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Is it inherently problematic to quantize classical theories with local symmetries? For example, consider the action of EM but now interpret $A_\mu$ as physical. At a classical level, there is nothing prevent me from doing that (although it might give one an uncomfortable feeling). Is there anything preventing me from making this into a quantum theory? Indeed, it seems I can just write down the path integral of this, which will in fact coincide with the gauge-invariant path integral of standard EM --- up to a global (irrelevant) constant which measures the volume of the gauge orbits (which is just the constant volume of the gauge group).

Note: I am aware that the usual procedure is problematic since the propagator is not well-defined. That is not an intrinsic problem, though, and is rather a short-coming of the perturbative point of view. Indeed, lattice gauge theory can quantize theories without fixing a gauge, circumventing the aforementioned propagator-issue.

In fact, if I simply take a $U(1)$ lattice gauge theory and ignore the constraint, it seems I have a perfectly well-defined quantum theory with a local symmetry? At the same time, it seems surprising from a conceptual point of view: symmetries which are local in time are already very weird at a classical level since it implies an underdeterminedness at the level of $A_\mu$ (even fixing initial and final values, one can smoothly deform $A_\mu$ at intermediate times). This would suggest that a Hamiltonian formulation (where $A_\mu$ is taken to be physical) would be problematic. If I take a lattice gauge theory and ignore any gauge constraints, is the theory underdetermined? (A counter-example seems to be the toric code $H = - K \sum A_v - \sum B_p$: for $K\to \infty$ this is a gauge theory with constraint $A_v = 1$, however for $K$ finite this can be interpreted as a physical model with a local symmetry, yet it is not undetermined. A possible way out: explicitly adding $A_v$ to the Hamiltonian might be equivalent ---in a Lagrangian picture--- to destroying the symmetry that was local in time (clearly keeping the symmetry which is local in space)?)

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  • $\begingroup$ Note that in general, gauge $\equiv $ local. "Local (non-gauge!) symmetry" is an oxymoron. Note also that the gauge d.o.f. always have negative norm, so they break unitarity, which is obviously no good. $\endgroup$ – AccidentalFourierTransform Jan 3 '18 at 18:26
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    $\begingroup$ Not all local symmetries are gauge symmetries. For example, 2d conformal symmetries are local, but not gauge. $\endgroup$ – user1504 Jan 3 '18 at 18:31
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    $\begingroup$ @AccidentalFourierTransform - In many cases, there are local symmetries which are not gauge. Asymptotic symmetries in AdS$_3$ are an example. More recently, there's been discussion of the BMS group which is a non-gauge diffeomorphism symmetry of general relativity in flat spacetime. $\endgroup$ – Prahar Jan 3 '18 at 18:34
  • $\begingroup$ For most people, "gauge" and "local" are synonyms. What is the definition of "gauge" other than "symmetry parameter depends on $x$"? But this is hardly relevant to the question in the OP anyway, so nevermind. $\endgroup$ – AccidentalFourierTransform Jan 3 '18 at 18:36
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    $\begingroup$ @AccidentalFourierTransform 'gauge' means 'redundancy in description'. This for example affects the measure one uses in the path integral. Note that local symmetries need not be gauge (consider CFTs, or more simply the quantum Ising chain $H= \sum_n \sigma^x_n \sigma^x_{n+1}$ which ---in this limit--- has local symmetries $P_n = \sigma^x_n$). Moreover, gauge symmetries need not be local (e.g. fermionic parity 'symmetry' is commonly viewed as a global gauge symmetry, since only states which well-defined parity are physical). $\endgroup$ – Ruben Verresen Jan 3 '18 at 19:36
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Why local transformations must be gauge transformations:

Traditionally, quantization is recipe in which the phase space of a classical system is replaced by a Hilbert space of a quantum system; and functions on the phase space representing the observables are replaced by operators on the Hilbert space. Also, the action of the classical observables on the phase space is replaced by a quantum action of their quantum counterparts weighted by a parameter $\hbar$ such that in the limit $\hbar \rightarrow 0$, the action coincides with the classical action (correspondence principle).

