Thermal Operator Representation of Finite-Temperature Amplitudes in the Presence of Chemical Potential

Abstract

In a recent paper [Phys. Rev. D {\bf 72}, 085006 (2005)], Brandt {\em et al}. deduced the thermal operator representation for a thermal $N$-point amplitude, both in the imaginary-time and real-time formalisms. In the case when a chemical potential present, however, the representation is not as simple as in the case with vanishing chemical potential. We propose a much simpler and transparent representation for the case of non-zero chemical potential.