### Abstract

For a system of 2*n* fermions it is shown that the occupation number *n* _{α} of any fermion‐pair state is , where ρ_{2}, assumed normalized to unity, is the two‐particle density matrix. The known upper bound on the largest eigenvalue of ρ_{2} implies that *n* _{α} ≤ ½ in the thermodynamic limit *n* → ∞, equality being approached, with suitable φ_{α}, only for certain limiting BCS states ψ_{BCS}. Bose condensation of fermion pairs, in the sense of macroscopic occupation of any *n* _{α}, is impossible. Fermi condensation into φ_{α} is defined to be present if *n* _{α} > 0 in the thermodynamic limit. It occurs, for example, for suitable BCS states, but for a normal Fermi system all *n* _{α} are of order *n* ^{−1} or smaller. It is argued that the physical criterion for ψ_{BCS} exhibiting Fermi condensation is that the pair state from which it is constructed must have a bound‐state component of range , where *k*_{F} is the Fermi momentum (3π^{2}ρ)^{⅓}. The maximally‐occupied pair state is the eigenfunction of ρ_{2} belonging to the largest eigenvalue, associated with off‐diagonal long‐range order of the type defined by Yang. A formula for *n* _{0} for the original BCS state is derived. Some remarks are made concerning the interpretation of the Fermi condensation as a superconducting transition. The analysis is generalized to occupation of *l*‐fermion states. It is conjectured that, when Fermi condensation first sets in at a given even *l*, this is associated with the formation of bound states of *l* fermions, and a formula for the maximal occupation of such states is exhibited. The implications, for the theory of liquid ^{4}He, of the fact that a helium atom contains electrons are examined. It is shown that Bose condensation into a single‐^{4}He‐atom state is impossible, but a Fermi condensation similar in some respects to that in a superconductor can and probably does occur. It is argued that the mechanism preventing Bose condensation in superconductors and liquid ^{4}He lies in the effect of collisions, acting via the exclusion principle, in causing virtual internal excitations.

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