2 views

Skip to first unread message

Mar 16, 2004, 3:48:30 AM3/16/04

to

"Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote in message news:<404f064e$1...@news.sentex.net>...

"I believe that LQG-like quantization of gravity could make

sense for some V=complicated correction. Maybe.

But how should we find this V? And is anyone looking for it?

[...]

Ok, so it seems that I am proposing a new LQG program:

Find a V=something quantum correction to the diffeomorphism

group such that a sensible semiclassical limit is obtained. "

>

> I'll admit or deny authorship after having seen the reaction of the

> experts... ;-)

>

I will come back to this point, because I want you to ponder the logical

consequences of your suggestion.

I think you mean that LQG should treat the diffeo constraint in a

similar way as string theory handle its conformal worldsheet symmetry.

In particular, the diffeo group should acquire some kind of quantum

correction, at least if you restrict yourself to the fields. There

should perhaps also be a negative contribution from ghosts, making the

total anomaly zero, but the field contribution should be non-zero.

If this is not what you mean, then you should please correct me.

However, if I have indeed understood you correctly, one question

immediately arises: why just LQG? The main complaint about Thomas

Thiemann's string quantization is that he gets a different conformal

anomaly than string theory does. If there is essentially a unique

way to quantize the string, as I believe, then there should be a

single correct value for the central charge (for the fields alone).

If so, shouldn't there also be a single correct value for a diffeo

anomaly, to be found in any correct quantization of gravity?

In short: if the diffeo group should be anomalous in LQG, shouldn't it

be anomalous in any correct formulation of QG, based on strings, fields,

loops, or whatever?

Mar 16, 2004, 4:08:42 AM3/16/04

to

"Thomas Larsson" <thomas_l...@hotmail.com> schrieb im Newsbeitrag

news:24a23f36.04031...@posting.google.com...

> I think you mean that LQG should treat the diffeo constraint in a

> similar way as string theory handle its conformal worldsheet symmetry.

The Polyakov action is nothing but 1+1 dimensional gravity coupled to scalar

fields. It would seem that by looking at it from this point of view one can

learn something about the quantization of 3+1d gravity. (Of course some

things in 1+1 d are unique to these numbers of dimensions, like the

classical conformal invariance.)

> In particular, the diffeo group should acquire some kind of quantum

> correction, at least if you restrict yourself to the fields. There

> should perhaps also be a negative contribution from ghosts, making the

> total anomaly zero, but the field contribution should be non-zero.

>

> If this is not what you mean, then you should please correct me.

As far as my current understanding goes this is what I mean. Often when I

say something outright wrong Aaron Bergman is so kind to point out the

mistake. So let's see if he does! :-)

> However, if I have indeed understood you correctly, one question

> immediately arises: why just LQG? The main complaint about Thomas

> Thiemann's string quantization is that he gets a different conformal

> anomaly than string theory does.

I wouldn't put it this way. The main complaint is rather that the procedure

by which he arrives at that result is not what is commonly called 'canonical

quantization'. It does for instance neither follow from path integral

quantization, BRST quantization or Dirac/Gupta-Bleuler (OCQ) quantization.

> If there is essentially a unique

> way to quantize the string, as I believe, then there should be a

> single correct value for the central charge (for the fields alone).

How about (D/12)(m^3 - m) ? :-)

> If so, shouldn't there also be a single correct value for a diffeo

> anomaly, to be found in any correct quantization of gravity?

Yes, maybe, that's at least what I currently expect based on my current

understanding.

> In short: if the diffeo group should be anomalous in LQG, shouldn't it

> be anomalous in any correct formulation of QG, based on strings, fields,

> loops, or whatever?

Well, this is indeed the question which I posed a few days ago in

http://groups.google.de/groups?selm=4051f3fc%241%40news.sentex.net . I have

also asked Ashtekar about this, but he didn't know. I should talk to a

string expert.

In http://golem.ph.utexas.edu/string/archives/000330.html#c000786 I

speculate about what would happen if LQG tried to handle the BRST extended

(including ghost sector) version of the EH action.

But it is not clear to me yet that extrapolations from quantization of 1+1d

gravity to the background equations motion found in string theory is

possible. I am not even sure if it makes sense to quantize the effective

action of striung backgrounds.

I have once talked about this with Thomas Mohaupt and his group, who are

working on string cosmology questions. Thomas Mohaupt himself told me that

he wasn't sure if it makes sense to quantize the string background effective

action, or if it must rather be regarded already as a quantized theory, the

quantum corrections being all the higher-order terms in alpha'. So perhaps

the answer is that in string theory the gravitational field is on a

completely different footing than is assumed in other approaches of QG, like

LQG or minisuperspace quantum cosmologies. I don't know. Comments are

appreciated.

