On a limiting procedure for obtaining physical states for an infinite Fermi system

Abstract

We propose a limiting procedure for obtaining physical states for an infinite non-relativistic Fermi system. We take the thermodynamic limit of vector states in the Fock representation of the C.A.R. algebra, representing a condensate state of “atoms” each of which is formed by 4 fermions. In a simplified example considered in detail, the limit state has a simple decomposition into the product of two B.C.S. states. IfB + is the operator creating the “atom” from the vacuum |φ0F 〉, it is proved that the states obtained by taking the thermodynamic limit of the vector states corresponding to (B +) n |φ0F 〉 and $$\sum\limits_{n = 0}^\infty {{z}^{n/2}} {{(B^+)^n}/ {(n!)^2 }}|\psi _{0,F} \rangle $$ respectively, coincide on the gauge-invariant elements of the algebra for a suitable value ofz.