76 views

Skip to first unread message

May 25, 2001, 12:47:07 AM5/25/01

to

Is there a way to reconcile M-theory with loop quantum gravity, combining

them into a unified theory?

them into a unified theory?

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

to

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...

May 29, 2001, 2:48:25 PM5/29/01

to

In article <9ehndj$jb2$1...@uwm.edu>,

Mark William Hopkins <whop...@alpha2.csd.uwm.edu> wrote:

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

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?

> 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.

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.

> 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

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:

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.

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;'

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.

>

>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/>

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:

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.

Jun 7, 2001, 8:42:09 PM6/7/01

to

In article <slrn9hsmrm....@cardinal0.Stanford.EDU>,

Aaron Bergman <aber...@princeton.edu> wrote:

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.)

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!". ;-)

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... ;-)

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

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

Jun 11, 2001, 5:07:00 PM6/11/01

to

A few random comments:

In article <Pine.SOL.4.10.101060...@strings.rutgers.edu>,

Lubos Motl <mo...@physics.rutgers.edu> wrote:

>Supersymmetry is an important feature of theories connected with string

>theory. It is seen perturbatively but holds also exactly [...]

Except, of course, in the real world. This is what makes me

nervous. Somehow supersymmetry must be broken to obtain a

theory that's consistent with experiment. I haven't heard

any good explanation of how this is supposed to work. What

is your attitude towards this issue?

>On the contrary, I would emphasize that it

>is very easy to construct some "nonperturbative physics" such as Loop

>Quantum Gravity that looks covariant to some extent etc. The difficult

>part of the task is to derive the low-energy gravity from such a quantum

>theory.

Yup. I've spent a lot of time talking to Ashtekar, Rovelli,

Smolin, and other people who work on loop quantum gravity.

They all agree that this is the big task. Most of them are

working on it in one way or another. (Smolin is probably

spending more time trying to forge connections with string

theory.)

>If someone believes that the correct low-energy physics is

>guaranteed just because we have put the most beautiful (Einstein's)

>equations in, he/she is a victim of a completely unjustified religion.

I agree. I don't know any such people - do you?

>... what I have problem with is the relatively simple

>machinery of Loop Quantum Gravity, with Einstein-Hilbert action postulated

>at all distance scales with no corrections at all, written in some simple

>Yang-Mills variables - and with people who claim that this is the ultimate

>solution of Quantum Gravity.

Do you know anyone who has made such a claim? I don't. It's possible

you've just talked to people who say that some *other* people make this

claim.

As for "relatively simple machinery", I think that's good rather than

bad.

As for "the Einstein-Hilbert action postulated at all distance scales

with no corrections at all", I've explained in a reply to Aaron Bergman

that this is almost surely NOT how things will work.

>String theory is different. Some people complain that it does not predict

>- but it in fact predicts a lot. To include gravity consistently to a

>quantum theory, we were forced to go to 10 dimensions, to discover

>supersymmetry, to allow topology change etc.

The remaining big task is to explain why the world looks 4-dimensional,

why it doesn't look supersymmetric, why we don't see topology change,

etc.. That is, why the low-energy effective limit matches the world

we see around us. Note that this task is very much like the remaining

big task of loop quantum gravity!

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.

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.

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

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.

quantum gravity and string theory, and a much more specific

question about quantization of volumes in string theory. I'm

here.

Lubos Motl <mo...@physics.rutgers.edu> wrote:

> Good that someone - even someone famous - (tries to) defend(s)

> Loop Quantum Gravity so that we enjoy plurality here. :-)

I'm not actually ``defending loop quantum gravity'' in the sense of

arguing that it's right. It's more that I have a stronger feeling of

uncertainty and ignorance than you do: I don't know the right way

to quantize gravity, and I suspect that no one else does either.

This comes in part from work in (2+1)-dimensional quantum

gravity---certainly a hugely oversimplified model, but one that

might tell us something. Our experience there is that

(1) There are lots of ways to quantize gravity, and they're not

all equivalent;

(2) It's not necessary to make the leap to string theory in order

to get a quantum theory; and

(3) The big problems are the ``conceptual'' ones that come from

trying to quantize a theory of the structure of spacetime---the

``problem of time,'' the problem of reconstructing a classical

limit from nonlocal diffeomorphism-invariant observables,

the problem of finding good observables to begin with, etc.

[Note that I'm *not* saying that the way to solve ``conceptual

issues'' is to think vague philosophical thoughts about the

concepts!]

Now, it may well be that (3+1)-dimensional quantum gravity is

fundamentally different, and that there are a whole new set of

issues that don't show up in the lower-dimensional model. But,

first, I think this is an open question, and, second, even if it's

true, it's not going to let you escape the problems that are already

there in 2+1 dimensions.

>The task to construct a quantum theory that behaves as general

> relativity at long distances (semiclassically) was probably the

> biggest problem that theoretical physics ever faced. The central

> problem of physics of the 20th century, some people say.