Mar 16, 2004, 11:16:18 AM3/16/04

to

"Thomas Larsson" <thomas_l...@hotmail.com> schrieb im Newsbeitrag

news:24a23f36.04031...@posting.google.com...

news:24a23f36.04031...@posting.google.com...

> I think you mean that LQG should treat the diffeo constraint in a

> similar way as string theory handle its conformal worldsheet symmetry.

> In particular, the diffeo group should acquire some kind of quantum

> correction, at least if you restrict yourself to the fields.

I have now talked with Hermann Nicolai about this point. He tells me that

this is more or less what he would expect. He says that he does not like the

space+time splitting of LQG and that he would rather see LQG handle the

constraints of 3+1d gravity in precisely the same (or at least analogous)

way as is done for 1+1d gravity in standard quantization of the Polyakov

action.

(My own opinion about that is, in case anyone cares to know, that the

space-time splitting itself and as such is not the problem. When doing

canonical quantization of the string in the Schroedinger picture instead of

in the Heisenberg picture, which is usually used in field theory, there is

also a space/time split on the worldsheet. But since this is just a question

of quantum pictures this does not affect in any way the resulting formulas

and results, as can easily be checked explicitly.)

I told H. Nicolai that you told me that all possible anomalies for the

diffeo algebra in arbitrary dimensions have been classified and understood.

(Hope that's about right.) He told me that he instead had been told that

there are infinitely many possibilities and that people don't know what to

do with them.

Unfortunately I couldn't say more regarding this point than what I knew

second-hand. Since I didn't have any references with me you should perhaps

contact Nicolai and point him to the respective papers. I'll be visiting the

AEI in April and will maybe have time to further clarify this then.

I also asked him if and how he thinks the equivalent problem is dealt with

in string theory. I.e. I asked if he thinks that it even makes sense to try

to canonically quantize the effective background action.

He said that he wouldn't try that, because nobody knows how to do it.

Rather, one should try to understand closed string field theory. Of course

there are proposals for how this could be done (using limits of open string

field theory or of Matrix Models), but it has not been done yet.

He didn't say so explicitly, but my summary of what he said is that if we

were able to do closed string field theory we would probably see a spacetime

diffeomorphism anomaly with respect to the physical fields which vanishes

when the ghosts are taken into account.

> There

> should perhaps also be a negative contribution from ghosts, making the

> total anomaly zero, but the field contribution should be non-zero.

Yes, that's the impression I got about what Hermann Nicolai is thinking, and

it sounds pretty plausible to me. (But please note very well that everything

I report here is only my possibly incomplete and maybe unintentionally

distorted summary of our conversation.)

> In short: if the diffeo group should be anomalous in LQG, shouldn't it

> be anomalous in any correct formulation of QG, based on strings, fields,

> loops, or whatever?

Ok, I hope I have successfully reported a possible answer to that! :-)

Mar 17, 2004, 9:09:20 AM3/17/04

to

"Urs Schreiber" <Urs.Sc...@uni-essen.de> wrote in message news:<40572852$1...@news.sentex.net>...

>

> I told H. Nicolai that you told me that all possible anomalies for the

> diffeo algebra in arbitrary dimensions have been classified and understood.

> (Hope that's about right.) He told me that he instead had been told that

> there are infinitely many possibilities and that people don't know what to

> do with them.

>

We probably compare apples with oranges. What has been classified are

abstract extensions of the diffeo algebra; more precisely, the algebra

of polynomial vector fields in N dimensions by modules of tensor

fields. There are many concrete nonlinear realizations of each abstract

cocycle using a given set of fields.

However, the anomaly by itself is nothing. What singles out the Virasoro

algebra in any dimension is that it has nice representations, which

essentially come from canonical quantization. One starts from a classical

oscillator realization, introduce a Fock vacuum, and normal order w.r.t.

this vacuum. However, when N > 1 one must start from a particular type of

nonlinear realization, which makes the extension non-central.

It is instructive to look at the algebra of gauge transformations instead.

It has an abstract cocycle of the form

[J(X), J(Y)] = J([X,Y]) + K^{abij}(d_i X^a d_j Y^b),

where X and Y are g-valued functions, d_i = d/dx^i and the extension

K^{abij} transforms as the indices indicate: as adjoint x adjoint under

g and as a two-tensor under diffeos. This abstract cocycles can be

concretely realized in 3D as

K^{abij}(F_{abij}) = e^ijk d^abc \int d^3x F_{abij}(x) A_ck(x)

for any smearing function F_{abij}. Here e^ijk is the epsilon tensor

(hence this only works in 3D), d^abc are the totally symmetric

structure constants associated to the third Casimir, and A_ck is an

adjoint-valued vector. In higher dimensions, the same abstract cocycle

can be realized in other ways, see http://www.arxiv.org/abs/hep-th/9401027 .

