Manifestly Covariant Lagrangians, Classical Particles with Spin, and the Origins of Gauge Invariance

TL;DR

This paper develops a covariant Lagrangian framework for classical particles with spin, revealing the origins of gauge invariance and connecting classical particle classifications to relativistic quantum theory.

Contribution

It introduces a manifestly covariant Lagrangian method that naturally explains gauge invariance and classical particle types, bridging classical and relativistic quantum descriptions.

Findings

Gauge invariance emerges naturally for massless particles with spin.

Classical particle classification mirrors Wigner's quantum classification.

Derived a classical analogue of the Higgs mechanism.

Abstract

In this paper, we review a general technique for converting the standard Lagrangian description of a classical system into a formulation that puts time on an equal footing with the system's degrees of freedom. We show how the resulting framework anticipates key features of special relativity, including the signature of the Minkowski metric tensor and the special role played by theories that are invariant under a generalized notion of Lorentz transformations. We then use this technique to revisit a classification of classical particle-types that mirrors Wigner's classification of quantum particle-types in terms of irreducible representations of the Poincar\'e group, including the cases of massive particles, massless particles, and tachyons. Along the way, we see gauge invariance naturally emerge in the context of classical massless particles with nonzero spin, as well as study the massless limit of a massive particle and derive a classical-particle version of the Higgs mechanism.