Gauge invariance of parametrized systems and path integral quantization

TL;DR

This paper enhances gauge invariance in parametrized systems by adding surface terms to the action, applies the method to relativistic particles and toy universes, and explores their path integral quantization and relation to constraint surfaces.

Contribution

It introduces a systematic way to improve gauge invariance via surface terms and applies it to quantize relativistic particles and toy universes using path integrals.

Findings

Surface terms improve gauge invariance in parametrized systems.

Quantization of relativistic particles and toy universes is achieved through path integrals.

The relation between constraint surface geometry and quantum states is analyzed.

Abstract

Gauge invariance of systems whose Hamilton-Jacobi equation is separable is improved by adding surface terms to the action fuctional. The general form of these terms is given for some complete solutions of the Hamilton-Jacobi equation. The procedure is applied to the relativistic particle and toy universes, which are quantized by imposing canonical gauge conditions in the path integral; in the case of empty models, we first quantize the parametrized system called ``ideal clock'', and then we examine the possibility of obtaining the amplitude for the minisuperspaces by matching them with the ideal clock. The relation existing between the geometrical properties of the constraint surface and the variables identifying the quantum states in the path integral is discussed.