Microscopic Theory of Superfluid Helium

Abstract

This paper discusses the properties of an interacting condensed Bose system, emphasizing those aspects which do not depend on the weakness of the potential, and which therefore apply to superfluid helium. A physical and mathematical characterization of Bose condensation is presented in terms of an additional macroscopic quantity, the wave function of the condensed mode, which is defined in terms of microscopic quantities. It is shown that in equilibrium the assumption of Bose condensation leads to a two-fluid model. The hydrodynamic generalization applicable to slowly varying disturbances from equilibrium is then discussed and rigorous microscopic expressions are derived for the parameters of this theory (including dissipative coefficients). The elementary excitation spectrum in this collision dominated regime is exhibited. The Landau quasi-particle theory is examined, as well as the relation of condensation and the excitation spectrum, to the property of superfluidity. Under certain regularity assumptions the form of the long-wavelength excitation spectrum at vanishing temperature is deduced. The corresponding derivation at finite temperature is presented and criticized. Finally, techniques are discussed for evaluating properties of the Bose system starting from the interaction potential. Approximation schemes consistent with the conservation laws and with the absence of a gap in the elementary excitation spectrum are discussed. Previous approximations for the weakly interacting Bose gas are classified and summarized and additional approximations are examined.