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

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).