Underlying gauge symmetries of second-class constraints systems

TL;DR

This paper explores the gauge symmetries underlying second-class constrained systems, demonstrating their equivalence to gauge-invariant systems and quantizing specific mechanical and field theory examples.

Contribution

It reveals the existence of underlying gauge symmetries in second-class systems and provides methods for their quantization using gauge invariance and explicit constraint solutions.

Findings

Second-class systems have gauge-invariant counterparts within original phase spaces.

Quantization of a mechanical pendulum on a sphere and the O(n) sigma model is achieved via gauge symmetry.

Dirac's conditions are reformulated in terms of Wigner functions with broad physical equivalence.

Abstract

Gauge-invariant systems in unconstrained configuration and phase spaces, equivalent to second-class constraints systems upon a gauge-fixing, are discussed. A mathematical pendulum on an $n-1$-dimensional sphere $S^{n-1}$ as an example of a mechanical second-class constraints system and the O(n) non-linear sigma model as an example of a field theory under second-class constraints are discussed in details and quantized using the existence of underlying dilatation gauge symmetry and by solving the constraint equations explicitly. The underlying gauge symmetries involve, in general, velocity dependent gauge transformations and new auxiliary variables in extended configuration space. Systems under second-class holonomic constraints have gauge-invariant counterparts within original configuration and phase spaces. The Dirac's supplementary conditions for wave functions of first-class constraints systems are formulated in terms of the Wigner functions which admit, as we show, a broad set of physically equivalent supplementary conditions. Their concrete form depends on the manner the Wigner functions are extrapolated from the constraint submanifolds into the whole phase space.