nonperturbatively it is eleven in the context of M-theory.
Yes, but the distinction between large and compact dimensions is
something that was only needed when it turned that string theory only
works if the total number of dimensions is eleven (or ten, but
certainly not four).
The number of compact dimensions is a (discrete) parameter that has to
take a particular value to get string theory to have the right
semiclasical limit. The obvious (though not necessarily correct) value
for that parameter is 0.
> > (Note: I don't know of any theory that does). A less fair (but valid
> > if your argument above is) statement would be "string theory makes a
> > prediction for the dimensionality of space-time that is off by a
> > factor of 11/4". What that means depends on how "obvious" 4 dimensions
> > is to you.
>
> :-) This is based on your incorrect belief that the number 10 or 11 of the
> total number of dimensions is wrong. But this number is probably correct.
I have no belief whatsoever on that matter. The total number of
dimensions may well be 10 or 11. However, it is also possible that
ln(2)/(PI Sqrt(3)) is the correct value of the Barbero-Immirzi
parameter. In both cases it is not the OBVIOUS value, but it is the
value that lets the theory work.
By the way, I *DO* expect a theory of quantum gravity to predict the
number of large dimensions. Of course that makes the current number of
good theories zero.
> The extra six or seven dimensions at every point must be taken seriously,
> we think. It is hard to get used to this pretty radical change but this is
> certainly a general change of paradigm that string theory brought us, just
If you had used the word "paradigm" sooner, I could have saved a lot
of time (by not reading you posts) :-). Be aware that the word has
been so abused that it will instantly put some people into a trance.
It's sort of like the "org chart test". If the presenter puts up a
slide with an organizational chart of his lab, it is safe to go to
sleep; nothing interesting will be said.
Ralph Hartley
zirkus
> For a nice easy introduction, try this:
> Abhay Ashtekar, Quantum Mechanics of Geometry, available
> as http://xxx.lanl.gov/abs/gr-qc/9901023
Thanks for this reference. I have started reading it and it is not
hard to follow.
> Not really. Majid and others replace the commutative algebra
> of functions on space by a noncommutative algebra. Ashtekar and
> company take the kinematical phase space of general relativity
> and quantize that. This is more directly motivated by physics.
Majid's NCG and LQG are different in origin but it turns out that they
might have a nontrivial connection (unlike, it seems, with the case of
Connes' NCG). I took a look at a variant [1] of the first paper I
cited, and here Majid suggests that his treatment of quantum
differential forms could bring us "towards understanding how
macroscopic differential geometry arises out of the loop gravity and
spin network formalism". His conjecture seems to depend on whether the
q-deformed enveloping algebras U_q(g) can describe the NCG coming from
LQG (which is a LQG question I don't know anything about). Also, here
he refers to a black hole entropy computation by Ashtekar et al.
(which is also something I'm not familiar with). Here, I can't
paraphrase Majid's conjecture because I don't really understand it,
but if anyone is interested in seeing it then look at pages 42-43 of
[1], especially the second paragraph on page 43.
Lubos Motl
> I'm curious about the second of these two statements: Do you mean
> that we can use the "anthropic technique" to guess which region of the
> vacuum moduli space corresponds to our world by eliminating from
> consideration vacua which clearly do not correspond to reality? I hope
To avoid a misunderstanding: string theory does not seem to allow exact
(continuous) moduli spaces in the phenomenologically interesting context.
A N=1 supersymmetric theory generates a potential and it has isolated
minima only. At least generically. Even if we had exactly flat directions
(where the potential is constant), the corresponding massless scalar
fields (moduli) would have dramatic low-energy consequences. String theory
contains no adjustable continuous constant parameters; all such
"parameters" become dynamical scalar fields.
Therefore it is likely that the final formulation can lead to a discrete
collection of vacua only. I agree that one could not use anthropic
considerations to calculate the continuous constants precisely! The fine
structure constant changed by a little bit would still allow life. I think
that I have mentioned that already. But among the large collection of
candidate vacua, the anthropic principle should be used to pick the
correct one (or at least rule out most of the candidates). This is what
the anthropic principle is all about.
Nevertheless, I hope that people will not be forced to use such dirty
arguments such as the anthropic principle. At least not too much. ;-)
Dirk Bruere
news:Pine.SGI.4.33.0106221...@hep.uchicago.edu...
>
> I'm curious about the second of these two statements: Do you mean
> that we can use the "anthropic technique" to guess which region of the
> vacuum moduli space corresponds to our world by eliminating from
> consideration vacua which clearly do not correspond to reality? I hope
> you're not claiming that -- as a last resort -- we should try to use the
> anthropic principle to figure out which vacuum is ours. The anthropic
> principle is fairly lousy as a physical principle, because it can't make
> exact predictions; it can only eliminate wrong answers. The problem is
> that no one will ever notice if Newton's constant changes by a factor of
> 1.0000000000000000000000000000000000002.
