Gribov (gauge fixing) ambiguities in quantum gravity

[tom.stoer]

Recently I started a new thread regarding Gribov (gauge fixing) ambiguities in quantum field theory, especially in QCD https://www.physicsforums.com/showthread.php?t=429759.

Of course every theory incorporating a local gauge symmetry may have this gauge fixing issue - therefore in GR and all approaches towards a theory of quantum gravity the same problem should arise.

Has anybody seen something like gauge fixing ambiguities in LQG (the method is totally different from QFT, therefore I cannot even say where to look for these issues), in asymptotic safety (there is an overcounting of metrics which is not gauge but diffeo - perhaps the same applies here) or any other approach towards QG?

If these gauge fixing ambiguities can be avoided, why isn't this possible in non-abelian QFTs?

Rearding LQG it seems to be related to the loops which are gauge invariant. The non-separable Hilbert space can be reduced via the diffeo-constraint - which is not available in QFTs; is this the main difference?

 

[sheaf]

No idea what the answer is - just more questions:

Can we start by asking this question purely in the classical domain ? i.e. does a Gribov ambiguity occur when the gauge theory bundle is non trivial ?

If so, then in the classical picture from which LQG traditionally arises, then the frame bundle from which the SU(2) gauge transformations arise can in principle be non trivial I guess.

However, when we "skeletonize" the space with a spin-network style graph, isn't the information about that non-triviality still encoded in the holonomies of the connection ?

 

[tom.stoer]

The question arises (if it arises at all) already on the classical level.

The main difference between LQG and QCD is that in QCD you specify a gauge condition f[A]=0 which (afaik) always introduces gauge ambiguities or other issues. In LQG you never specify any gauge fixing conditions, but you always work with gauge invariant holonomies. So there are no gauge issues at all. You translate the theory into a theory involving a huge number of unphysical degrees of freedom (the loop-space is over-complete) and later eliminate them via cylinder functions and their representatives = spin networks. I guess this is posible only in LQG due to diff.-inv.; so it doesn't work in QCD (I have talked to some QCD guys 15 years ago and we found that counting degrees of freedom in loop space gives the wrong result).

Now the problem for me is the following: in QCD with explicit gauge fixing we find Gribov ambiguities; in LQG with "implicit" gauge fixing we do not see any Gribov ambiguities. So where are they? what happened during gauge fixing?

 

[genneth]

No answers either, also more questions:

What happens in lattice QCD? In other words, if one is careful about the process of taking limits (IR and UV completion) does this still occur? I was under the impression that gauge theories on finite lattices are perfectly well-defined, and occasionally exactly solved?

 

[sheaf]

Right, I need to go away and try to understand better GR as a gauge theory. It's often said that Ashtekar variables reformulate GR as an SU(2) gauge theory, but I guess the diffeomorphism group is the true gauge group.

 

[tom.stoer]

What happens in lattice QCD? In other words, if one is careful about the process of taking limits (IR and UV completion) does this still occur?

afaik the Gribov ambiguities play a role in lattice QCD as well

 

[tom.stoer]

It's often said that Ashtekar variables reformulate GR as an SU(2) gauge theory, but I guess the diffeomorphism group is the true gauge group.

No, the SU(2) gauge group arises due to the dreibein in tangent space; the gauge symmetry is the local rotation of the dreibein; it disappears when calculating the metric as it is bilinear in the dreibein.

The diffeomorphism is an independent feature which is visiblein the metric formulation as well.

 

[sheaf]

No, the SU(2) gauge group arises due to the dreibein in tangent space; the gauge symmetry is the local rotation of the dreibein; it disappears when calculating the metric as it is bilinear in the dreibein.

The diffeomorphism is an independent feature which is visiblein the metric formulation as well.

I agree that is where SU(2) gauge freedom comes into the Ashetekar picture, but what I was getting at is:

is it not true that diffeomorphism invariance is also regardable as a gauge invariance in the sense that it partitions the constraint surface into gauge orbits ? So isn't there a potential for a Gribov question to be asked there too ? (actually this question stands even if we don't use the connection formalism for GR).

Rightly or wrongly I tend to think that diff invariance is somehow more "fundamentally general relativistic" than SU(2) invariance, which we kind of put in by hand when we chose to use the dreibeins.

