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Journal of Mathematical Physics

Noninvariance Groups in the Second-Quantization Picture and Their Applications

M. Moshińsky, C. Quesne

Published May 1, 1970• 141 citations
Publisher

Abstract

We investigate the existence of noninvariance groups in the second-quantization picture for fermions distributed in a finite number of states. The case of identical fermions in a single shell of angular momentum j is treated in detail. We show that the largest noninvariance group is a unitary group U(22j+1). The explicit form of its generators is given both in the m scheme and in the seniority—angular-momentum basis. The full set of 0-, 1-, 2-, ⋯, (2j + 1)-particle states in the j shell is shown to generate a basis for the single irreducible representation [1] of U(22j+1). The notion of complementary subgroups within a given irreducible representation of a larger group is defined, and the complementary groups of all the groups commonly used in classifying the states in the j shell are derived within the irreducible representation [1] of U(22j+1). These concepts are applied to the treatment of many-body forces, the state-labeling problem, and the quasiparticle picture. Finally, the generalization to more complex configurations is briefly discussed.

DOI
10.1063/1.1665304

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We investigate the existence of noninvariance groups in the second-quantization picture for fermions distributed in a finite number of states. The case of identical fermions in a single shell of angular momentum j is treated in detail. We show that the largest noninvariance group is a unitary group U(22j+1). The explicit form of its generators is given both in the m scheme and in the seniority—angular-momentum basis. The full set of 0-, 1-, 2-, ⋯, (2j + 1)-particle states in the j shell is shown to generate a basis for the single irreducible representation [1] of U(22j+1). The notion of complementary subgroups within a given irreducible representation of a larger group is defined, and the complementary groups of all the groups commonly used in classifying the states in the j shell are derived within the irreducible representation [1] of U(22j+1). These concepts are applied to the treatment of many-body forces, the state-labeling problem, and the quasiparticle picture. Finally, the generalization to more complex configurations is briefly discussed.
The representation of an N-particle Hamiltonian on an Nth rank orbital product space is exactly modeled by a second-degree polynomial in the infinitesimal generators of the unitary group. Symmetry adaptation of the product space with respect to the symmetric group yields Gel'fand states which are bases for the irreducible representations of the unitary group. These exist in closed form, as, in consequence, does the representation of the model Hamiltonian. The exclusion principle is imposed by restricting the irreducible representations of the unitary group. The unitary group formulation is equivalent to standard many-body theory for the particle-conserving case.
The n-dimensional Euclidean group—the group of rotations and translations of Rn —is the natural canonical group for quantizing a system whose configuration space is the sphere Sn−1, n≥2. Since every unitary irreducible representation of E(n) is nonsquare integrable it follows that coherent states involving the whole group do not exist for such representations. Instead, coherent states are defined based on unitary, reducible representations of the group, and it will be shown that, with such states, quantum dynamics on the sphere admits a path-integral representation involving novel first-order classical actions defined on E(n) as well as integrations over the entire group manifold at each time slice. Irreducibility of the quantization is achieved through a final limiting process.
Few, if any, applications of quantum technology are as widely known as the quantum simulation of quantum matter. Consequently, many interesting questions have been sparked at the intersection of condensed matter, quantum chemistry, and quantum computing. Given the common mathematical foundation of these subjects, we walk through the necessary group theory and representation theory serving as background in all of these fields. Our discussion will include a full development of the mathematics of identical particles and the mechanics of describing systems of identical particles in both first and second quantization schemes. This chapter is an offshoot of a larger work that provides a graduate-level introduction to quantum information science. This chapter is being released separately because it is not explicitly focused on quantum information. It has grown beyond a short digression into a full-fledged journey into the symmetries and representations of identical particles that we invite you, the reader, to join.
Abstract The representation of an N‐electron Schrödinger Hamiltonian on an orthonormal, spin‐free (freon) orbital product space is exactly modeled by a second‐degree polynomial in the infinitesimal generators of the unitary group. Symmetry adaptation of the orbital product space with respect to the symmetric group yields Gel'fand states which provide base for irreducible represntations of the unitary group. These exist in closed form as, in consequence, does the representation of the model Hamiltonian in this same basis. We retain as physical only those states characterized by tableaux with no more than two columns for which the spin labelling is one‐half the difference in the lengths of the two columns. The unitary group formulation is equivalent to standard, number‐conserving, second‐quantized, many‐body theory.
For an nnn-fermion system in which each particle can occupy any one of kkk states, an "irreducible bbb-body operator" is defined as an operator belonging to [2b{2}^{b}2b, ${1}^{k\ensuremath{-}2b}$] of SUk{\mathrm{SU}}_{k}SUk​. It is shown that such an operator can always be written as a bbb-body operator multiplied by a function of nnn. A measure of the failure of a basis to diagonalize the Hamiltonian is constructed. If the Hamiltonian is analyzed into irreducible bbb-body parts, the measure of error for nnn particles can be calculated from two-particle parameters. An upper limit on the error of a group theoretically defined basis can be found without having to calculate any nnn-particle interaction matrix elements. A measure of the magnitude of an interaction is defined, and shown to depend differently on nnn for irreducible 0-body, 1-body, and 2-body interactions. The effect of the Pauli principle on the formation of shell-model potentials is discussed from this point of view.
All bound-state solutions of the Dirac Coulomb problem are shown to provide a Hilbert space for irreducible representations of the de Sitter group O(4,1)O(4, 1)O(4,1) Selected solutions of this problem provide the basis for other representations of O(4,1)O(4, 1)O(4,1) These results are compared to analogous results previously obtained for the Schr\"odinger Coulomb problem. Operator forms of the generators of O(4,1)O(4, 1)O(4,1) are explicitly constructed as functions of p and r which operate on the Hilbert space provided by the Dirac problem.