 Complex Proca Hamiltonian

For Minkowski metric $(+---)$ and $\mathcal{L}=-G_{\mu \nu }^{\ast }{G}^{\mu \nu }/2+M^2{B}_{\mu }^{\ast }{B}^{\mu }$, use coordinates $B^{\mu },B^{\ast \mu }$. The spatial canonical momenta are ${\pi }_i=G^{\ast 0i}$, ${\pi }_i^{\ast }=G^{0i}$; the temporal momenta vanish. Constraint preservation gives ${\partial }_i{\pi }_i=M^2{B}_0^{\ast }$. Eliminating $B_0$ and its conjugate gives the positive reduced Hamiltonian density displayed above, up to a spatial surface term. In particular the massive Gauss law has a mass-dependent right-hand side; setting it to zero removes a physical longitudinal particle polarization.