Second quantization and the spectral action

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