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 $\bar{\theta}$ must be $0$ or $\pi$. We find that for $P$ to be a physical symmetry, it must leave the Hilbert space $\mathcal{H}_\theta$ associated with the $\theta$-vacuum invariant ($P: \mathcal{H}_\theta \rightarrow \mathcal{H}_\theta$), which is possible only for $\theta = 0$ or $\pi$. We also show that forming linear combinations of states from different $\theta$-sectors produces only classical statistical mixtures, consistent with superselection rules, confirming that $\mathcal{H}_\theta$ is the most general Hilbert space for the quantum theory. Furthermore, we demonstrate that requiring $[P,\Omega]=0$, where $\Omega$ is the generator of large gauge transformations, independently enforces $\bar{\theta}=0$ (mod $\pi$), and that for complex quark mass matrix $M$, if a generalized parity operator $\mathcal{P}$ is a symmetry, then the value of $\theta$ gets determined so that it exactly cancels $Arg Det M$, again giving $\bar{\theta}=0$ (mod $\pi$). These results establish the equivalence of the Hamiltonian and Lagrangian approaches to the strong CP problem.