Has anyone any estimate as to how much any of the constants would have to
change to eliminate (current) life?
Of course, another problem is that its a good bet that our 'constants' are
actually related (the connection lying somewhere in a TOE).
Finally, is there any indication of jitter in any constant we measure eg is
c=c true or approximate?
Dirk
John Baez
zirkus <zir...@my-deja.com> wrote:
>ba...@galaxy.ucr.edu (John Baez) wrote:
>> Abhay Ashtekar, Quantum Mechanics of Geometry, available
>> as http://xxx.lanl.gov/abs/gr-qc/9901023
>Thanks for this reference. I have started reading it and it is not
>hard to follow.
Yes, it's a great place to start. It's written in honor of
Varadarajan, I think, and the title is a clever twist on
that of Varadarjan's book "Geometry of Quantum Mechanics".
>Majid's NCG and LQG are different in origin but it turns out that they
>might have a nontrivial connection (unlike, it seems, with the case of
>Connes' NCG). I took a look at a variant [1] of the first paper I
>cited, and here Majid suggests that his treatment of quantum
>differential forms could bring us "towards understanding how
>macroscopic differential geometry arises out of the loop gravity and
>spin network formalism". His conjecture seems to depend on whether the
>q-deformed enveloping algebras U_q(g) can describe the NCG coming from
>LQG (which is a LQG question I don't know anything about).
People in loop quantum gravity believe that q-deformed spin networks
are relevant to quantum gravity with nonzero cosmological constant.
Thus in this context, q-deformation goes along not with nonzero hbar,
but with nonzero cosmological constant. Of course you need all hbar,
c, G *and* a cosmological constant to build a dimensionless number
like the "q" in q-deformation, so you could argue that q-deformation
is some funny combination of letting hbar be nonzero, G be nonzero,
1/c be nonzero and the cosmological constant be nonzero. But anyway,
loop quantum gravity has no apparent relation to q-deformation until
the cosmological constant shows up.
>Also, here
>he refers to a black hole entropy computation by Ashtekar et al.
Yeah, I'm one of the "al". The others are Alex Corichi and Kirill
Krasnov.
Stephen Selipsky
Simultaneous E and B confinement goes against my intuition from
supersymmetric and broken-SUSY gauge theories, where magnetic-electric
duality allows a quantitative description of magnetic monopole
condensation. (This is just like the electrically charged Cooper pair
condensate leading to superconductivity and the Meissner effect.)
To confine *both* B and E, you would need condensates of *both* electric
and (dual variable) magnetic monopoles. Unless some external (non-gauge)
forces stabilize the condensates, you'd expect confinement of one to damp
out forces leading to condensation and confinement of the other. (See
e.g. http://arXiv.org/find/hep-th/9709177 , though there must have been
progress since then.)
Theories with dyons (objects with BOTH magnetic and electric charge)
may provide a loophole, if they exhibit condensates of dyons and monopoles.
That's hard to describe because the degrees of freedom aren't simultaneously
local; see http://xxx.lanl.gov/e-print/hep-th/9606191 and
http://xxx.lanl.gov/e-print/hep-th/9604004 .
I remember at the time worrying about the approximations breaking down
just where the condensate occured, but we should get the full story from
Jacques Distler, who participated in the work above-- I left research
three years ago...
-- Stephen Selipsky
stephen....@iname.com
Lubos Motl
> Yes, but the distinction between large and compact dimensions is
> something that was only needed when it turned that string theory only
> works if the total number of dimensions is eleven (or ten, but
> certainly not four).
People started to study compactification of dimensions because they
realized that it was a perfectly meaningful procedure to get new
consistent vacua in string theory - and by the way, we certainly do not
observe 10 very large dimensions and therefore the remaining dimensions
must be compactified or dealt with in a similar way. It took some time to
realize why the extra dimensions are a *good* thing, not bad.
But what I want to emphasize is that string theory with compactified
dimensions is equally well-defined as string theory in 10 flat Minkowski
dimensions. There is also no experimental evidence that the world does not
contain compactified dimensions shorter than some scale. On the contrary,
there is an overwhelming theoretical evidence that the world must have
more than four dimensions, in order to contain quantum gravity.
In fact, most of the papers in string theory study the backgrounds with
some dimensions compactified. Not because someone wants to match the
reality immediately; simply because compactification is an extremely
important part of the whole theory.
> The number of compact dimensions is a (discrete) parameter that has to
> take a particular value to get string theory to have the right
> semiclasical limit. The obvious (though not necessarily correct) value
> for that parameter is 0.
Nope. The other numbers are equally stringy, equally correct and equally
consistent. And understanding compactification as well as D-branes of
various dimensions etc. represents a very large fraction of string theory.
Not because we need it; rather because mathematics dictates so.
