Correlations in quantum plasmas. I. Resummations in Mayer-like diagrammatics

Abstract

For a system of point charges that interact through the three-dimensional electrostatic Coulomb potential (without any regularization) and obey the laws of nonrelativistic quantum mechanics with Bose or Fermi statistics, the static correlations are expressed in terms of Mayer-like diagrams. The exchange effects are taken into account systematically and the long-range Coulomb divergencies are exactly resummed in order to get finite diagrams. For this purpose, in the framework of the grand canonical ensemble, the matrix elements of the imaginary-time evolution operator are represented by the Feynman-Kac functional integral according to Ginibre's idea [J. Math. Phys. 6, 238 (1965); 6, 252 (1965), 6, 1432 (1965)] and we exhibit a correspondence between the correlations in the quantum system of point particles and the distribution functions in a classical fluid of "exchange" loops with Brownian random shapes. The size of a loop, which corresponds to the number of particles involved in the corresponding cyclic permutation, and the shape of a loop, which describes the quantum fluctuations, play the part of internal degrees of freedom that must be integrated over when calculating the distributions relative to the quantum point charges. The loops interact through a two-body potential that is different from the electrostatic interaction between two charged curves: each line element of a loop interacts only with a discrete number of line elements in every other loop. The linear response theory to an infinitesimal external charge distribution can be written in this formalism by a formula analogous to that of classical statistics, and the loop-fugacity and loop-density expansions of the loop-distribution functions are derived by the usual techniques of Mayer diagrams generalized to extended objects. At large distances $r$, the loop potential behaves as $\frac{1}{r}$ and, as in the classical case, every Mayer diagram diverges. However, this nonintegrable tail is independent of the shape of the loops and it is explicitly and exactly resummed by a generalization of the method developed by Meeron [J. Chem. Phys. 28, 630 (1958); Plasma Physics (McGraw-Hill, New York, 1961)], for classical fluids of point entities. Auxiliary $\frac{1}{r}$ bonds are introduced and subdiagrams involving chains of $\frac{1}{r}$ bonds are integrated over first in a systematic way. The new diagrams contain bonds between loops that decay either exponentially or algebraically, with a $\frac{1}{{r}^{3}}$ leading term, and the new diagrams are at least conditionally integrable. The part of the quantum particle-particle correlation arising directly from exchange, which is derived from the density of loop, decays faster than any inverse power law, whereas, as shown in the following paper [Phys. Rev. E 53, 4595 (1996)], the whole quantum particle-particle correlation, which also involves the correlations between loops, decays only as $\frac{1}{{r}^{6}}$.