Strong CP phase and parity in the Hamiltonian formalism

We show using the Hamiltonian formalism that if parity is a good symmetry of QCD, then the strong CP phase θ¯ must be 0 or π. We find that for P to be a physical symmetry, it must leave the Hilbert space Hθ associated with the θ-vacuum invariant (P: Hθ→Hθ), which is possible only for θ=0 or π. We also show that forming linear combinations of states from different θ sectors produces only classical statistical mixtures, consistent with superselection rules, confirming that Hθ is the most general Hilbert space for the quantum theory. Furthermore, we demonstrate that requiring [P,Ω]=0—where Ω is the generator of large gauge transformations—independently enforces θ¯=0 (mod π), and that for complex quark mass matrix M, if a generalized parity operator P is a symmetry, then the value of θ gets determined so that it exactly cancels ArgDetM, again giving θ¯=0(modπ). These results establish the equivalence of the Hamiltonian and Lagrangian approaches to the strong CP problem.