Note on Gauge Invariance and Causal Propagation

TL;DR

This paper demonstrates that gauge invariance alone does not guarantee causal propagation in gauge theories, using a massless spin-3/2 field coupled to electromagnetism as an example, highlighting the importance of causality as a separate consistency criterion.

Contribution

The paper provides an explicit example showing gauge invariance may not ensure causality, emphasizing the need for causality analysis in gauge field interactions.

Findings

Gauge invariance does not guarantee causal propagation.

Certain gauge-invariant couplings allow superluminal motion.

Causality constraints can exclude otherwise permissible interactions.

Abstract

Interactions of gauge-invariant systems are severely constrained by several consistency requirements. One is the preservation of the number of gauge symmetries, another is causal propagation. For lower-spin fields, the emphasis is usually put on gauge invariance that happens to be very selective by itself. We demonstrate with an explicit example, however, that gauge invariance, albeit indispensable for constructing interactions, may not suffice as a consistency condition. The chosen example that exhibits this feature is the theory of a massless spin-3/2 field coupled to electromagnetism. We show that this system admits an electromagnetic background in which the spin-3/2 gauge field may move faster than light. Requiring causal propagation rules out otherwise allowed gauge-invariant couplings. This emphasizes the importance of causality analysis as an independent test for a system of interacting gauge fields. We comment on the implications of allowing new degrees of freedom and non-locality in a theory, on higher-derivative gravity and Vasiliev's higher-spin theories.