Past exam of the theoretical physics major option course of the University of Oxford / 2022 / 4 / e / Solution

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.