What about GR? Is the situation analogous? Can we find a combination of constraints that behave nicely?

Does generate a normal coordinate transformation? no

What about . no

So not a gauge transformation. Uncoordinated effort does not generate gauge transformation.

Many examples follow of first-class constraints that do not generate gauge transformations.

Bergmann and Anderson 1951 say sometimes one needs to take a Poisson bracket of a velocity term. Their motivation is unclear. For Pitts it is necessary to make all of this stuff go through.

Get a bad physical change with a secondary first-class constraint.

Can we find a team effort? Yes by combining the primary and secondary first-class constraints.

Now possibility of change re-emerges in GR. Not exactly clear from this, but at least door open.

What goes wrong here? Dirac makes an error in his book – forgetting about the hidden secondary constraint living inside transformed Hamiltonian. Then this error widely propagated in the literature.

Can find the root of much trouble on page 21 of Dirac’s book.

Can do some pretty easy tests in classical electrodynamics (em) and just find out. Can show by direct calculation that first class constraint does not, generally, generate a gauge transformation. 

And how has this not been noticed?!?

Confusion seems to arise out of failure of associating between Hamiltonian and Lagrangian formulations.

What is link between first-class constraints and gauge freedom? Team effort or individual? two answers in literature: Bergmann and Anderson argue for former. And that’s correct. G comes from constraints acting in concert. Latter is wrong says Pitts. Can see this by looking at the actual Poisson brackets to find out whether commuting with Hamiltonian entails making no physical change – i.e. generates a gauge transformation.

A First Class Constraint Generates Not a Gauge Transformation, But a (Bad) Physical Change: The Cases of Maxwell and GR

A project in recovery of Anderson and Bergmann that were clear in the 50s but lost since. Question whether first-class constraints can do their work in isolation, or rather they must work as a team. Idea will be that, for example, change is not lost in GR when we understand better the connection between gauge transformations and first class constraints.

Question from Oriti: asymptotic safety people don’t suggest that safety leads to fundamentality. Huggett says he and Weingard argued same thing before. Consensus seems to be that safety does not lead to fundamentality. Butterfield objects that QCD fundamental from perspective that excludes gravity. Fundamental is to be taken wrt to the perspective of the analysis.

Question about why relativity of locality: Baker thinks bad for a theory. Advocates see observations of momentum stuff as more fundamental. Butterfield thinks claims applies to classical stuff as well. Darboux’s theorem seems relevant here. Bain says worth thinking about.

Oriti offers motivations: worries about lorentz invariance from fundamental length might lead to attempt to deform the invariance – and that can be done with momentum space curvature; also in simple qg models, can couple point particles to gravity – end up with curved momentum space. These realize relative locality. Bain adds that reason you think you observe locality properties is because the momentum is very low. So only seem like position measurements.

Huggett returns to asymptotic safety concerns about why should have same fixed points in GR and QCD. Latter is gaussian, but GR’s should not be. Bain’s concerns orthogonal to this question. 

Questions have been raised about appropriateness both of asymptotic safety and relativity of locality.

Motivating phase space realism:

Descriptions of systems in terms of energy and momentum more fundamental that spatiotemporal properties. (Amelino-Camelia especially.)

Analogy between relativity of simultaneity and relativity of locality. Latter explained by invariant phase space interval with observer-dependent slicing of phase space.

2 concerns with latter motivation: doesn’t tell us phase space absolute features more fundamental than decomposed bits; global vs local quantity distinction rather than absolute vs relative quantities.

At most uphold momentum space realism rather than phase space realism.

Oriti asks whether there is a relation to actual non-commutativity of geometry. Bain says not that he’s aware.