The Hamiltonian and Generating Functional for Representations of a Non-Relativistic Local Current-Algebra

Bogolubov, Prykarpatsky

The non-relativistic current algebra approach is analyzed subject to its application to studying the distribution functions of many-particle systems at the temperature equilibrium and their stability properties.We show that the classical Bogolubov generating functional method is a very effective tool for constructing the irreducible current algebra representations and the corresponding different generalized measure expansions including collective variables transform.The effective Hamiltonian operator construction and its spectrum peculiarities subject to the stability of equilibrium many-particle systems are discussed.

Tengiz M. Bibilashvili

Gerald A. Goldin, S Sarkar

Ralph Menikoff

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.

Ralph Menikoff, David H. Sharp

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.

E M Chabab, Gerald A. Goldin, Sarben Sarkar