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Why Gauges? Gauge symmetries for the classification of the physical states

Franco Strocchi

Abstract

This note focuses the problem of motivating the use of gauge symmetries (being the identity on the observables) from general principles, beyond their practical success, starting from global gauge symmetries and then by emphasizing the substantially different role of local gauge symmetries. In the latter case, a deterministic time evolution of the local field algebra, necessary for field quantization, requires a reduction of the full local gauge group $\cal{G}$ to a residual local subgroup $\cal{G}_r$ satisfying suitable conditions. A non-trivial residual local gauge group allows for a description/construction of the physical states by using the vacuum representation of a local field algebra, otherwise precluded if $\cal{G}$ is reduced to the identity. Moreover, in the non-abelian case the non-trivial topology of the a residual $\cal{G}_r$ defines the (gauge invariant) topological invariants which classify the vacuum structure with important physical effects; furthermore, it provides a general mechanism of spontaneous symmetry breaking without Goldstone bosons.

Why Gauges? Gauge symmetries for the classification of the physical states

Abstract

This note focuses the problem of motivating the use of gauge symmetries (being the identity on the observables) from general principles, beyond their practical success, starting from global gauge symmetries and then by emphasizing the substantially different role of local gauge symmetries. In the latter case, a deterministic time evolution of the local field algebra, necessary for field quantization, requires a reduction of the full local gauge group to a residual local subgroup satisfying suitable conditions. A non-trivial residual local gauge group allows for a description/construction of the physical states by using the vacuum representation of a local field algebra, otherwise precluded if is reduced to the identity. Moreover, in the non-abelian case the non-trivial topology of the a residual defines the (gauge invariant) topological invariants which classify the vacuum structure with important physical effects; furthermore, it provides a general mechanism of spontaneous symmetry breaking without Goldstone bosons.

Paper Structure

This paper contains 3 sections, 2 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. The operational and the mathematical description of a physical system
  3. Gauge symmetries and local field algebras for the description of physical states

Key Result

Proposition 3.1

The condition of deterministic time evolution of the field algebra requires that the gauge functions ${\cal U}$ which define the residual group ${\cal G}_r$, satisfy a sort of deterministic structure, namely if two gauge functions coincide at a given time $t_0$, together with their first order time

Theorems & Definitions (2)

  • Proposition 3.1
  • Proposition 3.2