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

Click here to download the paper.

Abstract: In Dirac-Bergmann constrained dynamics, a first-class constraint typically does not alone generate a gauge transformation. By direct calculation it is found that each first-class constraint in Maxwell’s theory generates a change in the electric field E by an arbitrary gradient, spoiling Gauss’s law. The secondary first-class constraint {p^{i}}_{,i} = 0 still holds, but being a function of derivatives of momenta, it is not directly about E (a function of derivatives of A_{\mu}). Only a special combination of the two first-class constraints, the Anderson-Bergmann (1951)-Castellani gauge generator G, leaves E unchanged. This problem is avoided if one uses a first-class constraint as the generator of a canonical transformation; but that partly strips the canonical coordinates of physical meaning as electromagnetic potentials and makes the electric field depend on the smearing function, bad behavior illustrating the wisdom of the Anderson-Bergmann (1951) Lagrangian orientation of interesting canonical transformations.

The need to keep gauge-invariant the relation \dot{q} - \frac{\delta H}{\delta p} = -E_{i} - p^{i} = 0 supports using the total Hamiltonian rather than the extended Hamiltonian. The results extend the Lagrangian-oriented reforms of Castellani, Sugano, Pons, Salisbury, Shepley, etc. by showing the inequivalence of the extended Hamiltonian to the total Hamiltonian (and hence the Lagrangian) even for observables, properly construed in the sense implying empirical equivalence.

Dirac and others have noticed the arbitrary velocities multiplying the primary constraints outside the canonical Hamiltonian while apparently overlooking the corresponding arbitrary coordinates multiplying the secondary constraints inside the canonical Hamiltonian, and so wrongly ascribed the gauge quality to the primaries alone, not the primary-secondary team G. Hence the Dirac conjecture about secondary first-class constraints rests upon a false presupposition. The usual concept of Dirac observables should also be modified to employ the gauge generator G, not the first-class constraints separately, so that the Hamiltonian observables become equivalent to the Lagrangian ones such as the electromagnetic field F_{\mu\nu}.

An appendix discusses analogous calculations for GR and sketches their conceptual consequences.

2013-09-29 10:25:50: 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.

2013-09-29 10:30:48: 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.

2013-09-29 10:33:13: 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.

2013-09-29 10:34:55


2013-09-29 10:37:49: 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.

2013-09-29 10:48:42: What about GR? Is the situation analogous? Can we find a combination of constraints that behave nicely?

Does H_0 generate a normal coordinate transformation? no

What about H_i. 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.

2013-09-29 10:53:49: Result: Change is having no time-like Killing field, and we can get that in homogeneous GR.

2013-09-29 10:55:36: So Pitts gets Earman’s fondness for Hamiltonian formulation and Maudlin’s desire for change to live happily together. Should have been obvious, he says, because Lagrangian and Hamiltonian formulations should be equivalent.

2013-09-29 10:59:07: Pitts tells us that perhaps there is no coherent notion of Bergmann observables. On the other hand we can tell what Dirac observables are supposed to be – so his conception can be fixed.

2013-09-29 11:14:49: Question about connection with observables and diffeomorphisms. Replies never used diff talk. Rejects invariance in favor of covariance. See Trautmann: need actual points, and never move them alone, only individuated by fields.

Is main idea that we introduce points? No. Doesn’t introduce them primitively. Thinks active diffs don’t make much sense: like building a building, seeing you don’t need it, and knocking it down.

Accused of adding points and having more information. Responds that you only get them after the fact. But what if have two manifolds? Why? Question continues: moving points around gives same physical situation. Pitts: we all agree that points exist in GR, just not individuated until have the metric.

This was a confusing QA session. Not sure what the actual worry of the questions was. Butterfield tries to redirect from point type worries to the actual issue of whether we need to coordinate the constraints in order to generate gauge transformations.

I could not really follow Dittrich’s point. If anyone is reading this, perhaps that can be filled in here.

Argument at length about the connection between phase space and observables generally. Chair cut it off.


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