Density Matrix of the Fermionic Harmonic Oscillator

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: ${\rho }_F(\beta )=c^{\ast }(\beta )c(\beta )\pm c^{\ast }(\beta )c(\beta )e^{-\beta \omega }$, 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.