Loop Quantum Gravity vs. M-Theory.

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Mark William Hopkins

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May 25, 2001, 12:47:07 AM5/25/01
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Is there a way to reconcile M-theory with loop quantum gravity, combining
them into a unified theory?

Demian Cho

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May 27, 2001, 12:13:43 PM5/27/01
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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

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May 29, 2001, 2:48:25 PM5/29/01
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In article <9ehndj$jb2$1...@uwm.edu>,

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

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Jun 4, 2001, 1:36:31 PM6/4/01
to Demian Cho
> Is there a way to reconcile M-theory with loop quantum gravity, combining
> 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

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Jun 6, 2001, 12:14:16 AM6/6/01
to
Lubos Motl <mo...@physics.rutgers.edu> wrote:

> 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

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Jun 6, 2001, 11:34:36 AM6/6/01
to
In article <Pine.SOL.4.10.101060...@strings.rutgers.edu>,
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

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Jun 6, 2001, 6:53:42 PM6/6/01
to ba...@math.ucr.edu
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 ???
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

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Jun 6, 2001, 6:51:53 PM6/6/01
to
In article <9fliic$kd2$1...@news.state.mn.us>, John Baez wrote:
>
>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

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Jun 7, 2001, 8:11:45 PM6/7/01
to
In article <m2puchm...@pusch.IntegratedGenomics.com>,

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

unread,
Jun 7, 2001, 8:42:09 PM6/7/01
to
In article <slrn9hsmrm....@cardinal0.Stanford.EDU>,
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

unread,
Jun 8, 2001, 8:43:45 PM6/8/01
to
Good that someone - even someone famous - (tries to) defend(s) Loop
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

unread,
Jun 8, 2001, 10:16:34 PM6/8/01
to
Lubos Motl wrote in part:

>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

unread,
Jun 9, 2001, 4:48:39 PM6/9/01
to
Some poor soul whom Demian Cho failed to properly cite wrote:

> 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

unread,
Jun 11, 2001, 5:07:00 PM6/11/01
to
A few random comments:

>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

unread,
Jun 11, 2001, 10:29:33 PM6/11/01
to
Hi Lubos,

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

unread,
Jun 11, 2001, 10:37:12 PM6/11/01
to
This is a follow-up to a post by Lubos Motl headed ``Loop Quantum
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

unread,
Jun 12, 2001, 12:09:55 PM6/12/01
to
There are two different issues here: 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 will say a little about the general issues
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

unread,
Jun 12, 2001, 12:14:42 PM6/12/01
to
On 11 Jun 2001 21:07:00 GMT, John Baez wrote (in
<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

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Jun 12, 2001, 5:51:07 PM6/12/01
to
This discussion, mostly between Steve Carlip and me, Lubos Motl, is about
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

unread,
Jun 12, 2001, 5:51:17 PM6/12/01
to
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.

> 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

unread,
Jun 12, 2001, 5:51:32 PM6/12/01
to
On 12 Jun 2001, Squark wrote:

[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
,-----------------------------------------------------------------------------.
> | www.het.brown.edu www.cecm.usp.br <
> Daniel Doro Ferrante | <
> danieldf@olympus | This signature was automatically generated with <
> | Signify v1.06. For this and other cool products, <
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Aaron Bergman

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Jun 12, 2001, 6:52:19 PM6/12/01
to
In article <5WqV6.5324$pb1.2...@www.newsranger.com>, Squark wrote:
>
>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

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Jun 12, 2001, 6:06:53 PM6/12/01
to
I separate this (important?) issue from the rest of the discussion. The
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

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Jun 12, 2001, 5:13:10 PM6/12/01
to
On Sat, 9 Jun 2001, Demian H.J. Cho wrote:

> 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

unread,
Jun 12, 2001, 6:49:45 PM6/12/01
to
A separate text on SUSY breaking has been posted...

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

unread,
Jun 13, 2001, 2:01:05 PM6/13/01
to
I separate this (important?) issue from the rest of the discussion. The
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

unread,
Jun 13, 2001, 2:01:23 PM6/13/01
to
A separate text on SUSY breaking has been posted...

> >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

unread,
Jun 13, 2001, 2:03:18 PM6/13/01
to
On Tue, 12 Jun 2001, Lubos Motl wrote:
> 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

unread,
Jun 13, 2001, 2:07:49 PM6/13/01
to
On Tue, 12 Jun 2001, Lubos Motl wrote:

> 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

unread,
Jun 14, 2001, 12:10:15 PM6/14/01
to

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

unread,
Jun 14, 2001, 12:22:01 PM6/14/01
to
Hi A.J.!

> 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

unread,
Jun 14, 2001, 2:48:39 AM6/14/01
to
Lubos Motl wrote in small part:

>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

unread,
Jun 12, 2001, 11:38:09 PM6/12/01
to
On 12 Jun 2001, Squark wrote:

> 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

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Jun 13, 2001, 11:05:31 PM6/13/01
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John Baez writes:

> 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 f