> I have no belief whatsoever on that matter. The total number of
> dimensions may well be 10 or 11. However, it is also possible that
> ln(2)/(PI Sqrt(3)) is the correct value of the Barbero-Immirzi
> parameter. In both cases it is not the OBVIOUS value, but it is the
> value that lets the theory work.
The difference is that the dimension of spacetime in string theory can be
calculated in hundreds of inequivalent ways - and there are no doubts
about them - while the specific value of the Barbero-Immirzi parameter is
forced on us by a single result only.
> By the way, I *DO* expect a theory of quantum gravity to predict the
> number of large dimensions. Of course that makes the current number of
> good theories zero.
I do not and I am pretty sure that the vacua in higher dimensions such as
10 or 11 (with unbroken supersymmetries) are perfectly consistent and
therefore the corresponding Universes do exist "somewhere" although there
can be probably no life over there, so to say.
> If you had used the word "paradigm" sooner, I could have saved a lot
> of time (by not reading you posts) :-). Be aware that the word has
> been so abused that it will instantly put some people into a trance.
Well, I do not write my posts for the people who are instantly put into a
trance by a single word. I expect the posts to be read by people who try
to understand the contents - and not just emotionally explode while seing
a sequence of eight letters. I do not care whether someone abused a word.
I just know that the word "paradigm" is an appropriate word to describe
our insight that the world has more dimensions than people used to think
before the 1980s (and some of them think even today).
Chris Hillman
On 19 Jun 2001, Alfred Einstead wrote:
> So, a long time ago, I got this really neat idea:
[snip]
> But, by analogy, the same thing holds true. If you parallel
> transport a x-y-z coordinate frame in a circle along the (i-j) plane
> (i,j=1,2,3) then the frame will come back with a fixed
> rotation ONLY IF the loop links one of these singular
> lines or circles. And, like before, one finds that associated
> with each singularity is a characteristic value which
> indicates by how much a general frame will be transformed
> for each cycle it takes around one of these singularities.
> This will be a function mapping (i-j)'s into rotations, or:
>
> Characteristic Property: Bivectors -> Rotations
>
> This, of course, is just the Riemannian tensor, concentrated
> on lines.
I'm surprised no-one has yet mentioned the Regge calculus, which
reformulates gtr in terms of deficits around the two-simplices of a
simplicial approximation to the spacetime M. I think you are essentially
talking about a 3+1 splitting of the Regge calculus, for which see the
chapter in MTW. That is, you propose to describe a Riemannian three
manifold via the angular deficits around the "struts" of a simplicial
decomposition, and then (I guess) to evolve this "in time" following the
model of the ADM constraint/evolution reformulation of the EFE. It would
be interesting to work this out in detail, if it hasn't already been done!
By the way, regarding curvature concentrated in curves in three
manifolds-- did you know that a particularly striking explicit example is
afforded by S^3, the higher dimensional analogue of the S^2 with which you
began your post? Consider the Hopf fibration
S^1 >--> S^3 >-->> S^2
and a natural "affinization" of S^3 in which you take a tetrahedron and
identify faces in the obvious way, with the result being a topological S^3
with all its curvature concentrated in two linked circles (which before
identifying faces were two "opposite" edges in the tetrahedron)!
You may know that a major industry in Riemannian geometry these days
involves evolving an arbitrary three-manifold into a "canonical form" by
trying to smooth out the curvature using an appropriate evolution
operator. People have also been doing something analogous with knots in
E^3. Greg Kuperberg could probably say a lot more about this... I was
last very modestly well-informed ;-/ about this stuff about eight years
ago.
Anyway, we are talking about going the other way: trying to -concentrate-
the curvature in a way which results in a simple topological description
of M along the lines of the above example. Unfortunately, this would
appear to be much harder to carry out systematically, and AFAIK no-one has
ever investigated this. In the case of S^3, the Hopf decomposition gives
a family of circular fibers, with every pair being linked, organized a
pair in which we wish to concentrate the curvature. In the very special
case of S^3, the Hopf fibration shows one way to do that by continuously
and homeotopically deforming an S^3 in E^4.
(Homeotopy: a homeo of E^4 which has the desired effect on S^3 as a
topological submanifold; the idea is that a homeotopy preserves the
topological nature of a natural embedding as well as the topology of M.
The knot theory stuff I mentioned above involves evolving knots into a
canonical form by a continuous family of homeotopies. Of course, it would
be more natural to discuss homeotopies of topological S^3 embedded as a
topological submanifold of E^5, not E^4...)
Chris Hillman
Home Page: http://www.math.washington.edu/~hillman/
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
NOTE: Since I post under my real name, as an anti-spam measure, I have
installed a mail filter which deletes incoming messages not from the
"*.edu" or "*.gov" domains or overseas academic domains.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Dirk Bruere
"Ralph E. Frost" <ref...@dcwi.com> wrote in message
news:tjeto7b...@corp.supernews.com...
