Density matrices Fourier representation

In coordinate space, the diagonal elements of the canonical density matrix in the Fourier path integral representation are given by [20]... 

If we are interested in the off-diagonal elements of the density matrix, that is p(x, x j3), the paths are no longer cyclic but begin at x and terminate at x1. Then the Fourier representation of the path is... 

Figure 5.2. The relationships among the r-space density matrix F, the p-space density matrix n, the Wigner representation W, and the Moyal representation A. Two-headed arrows with a T beside them signify reversible, three-dimensional, Fourier transformations.

Throughout this section, the canonical density matrix and the Feynman propagator can be used interchangeably, the transformation P = it taking C into the propagator K, with t the time. While most frequently we shall use the coordinate representation r and r, it will be convenient in this section to work in k or momentum representation, by taking a double Fourier transform with respect to r and r. ... 

These operators enable a compact Fourier matrix representation of the Hamiltonian and the density matrix ... 

The DPI representation of the path integral that was developed in the preceding section is not unique. Another path-integral representation is often used that has come to be known as the Fourier representation [33,34,36-42,44,85]. Like the DPI representation, the Fourier representation transforms the path integral into an infinite-dimensional Riemann integral. In this formalism, we consider the paths to be periodic signals that can be represented as a Fourier series. Consider the density matrix p(x, x j8). Since the partition function is the trace of the density matrix, we have... 

Finally, one notes the important role played by the momentum distribution, which is related to the one-body density matrix in the coordinate representation through a Fourier transform. The one-body density matrix in this case contains the correlations between the ends of the PI particle path which, as a result of bosonic exchange, becomes an open string. For details on PI calculations of properties such as the momentum distribution, condensate fraction, and superfluid density the reader is referred to Refs. 28,70,215,231,232. 

At this stage in our discussion it becomes convenient to represent the Hamiltonian and the density operator in a Fourier matrix representation defined by a set of dressed Fourier states n,k In this infinite representation we... 

The usual way of solving eqn (7) requires its transformation into the interaction representation (Dirac picture) that is often called rotating frame for a particular case, when static part of the spin Hamiltonian is restricted to the electron Zeeman interaction. In the Dirac picture only the stochastic dipolar interaction is left in the spin Hamiltonian, its matrix elements get additional oscillatory factors due to the static Hamiltonian transitions. The integral on each matrix element of the double commutator in eqn (7) thus evolves into the Fourier transform /(co ) of the correlation function for the corresponding stochastic process. This Fourier transform is often called spectral density of the stochastic process and it is to be taken at a frequency co of a particular transition of the static Hamiltonian operator, driven by a single transition operator ki ... 

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