Revisiting the gauge principle: enforcing constants of motion as constraints

TL;DR

This paper proposes a reformulation of the gauge principle by focusing on promoting constants of motion to constraints, connecting rigid symmetries with gauge symmetries, and extends this approach from mechanics to field theory.

Contribution

It introduces an alternative gauge principle formulation emphasizing generators of symmetries as constraints, applicable to both mechanical systems and field theories.

Findings

Reformulation of gauge principle via constants of motion

Application to relativistic particle, string, and field theory

Extension using De Donder--Weyl formalism

Abstract

In this paper we examine an alternative formulation of the gauge principle in which the emphasis is shifted from the symmetry transformations to their generators. We show that the gauge principle can be entirely reformulated in terms of promoting constants of motion - which generate rigid symmetries - to constraints - which generate gauge symmetries. In our exposition we first explain the basic philosophy on mechanical systems, and then with the help of De Donder--Weyl formalism we extend our scenario also to a field-theoretical setting. To put some flesh on bare bones we demonstrate our method in numerous examples, including the massive relativistic particle, the Nambu--Goto string and relativistic field theory.