Spectral actions for q-particles and their asymptotics

TL;DR

This paper analyzes spectral actions for general free q-particles in thermal equilibrium, deriving asymptotic expansions and examining phenomena like condensation, with distinctions between massless, massive, and relativistic cases.

Contribution

It provides the first detailed asymptotic analysis of spectral actions for q-particles, including the effects of different mass regimes and the transition to infinite systems.

Findings

Asymptotic expansions differ for massless and massive particles.

Condensation phenomena are identified for Bose-like q-particles.

Comparison between finite and infinite volume cases reveals spectral effects.

Abstract

For spectral actions consisting of the average number of particles and arising from open systems made of general free $q$-particles (including Bose, Fermi and classical ones corresponding to $q=\pm 1$ and $0$, respectively) in thermal equilibrium, we compute the asymptotic expansion with respect to the natural cut-off. We treat both relevant situations relative to massless and non relativistic massive particles, where the natural cut-off is $1/\beta=k_{\rm B}T$ and $1/\sqrt{\beta}$, respectively. We show that the massless situation enjoys less regularity properties than the massive one. We also treat in some detail the relativistic massive case for which the natural cut-off is again $1/\beta$. We then consider the passage to the continuum describing infinitely extended open systems in thermal equilibrium, by also discussing the appearance of condensation phenomena occurring for Bose-like $q$-particles, $q\in(0,1]$. We then compare the arising results for the finite volume situation (discrete spectrum) with the corresponding infinite volume one (continuous spectrum).