The Hamiltonian and generating functional for a nonrelativistic local current algebra

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 $\exp \mathrm{[i\; \int d}\mathbf{x}\mathrm{\rho (}\mathbf{x}\mathrm{)f(}\mathbf{x}\mathrm{)].}$ 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.