Generalized quantum mechanics of nonabelian gauge theories

TL;DR

This paper applies Hartle's generalized quantum mechanics to nonabelian gauge theories, analyzing constraints and decoherence, and providing a Lorentz-invariant formulation with new insights into the behavior of gauge constraints.

Contribution

It extends the sum-over-histories formalism to nonabelian gauge theories, examining constraint behavior and decoherence, and offers a Lorentz-invariant path integral approach.

Findings

Vanishing of the momentum space constraint confirmed.

Coarse grainings by the constraint predict its vanishing or lack of decoherence.

Certain coarse grainings yield definite predictions that challenge the assumption of constraint vanishing.

Abstract

Hartle's generalized quantum mechanics in the sum-over-histories formalism is used to describe a nonabelian gauge theory. Predictions are made for certain alternatives, with particular attention given to coarse-grainings involving the constraint. In this way, the theory is compared to other quantum-mechanical descriptions of gauge theories in which the constraints are imposed by hand. The vanishing of the momentum space constraint is seen to hold, both through a simple formal argument and via a more careful description of the Lorentzian path integral as defined on a spacetime lattice. (Incidentally, the treatment of the time slicing in the path integral may be of general technical interest.) The configuration space realization of the constraint is shown to behave in a more complicated fashion. For some coarse grainings, we recover the known result from an abelian theory, that coarse grainings by values of the constraint either predict its vanishing or fail to decohere. However, sets of alternatives defined in terms of a more complicated quantity in the abelian case are exhibited where definite predictions can be made which disagree with the assumption that the constraints vanish. Finally, the configuration space sum-over-histories theory is exhibited in a manifestly Lorentz-invariant formulation.