Reduced phase space: quotienting procedure for gauge theories

TL;DR

This paper introduces a reduction method for gauge theories by quotienting out the kernel of the presymplectic form, showing its equivalence to the Dirac method under certain conditions, and discussing its advantages and limitations.

Contribution

It develops a quotienting reduction procedure for gauge theories, clarifies its relation to the Dirac method, and analyzes cases with ineffective constraints.

Findings

The reduction procedure is equivalent to the extended Dirac theory when the Dirac conjecture holds.

The method can handle phase spaces with an odd number of physical degrees of freedom.

Differences from the standard Dirac method occur with ineffective constraints, favoring the standard approach in such cases.

Abstract

We present a reduction procedure for gauge theories based on quotienting out the kernel of the presymplectic form in configuration-velocity space. Local expressions for a basis of this kernel are obtained using phase space procedures; the obstructions to the formulation of the dynamics in the reduced phase space are identified and circumvented. We show that this reduction procedure is equivalent to the standard Dirac method as long as the Dirac conjecture holds: that the Dirac Hamiltonian, containing the primary first class constraints, with their Lagrange multipliers, can be enlarged to an extended Dirac Hamiltonian which includes all first class constraints without any change of the dynamics. The quotienting procedure is always equivalent to the extended Dirac theory, even when it differs from the standard Dirac theory. The differences occur when there are ineffective constraints, and in these situations we conclude that the standard Dirac method is preferable --- at least for classical theories. An example is given to illustrate these features, as well as the possibility of having phase space formulations with an odd number of physical degrees of freedom.