Propagators. The polarization propagator

In the general treatment of propagators (see e.g. Zubarev 1960, 1964) it is usual to define all operators in the so-called Heisenberg picture this picture (implicit in the elementary treatment (Section 12.1) of time-dependent perturbations) must therefore be made explicit. To do this, we momentarily add a superscript S to distinguish wavefunctions and operators in the usual Schrodinger representation, and start from the time-dependent Schrodinger equation, which now becomes 

When is the usual time-dependent Hamiltonian of the (unperturbed) system, a formal solution may be written  

When f (0) is a (time-independent) eigenfunction of H , the exponential in (13.2.2) becomes the trivial time factor for a stationary state of given energy . 

In the Heisenberg picture we simply make a unitary transformation in which wavefunctions and operators are replaced (cf. Section 2.2) according to 

The resultant wavefunction in the Heisenberg picture is thus reduced to rest , becoming UV)U(t)y (0) = f (0) but all operators (even those, like the Hamiltonian, that do not depend on time in the Schrddinger picture) in general acquire a time dependence. Thus a general Heisenberg operator is 

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