Spin supplementary condition in quantum field theory: covariant SSC and physical state projection

TL;DR

This paper links classical spin supplementary conditions to quantum projection constraints, demonstrating how they remove unphysical polarizations in massive spinning particles across different quantum field theories.

Contribution

It establishes a correspondence between classical covariant spin conditions and quantum projection constraints, clarifying the physical spectrum of massive spinning particles.

Findings

Projection conditions remove unphysical polarizations.

Examples include Dirac, Proca, and worldline QFT.

Resolves discrepancies in effective field theories.

Abstract

The spin supplementary conditions are constraints on spin degrees of freedom in classical relativity which restricts physical degrees of freedom to rotations. It is argued that the equivalent constraints in quantum field theory are the projection conditions on polarisation tensors, which remove timelike/longitudinal polarisations from the physical spectrum. The claim is supported by three examples of massive spinning particles coupled to electromagnetism: Dirac and Proca fields in quantum field theory, and $\mathcal{N} = 1$ worldline QFT for classical worldline theory. This suggests a resolution to the apparent discrepancy between effective field theory description of massive higher-spin fields~\cite{Bern:2020buy,Bern:2022kto} and post-Newtonian effective field theory of spinning classical particles~\cite{Levi:2015msa}, where the former admits more unfixed parameters compared to the latter; the additional parameters are fixed by projection conditions and therefore are not tunable parameters.