 One-body density matrix

The one-body density matrixuses a creation operator and an annihilation operator to describe first-order spatial quantum coherence. Its diagonal is particle-number density and its trace is the expectation value of the particle number operator, $\mathrm{Tr}{G}^{(1)}=⟨N⟩$. Dividing by $⟨N⟩$ instead gives a unit-trace convention, so the normalization matters when interpreting its eigenvalues.

For bosons, an eigenvalue that remains proportional to particle number in the thermodynamic limit identifies Bose-Einstein condensation. This criterion extends occupation of a single-particle mode to interacting systems.^[ref]

In a homogeneous system of volume $V$, an occupied plane-wave mode contributesA constant long-distance limit therefore corresponds to a zero-momentum condensate; a moving condensate can retain a spatial phase. Finite local density or short-distance quantum coherence alone does not establish off-diagonal long-range order.