Interactions of Particles with "Continuous Spin" Fields

Powerful general arguments allow only a few families of long-range interactions, exemplified by gauge field theories of electromagnetism and gravity. However, all of these arguments presuppose that massless fields have zero spin scale (Casimir invariant) and hence exactly boost invariant helicity. This misses the most general behavior compatible with Lorentz symmetry. We present a Lagrangian formalism describing interactions of matter particles with bosonic "continuous spin" fields with arbitrary spin scale $\rho$. Remarkably, physical observables are well approximated by familiar theories at frequencies larger than $\rho$, with calculable deviations at low frequencies and long distances. For example, we predict specific $\rho$-dependent modifications to the Lorentz force law and the Larmor formula, which lay the foundation for experimental tests of the photon's spin scale. We also reproduce known soft radiation emission amplitudes for nonzero $\rho$. The particles' effective matter currents are not fully localized to their worldlines when $\rho \neq 0$, which motivates investigation of manifestly local completions of our theory. Our results also motivate the development of continuous spin analogues of gravity and non-Abelian gauge theories. Given the correspondence with familiar gauge theory in the small $\rho$ limit, we conjecture that continuous spin particles may in fact mediate known long-range forces, with testable consequences.