Heisenberg Green Functions

The average value of any operator O can be written as (O) = (t Os t) in the Schrodinger representation or (O) = (0 Off(t) 0) in the Heisenberg representation, where 0) is some initial state. This initial state is in principle arbitrary, but in many-particle problems it is convenient to take this state as an equilibrium state, consequently without time-dependent perturbation we obtain usual equilibrium Green functions. 

We start from the general definition of a Green function as the average of two Heisenberg operators A(t) and B(t), denoted 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... 

The standard NEGFT formulated in terms of the four Hilbert space Green functions time ordered (G ), anti-time ordered (G ), greater (G ) and lesser (G ) [23,25], These are defined in the Heisenberg picture as... 

The basic quantities required for describing the coupled molecule-lead system are the one particle electron and the phonon Green functions. Following the steps outlined in Section 14.2, the time development for various superoperators (Heisenberg equations) is (all primed indices should be summed over)... 

This postulate allows the consistent formulation of the Schrodinger, Interaction, and the Heisenberg pictures, the treatment of time dependent perturbations, as well as the description of the quantum events by means of the so-called propagators (Green functions) linking them in a causal ... 

Born approximation Green function Heisenberg picture hydrogenic states interaction picture quantum electric resistance quantum ground level quantum mechanics postulates quantum resonance... 

Swendsen (1972) has shown theoretically in a Green s function analysis of a Heisenberg ferromagnet, including also dipolar interactions, that TqIO cannot be larger than 0.9. As well for EuO as for EuS the data of table 19.2 show that this ratio is 0.89. 

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