Could GR's background independence be a theoretical artifact?

[Bob_for_short]

You see - due to geometrizations of gravity we lost everything. Is it practical, advantageous, advancing? No. That is why I think we have to preserve the flat space-time background in the theory construction explicitly. It is well possible (RTG), so why to put everything in a curved space-time? It is better to put gravity in the Minkowsky world and consider it as physical forces, as the other interactions.

 

[Thomas Larsson]

Background independence, diffeomorphism invariance and gauge symmetry are in some sense always a theoretical artefacts.

This statement needs some qualification. A gauge symmetry in the literal sense is a local symmetry, which vanishes at spatial infinity. However, a typical gauge symmetry naturally determines a global symmetry, which does act nontrivially on physical states. E.g., local U(1) in EM requires global U(1), and local SU(2) in weak interaction requires global SU(2). These global symmetries implied by local symmetries are not theoretical artefacts; EM and weak charges are observed.

This observation can be taken one step further. Instead of considering just local symmetries, which vanish at infinity, and global symmetries, which have a finite limit at infinity, one could also consider divergent symmetries, which go to infinity at infinity. In the presence of divergent symmetries, local symmetires can no longer be factored out, because

[local, divergent] = global,

and the RHS is nonzero if charge is so.

This situation typically arises if the symmetry algebra is expanded in a Laurent series. E.g., consider conformal symmetry in 2D, with generators L_m = z^{m+1} d/dz. The generators fall into three classes, according to the value of m:

local: m < 0
global: m = 0
divergent: m > 0

If we only consider local+global generators, there is only one physical state |0>, which is annihilated by the local symmetry:

L_-m |0> = 0,
L_0 |0> = h |0>.

However, local generators are no longer trivial in the presence of divergent generators, because

[L_-m, L_m] = 2 m L_0 != 0.

This is not a contradiction, because the divergent operators generate new gauge-noninvariant states like L_1 L_2 |0>, which do not belong to the original, one-dimensional, physical Hilbert space.

 

[tom.stoer]

This statement needs some qualification. A gauge symmetry in the literal sense is a local symmetry, which vanishes at spatial infinity. However, a typical gauge symmetry naturally determines a global symmetry, which does act nontrivially on physical states. E.g., local U(1) in EM requires global U(1), and local SU(2) in weak interaction requires global SU(2). These global symmetries implied by local symmetries are not theoretical artefacts; EM and weak charges are observed.

If the local symmetry (of a gauge field A) is expressed in a path integral formalism then it's simply the statement that the measure DA and the action S[A] are invariant under the change A => A'. This is a change of coordinates (in the fibre bundle specified by the gauge group) along the gauge orbit.

I do not see why the gauge symmetry must vanish at spatial infinity. You may need some boundary conditions, but gauge theories can be formulated (e.g.) in compact space as well; then periodic boundary conditions will do the job. I don't think this is of any relevance in the present context.

Your most interesting point is the global symmetry. In my opinion the global symmetry is nothing else but a special sub-sector of the local symmetry. The charge conservation can be derived from the local symmetry. So you are right, charges (from a local gauge symmetry) must be conserved, otherwise the theory becomes ill-defined due to anomalies, but I do not see why you need the global symmetry.

Look at QED (QCD): the operator that generates gauge transformations is the abelian (non-abelian) Gauss law. The conserved charges are dervived from the Gauss law via integration over 3-space.

This situation typically arises if the symmetry algebra is expanded in a Laurent series. E.g., consider conformal symmetry in 2D, with generators L_m = z^{m+1} d/dz. The generators fall into three classes, according to the value of m:

local: m < 0
global: m = 0
divergent: m > 0

If we only consider local+global generators, there is only one physical state |0>, which is annihilated by the local symmetry:

L_-m |0> = 0,
L_0 |0> = h |0>.

However, local generators are no longer trivial in the presence of divergent generators, because

[L_-m, L_m] = 2 m L_0 != 0.

This is not a contradiction, because the divergent operators generate new gauge-noninvariant states like L_1 L_2 |0>, which do not belong to the original, one-dimensional, physical Hilbert space.

Unfortunately I am not an expert in conformal field theories; so what does that mean if you express such a theory in the path integral formalism? If you use complex coordinates for spacetime isn't it then artificial to distinguish between m < 0 and m > 0? You have holomorphic functions, that's all you need.

In non-abelian gauge theories it's rather simple: Once you factor away the gauge degrees of freedom (you can do that in the PI formalisms as well as in the canonical formalism) what is left is a theory with gauge invariant fields or operators only.