> Can you expand on what you mean when you refer to the *weak* and *strong*
> varieties of anthropic principles, please? I looked for "anthropic" in
> the FAQ and didn't find anything.
IIRC strong is 'the universe exists because we do', and
weak is 'the universe is as we see it because otherwise we would not be
here'.
Dirk
Lubos Motl
> Has anyone any estimate as to how much any of the constants would have to
> change to eliminate (current) life?
Yes, people have made rough calculations like that. If you want to destroy
life of any sort, it is enough to change the strengths of the fundamental
forces by a very small amount (a few per cent or so) and galaxies won't
form anymore. This would imply that no life remotely similar to ours would
be possible. There are at least 19 parameters in the Standard Model, so it
is hard to describe the region of the parameter space (that allows life)
precisely. But for example, the cosmological constant must be between -1
and 100 times the currently observed value to allow the galaxy formation.
If you are interested in this particular life based on "our" DNA etc., the
changes capable to kill such life are even much smaller. The energies of
various biomolecules must be fine-tuned quite precisely to make the
mechanisms of life work. And a slightly different mass of the electron
would damage everything.
> Of course, another problem is that its a good bet that our 'constants' are
> actually related (the connection lying somewhere in a TOE).
Right, in fact, TOE is expected to have the capacity to compute all of
them.
> Finally, is there any indication of jitter in any constant we measure eg is
> c=c true or approximate?
The identity c=c is certainly exact. ;-) More seriously, TOE will explain
only the *dimensionless* constants, i.e. the constants that do not depend
on our choice of units because they do not have any units: for example,
the fine structure constant e^2/hbar.c.4.pi.epsilon0 = 1/137.036....
Constants such as c, hbar, Gnewton etc. are dimensionful and "adult"
physicists often use units where these constants are set equal to one.
For example, distance is measured in the same units as time (light year
can be called a "year" of distance).
Recall that centuries ago, people used different units for work and energy
and for heat. Then Joule realized that one can be converted to the other.
And today we know why to use the same units both for energy and for heat.
Similarly, Einstein realized that time and space are different projections
of the same thing and we should use the same units for them. Quantum
mechanics shows that the (angular) frequency and energy (of a photon, but
in fact of anything) is also the same parameter (in the correct units,
where hbar=1).
One second was originally defined as 1/86400 of a (adjectives...) solar
day. We have been using more exact definitions counting periods of a very
specific radiation, for several decades.
One meter was defined originally as 1/40,000,000 of the circumference of
the maximal circle on the Earth (that go through the poles). Later it was
found out that the expected size of our planet did not really match our
idea what a "meter" should be. Meter was then defined as the length of the
international prototype in France (made of platinum).
A more exact definition then said that meter is some number of wavelengths
of another radiation (different than the light defining one second).
Therefore the speed of light was not clearly defined. People measured the
speed of light using these "spectral" definitions of one meter and one
second and by 1980 they got 299 792 458 plus minus 1 meter per second.
This was exact enough to change the definition of meter: one meter is
today 1 / 299 792 458 of a light second (in vacuum). Therefore the speed
of light is (constant) 299,792,458 m/s by definition. This is good because
the speed of light is one of the most universal parameters in the Universe
and it is better to know its value exactly. The wavelength of the
radiation of a specific atom etc. is not that fundamental and universal.
Kevin A. Scaldeferri
replying to. Where I feel that it makes the discussion clearer, I
have restored them.
In article <Pine.SOL.4.10.101062...@strings.rutgers.edu>,
Lubos Motl <mo...@physics.rutgers.edu> wrote:
>On 19 Jun 2001, Kevin A. Scaldeferri wrote:
>>> String theory on the contrary tells you
>>> completely rigorously that gravity fundamentally lives in 10 or 11
>>> (nonperturbatively) dimensions.
>> This statement is, of course, completely false.
>I do not think that one must be a physicist to know *what* is the
>dimension of spacetime according to string theory. It is enough to read
>the press and popular books. A theoretical physicist should be at least
>able to reproduce the calculation that shows that perturbative string
>theory requires total of 10 dimensions, i.e. to answer *why*. I can do it.
>If you are interested, just ask! ;-)
I know what dimensions string theory requires to work. I've seen the
calculations and could probably reproduce them if I spent a couple
weeks brushing up. That's not the point. What you said was that
string theory rigorously tells you that gravity lives in 10 or 11
dimensions. This is false. String theory tell you that string theory
lives in 10/11 dimensions, but because string theory makes few, if
any, statements about QG in general, it doesn't tell you that QG is
constrained to 10/11 dimensions
>> Beyond the fact that it is false on its surface, that little "or" is a
>> good indicator of why one should be skeptical of such pronouncements.
