Propagators, Greens functions and density matrices

One choice of the operators A and B in the propagator equations has been explored already that in which both are of 1-body form, so that 

Another very important choice occurs when A = al(t ), B = and, since these are not number-conserving operators, ij = — 1. The time form of the corresponding electron propagator is then, with T = t — t as in (13.2.9), 

This equation reveals an important connection between propagators and density matrices, since in Chapter S (p. 132) we noted that p , a density-matrix element in the discrete representation provided by a set of spin-orbitals V r(x) could be expressed quite generally as an expectation value of ajar and wc let t— 0 from below, the first term in the braces, in (13.3.2), will be zero while the second will yield (ajar)- Thus we find 

In other words, the electron propagator is a generalization of the 1-electron density matrix and if we can find an approximation to the propagator then we can determine p (and hence all 1-electron expectation values) simply by taking a suitable limit. 

It is often convenient to express the above result in a slightly different way using the form (5.3.5), the density matrix p(x x ) may be expressed as 

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