Thermal operator representation of finite temperature graphs

Abstract

Using the mixed space representation t;p in the context of scalar field theories, we prove in a simple manner that the Feynman graphs at finite temperature are related to the corresponding zero temperature diagrams through a simple thermal operator, both in the imaginary time as well as in the real time formalisms. This result is generalized to the case when there is a nontrivial chemical potential present. Several interesting properties of the thermal operator are also discussed.

Forward scattering amplitudes and the thermal operator representation

“…(Unlike in our earlier papers [1,2,7], here we will follow the simple convention of representing quantities at zero temperature without any superscript (T = 0). We will reserve the superscript (T ) only for quantities at nonzero temperature in order to simplify the notation.…”

“…Let us recall that in the closed time path formalism, the thermal propagator for a scalar field can be related to the zero temperature one through the thermal operator as [1] …”

“…We know that any Feynman graph at finite temperature is related to the corresponding zero temperature graph through a thermal operator [1,2] that can be built out of the basic thermal operator in (32). For example, the integrand of a graph (after the internal time integrations are done in the mixed space or the energy integrations are done in the energy-momentum space) with N scalar propagators carrying energy E i , i = 1, 2, · · · , N at finite temperature is related to the integrand of the corresponding graph at zero temperature by the thermal operator…”

“…As we have already argued, the thermal operator representation [1,2,7] provides a powerful method for obtaining results at finite temperature starting from zero temperature and in this paper we show how the forward scattering amplitudes for retarded thermal n-point functions can be derived through the use of the thermal operator representation. The thermal operator representation is clearly meaningful in studying this question if there exists a forward scattering description at zero temperature.…”

“…In a series of papers [1,2,3,4], the idea of a thermal operator representation [5,6] has been developed extensively in both the imaginary time formalism as well as the real time formalism of closed time path. In simple terms, the thermal operator representation relates a Feynman graph at finite temperature (with or without a chemical potential) to the corresponding graph at zero temperature.…”