>The total number of dimensions is not a completely well-defined
>notion.
[snip elaboration on this point]
All I'm saying is that 10 years ago you would have left out the "or
11" from that sentence.
As an aside, I know what the dimensionality of a manifold is, and it
is a well-defined notion. I'm not sure what the dimension of
non-manifold structures means in general, but given that you yourself
say it is not a well-defined notion, I'm not sure how you can claim
that string theory makes "completely rigorous" statements about it.
>> >String theory is full of such examples, therefore it smells like God. But
>> >LQG smells like Man just because its checks are really not of "the same"
>> >sort as you say very well.
>> Facetiously, I will point out that people frequently do not notice
>> their own BO. Also that while there is a long history of people
>> finding evidence of God in numerical coincidence, most scientists are
>> not impressed by such activities.
>Well, then one should note that the "miracles" that string theory shows
>are slightly too huge and unbelievable phenomena to be called
>"coincidences". Most fascinating things in physics have the character of
>such "coincidences". For example, the experiment does not tell you
>"directly" that you should compute loop diagrams. Feynman (and people
>before him) had to figure out the rules theoretically. And then the
>higher-order calculation of the electron's magnetic moment agrees to 12
>decimals or so with the experiment. Is this also just a "coincidence that
>most scientists are not impressed by"?
No, it is an impressive agreement between theory and experiment. If
you can't see how this is different from purely theoretical results,
we are going to have a hard time.
One could argue that experiment does tell you that you must compute
loop diagrams, though. There are phenomena that are not present at
tree level that are experimentally observed.
> ;-) The situation in string theory
>is analogous but the "experiments" have a theoretical character, too. The
>technical situation of the current civilization simply does not allow to
>do the relevant experiments for any theory going beyond the Standard
>Model.
This misfortune of economics does not justify redefining the word
"experiment".
>>>String theory does not depend on your choice of the action.
>> Really? So, string theory follows from, say, the Dirac or
>> Klein-Gordon action?
>No, on the contrary. String theory is more fundamental than the Dirac or
>Klein-Gordon action. You can derive both as effective descriptions of
>parts of your physics from string theory.
Now that I've restored your original statement, I can point out that
your claimed that string theory does not depend on the action. I just
pointed out that this is clearly a false statement. I don't want to
badger you about this because you undoubtedly didn't mean what you
wrote, but I did want to restore the context of my remark.
>> >String theory is the only known consistent theory that can predict a world
>> >containing all the forces and particles we see.
>> The last time I asked the question, does anyone know of a
>> compactification of string theory that yields the Standard Model (or,
>> even, the MSSM) the answer I got was no. Are you claiming that this
>> is no longer the case?
>Are you kidding? There are thousands of choices. In fact, too many, I
>would be personally happier if the number was smaller. Initially people in
>the 80s studied HE heterotic strings on Calabi-Yau 3-folds. Even the
>simplest, orbifold models can give GUT theories (or directly MSSM, if
>broken in the proper way) plus gravity at low energies (sometimes plus a
>couple of exotics). Then I personally liked the free fermionic heterotic
>models - there are about 100 papers like
Well, this is somewhat nice since, as I said, I've never gotten a
straight answer to this question. I'll track down the paper you
mentioned before passing final judgement, though ;)
--
======================================================================
Kevin Scaldeferri Calif. Institute of Technology
The INTJ's Prayer:
Lord keep me open to others' ideas, WRONG though they may be.
Steve Carlip
> String theory on the contrary tells you completely rigorously
> that gravity fundamentally lives in 10 or 11 (nonperturbatively)
> dimensions.
This is not so obvious to me, though perhaps it's a question of
how you define ``dimension.'' In particular:
> Why 10? The bc ghosts have central charge 1-3.3^2=-26. They
> are cancelled by 26 dimensions in the bosonic string, but in the
> case of superstring you add beta-gamma superconformal ghosts
> (to have worldsheet supersymmetry) with central charge 3.2^2-1=11.
> -26+11=-15. So we need to cancel remaining central charge of 15.
> This is the central charge of 10 bosons and their 10 fermionic
> superpartners (10/2=5). This is why perturbative quantum gravity
> lives in 10 dimensions.
It's certainly true that *one* way to get the remaining c=15 is to add
ten bosons and their fermionic superpartners. But at this level of
the argument, at least, you can also add any conformal field theory
with c=15. Take, for example, 4 free bosons, 4 fermions, 10 copies
of the three-state Potts model, and two copies of the Ising model.
That will also give you c=15, but it's certainly not clear that there
are ten ``dimensions'' in any sensible meaning of the word. (A
three-state Potts model has c=4/5; you don't want to argue that it's
four-fifths of a dimension, do you?)