Even though in many applications it is not explicitly pronounced, a quantization procedure should start from a phase space. The basic meaning of a phase space is the space of all possible initial conditions (space of initial data). On the basic level, we deal with systems whose equations of motion satisfy the property of existence and uniqueness of solutions; thus each initial condition corresponds to a unique solution. Therefore we can think of the phase space as the space of all classical solutions. The latter definition of phase-space has advantages as it doesn't need to separate the time from the other coordinates and allows a covariant definition of the phase space. In the physical literature, it is known by the Crnković-Witten formalism.

When local symmetries exist, the property of uniqueness of solutions is lost and there are combinations of coordinates or fields in the Lagrangian which are not controlled by the equations of motion and can assume arbitrary values. The theory cannot say anything about them. On the other hand, the combinations which are controlled are exactly the gauge invariant combinations. This is one of the consequences of Noether's second theorem.

Remembering the basic definition of the phase space as the space of initial conditions; the best we can do is to work with the subspace of initial data which the theory can control; i.e. the space of gauge invariant observables. These observables generate the reduced phase space, i.e. a phase space in which the local symmetry is gauged away.

This space in general is not a manifold. It contains points of singularity, which make it hard to quantize even is simple quantum mechanical systems; please see Emmrich and Römer. This is why methods like BRST are used to impose the gauge symmetry after quantization.

On the Lattice:

On the lattice, only correlators of gauge invariant observables are computed. In this case the gauge redundancy is manifested by a multiplicative constant of the volume of the discretized gauge group in both the denominator and the numerator. We would be committing an error if we had computed correlators of gauge non-invariant quantities which are not controlled by the theory. Their correlators will not depend on any parameter of the theory (such as coupling constants) that we want to study and they would produce just a random result, very sensitive to the method that we chose to interpret them as quantum observables.

Also, it is much more convenient to work on unreduced phase space on the lattice. It would be extremely hard if we had worked on the reduced phase space which as explained above a very complicated space.

Global symmetry

In contrast to the case of gauge symmetry, the theory does not constrain how we treat global symmetries. As long as they are not anomalous, we have, in principle, the liberty to gauge away global symmetries or leave them as symmetries of the system (classical and quantum). In the first case we interpret configuration related by a symmetry operations as the same physical state, while in the second, we interpret them as distinct physical states related by symmetry. A special case of these symmetries is the large gauge symmetries which in many examples act as symmetries and not redundancies as they connect between physically distinct states. This subject was discussed many times here in Physics stack exchange; please see the following answer and the references therein.

Asymptotic symmetries

Asymptotic symmetries are "gauge" symmetries on noncompact spaces or spaces with a special closed surface leaving the boundary conditions invariant modulo those connected to the identity component. These symmetries generate, in certain cases infinite dimensional Lie groups, are also not included in Noether's second theorem which assumes compact support of the variation. These symmetries give rise to physical charges such as electric and magnetic charges. Hence, they should also be considered as global symmetries from the point of view of the quantization process.

Summary

Asymptotically trivial local symmetries must be considered as gauge transformations

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  • $\begingroup$ A mathematical caveat: To violate the uniqueness of time evolution, it must be possible to construct a symmetry which vanishes on the initial data. This is easy in Yang-Mills theory, but not guaranteed in general. $\endgroup$ – user1504 Jan 4 '18 at 15:52
  • $\begingroup$ @user1504 you mean a symmetry, i.e. a transformation commuting with the Hamiltonian, which in addition acts trivially on the manifold of initial data? $\endgroup$ – David Bar Moshe Jan 4 '18 at 16:09
  • $\begingroup$ Yes. Easy to make that happen with smooth functions. Impossible with holomorphic. $\endgroup$ – user1504 Jan 4 '18 at 17:10
  • $\begingroup$ @user1504 But if the symmetry is trivial on the initial data and commutes with the Hamiltonian, then it will be trivial at all times, then why is it not just represented by the identity map\ the identity operator? $\endgroup$ – David Bar Moshe Jan 4 '18 at 17:18

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