Thermodynamics of Continuous Spin photons

Special relativity allows massless particles to have states of different integer (or half-integer) helicities that mix under boosts, much like the spin-states of a massive particle. Such massless particles are known as continuous spin particles (CSPs), a term coined by Wigner, and they are notable for their infinite tower of spin polarizations. The mixing under boosts is controlled by a spin-scale $\rho$ with units of momentum. Normally, we assume $\rho=0$. The interactions of CSPs are known to satisfy certain simple properties, one of which is that the $\rho \rightarrow 0$ limit generically recovers familiar interactions of massless scalars, photons, or gravitons, with all other polarizations decoupling in this limit. Thus, one can ask if the photon of the Standard Model is a CSP at small but non-zero $\rho$. One concern about this possibility -- originally raised by Wigner -- is that the infinite tower of polarizations could pose problems for thermodynamics. To address this question, we study the thermal evolution of a CSP photon gas coupled to isothermal matter, across CSP helicity modes and phase space. We find that the structure of the interactions dictated by Lorentz symmetry imply well behaved thermodynamics. When the CSP photon's interactions to charged matter are turned on, the primary $h=\pm 1$ helicity modes thermalize quickly, while the other modes require increasingly long time-scales to thermalize, set by powers of $T/\rho$. In familiar thermal systems, the CSP photon behaves like the QED photon with small $\rho$- and time- dependent corrections to its effective relativistic degrees of freedom. Sizable departures from familiar thermal behavior arise at energy scales comparable to $\rho$ and could have testable experimental consequences.