Generating functionals determining representations of a nonrelativistic local current algebra in the N/V limit

Abstract

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).