Density Matrix of the Fermionic Harmonic Oscillator

TL;DR

This paper derives a density operator expression for the fermionic harmonic oscillator using path integral techniques and Grassmann variables, enabling calculation of its partition function in thermal equilibrium.

Contribution

It introduces a novel expression for the fermionic density operator in terms of Grassmann variables, applicable to periodic and antiperiodic orbits, advancing the theoretical understanding of fermionic systems.

Findings

Derived the fermionic density operator using path integrals

Obtained the fermionic partition function in thermal equilibrium

Presented the graded fermionic partition function for periodic orbits

Abstract

The path integral technique is used to derive a possible expression for the density operator of the fermionic harmonic oscillator. In terms of the Grassmann variables, the fermionic density operator can be written as: $\rho_F (\beta)=c^* (\beta)c(\beta) \pm c^*(\beta)c(\beta)e^{-\beta\omega}$, where +(-) means that the sum over all antiperiodic (periodic) orbits. Our density operator is then used to obtain the usual fermionic partition function which describes the fermionic oscillator in thermal equilibrium. Also, according to the periodic orbit $c(\beta)=c(0)$, the graded fermionic partition function is obtained.