It turns out that Jacobi identities still hold if A_ck is a connection

rather than an adjoint vector. The algebra is then known as the

Mickelsson-Faddeev algebra, which pertains to the Hamiltonian formulation

of the chiral anomaly. Some kind of no-go theorem has been proven for

this algebra, which essentially states that it has no good representation,

see

D. Pickrell, On the Mickelsson-Faddeev extensions and unitary representations,

Comm. Math. Phys. 123 (1989) 617.

There is also a compelling physical reason to avoid this kind of anomaly:

the fermion content of the standard model is chosen to make d^abc = 0.

The gauge algebra also has a different kind of anomaly, which generalizes

affine Kac-Moody algebra to higher dimensions:

[J(X), J(Y)] = J([X,Y]) + delta^ab S^i(X^a d_i Y^b)

S^i(d_i F) = 0.

delta^ab is the Killing metric proportional to the second Casimir, which

does not vanish in the standard model, so the usual demand for anomaly

cancellation does evidently not apply here. S^i not quite a tensor field,

due to the last condition, but rather dual to a closed one-form. This

is the kind of extension that arises from normal ordering and has a

nice representation theory.

Let me also explain how to build representations. The critical idea is

to expand all fields in a Taylor series, written in 1D for simplicity:

f(x) = sum_k f_k (x-q)^k.

Classically at least, we can express everything in terms of Taylor data

q, f_0, f_1, f_2, ...

instead of field data f(x). This may be inconvenient and there may be

problems with convergence, but I see no conceptual problems here. Diffeos

evidently act on the Taylor data as well. One may ask what such a trivial

reformulation is good for. The answer is that the cocycle S^i depends on

the expansion point q. There is no way to express the cocycle in terms of

the field f(x) which is indendent of q.

This is in agreement with the well-known fact that the only anomalies

in 4D is a pure gauge anomaly proportional to the third Casimir and a

mixed gravity-U(1) anomaly proportional to the first Casimir, see e.g.

Weinberg II, chapter 22. To express the Kac-Moody anomaly proportional

to the second Casimir, or the pure gravitational Virasoro anomaly, you

need to use Taylor data, which evades Weinberg's axioms.

The situation is slightly more complicated. Instead of expanding around

a point q we must expand around a closed 1D line q^i(t); the Taylor

coefficients are then replaced by functions f_k(t). We must also

truncate the Taylor series at some finite order p. This can be viewed as

a kind of regularization, with the very peculiar property that diffeos

act in a natural manner. We now have a realization on finitely many

functions of a single variable t, which is exactly the situation where

normal ordering works.

The cocycle takes the explicit form

S^i(F_i) = \int dt dq^i(t)/dt F_i(q(t)).

The closedness condition follows from

S^i(dF) = \int dt dF(q(t))/dt = 0,

because there are no endpoints. The cocycle is central in 1D because

there is a single circle along which to integrate.

>

> Ok, I hope I have successfully reported a possible answer to that! :-)

I am very grateful for your efforts.

Mar 22, 2004, 10:51:25 AM3/22/04

to

Urs Schreiber <Urs.Sc...@uni-essen.de> skrev i

diskussionsgruppsmeddelandet:4056c41a$1...@news.sentex.net...

> I have once talked about this with Thomas Mohaupt and his group, who are

> working on string cosmology questions. Thomas Mohaupt himself told me that

> he wasn't sure if it makes sense to quantize the string background effective

> action, or if it must rather be regarded already as a quantized theory, the

> quantum corrections being all the higher-order terms in alpha'. So perhaps

> the answer is that in string theory the gravitational field is on a

> completely different footing than is assumed in other approaches of QG, like

> LQG or minisuperspace quantum cosmologies. I don't know. Comments are

> appreciated.

If gravity is on a different footing from the other fields, you would

lose your case against LQG, wouldn't you? LQG quantization of the

string

may be wrong (perhaps), but the real issue is whether LQG quantization

of gravity is right. If gravity is different (and 4D gravity is very

different from 2D gravity), LQG still may be ok. Or maybe gravity

shouldn't be quantized at all, as suggested in other threads on this

newsgroup.

Reply all

Reply to author

Forward

0 new messages

Search

Clear search

Close search

Google apps

Main menu