Extended observables in theories with constraints

TL;DR

This paper develops formulas for extended observables in constrained Hamiltonian systems, enabling covariant quantization by representing constrained observables in an unconstrained phase space.

Contribution

It introduces general formulas for extended observables in theories with second and first class constraints, facilitating covariant quantization.

Findings

Formulas for extended observables in second class constrained systems.

Construction of gauge-dependent and gauge-invariant extended observables.

Proposal of a covariant quantization mapping procedure.

Abstract

In a classical Hamiltonian theory with second class constraints the phase space functions on the constraint surface are observables. We give general formulas for extended observables, which are expressions representing the observables in the enveloping unconstrained phase space. These expressions satisfy in the unconstrained phase space a Poisson algebra of the same form as the Dirac bracket algebra of the observables on the constraint surface. The general formulas involve new differential operators that differentiate the Dirac bracket. Similar extended observables are also constructed for theories with first class constraints which, however, are gauge dependent. For such theories one may also construct gauge invariant extensions with similar properties. Whenever extended observables exist the theory is expected to allow for a covariant quantization. A mapping procedure is proposed for covariant quantization of theories with second class constraints.