Fourier representation path integrals

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

Edwards approach (Ref. [10]) is based on field-theoretic path-integral representation of the partition function Wn(R0, RN, N) defining the probability density of the fact that end points of an JV-link chain are placed at the points R0 and Rn, respectively, and the chain turns n times around the string (the obstacle). The same problem in a slightly different way was considered by Prager and Frisch by using the combinatorial methods [11] and later by Saito and Chen by employing Fourier analysis [12]. 

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... 

The S-matrix may be expressed through the conventional path integral in coordinate representation by the Fourier transform in Pf j (Bogdanov et al. 1989) ... 

The quasiclassical amplitude in the momentum representation does not suffer from this feature, because boundary conditions Pz h) = Pu> Pzitz) = P2z determine the unique classical trajectory for the typical scattering potentials. Such amplitude cannot be obtained as a Fourier transform of the quasiclassical propagator in coordinate representation. So it is necessary to modify the stationary phase method for the evaluation of the path integral in momentmn representation. 

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