Now, it's true that many conformal field theories can be obtained by
suitably compactifying some ordinary dimensions. But as far as I
know, there's no good argument that *all* can. The best I know
of is work by Moore and Seiberg showing that most, perhaps all,
*rational* CFTs can be gotten from Chern-Simons theory; this
is probably related, but nowhere near a complete answer.
You might get extra restrictions by, for example, demanding that
there be no tachyons in the spectrum. But still, it's at least quite
unobvious that this would be enough.
Am I missing something here?
Steve Carlip
John Baez
Lubos Motl <mo...@physics.rutgers.edu> wrote:
>On Wed, 27 Jun 2001, Dirk Bruere wrote:
>> Of course, another problem is that its a good bet that our 'constants' are
>> actually related (the connection lying somewhere in a TOE).
>Right, in fact, TOE is expected to have the capacity to compute all of
>them.
Yeah - and the reason is not very deep: if it couldn't, it wouldn't be
the Theory of Everything. It would just be a Theory of Something.
Charles Francis
writes:
>In article <e58d56ae.01061...@posting.google.com>,
>Alfred Einstead <whop...@csd.uwm.edu> wrote:
>>So, a long time ago, I got this really neat idea: let's create
>>a particle-based theory of gravity by pretending that all of
>>space-time curvature actually resides at points and that
>>these vertices are actually the particles of gravity.
Before you go too far with this idea, you have to make sure you are
working with a correct interpretation of quantum mechanics. The
particles may be point-like, but uncertainty means they certainly don't
look like points. This can only be so if the empty space between the
particles isn't really "there", but is a figment of way in which we
organise the data coming to us by observation.
>This idea works really well in 2+1-dimensional gravity,
>since in this dimension, the Einstein equations say that
>spacetime is flat in the vacuum, and point particles give
>conical singularities in the metric.
>
>If you want a truckload of references on how people have
>used this to study quantum gravity coupled to point particles,
>let me know. The results are very cool, but there is much
>more left to do. My pal Kirill Krasnov is working on this
>topic - he's a postdoc at U. C. Santa Barbara.
Yes please. I have been working on this idea for years, and I still
can't make it go wrong. You have given me some references, like Fulling,
and Wald, but so far in these and every other book I have read all I can
find are hand wavy arguments that it can't be particles based on
misconceptions about the interpretation of quantum mechanics and a
failure to recognise the fundamental role of measurement by physical
rods and clocks in Einstein's theory.
>>Of course, as you know, this isn't possible. Once you move up
>>an extra dimension, a circle no longer objectively encloses
>>a point, so there are no longer any characteristic measures
>>for point vertices.
>Right. A somewhat related sad fact is that in 3+1 dimensions,
>the Einstein equations no longer say that spacetime is flat in
>the vacuum.
???. And what does Einstein's field equation in a vacuum
R=0
say if it is not a statement of zero curvature? Is zero curvature not
what we mean by flat? Or are you referring to the cosmological constant?
Regards
--
Charles Francis
Dirk Bruere
"Lubos Motl" <mo...@physics.rutgers.edu> wrote in message
news:Pine.SOL.4.10.101062...@strings.rutgers.edu...
> Some poor soul whom Dirk Bruere forgot to cite wrote:
> > Finally, is there any indication of jitter in any constant we measure,
> > e.g. is c=c true or approximate?
> The identity c=c is certainly exact. ;-) More seriously, TOE will explain
> only the *dimensionless* constants, i.e. the constants that do not depend
....
So, no getting a particular value for c? (or the metre, see below)
> A more exact definition then said that meter is some number of wavelengths
> of another radiation (different than the light defining one second).
> Therefore the speed of light was not clearly defined. People measured the
> speed of light using these "spectral" definitions of one meter and one
> second and by 1980 they got 299 792 458 plus minus 1 meter per second.
>
> This was exact enough to change the definition of meter: one meter is
> today 1 / 299 792 458 of a light second (in vacuum). Therefore the speed
> of light is (constant) 299,792,458 m/s by definition. This is good because
> the speed of light is one of the most universal parameters in the Universe
> and it is better to know its value exactly. The wavelength of the
> radiation of a specific atom etc. is not that fundamental and universal.
Anyway, my question was - is there any jitter associated with the
measurement that cannot be attributed to experimental error? [I guess not,
since it would be big news]
And, is there any expected [I guess not...]
And, what if there were?
Dirk
Urs Schreiber
> On Wed, 27 Jun 2001, Dirk Bruere wrote:
> > Has anyone any estimate as to how much any of the constants would have to
> > change to eliminate (current) life?
> Yes, people have made rough calculations like that. If you want to destroy
> life of any sort, it is enough to change the strengths of the fundamental
> forces by a very small amount (a few per cent or so) and galaxies won't
> form anymore. This would imply that no life remotely similar to ours would
> be possible.
