Different forms of the Kadanoff–Baym equations in quantum statistical mechanics

Spectral Functions and Properties of Nuclear Matter

“…For the study of nonequilibrium phenomena, the method can be used with constraints to define an initial condition. It should be stated that, in the case of nonequilibrium phenomena, different forms of the Kadanoff-Baym equations can be preferable for different physical systems [18,19].…”

“…for all values of momentum p. This result follows directly from the commutator relations for field operators and can serve as a keystone for checking all approximations to the spectral function (1). It was shown in [1,2] that, in the case of slowly varying in space and time disturbances, after the transition to the Wigner coordinates and Fourier transform with respect to the difference of the initial time and space variables, the spectral function obeys (9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27) in [1,2] ((10) in [10]) and can be represented by the same equation (1), and only all the quantities entering (1) become also the functions of space and time variables 𝑅 and 𝑇. This property is one more keystone for checking the validity of different approximations to the spectral function.…”

“…Although the uniform convergence of the Taylor series resulting from the expansion of the exponent in ( 18) is absent, the term-by-term integration, as we will see, leads to the reasonable result. The terms of the expansion containing the odd powers of Γ in correspondence with (19) generate a geometric sequence with the common ratio 𝑟:…”