 

[tom.stoer]

I agree that is where SU(2) gauge freedom comes into the Ashetekar picture, but what I was getting at is:

is it not true that diffeomorphism invariance is also regardable as a gauge invariance in the sense that it partitions the constraint surface into gauge orbits ? So isn't there a potential for a Gribov question to be asked there too ? (actually this question stands even if we don't use the connection formalism for GR).

Rightly or wrongly I tend to think that diff invariance is somehow more "fundamentally general relativistic" than SU(2) invariance, which we kind of put in by hand when we chose to use the dreibeins.

I agree that fixing diff.-inv. may cause ambiguities as well. But in LQG first gauge inv. is fixed by using holonomies (loops), later diff.-inv. is used to reduce the overcomplete loop-space.

So this is even more involved.

 

[genneth]

afaik the Gribov ambiguities play a role in lattice QCD as well

What about even simpler systems? Say, SU(2) lattice gauge theory on a finite square 2D lattice. I think the state can be unambiguously characterised through holonomies?

 

[tom.stoer]

Yes, I think this is correct. But you have to count physical degrees of freedom and afaik the loop space has too many degrees of freedom; it is overcomplete.

Take a 3-torus as an example.

You know that the gluon has two physical polarizations and you know that plane wave states on a torus are restricted by periodic boundary conditions. So you have countably many degrees of freedom which is reflected by the usual Fock space structure.

If you take the space of holonomies you immediately see that the loops (w/o base point) form an uncountable set.

I can't remember the details but I think that you need an additional symmetry to reduce this space to a physical subspace. In LQG diff.-inv. is available to allow one to define something like "physical subspace = loopspace / diff.-inv." This is missing in ordinary QCD that's why the loop space cannot be used for standard quantization.

So perhaps instead of solving the gauge ambiguity directly you use loops, introduce an overcomplete set of gauge invariant objects and reduce them to a physical sub space by using diff.-inv.

 

[genneth]

And thus my original question: this sounds like an issue of UV completion. My point of view is that if there is a finite theory which works, with clear physical interpretation, but the UV completion doesn't, then the problem resides in the completion procedure. So I'm a bit surprised that such problems occur in lattice QCD; I guess your statement is that for practical lattice QCD computations, people employ gauge-fixing rather than use holonomies as the fundamental degrees of freedom?

 

[tom.stoer]

Do you really think that UV completion is already an issue in classical field theory?

Gribov ambiguities are discussed in the context of quantizing non-abelian gauge theories, but their origin is purely classical. So there is no "completion". You write down a classical action with a certain symmetry (using a vector bundle) - that's all. You could even say that it'snothing else but the property of a set of couples, non-linear differential equations.

I am not an expert in lattice QCD; my example only shows that using loops in the continuum formaulation of gauge theories may solve the problem of gauge ambiguities, but it creates a new problem which is not easier to solve.

But I thin the main problem is to understand how LQG avoids the gauge ambiguity issue = how it can be solved in loop space

 

[sheaf]

Here's a reference on the issue of gauge fixing in the Ashtekar formalism:

http://arxiv.org/abs/math-ph/0007001"

 

[tom.stoer]

I checked Thiemann's paper

http://arxiv.org/abs/gr-qc/0110034
Introduction to Modern Canonical Quantum General Relativity
Thomas Thiemann
(Submitted on 5 Oct 2001)
Abstract: This is an introduction to the by now fifteen years old research field of canonical quantum general relativity, sometimes called "loop quantum gravity". The term "modern" in the title refers to the fact that the quantum theory is based on formulating classical general relativity as a theory of connections rather than metrics as compared to in original version due to Arnowitt, Deser and Misner. Canonical quantum general relativity is an attempt to define a mathematically rigorous, non-perturbative, background independent theory of Lorentzian quantum gravity in four spacetime dimensions in the continuum. The approach is minimal in that one simply analyzes the logical consequences of combining the principles of general relativity with the principles of quantum mechanics. The requirement to preserve background independence has lead to new, fascinating mathematical structures which one does not see in perturbative approaches, e.g. a fundamental discreteness of spacetime seems to be a prediction of the theory providing a first substantial evidence for a theory in which the gravitational field acts as a natural UV cut-off. An effort has been made to provide a self-contained exposition of a restricted amount of material at the appropriate level of rigour which at the same time is accessible to graduate students with only basic knowledge of general relativity and quantum field theory on Minkowski space.