A more general and more difficult question comes to mind:
Which values of the constants might lead to universes that
allow (or are bound to develop) *any* complex structures
capable of self-organization, complex enough to allow for
something like conciousness?
I remember a sci-fi story where a "living" cloud of
interstellar dimensions endangers terrestric life while
restocking its energy supplies by enveloping the sun (sort of
a living Dyson sphere :-).
So: If no galaxies are to form, does this rule out highly
complex dissipative systems of any sort?
This seems to have some relevance to the value of the
anthropic principle (in case we want to, or need to, apply
it): If we should ever discover a principle arbitraryness in
nature's constants, can we fix the remaining freedom by
considering the *unique* form of concious life (carbohydrate
based fellows like us), or do we still have to be puzzled why
we did not turn out to be some sort of telepathic cosmic dust?
Urs Schreiber
Martin Hardcastle
Lubos Motl <mo...@physics.rutgers.edu> wrote:
>Yes, people have made rough calculations like that. If you want to destroy
>life of any sort, it is enough to change the strengths of the fundamental
>forces by a very small amount (a few per cent or so) and galaxies won't
>form anymore. This would imply that no life remotely similar to ours would
>be possible. There are at least 19 parameters in the Standard Model, so it
>is hard to describe the region of the parameter space (that allows life)
>precisely. But for example, the cosmological constant must be between -1
>and 100 times the currently observed value to allow the galaxy formation.
Apropos of this sort of claim, people who are interested may want to
look at an interesting paper by Aguirre, astro-ph/0106143, which
discusses anthropic constraints on *cosmological* parameters.
From the abstract:
In this paper, I explicitly show that the requirement that the
universe generates sun-like stars with planets does not fix these
parameters, by developing a class of cosmologies (based on the
classical `cold big-bang' model) in which some or all of the
cosmological parameters differ by orders of magnitude from the
values they assume in the standard hot big-bang cosmology, without
precluding in any obvious way the existence of intelligent life.
Martin
--
Martin Hardcastle Department of Physics, University of Bristol
Never attribute to malice what can adequately be explained by stupidity
Squark
<3B390DF4...@iname.com>):
> To confine *both* B and E, you would need condensates of *both* electric
>and (dual variable) magnetic monopoles. Unless some external (non-gauge)
>forces stabilize the condensates, you'd expect confinement of one to damp
>out forces leading to condensation and confinement of the other.
So all you are saying, is that it can't happen in pure gauge theory? Can we
have spontaneous symmetry breaking (confinement of B) in pure gauge theory?
Actually, I remember an article suggesting this possibilty, but I never got
to reading it, and I don't remember its name...
Best regards,
Squark.
--------------------------------------------------------------------------------
Write to me at:
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dSdqudarkd_...@excite.com
Urs Schreiber
news:U342JZAX...@clef.demon.co.uk...
Ricci has to vanish in vacuum, but Riemann need not, as is e.g. the case for
the Schwarzschild metric.
Gordon D. Pusch
> ???. And what does Einstein's field equation in a vacuum
>
> R=0
>
> say if it is not a statement of zero curvature?
A zero curvature-scalar is not the same as zero curvature!
> Is zero curvature not what we mean by flat?
Exactly --- but ``zero curvature'' means R^i_{jkl} vanishes, not that
the curvature scalar R vanishes. In Riemannian spaces with more than
two dimensions, it is possible for R to be zero even though R^i_{jkl}
is non-zero, just as it is possible for a non-vanishing tensor to have
a vanishing trace.
-- Gordon D. Pusch
perl -e '$_ = "gdpusch\@NO.xnet.SPAM.com\n"; s/NO\.//; s/SPAM\.//; print;'
Kevin A. Scaldeferri
Lubos Motl <mo...@physics.rutgers.edu> wrote:
>
>> Has anyone any estimate as to how much any of the constants would have to
>> change to eliminate (current) life?
>
>changes capable to kill such life are even much smaller. The energies of
>various biomolecules must be fine-tuned quite precisely to make the
>mechanisms of life work. And a slightly different mass of the electron
>would damage everything.
This is claimed frequently, but I'm not sure how often people who
are truly knowledgable about molecular biology are consulted.
Although, I'm not sure that anyone is knowledgable about molecular
biology in the necessary way.
To be more specify, it is true that minor changes in binding energies
and the like would totally disrupt various molecular pathways of
critical importance, the folding of proteins, etc. However, it isn't
as if all possible enzymes and proteins are used in life as we know
it. It is easy to modify things so that the wrong proteins are made
-- things that don't fold or that fold in the wrong way.
The combination of these makes me skeptical that "life as we know it"
wouldn't be possible if you made small tweaks to physical constants.
While the precise combinations of molecules that we use would be
messed up, another set might start to work.
Of course, this is all just speculation. Theoretical biology is
really in its infancy and there is too much to understand about the
world as it is for people to get really excited about the world as it
isn't.
