Schwinger-Keldysh

Since the non-tilde operators describe physical variables, G(k (3)11 is the physical propagator to be used to treat the properties of the thermal bosonic system. It is interesting to observe that, except for the non-diagonal elements, this TFD-propagator is equal to the one introduced in the Schwinger-Keldysh approach, which is claimed to be (in this equivalence with TFD) a thermal theory describing linear-response processes only (H. Chu et.al., 1994). 

Schwinger-Keldysh time contour and contour functions... 

The problem can be generally resolved by using the EOM on the Schwinger-Keldysh time contour. Contour-ordered Green function is defined as... 

To describe nonequilibrium phase transitions, there have been developed many methods such as the closed-time path integral by Schwinger and Keldysh (J. Schwinger et.al., 1961), the Hartree-Fock or mean field method (A. Ringwald, 1987), and the l/lV-expansion method (F. Cooper et.al., 1997 2000). In this talk, we shall employ the so-called Liouville-von Neumann (LvN) method to describe nonequilibrium phase transitions (S.P. Kim et.al., 2000 2002 2001 S.P. Kim et.al., 2003). The LvN method is a canonical method that first finds invariant operators for the quantum LvN equation and then solves exactly the... 

The effect of thermal pion fluctuations on the specific heat and the neutrino emissivity of neutron stars was discussed in [27, 28] together with other in-medium effects, see also reviews [29, 30], Neutron pair breaking and formation (PBF) neutrino process on the neutral current was studied in [31, 32] for the hadron matter. Also ref. [32] added the proton PBF process in the hadron matter and correlation processes, and ref. [33] included quark PBF processes in quark matter. PBF processes were studied by two different methods with the help of Bogolubov transformation for the fermion wave function [31, 33] and within Schwinger-Kadanoff-Baym-Keldysh formalism for nonequilibrium normal and anomalous fermion Green functions [32, 28, 29],... 

The external potential is included in these equations. The self-energy is defined as a function or operator in the Keldysh space according to Eq. (20). obeys the Kubo-Martin-Schwinger boundary conditions (16). The ( )-projections of the Kadanoff-Baym equations can written as... 

The operators P and obey the usual equal time anticommutation relations. The time-dependence of the field operators appearing here is due to the Heisenberg representation in the L-space. In view of the foregoing development which parallels the traditional Schrodinger quantum theory we may recast the above Green function in terms of the interaction representation in L-space. This leads to the appearance of the S-matrix defined only for real times. We will now indicate the connection of the above to the closed-time path formulation of Schwinger [27] and Keldysh [28] in H-space. Equation (82) can be explicitly... 

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