Journal of Statistical Physics, Vol. 77, Nos. 1/2, 1994. p. 77-87.
The hydrodynamic regime of superfluids is dominated by a Goldstone mode corresponding to a spontaneously broken U(1) symmetry. In this article we map the Kawasaki-lsing model for a classical lattice gas into a quantum model for a superfluid and establish a connection between the normal density fluctuations of the first and the Goldstone mode of the second. The fact that the quantum model we obtain describes a superfluid derives from an inequality by Penrose and Onsager which gives a lower bound to the Bose-Einstein condensate density. Mathematically, the Goldstone mode can be described by means of a "quantum" extension of the local algebra of the Ising model. The classification of its irreducible representations requires an additional U(l) phase factor and the corresponding U(l) gauge symmetry is spontaneously broken for all finite values of the temperature and of the density.