= Canonical momentum sign in the affine particle action
{title2=$P_\mu^c=-m g_{\mu\nu}\dot x^\nu$}
For the affine action $S=-\tfrac m2\int g_{\mu\nu}\dot x^\mu\dot x^\nu d\tau$ with signature $(-+++)$, differentiating with respect to velocity gives the displayed <canonical momentum>. Future-oriented proper time in a conformally flat metric yields $P_i^c=-a\gamma mv_i$, whereas the conventional future-directed mechanical momentum has $q_i=a\gamma mv_i$. Changing the action's overall sign interchanges these canonical labels but leaves its classical geodesics unchanged. Both labels are conserved by spatial <translation invariance>, and either gives the same physical momentum magnitude proportional to $a^{-1}$. The sign should be stated when comparing a worldline canonical covector with a positive-frequency field-mode label.
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