Perturbation Expansion for a Fermi Superfluid. I. Basic Formalism and Angular Momentum

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Abstract
A theory of a Fermi superfluid is formulated based on the condensate wave function of the configuration space. First, two basic ingredients are extracted from the wave function, i.e. the effective condensate wave function v ( x 1 , x 2 ) and the quasiparticle field γ( x ) with x denoting the space and spin coordinates r α. After transforming a Hamiltonian into a normal-ordered product of γ, a perturbation expansion with respect to the quasiparticle interaction is formulated for both the thermodynamic potential Ω and the one-particle density matrix ρ. The thermodynamic potential thus obtained includes the normal-state expression of Luttinger and Ward as the limit of v → 0. The density matrix is written as a sum of the coherent part ρ ( c) and the quasiparticle part ρ ( q) , where ρ ( c) is expressed only with respect to v ( x 1 , x 2 ). It is shown that v and γ are responsible for the coherent and thermodynamic properties respectively. Thus the present formalism gives a microscopic foundation for the two-fluid model. Connections with the Gor'kov equations and the Bogoliubov-de Gennes equations are also clarified. It is found that the one-particle density matrices of those formalisms are not equivalent to ρ of the present formalism. The origin of the angular momentum paradox of 3 He - A is traced to this difference.