By S. Doniach

"Green's capabilities for reliable nation Physicists" has helped introduce numerous generations of condensed subject physics graduate scholars to the very strong rules of quantum many physique idea and a few in their purposes, quite these within the physics of itinerant magnetism and superconductivity. it truly is pedagogical in philosophy, and relies at the notion of top the reader to knowing subtle formalism during the examine of easy examples which exhibit the formal issues. within the reprint version, new chapters were additional to the unique textual content to supply an advent to the hot advancements within the physics of correlated electron platforms and, specifically, of high-T[subscript c] superconductivity.

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4) 35 The Phonon Temperature Green’s Function The brackets ( - . now denote the thermal average, defined as in Eq. 2), of the operator product at inverse temperature 0,and the time variable in the argument of the operators has been replaced by a “temperature variable” U. 6) The T symbol orders the operators according t o the value of u with the larger on the left. The temperature Green’s function is a function of the variables u and u’ and, if H is independent of time, it depends only on the difference u - U’ (this follows, as at T = 0, from the analysis of Appendix 2).

16 Green’s Functions for Solid State Physicists In fact, t o obtain G o ( t )explicitly it is simplest not t o use the differential equation, but t o go back t o the original definition of the Green’s function. 2) with This is a standard harmonic oscillator problem, and we introduce annihilation and creation operators Bi, Bit in the usual way [Messiah (1961, Chap. XII)]. , B F ] = 0, and the hamiltonian becomes (1 3 . 5 ) The time dependence of the B,, B? 7) 17 The Iteration Solution for G where Bi = B i ( 0 ) ,Bit = B:(O).

We therefore now consider a powerful alternative method of developing the perturbation theory for the Green’s function, which is based more directly on the methods of relativistic quantum field theory. This is the Feynman-Dyson expansion [ Feynman (1949), Dyson (1949)l. In contrast to methods based on equations of motion, the idea is t o expand the operators in the original form of the Green’s function before one attempts to evaluate expectation values of operators. This leads t o systematic rules for obtaining terms of arbitrary order in the perturbation series.

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