Second Quantization and the Spectral Action

TL;DR

This paper explores how spectral actions in second quantization of spectral triples relate to entropy and energy, revealing explicit spectral coefficient formulas involving Bessel functions and zeta functions, and connecting to prior theoretical results.

Contribution

It introduces a spectral action framework for bosonic and fermionic second quantization with chemical potential, deriving explicit spectral coefficients and linking to known mathematical functions.

Findings

Spectral action coefficients are expressed via modified Bessel functions.

Fermionic spectral coefficients for von Neumann entropy relate to Riemann zeta functions as chemical potential approaches zero.

The work recovers and extends results by Chamseddine, Connes, and van Suijlekom.

Abstract

We consider both the bosonic and fermionic second quantization of spectral triples in the presence of a chemical potential. We show that the von Neumann entropy and the average energy of the Gibbs state defined by the bosonic and fermionic grand partition function can be expressed as spectral actions. It turns out that all spectral action coefficients can be given in terms of the modified Bessel functions. In the fermionic case, we show that the spectral coefficients for the von Neumann entropy, in the limit when the chemical potential $\mu$ approaches $0,$ can be expressed in terms of the Riemann zeta function. This recovers a result of Chamseddine-Connes-van Suijlekom.