Urs Schreiber
>
> "Ralph E. Frost" <ref...@dcwi.com> wrote in message
> news:tjeto7b...@corp.supernews.com...
>
> > varieties of anthropic principles, please? I looked for "anthropic" in
> > the FAQ and didn't find anything.
>
This sounds like "Cogito ergo sum." promoted to "Ontology ergo
Cosmology." :-)
But seriously: Does it make sense to consider a principle that
supposedly explains any existence as such? Isn't that bound to
be circular?
Urs Schreiber
Squark
>and (dual variable) magnetic monopoles. Unless some external (non-gauge)
>forces stabilize the condensates, you'd expect confinement of one to damp
>out forces leading to condensation and confinement of the other.
So all you are saying, is that it can't happen in pure gauge theory? Can we
zirkus
> Finally, is there any indication of jitter in any constant we measure
Please read the article below for an intro to variable-speed-of-light
(or VSL) cosmology. (Also, the article at the bottom of the page is
kind of interesting because it implies that if there are extra
dimensions then they have to be smaller than .2 millimeter.)
Dirk Bruere
news:29b67412.01062...@posting.google.com...
> "Dirk Bruere" <art...@kbnet.co.uk> wrote:
> > Finally, is there any indication of jitter in any constant
> >we measure [...]
> Please read the article below for an intro to variable-speed-of-light
> (or VSL) cosmology. (Also, the article at the bottom of the page is
> kind of interesting because it implies that if there are extra
> dimensions then they have to be smaller than .2 millimeter.)
>
> http://www.spacedaily.com/news/physics-01c.html
I'm familiar with this.
By jitter, I mean over small timescales - days at most.
Dirk
Dirk Bruere
news:3B3C54A0...@uni-essen.de...
> Some person whom Dirk Bruere forgot to cite wrote:
> > Some other person whom Dirk Bruere forgot to cite wrote:
> > > Can you expand on what you mean when you refer to the *weak* and
> > > *strong* varieties of anthropic principles, please? I looked
> > > for "anthropic" in the FAQ and didn't find anything.
> > IIRC strong is 'the universe exists because we do',
> But seriously: Does it make sense to consider a principle that
> supposedly explains any existence as such? Isn't that bound to
> be circular?
Isn't any TOE that wraps up everything going to be the same?
Dirk
John Baez
Lubos Motl <mo...@physics.rutgers.edu> wrote:
>Today we know that the two heterotic string theories are related by
>various dualities. For example, in 17+1 dimension, the lattices Gamma16
>and Gamma8+Gamma8, with an added Lorentzian Gamma_{1,1}, become
>isometric. There is a single even self-dual lattice in 17+1 dimensions,
>Gamma_{17,1}. This is the reason why two heterotic string theories are
>T-dual to each other. The compactification on a circle adds two extra
>U(1)s (from Kaluza-Klein graviphoton and the B-field), and with
>appropriate Wilson lines, a compactification of one heterotic string
>theory on radius R is equivalent to the other on radius 1/R, using
>correct units.
Hey, that's COOL! I've been wondering for a long time where 18-dimensional
spacetime showed up in string theory... and now I know! Thanks, Lubos.
>From "week93":
For starters I just want to explain why dimensions of the form 8k + 2 are
special. Notice that if we take k = 0 here we get 2, the dimension of the
string worldsheet. For k = 1 we get 10, the dimension of spacetime in
"supersymmetric string theory". For k = 3 we get 26, the dimension of
spacetime in "purely bosonic string theory". So these dimensions are
important. What about k = 2 and the dimension 18, I hear you ask? Well,
I don't know what happens there yet... maybe someone can tell me! All I
want to do now is to explain what's good about 8k + 2.
[and then I went on to talk about how Majorana-Weyl spinor and self-dual
even lattices only occur in Minkowski spacetimes of dimension 8k + 2.]
Ace Schallger
> Your further remarks about bivectors and curvature were also
> dear to my heart. This connection is why "quantum bivectors"
> play a fundamental role in loop quantum gravity, and especially
> spin foam models. The fact that bivectors are also handy for
> describing angular momentum is what puts the "spin" in "spin
> network".
Uh, care to elucidate the nature of a 'bivector' in some sort of simple
layman's terminology?
Ace Schallger.
John Baez
Dirk Bruere <art...@kbnet.co.uk> wrote:
>"Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote in message
>news:3B3C54A0...@uni-essen.de...
>> But seriously: Does it make sense to consider a principle that
>> supposedly explains any existence as such? Isn't that bound to
>> be circular?
>Isn't any TOE that wraps up everything going to be the same?
The goal of a TOE is not to "explain existence as such": it is
to give us recipes for making lots and and lots of quantitative
predictions...
... and in some vague sense which I do not want to make
precise, it should be the ultimate recipe of this sort.