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Varying the vector field gives the Proca equation,Its divergence uses antisymmetry of $F^{\mu \nu }$ to cancel the first term. Since $m_A\ne 0$,For an on-shell momentum this imposes $p_{\nu }{ϵ}^{\nu }=0$ on the four polarization components, leaving three independent solutions. In the rest frame, $p=(m_A,0)$ makes ${ϵ}^0=0$, leaving precisely the three spatial polarizations of a massive spin-one particle.

There is no fourth independent oscillator hidden in $A^0$. Its canonical momentum is zero because the Lagrangian has no ${\dot{A}}^0$, and the temporal equation constrains $A^0$ in terms of the spatial data. Equivalently the primary and secondary second-class constraints remove two variables from the eight-dimensional phase space, leaving six phase-space variables, or three physical degrees of freedom.

The naive equal-time canonical commutation relations would bewith all coordinate-coordinate, momentum-momentum and mixed conjugate-sector commutators zero before enforcing constraints. But the $\mu =\nu =0$ relation contradicts ${\pi }_0=0$ as an operator identity: its left side would vanish and its right side would not.

The primary zero-momentum constraints and the secondary massive Gauss laws form second-class constraints; their Poisson bracket matrix contains nonzero terms proportional to $M^2$. They must be eliminated, or treated using a Dirac bracket, before quantization. In a reduced description the three complex spatial coordinates and their momenta are canonical, whileare dependent operators. Substituting them in the integrated Hamiltonian gives the positive density, up to a surface term,Thus the problem is a constrained choice of canonical variables, not an absence of a consistent free quantum field theory.