Factorization of the bosonic partition function

The factorization formula in the non-interacting quantum field theories that relates the fermionic partition function to the bosonic partition function considered recently by Chair (2013) [3] is obtained for the harmonic oscillator using the path integral formulation. By using the latter, the fermionic partition function turns out to be the ratio of two determinants of the same operator (∂τ+ω), whose eigenmodes being both periodic on the imaginary time intervals [0,2β], [0,β]. The natural generalization of the factorization formula when β→2mβ is derived, such a factorization implies that the bosonic oscillator at temperature β can be seen as a non-interacting mixture of a bosonic oscillator at temperature 2mβ and m-fermionic oscillators at different temperatures 2m−kβ, k=1,2,…,m. As a consequence, a general relationship between the bosonic and fermionic thermal zeta functions is deduced.