Perhaps a look at the canonical formalism is interesting. Implementing the Gauss law constraint is rather complex (especially in the non-abelian case), but formally it is equivalent to the treatment of a two-particle system with potential V(x,y) = V(x-y). You can introduce (via a unitary transformation) two new coordinates r and R with momenta p and P. The new Hamiltonian looks like H(r,p) + H'(P). R does neither appear in H nor in H'. Then you can chose to solve the theory entirely in the P=0 subspace, so from all momentum eigenstates |p,P> you select |p,0> with P|p,0> = 0. You are allowed to do this because H' commutes with H, therefore P is conserved. From that time on you call H(r,p) the physical Hamiltonian with physical fields (r,p) and H'(P) the unphysical Hamiltonian with an unphysical field (P). So the original gauge freedom generated by the Gauss law is something like the coordinate transformation R => R' generated by the momentum P. (Unfortunately there's no good reference on arxiv, but I can find some papers for, if you like).

 

[atyy]

If g is minkowski, than you definitely cannot reach say a FRW solution with a cc for instance. The perturbation 'h' would require infinite energy to reach the solution. Hence a different superselection sector.

Trying to connect two spacetimes by wondering say about how much radiation you have to bring in from infinity is also a hard story, and far more recent. That involves topics like global stability theorems, the positive energy theorem and the like.

I've read this on various blogs like Motl's and Rozali's, but don't really understand what it means - any recommendations for reading?

 

[Thomas Larsson]

I do not see why the gauge symmetry must vanish at spatial infinity. You may need some boundary conditions, but gauge theories can be formulated (e.g.) in compact space as well; then periodic boundary conditions will do the job. I don't think this is of any relevance in the present context.

In the context of Yang-Mills theory, the global charges (which vanish on compact manifolds, so I need to have spatial infinity) are like gauge transformations, except that they are constant over space. The algebra of gauge transformations is generated by all g-valued functions, where g is a finite-dimensional Lie algebra with generators T^a. If we consider 3D space and spherical coordinates, a gauge transformation is thus given by a function

f(r, theta, phi) T^a.

In contrast, the global charges are the g generators T^a.

Your most interesting point is the global symmetry. In my opinion the global symmetry is nothing else but a special sub-sector of the local symmetry.

In the Yang-Mills example, the global charge symmetry corresponds to constant functions f = 1. However, there is a crucial difference: local gauge generators annihilate physical states, but global charge does not (for charged states).

Look at QED (QCD): the operator that generates gauge transformations is the abelian (non-abelian) Gauss law.

My argument does not work for QCD, because there are no charged physical states due to confinement. But for QED everything is fine.

Let me make my point specifically for Yang-Mills theory. A useful basis for the space of Gauss law generators consists of

J^a(n,l,m) = r^n Y_lm(theta,phi) T^a,

where Y_lm are the spherical harmonics. However, only the basis elements with n < 0 are good gauge transformations, because the others do not vanish when r = infinity. In particular,

J^a(0,0,0) = T^a

are identified with global charges (more precisely, the Cartan subalgebra of g is), which are obviously nonzero on charged states.

My observation is now that not even the proper gauge transformations J^a(-n,l,m) can annihilate all physical states in the presence of divergent operators, because

[J^a(-n,0,0), J^b(n,0,0)] = i f^abc J^c(0,0,0) = i f^abc T^c

is a linear combinatinos of charge operators. Hence there must be some physical states such that

J^a(-n,0,0) |phys> != 0.

This does not contradict the usual picture, because the divergent operators generate new physical states, which lie outside the original Hilbert space. To have a well-defined action of the full gauge algebra, one must complete the Hilbert space by adding the new states. On the completed Hilbert space, the original gauge symmetry acts in a non-trivial way. It still annihilates the original subspace, of course.

There are different ways to react on this observation. One possibility is to say that divergent operators violate boundary conditions, and that thinking about them is prohibited. However, I fail to see why this is any worse than the situation in CFT; why is it ok that operators diverge when z -> infinity but not when r -> infinity? A more fruitful option is to study the action of the completed gauge algebra in the completed Hilbert space. Who knows, one may even learn something new by doing so...

In non-abelian gauge theories it's rather simple: Once you factor away the gauge degrees of freedom

Evidently, the gauge dofs cannot be factored out anymore in the completed Hilbert space.

 

[ConradDJ]

You see - due to geometrizations of gravity we lost everything. Is it practical, advantageous, advancing? No. That is why I think we have to preserve the flat space-time background in the theory construction explicitly. It is well possible (RTG), so why to put everything in a curved space-time? It is better to put gravity in the Minkowsky world and consider it as physical forces, as the other interactions.

This point of view is interesting to me... and not necessarily old-fashioned, or even incompatible with "background independence".

If we assume that to begin with we have no well-defined geometric background, and that geometry has to emerge out of a more primitive kind of connection-topology... why should it be gravity that has to emerge first? Given that the electromagnetic field is much simpler, why not suppose it's also more fundamental?

This may not be relevant to the discussion here, but it seems as though e/m field-structure is very closely tied to Minkowski spacetime. You don't get a "flat" metric out of e/m, I guess, but maybe some scaffolding on which the gravitational metric could emerge?