> And we still do not understand many consequences of the

> solution that some of us ;-) think is the only correct solution

> of this problem. [...]

> I would emphasize that it is very easy to construct some

> "nonperturbative physics" such as Loop Quantum Gravity

> that looks covariant to some extent etc. The difficult part of

> the task is to derive the low-energy gravity from such a quantum

> theory.

Well, I'm not at all sure that it's so easy to construct a covariant

nonperturbative theory that looks anything like general relativity.

But the more important point, I think, is that I believe it's too early

to call string theory ``the correct solution of this problem'': it's

more a promising program for finding an as yet unknown solution.

Let me explain by means of an analogy. We've actually known

for more than 30 years how to construct a quantum theory that

behaves like general relativity at long distances. Start with the

Einstein-Hilbert action and add a curvature squared term with

a small coefficient. This makes propagators go as 1/k^4 at high

momenta, and Stelle showed in 1977 that the resulting theory is

perturbatively renormalizable, and has general relativity as its

low energy limit.

There's a catch, of course: perturbatively, the theory has negative

norm states. For simpler theories, though, there's a known

mechanism (the Lee-Wick mechanism) for getting rid of such

states, and it's not unreasonable to hope for such a mechanism

in gravity.

Does this mean we've solved the problem? I, and I assume you,

would say, ``No.'' The answer is ``no'' because the ``solution''

depends on a postulated nonperturbative mechanism that hasn't

been shown to work for the case of physical interest. So R+R^2

gravity isn't a quantum theory of gravity; it's at best a program

for finding one by working out the existence of a nonperturbative

method of getting rid of negative norm states.

I would argue that as far as quantum gravity is concerned, the status

of string theory isn't so different. Perturbative string theory has

general relativity as a long distance limit, but the perturbation series

is not Borel summable, so the theory only really makes sense as an

asymptotic expansion of some nonperturbative theory. And even

though we have a bunch of hints about features of nonperturbative

string theory, we don't really know what it is, or even if it is.

Now, I happen to think that string theory is a very good guess for a

program that will eventually give us a quantum theory of gravity.

But I also know that there have been a lot of other guesses that

looked good at the time. So I think a bit of caution is in order.

> I think that string theory has already said a lot about "what

> replaces a smooth classical spacetime at short distances". T-duality,

> noncommutative geometry, topology transitions etc. are pretty

> radical insights in this direction.

T-duality certainly seems to provide a radical insight, though I'm

not quite sure what that insight is. The traditional interpretation

that the string scale provides a minimum length seems to be wrong,

since D0 branes can probe smaller distances. But there's certainly

*something* there.

Noncommutative geometry, on the other hand, arises even in

condensed matter physics (electrons restricted to their lowest

Landau level), but no one thinks that says something deep about

the nature of space and time. Maybe noncommutative geometry

in string theory is deeper, and I haven't read enough; the stuff

I've seen mostly refers to particular background configurations.

As for topology change, the idea's been around since the '50s.

It's certainly very nice that string theory can implement some

particular examples, but again, I don't think anything can be

said systematically yet. These are interesting directions; I just

don't think they're yet interesting *answers*.

> But what I have problem with is the relatively simple machinery

> of Loop Quantum Gravity, with Einstein-Hilbert action postulated

> at all distance scales with no corrections at all, written in some

> simple Yang-Mills variables - and with people who claim that

> this is the ultimate solution of Quantum Gravity.

Who claims that? This is the same kind of stereotype that some

loop quantum gravity people have about string theorists---that

they postulate a bunch of little loops running around on a fixed

background and claim that this is the ultimate solution of Quantum

Gravity. If you want to criticize loop quantum gravity, that's

great. But make the criticisms real: setting up straw men to knock

down doesn't do anyone any good.

> What also disturbs me is that as far as I know, no surprising

> consistency checks in LQG appeared (in deep contrast with

> string theory). In fact I do not see too many surprises at all.

Well, there's the fact that spin network states are eigenstates of

the area operator---something not at all obvious, and not really

put in by hand---and there's the equivalence of the analytic

``generalized connection'' approach and the combinatoric spin

network approach, again something that certainly wasn't

obvious a priori.

On the other hand, I agree that string theory has some amazing

consistency checks, of the sort that loop quantum gravity doesn't.

On the other hand, QED doesn't, either. In some sense, this

makes loop quantum gravity much more like an ordinary quantum

theory, and not, as you charge, ``a religion based on the "exceptional"

nature of classical Einstein's equations.

Steve Carlip

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.

<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

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.

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.

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. ;-)

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, <

> | check out http://www.debian.org/ <

`-----------------------------------------------------------------------------'

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?

>

>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.

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? ;-)

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!

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

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. ;-)

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? ;-)

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

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. ;-)

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.

> 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.

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.

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

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.

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

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.

Jun 13, 2001, 11:05:31 PM6/13/01

to

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