Canonical momentum sign in the affine particle action ($P_{\mu }^c=-m{g}_{\mu \nu }{\dot{x}}^{\nu }$)

For the affine action $S=-\frac{m}{2}\int g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }d\tau$ with signature $(-+++)$, differentiating with respect to velocity gives the displayed canonical momentum. Future-oriented proper time in a conformally flat metric yields $P_i^c=-a\gamma m{v}_i$, whereas the conventional future-directed mechanical momentum has $q_i=a\gamma m{v}_i$. 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.