He writes

"2) One fixes a gauge and solves the constraints. Years of research in the field of solving the Cauchy problem for general relativity reveal that such a procedure works at most locally, that is, there do not exist, in general, global gauge conditions. This is reminiscent of the Gribov problem in non-Abelian Yang-Mills theories."

…

Then he comments on Fleischhack's paper

"[he] also investigated the issue of Gribov copies in A [134] with respect to SU(2) gauge transformations. It should be noted that fortunately Gribov copies are no problem in our context: The measure is a probability measure and the gauge group therefore has finite volume. Integrals over gauge invariant functions are therefore well-defined."

…

"It is important to notice that in contrast to other measures on some space of connections
the “volume of the gauge group is finite”: The space C(A/G) is a subspace of C(A) …
We do not have to fix a gauge and never have to deal with the problem of Gribov copies."

So the ambiguities are there, but one does need to fix the gauge as (in the path integral) the measure produces a volume factor of the gauge group which is finite and therefore doesn't matter.

So Thiemann essentially says that one can avoid gauge fixing ambiguities by avoiding gauge fixing at all!

 

[sheaf]

"It is important to notice that in contrast to other measures on some space of connections
the “volume of the gauge group is finite”: The space C(A/G) is a subspace of C(A) …
We do not have to fix a gauge and never have to deal with the problem of Gribov copies."

So the ambiguities are there, but one does need to fix the gauge as (in the path integral) the measure produces a volume factor of the gauge group which is finite and therefore doesn't matter.

So Thiemann essentially says that one can avoid gauge fixing ambiguities by avoiding gauge fixing at all!

Have I understood this correctly - would you say that the avoidance of the need to gauge fix is achieved in LQG by the process of obtaining SU(2) gauge invariance (for the Hilbert space states) through averaging over the group, as defined in equation (3.26) of

http://arxiv.org/abs/1007.040"

 

[tom.stoer]

The link does not work.

But yes, something like that.

There's a simple example why you need gauge fixing. Think about an integral

$$Z[f] = \int d^2r f(x,y)$$

First let's study translational invariance in y:

$$Z[f] = \int dy \int dx f(x) = vol(y) Z[f] = \infty Z^0[f]$$

You need gauge fixing and introducing a delta function will do the job; Fadeev-Popov is nothing else but introducing a delta function for arbitrary fixing conditions.

Then let's study rotational invariance

$$Z[f] = \int d\phi \int dr r f(r) = vol(\phi) Z[f] = 2\pi Z^0[f]$$

You can gauge fix this integral by introducing

$$Z^0[f] = \int d^2r \delta(\phi - \phi_0) f(r)$$

Or you can do something strange like

$$Z^0[f] = \int d^2r J_G(\phi) \delta(G(\phi)) f(r)$$

Where I have introduced a factor J which compensates the gauge fixing G. If G has multiple zeros then (of course this is stupid) these are Gribov ambiguities. The difference between our example and local SU(2) is that we know that there are unique gauge fixing functions whereas in SU(2) you know that there are no gauge fixing functions w/o ambiguities.

But - as the rotational symmetry generates a finite volume, there is no need for gauge fixing. You can simply write

$$Z^0[f] = (2\pi)^{-1}\int d^2r f(r)$$

and that's it.

I think that this is what Thiemann has in mind

 

[sheaf]

Have I understood this correctly - would you say that the avoidance of the need to gauge fix is achieved in LQG by the process of obtaining SU(2) gauge invariance (for the Hilbert space states) through averaging over the group, as defined in equation (3.26) of

http://arxiv.org/abs/1007.040"

Bah, sorry, I messed up the link, the correct one is


http://arxiv.org/abs/1007.0402"

Nice clear explanation with the rotational invariance example Tom, thanks !

 

[tom.stoer]

I have to admit that you can find it on several web pages, lecture notes and books ...

 

[tom.stoer]

Have I understood this correctly - would you say that the avoidance of the need to gauge fix is achieved in LQG by the process of obtaining SU(2) gauge invariance (for the Hilbert space states) through averaging over the group, as defined in equation (3.26) of

http://arxiv.org/abs/1007.040"

Yes, exactly.