Strong CP Phase and Parity in the Hamiltonian Formalism

Abstract

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 ($P:{\mathcal{H}}_{\theta }\to {\mathcal{H}}_{\theta }$), 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 $ArgDetM$, again giving $\bar{\theta }=0$ (mod $\pi$). These results establish the equivalence of the Hamiltonian and Lagrangian approaches to the strong CP problem.