  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.
