= Renormalized momentum of a condensate
{title2=$\mathbf p$}
For a condensate with uniform bulk complex field one, its convergent localized <momentum> is
$$
\mathbf p=\frac1{2i}\int[(\psi^*-1)\nabla\psi-(\psi-1)\nabla\psi^*]d^d x.
$$
The subtraction removes the uniform-background phase-gradient contribution. If $\psi-1=O(r^{-1})$ and $\nabla\psi=O(r^{-2})$, its integrand is $O(r^{-3})$, integrable in two dimensions. Under decaying variations its <functional derivative> is $\delta p_j/\delta\psi^*=-i\partial_j\psi$; the subtraction is a boundary correction and does not change the bulk equation.
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