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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>ρ</a:mi></a:math> with units of momentum. Normally, we assume <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>ρ</c:mi><c:mo>=</c:mo><c:mn>0</c:mn></c:math>. The interactions of CSPs are known to satisfy certain simple properties, one of which is that the <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>ρ</e:mi><e:mo stretchy="false">→</e:mo><e:mn>0</e:mn></e:math> limit 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 nonzero <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mi>ρ</h:mi></h:math>. 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 implies well-behaved thermodynamics. When the CSP photon’s interactions with charged matter are turned on, the primary <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mi>h</j:mi><j:mo>=</j:mo><j:mo>±</j:mo><j:mn>1</j:mn></j:math> helicity modes thermalize quickly, while the other modes require increasingly long time scales to thermalize, set by powers of <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mi>T</l:mi><l:mo>/</l:mo><l:mi>ρ</l:mi></l:math>. In familiar thermal systems, the CSP photon behaves like the QED photon with small <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mi>ρ</n:mi></n:math>- and time-dependent corrections to its effective relativistic degrees of freedom. Sizable departures from familiar thermal behavior arise at energy scales comparable to <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mi>ρ</p:mi></p:math> and could have testable experimental consequences.