= One-body density matrix
{title2=$G^{(1)}(\mathbf r,\mathbf r')=\langle a^\dagger(\mathbf r)a(\mathbf r')\rangle$}
The <one-body density matrix>
$$
G^{(1)}(\mathbf r,\mathbf r')=\langle a^\dagger(\mathbf r)a(\mathbf r')\rangle
$$
uses 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>, $\operatorname{Tr}G^{(1)}=\langle N\rangle$. Dividing by $\langle N\rangle$ 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.<Bose-Einstein Condensation and Liquid Helium by Oliver Penrose and Lars Onsager>[Abstract and section 3, equations 3–4: extensive largest eigenvalue of the one-particle reduced density matrix]{ref}
In a homogeneous system of volume $V$, an occupied plane-wave mode contributes
$$
G^{(1)}_{\rm cond}(\mathbf r,\mathbf r')=\frac{N_0}{V}e^{i\mathbf k_0\cdot(\mathbf r'-\mathbf r)}.
$$
A 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>.
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