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I have seen different explanations to understand why there are no local gauge invariant observables in gravity.

Some of them explain that diffeomorphisms are a gauge symmetry of the theory and thus any observable evaluated at a spacetime point will be gauge dependent and therefore not an observable. This line of reasoning then argues for Wilson loops, or asymptotic charges as good (non-local) observables in gravity. This explanation, in my opinion, is purely classical, it doesn't rely on the uncertainty principle, or commutation relations, etc.

However, other explanations give the argument that any device trying to measure a local observable will have a finite size, and therefore a finite accuracy. If the device wants to probe local physics, it should be smaller. However, the uncertainty principle forces the device to collapse into a black hole before allowing the experiment to give us local information. Alternatively, it is also explained that the commutator of two (quantum) operators has to vanish for spacelike separations but that a metric that is dynamical and fluctuates will mess with the causality of the theory, therefore making the operators not observable. These arguments seem absolutely quantum mechanical.

I have a couple of very similar questions:

Is the statement "no local gauge invariant observables in gravity" true in classical GR, in quantum gravity or both?

If it is true in both, why do people treat the statement "no local gauge invariant observables in quantum gravity" as something special?

Do statements about observables in classical and quantum gravity mean different things?The arguments given to explain each one are pretty different and seem to involve different physics. The first one relies heavily on diffeomorhism invariance while the second one relies on holographic-flavoured arguments about how much information you can concentrate in a given volume before you form a black hole.

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  • $\begingroup$ If you formulate the Standard Model in curved space time, the Standard Model also enjoys diffeomorphism symmetry. So, according to your argument, there is no local gauge invariant observables in the Standard Model? $\endgroup$
    – MadMax
    Dec 4, 2023 at 16:43
  • $\begingroup$ I don't know! That's exactly what I'd like to figure out! I believe the fact that standard model calculations can only give results in terms of the S matrix is probably related to this? Since the S matrix is an asymptotically defined object that compares states at the past and future boundaries but not in the bulk,technically. $\endgroup$ Dec 5, 2023 at 20:28

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The statement "there are no local gauge invariant observables in gravity" is true in both classical and quantum mechanics. You are right to point out that it is wrong to regard this as something unique to quantum gravity. This statement follows just from the fact that general relativity is invariant under active diffeomorphisms, meaning we can drag all fields smoothly along the spacetime manifold without changing the equations of motion. Thus, in both the quantum and classical case, all "local" measurements must be done relative to some local object. And there's no a priori reason why defining an observable relative to some local object is more difficult in quantum mechanics than in classical mechanics.

The argument about local measuring devices needing to be smaller is silly. We could have a sphere of detectors all aimed at a point at the center of the sphere, and that sphere could be as large as we want. I don't know of any physical principle which says that a smaller sphere is better; if anything, it is worse since it will have less angular resolution.

As for a fluctuating metric messing with the causality of the theory: this seems like a separate issue from the issue of no local gauge invariant observables. Obviously anything in quantum gravity is speculative to some extent, but there are two options here:

  1. The spacetime manifold is still well-defined at all levels of granularity in the quantum theory, in which case nothing essential has changed from the classical theory as regards "local" observables. The fact that observables which might have commuted without gravity no longer commute is a statement about the nonlocality of the dynamics, not the nonlocality of observables.

  2. The spacetime manifold is not well-defined at all levels of granularity, in which case yes of course there will not be "local" observables in the same sense as classical gravity, but there's no reason to expect that we wouldn't recover approximately "local" observables in a large enough region of spacetime.