 

[Bob_for_short]

..If we assume that to begin with we have no well-defined geometric background, and that geometry has to emerge out of a more primitive kind of connection-topology... why should it be gravity that has to emerge first? Given that the electromagnetic field is much simpler, why not suppose it's also more fundamental?

Using gravity as a geometrical property of space-time is tempting because of its universality.

But I find it quite weird and impractical to have always changing geometry. The geometry is thought as a stable background, space of all possible events. Then one compares different gravity or e/m effects in one geometry due to forces rather than curvature.

This may not be relevant to the discussion here, but it seems as though e/m field-structure is very closely tied to Minkowski spacetime.

As you know, the 4-geometry was derived by H. Poincaré from the Maxwell equations being valid in all reference frames (i.e. as a mathematical sequence of an experimental fact). He also advanced the idea of validity of the other interactions, including gravity, in such a 4-world. As the gravity is universal, it can be implemented as an effective geometry for the matter (RTG of Logunov's). On the other hand, the Minkowski space-time should be naturally separated from the effective geometry. It is done in the gravitational filed equations that contain the "harmonicity" equations. In GR these conditions are additional, physically motivated but they contradict the GR spirit, so the results obtained in GR in the frame of harmonic coordinates do not belong to GR. In RTG these conditions are obligatory as filed equations so the Minkowski metric appears explicitly in the whole system description.

The advantage of that is preserving the relativistic energy-momentum conservation laws that seemingly work in all interactions.

 

[atyy]

After eq. (32) a ghost action is specified. The operator between the ghost fields C-bar and C will certainly have zero eigenvalues for some background metric g. These zero eigenvalues correspond to singular gauges. Therefore this theory suffers from the same type of problems as explained above.

Note: in many cases these problems do not show up. In QCD one can use the (singular) Lorentz gauge for deep inelastic scattering and will never run into trouble. But this is only a certain sector of the theory where due to asymptotic freedom g << 1 and A = 0 is valid for perturbation expansion; reason is that all gauge fields stay "far away" from the Gribov horizon.

If one is interested in asymptotic safe gravity "g << 1" is no longer reasonable. All coupling constants could potentially cross Gribov horizons which have to be identified first. If one does not specify the background field one has to investigate all different sectors (Gribov domains) and check how they can be patched together. This analysis is missing.

I searched gr-qc for "Gribov". There is one paper mentioning this problem:
http://arxiv.org/abs/quant-ph/9611026
Title: Coherent State Approach to Time Reparameterization Invariant Systems M. C. Ashworth
(Submitted on 14 Nov 1996)
Abstract: For many years coherent states have been a useful tool for understanding fundamental questions in quantum mechanics. Recently, there has been work on developing a consistent way of including constraints into the phase space path integral that naturally arises in coherent state quantization. This new approach has many advantages over other approaches, including the lack of any Gribov problems, the independence of gauge fixing, and the ability to handle second-class constraints without any ambiguous determinants. In this paper, I use this new approach to study some examples of time reparameterization invariant systems, which are of special interest in the field of quantum gravity.

Relevant?

http://arxiv.org/abs/0907.1828
Asymptotically free scalar curvature-ghost coupling in Quantum Einstein Gravity
Astrid Eichhorn, Holger Gies, Michael M. Scherer
"We consider the asymptotic-safety scenario for quantum gravity which constructs a non-perturbatively renormalisable quantum gravity theory with the help of the functional renormalisation group. We verify the existence of a non-Gaussian fixed point and include a running curvature-ghost coupling as a first step towards the flow of the ghost sector of the theory. We find that the scalar curvature-ghost coupling is asymptotically free and RG relevant in the ultraviolet. Most importantly, the property of asymptotic safety discovered so far within the Einstein-Hilbert truncation and beyond remains stable under the inclusion of the ghost flow."

 

[tom.stoer]

I do not see if and how they treat Gribov copies correctly.

In QCD these ambiguities in the path integral are not relevant in the UV, but they certainly affect the IR regime (confinement etc.)

So if one studies asymptotic safety maybe it's OK to neglect these Gribov copies w/o affecting physics in teh UV regime.

 

[atyy]

I do not see if and how they treat Gribov copies correctly.

In QCD these ambiguities in the path integral are not relevant in the UV, but they certainly affect the IR regime (confinement etc.)

So if one studies asymptotic safety maybe it's OK to neglect these Gribov copies w/o affecting physics in teh UV regime.

Maybe they don't, but at least they mention the problem. I think right now the computations from the various groups are all just very suggestive, far from establishing AS for sure.

 

[atyy]

AdS/CFT is a theory of gravity in some universes is defined using a fundamentally "background dependent" CFT.

Another approach which is fundamentally "background dependent" is Oriti's group field theory http://arxiv.org/abs/gr-qc/0607032 "On the one hand, in fact, GFTs are almost ordinary field theories, defined on a group manifold with fixed metric and topology, and thus background dependent, ...".