Spectral actions for q-particles and their asymptotics

Abstract

For spectral actions made 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 $q=\pm1$ 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 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 consider the passage to the continuum describing infinitely extended open systems in thermal equilibrium. We briefly discuss the appearance of condensation phenomena occurring for Bose-like $q$-particles, for which $q\in(0,1]$, after passing to the continuum. We also compare the arising results for the finite volume situation (discrete spectrum) with the corresponding infinite volume one (continuous spectrum).

Gas of Particles Obeying the Monotone Statistics

“…We point out that such a process of expansion seems to be a quite natural one in investigating and solving long-standing problems of quantum physics, see [23]. The reader is also referred to [12] for a situation more close to the topics of the present paper.…”

“…for all models in (12). Therefore, all such models obeying the various monotone statistics seem to be suitable for potential physical applications.…”

“…Recently, the attempt to systematically investigate the thermodynamic properties of such exotic particles has been carried out in the series of papers [10][11][12][13][14]. The first study was that concerning the q-particles or quons, q ∈ [−1, 1].…”

On the Thermodynamics of Particles Obeying Monotone Statistics

“…The second natural question to be addressed is the investigation of the free gas formed of monotone particles. This is connected with the degeneracy and can be addressed either by considering the free particles confined in a box and then "passing to the continuum", as in Sections 4 and 5 of [19], or by removing the harmonic potential (i.e., performing an appropriate limit as the Hook constant changes to 0). For d = 1, the second approach seems to be directly applicable because, as pointed out above, the energy levels are non-degenerate, whereas the first approach already involves a non-trivial degeneracy in the case d = 1.…”

“…We conclude by noting that Proposition 5 allows one to compute the asymptotics of Connes' spectral action (cf. [1]), associated with the average number of monotone particles, as described in [19] for q-particles, but with the condition described in (16). We postpone the investigation of this aspect for a forthcoming analysis.…”