Off-Diagonal Long-Range Order and Generalized Bose Condensation

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Abstract
It is shown for boson systems with periodic boundary conditions that the existence of generalized Bose condensation, of which the simple type typified by the ideal Bose gas is a special case, is equivalent to the existence of off‐diagonal long‐range order (ODLRO) in the single‐particle density matrix ρ1 provided that the two noncomuting limits involved in the criterion for ODLRO are taken in the proper order: first size of system → ∞, then interparticle separation → ∞. It is shown by means of an example that certain assumptions concerning the behavior of the single‐particle momentum distribution function involved in proving this equivalence are actually satisfied in thermal equilibrium for some dynamical models. The analysis is generalized to box‐enclosure boundary conditions by an extension of an argument due to Schafroth, according to which box‐enclosure conditions are equivalent to homogeneous Neumann conditions except for a thermodynamically negligible surface effect, provided that one second‐quantizes with respect to Hartree‐Fock orbitals rather than free‐particle orbitals. A general criterion for generalized Bose condensation in terms of eigenvalues of ρ1 is proposed. On the basis of the behavior of soluble models it is conjectured that for a boson system in thermal equilibrium subject to arbitrary boundary conditions, the existence of such Bose condensation is equivalent to the existence of ODLRO in ρ1. The two‐particle density matrix ρ2 is discussed briefly. By means of a simplified model it is shown that for a Bose system generalized condensation implies large eigenvalues of ρ2 and ODLRO of ρ2, just as for ρ1. It is pointed out that the question of the existence or nonexistence of generalized Bose condensation of fermion pairs ought to be investigated.