= 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> 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.
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