Spectral Functions at finite temperature and chemical potential

There are two formulations at non-zero chemical potential; one is the formulation that a Lagrangian includes a chemical potential, the other is the formulation that a Lagrangian does not include a chemical potential. The existence of two formulations makes a calculation complicated. The results from those formulations are not corresponding directly. This discrepancy exists in the imaginary time formalism and the real time formalism. However, since this is essentially caused by a difference of a frequency, the discrepancy vanishes by modifying the Fourier transform. We show a calculational procedure with a spectral function to understand this.