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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Representations of Local Nonrelativistic Current Algebras

J. Grodnik, D. H. Sharp

Published Mar 15, 1970• 25 citations
Publisher

Abstract

In this paper we show how to specify the particle content of a nonrelativistic quantum theory of $N$ identical spinless particles in terms of observables like the particle number density and the flux density of particles. Our approach to this problem is through a study of the irreducible representations of the local, equal-time current algebra. It is shown how these representations define a functional representation of the current algebra, and that the Hamiltonian can be written in terms of the currents in a nonsingular fashion in any irreducible representation.

DOI
10.1103/physrevd.1.1531

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Physical Review A•Oct 1, 1974•25 citations
In this paper we show how to specify the particle content of a nonrelativistic quantum theory of NNN identical spinless particles in terms of observables like the particle number density and the flux density of particles. Our approach to this problem is through a study of the irreducible representations of the local, equal-time current algebra. It is shown how these representations define a functional representation of the current algebra, and that the Hamiltonian can be written in terms of the currents in a nonsingular fashion in any irreducible representation.
The nonrelativistic current algebra with conserved current consisting of ρ(x), the particle number density, and J(x), the flux density of particles, is studied. The Hamiltonian for any time reversal invariant system of spinless particles, interacting via a two-body interaction potential, is expressed as a Hermitian form in the currents. This leads to a functional equation for the generating functional, which is the ground state expectation value of exp[i ∫d xρ(x)f(x)]. In the N / V limit an expression for the generating functional in terms of correlation functions is given. Representations of the exponentiated current algebra which are translation invariant satisfy the cluster decomposition property, and those which have different Hamiltonians are shown to be unitarily inequivalent.
Representations will be given for the nonrelativistic local current algebra consisting of ρ(x), the particle number density, and J(x), the flux density of particles. These representations correspond to the N/V limit of (i) a free Bose gas, (ii) Bose gas in an external potential, (iii) free Fermi gas, and (iv) Bose gas (in one dimension) with a two-body interaction potential V(x)=2/x2. In each case the generating functional L(f), the ground state expectation value of exp[iρ(f)], determining the representation will be given. It will also be shown the generating functional satisfies a functional equation of the form [∇ − i∇f(x)](1/i) [δ/δf(x)]L (f) = A(x, (1/i)(δ/δf))L(f) and that the Hamiltonian written in terms of ρ and J has the form H = (1/8)∫dxK̃(x)† [1/ρ(x)] K̃(x) with K̃(x)=∇ρ(x) +2iJ(x)−A(x,ρ).
Local currents are used to describe nonrelativistic many-body quantum mechanics in the thermodynamic limit. The problem of determining a representation of the local currents corresponding to a given Hamiltonian is studied. We formulate the dynamics in such a way that one solves simultaneously for the ground state and the representation of the local currents. This leads to two coupled functional equations relating the generating functional to a functional which describes the ground state. Together these functionals determine a representation of the local currents in which the Hamiltonian is a well-defined operator. The functional equations are equivalent to a set of integro–differential equations for expansion coefficients of the two functionals.
It is possible that a complete physical theory can be written entirely in terms of operators such as current densities, rather than in terms of field operators. The current densities in these models correspond in general to distributions of unbounded operators. Such a theory is reviewed for the case of nonrelativistic quantum mechanics. It is found that one can exponentiate the current algebra to obtain a group, which may then be represented by unitary (hence bounded) operators in Hilbert space. This procedure is analogous to exponentiating the canonical commutation relations and obtaining the Weyl group. For nonrelativistic quantum mechanics without spin, the group is the semidirect product J∧K Schwartz space J(R3) with a group K of certain C∞ diffeomorphisms from R3 onto itself. For f ∈ J and φ ∈ K, the composition mapping (f,φ)→f∘φ defines the semidirect product law. The Gel'fand-Vilenkin formalism for ``nuclear Lie groups'' is suitable for the representation theory of such a group. Almost all of the physical information is contained in a cylindrical measure μ on J′, the dual of the nuclear space J. The Fourier transform of μ can be interpreted as an expectation functional with respect to the state of lowest energy. The nonrelativistic Fock representation (including the theory of n particles in a box) is examined in this formalism. Conditions on μ are systematically developed which suffice to recover all of the infinitesimal generators on a common, dense, invariant domain in the Hilbert space. For the Weyl group slightly weaker conditions than for the semidirect product case would suffice. Gaussian measures in J′, as well as the measures defining n-particle representations of J ∧K, satisfy these conditions.
It is now well established that current and density may be used as coordinates in describing a nonrelativistic many-body system. The current and density operators form a closed algebra as they obey a set of equal-time commutation relations. A particular choice of the algebra containing only the longitudinal component of the current describes a fluid in a simple way. For Bose systems, we show that there exists a special choice of this longitudinal current which is an irreducible representation of the algebra, but expressed in two different ways leading to the formulations of Bogoliubov and Zubarev (BZ) and that of Sunakawa and his coworkers (S). This demonstration proves the formal equivalence of the two formulations. The BZ formalism leads to a non-Hermitian Hamiltonian for which use of a certain mathematical technique recently proposed by us yields completely divergence-free results for Bose systems. A temperature-dependent matrix Green's-function theory is developed and the self-energy matrix calculated. From this, we also deduce the structure factor in a straightforward way. These results are applied here to compute numerically the excitation spectrum of the superfluid liquid helium using the experimental structure factor as the only input into the computation. Comparison of this with other calculations and with experiment is discussed in detail in this paper. The problem of a charged Bose gas is considered as a testing ground for all theories of Bose fluids since certain exact results are known for this system. We have applied our method using the BZ Hamiltonian to this problem and we are in complete agreement with the exact results. It thus appears that the current-algebra approach can be quite successfully applied in elucidating properties of interacting Bose systems.