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International Journal of Quantum Chemistry

Algebraic structure of fermion density matrices. II

G. G. Dyadyusha, Eugene S. Kryachko

Published Apr 1, 1981• 3 citations
Publisher

Abstract

Abstract The properties of the algebraic structure of fermion density matrices are studied. The algebraic structure of a density matrix leads to a more varied and detailed classification scheme than that offered by the usual shell structure approach. Investigation of the algebraic structure of fermion density matrices by the methods of algebraic topology leads to a classification scheme based on Betti numbers.

DOI
10.1002/qua.560190404

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Abstract The properties of the algebraic structure of fermion density matrices are studied. The algebraic structure of a density matrix leads to a more varied and detailed classification scheme than that offered by the usual shell structure approach. Investigation of the algebraic structure of fermion density matrices by the methods of algebraic topology leads to a classification scheme based on Betti numbers.
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We state in the following a structure theory of the density matrices from the standpoint of secind quantization. The difficulties involved in the quantum-mechanical many body problem have so far proved to be so tremendous that as yet some have been obliged to stop before exact treatments and others have retired to phenomenological model theories. There are two main tools to approach the problem, the theory of second quantization and the theory of density matrix. The former, in spite of its generality and compactness, has never revealed us its adaptability to the practical problems of quantum statistical mechanics. In fact, about the exchange effect, namely the statistical correlation of particles, the latter informs us the detailed features of it by application of the group theory, while the former lacks, at present, such an exposition. For a long time a more general and simpler formulation than the current group theoretic method has been desired in the theoretical researches of spectroscopy. A distinguished contribution has been made by P. Jordan (1934) and H. Ostertag (1937) from the standpoint of second quantization. H. Ostertag made use of the theory of hypercomplex ring and introdeced the original concept of the elementary algebra. (Part II) Since the latter part of his formulation, however, seems to be complicated, it will be better, for clearness' sake, to avoid this part. Accordingly the concepts of the elementary algebra and the symmetric algebra will be introduced as a remedy for it. Then we shall complete our theory to include the whole theory of symmetric group. These concepts will serve to clarify and generalize the previous formulations and to unify the several group theoretic standpoints of the spectroscopy. In §1 and §2 the representation of the quantized density of Bose or Fermi particles will reveal itself as the density matrix of Bose or Fermi particles or as the density matrix of many electrons with a certain term of spin multiplets. Indeed, the theory of the statistical operator, or the density matrix of J. von Neumann and P.A.M. Dirac, the equilibrium properties of which have been fully examined by one of the authors (1940), is the natural formalism of dealing with quantum gases. A contribution has been made by W. Kofink (1938) to introduce the intrinsic degrees of freedom of particles. This attempt has resulted in a fair seccess. His formulation consists in making use of the initial stages of the theory of spectroscopy, which will be reexamined in the second section and Part II. In §3, as an example at zero temperature, the Hartree-Fock equation for a prescribed orbital and spin configuration will be derived, and in §4 some properties of the reduced density matrix will be illustrated.
A possible method of classifying all density matrices for a N-state quantum system into different types, using the sameness of the characteristic polynomial of density matrices belonging to a given type, is discussed. The method employs the SU(N) generator expansion of density matrices. It is demonstrated that density matrices belonging to a given type, although different, share many common properties including the same eigenvalues, the same shape of the region in the parametric space in which lie the allowed parameter values, and the same global averages of the pertinent observables. Furthermore, we discuss the time evolution of a given type of density matrices and show that, in the appropriately defined time-dependent comoving matrix basis, a density matrix of a given type belongs, during entire unitary evolution, to the same type. In this way one is able to group and organize the multitude of different density matrices with certain analogous properties, into a smaller number of types, thus systematizing and cataloguing different mixed and pure states of a N-state system. The general discussion is illustrated throughout with the special case of a N=4-state quantum system. Two- and three-dimensional cross sections of the space of generalized Bloch vectors are determined by the Monte Carlo sampling method for several types of density matrices, providing some insight into the intricate and complex geometric structure of the space of density matrices.