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Here number-conserving means invariant under a global particle-number quantum phase rotation: the field mean and anomalous averages $⟨\psi \psi ⟩$ vanish. It does not mean a sharp total particle number. All normal correlations are obtained by Wick pairings of the one-body density matrix. Independent thermal bosonic modes satisfy this rule; generic nonvacuum bosonic number states do not. A Slater determinant is the analogous fermionic pairing state.

The one-body density matrix contains first-order spatial coherence and mode occupations of a many-particle state. Its trace is the particle number in the convention displayed. A macroscopic eigenvalue identifies Bose-Einstein condensation; in a homogeneous condensate the long-distance matrix approaches a nonzero constant. A finite density or local coherence alone does not imply this long-distance limit.

There is a state-assumption issue. For a number-conserving Gaussian quantum state with no anomalous contractions, put ${\psi }_i=\psi ({\mathbf{r}}_i)$, $n_i=⟨{\psi }_i^{†}{\psi }_i⟩$ and the one-body density matrix $g_{ij}=⟨{\psi }_i^{†}{\psi }_j⟩$. Wick's theorem pairs each creation operator with an annihilation operator. The direct pairing gives $n_1{n}_2$; the exchanged pairing gives $ϵg_{12}{g}_{21}$, with $ϵ=+1$ for bosons and $-1$ for fermions. Hencenormal ordering removes the equal-position delta-function self-counting term. Fermionic occupation-number product states are Slater determinants and obey this Wick theorem. Independent thermal bosonic modes also obey it, with $N_{\alpha }$ understood as mean occupations.

A bosonic Fock product state does not generally obey the asserted identity. The bosonic number-state corrections to Gaussian density correlations follow from $⟨a_{\alpha }^{†2}{a}_{\alpha }^2⟩=N_{\alpha }(N_{\alpha }-1)$ and giveFor one occupied mode this is $N(N-1)|{\phi }_1{|}^2|{\phi }_2{|}^2$, whereas the uncorrected Gaussian expression gives $2N^2|{\phi }_1{|}^2|{\phi }_2{|}^2$. This counterexample disproves the asserted identity for arbitrary bosonic number-product states. The formulas below give both the Gaussian result, with its additional state assumption, and the exact bosonic number-state alternative. The ensemble distinction is also explicit in a later primary treatment of ideal-gas correlations.^[ref]

For a number-conserving Gaussian pairing state, six permutations give this expression. Each transposition has sign $ϵ$, positive for bosons and negative for fermions; the two three-cycles have positive sign for either statistics. Equivalently it is the permanent or determinant of the three-by-three one-body density matrix. For bosonic number states, repeated-mode factorial corrections must be included instead of assuming that Wick theorem.