 Solution

Use the upper-index coordinates $B^{\mu },B^{\ast \mu }$ specified in the paper. Differentiation of the Lagrangian density gives the canonical momentaIn particular, ${\pi }_i=G^{\ast 0i}$ and ${\pi }_i^{\ast }=G^{0i}$. No time derivative of $B^0$ appears, so the zero momenta are primary constraints, not invertible equations for ${\dot{B}}^0$.

For spatial indices, $G^{0i}={\dot{B}}^i+{\partial }_i{B}_0$, so ${\dot{B}}^i={\pi }_i^{\ast }+{\partial }^i{B}_0$, with the conjugate relation for $B^{\ast i}$. AlsoThe Legendre transform for the complex Proca Hamiltonian, ${\mathcal{H}}_c={\pi }_i{\dot{B}}^i+{\pi }_i^{\ast }{\dot{B}}^{\ast i}-\mathcal{L}$ therefore givesRepeated spatial indices in the positive product ${\pi }_i^{\ast }{\pi }_i$ denote an ordinary three-component sum. A total Hamiltonian additionally includes multipliers for ${\pi }_0={\pi }_0^{\ast }=0$.

The Hamiltonian printed in the source cannot follow from the given complex Lagrangian density: its missing conjugations, mass coefficients and spatial mass sign are inconsistent. A concrete check is spatially uniform initial data $B^1=b\in \mathbb{R}$, $B^0=B^2=B^3=0$, with all canonical momenta zero. The correct Hamiltonian density is $M^2{b}^2$, while the printed expression gives $-M^2{b}^2/2$. These data satisfy the initial constraints and evolve as a spatially uniform massive oscillator. Thus the discrepancy is not a harmless surface term or a different complex-field normalization.