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  • $\begingroup$ The Standard Model formulated in curved space time is also invariant under active diffeomorphism. So, are you suggesting that there is no local gauge invariant observables in the Standard Model? $\endgroup$
    – MadMax
    Dec 4, 2023 at 16:50
  • $\begingroup$ MadMax: Yes, anywhere general relativity plays a role there will be no local gauge invariant observables, including the Standard Model in curved spacetime. That statement doesn't apply to the Standard Model in flat spacetime because in that case there is no gauge associated with diffeomorphisms. $\endgroup$
    – Travis
    Dec 5, 2023 at 11:19
  • $\begingroup$ Is Standard Model in flat spacetime just a specific case of Standard Model in curved spacetime? $\endgroup$
    – MadMax
    Dec 5, 2023 at 14:30
  • $\begingroup$ And on the surface of the earth, the spacetime is curved given the gravity from the earth. Are you suggesting that there is no local gauge invariant observables in the Standard Model if measured on the surface of the earth? $\endgroup$
    – MadMax
    Dec 5, 2023 at 16:11
  • $\begingroup$ Yes, that's correct. We implicitly pick a gauge when we decide what we're going to measure stuff relative to (this yard stick, that clock, etc.). If we measure stuff relative to other things, then we get different observables. $\endgroup$
    – Travis
    Dec 5, 2023 at 17:06
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It depends on how you define "local observable" and that - in turn - puts the focus of the question on which definition is physically relevant! The matter is discussed more generally in Local and gauge invariant observables in gravity, Khavkine.

I'm not keen on folklore results, which this has practically descended into, nor on hand-waving arguments, and the author strikes the wrong chord at the outset with "It is well known that" - which is the standard herald for folklore and (which I say as a reviewer:) is reason to return the paper for modification. However, the author did actually defer more in-depth discussion of the topic to their references 16 and 7 (respectively, Giddings', Marolf's and Hartle's Observables in effective gravity and Brunetti's, Fredenhagen's and Rejzner's Quantum gravity from the point of view of locally covariant quantum field theory).

The latter authors, particularly Brunetti and Fredenhagen, are well-known for having jumped on the causal perturbation theory bandwagon and taking it a step further with their microlocal analysis framework, and then attacking the problem of quantum gravity with this in a way that by-passes the issue of non-renormalizability. This looks like it's the latest in a series, but I haven't been paying attention on their progress since early this century. I might have to look at this more closely.

But, anyway, the key issue is: how do you actually define local observables? Khavkine is directly addressing the issue of physically relevant definitions, providing a better formulation that gets the job done. To quote the concluding part of their abstract:

Moreover, generalized local gauge invariant observables are sufficient to separate diffeomorphism orbits on this admissible subset of the phase space. Connecting the construction with the notion of differential invariants, gives a general scheme for defining generalized local gauge invariant observables in arbitrary gauge theories, which happens to agree with well-known results for Maxwell and Yang-Mills theories.

The predecessors to their idea, as they point out, go back to the early 1960's: Bergman and Komar. Section 2 of Khavkine lays out the standard definition that leads to the no-go result. Section 3 provides the modification that overcomes it. Briefly scanning of section 2, I see that everything is being framed in terms of $C^\ast$ algebras, which transcends the distinction between classical and quantum, so it looks like it's something that cuts across the board and is not specifically quantum.

The paper looks like it's an interesting read, because Khavkine is more closely in line with the standard treatment of field operators, i.e. one where points $x$ are smeared out as compactly-supported functions $f(x)$, with $φ(x)$ replaced by $φ(f) = \int φ(x) f(x) dx$. They're called "test functions" (a standard canard in theories of generalized functions, like Schwartz or Columbeau) but are better thought of as "smeared points". The discussion in section 1 will also help make Brunetti and Fredenhagen's work more accessible, since they use the kind of math in their work.

I'll probably have more to say on this later, either in comments or a later edit, after closely reviewing the paper and its references; since the whole issue of diffeomorphism symmetry and gravitation (particularly in the context of quantum theory) is something I actively sought out some information on, rather than something that came to me. It's a Pull Issue, with me, not a Push Issue.

In the context of quantum theory, one thing you should pay attention to is that any proposal for somehow "quantizing" geometry or even discretizing it, automatically breaks diffeomorphism symmetry, with loop quantum gravity being a borderline exception. In addition, all of the regularization approaches that I am aware of ... (discrete) causal sets, led by someone who I knew, who used to be at my alma mater; Regge calculus, which might be considered a predecessor of causal sets; Lattice Field Theory, which is used to regularize quantum field theory ... all break diffeomorphism symmetry. An example of something that might not (quite) break diffeomorphism symmetry would be a method of "third quantization" that quantizes the Second Noether Theorem (English version), instead of the geometry. That gets closer to the direction Brunetti and Fredenhagen were moving in, the last time I checked on their progress.

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