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 $\barθ$ must be $0$ or $π$. We find that for $P$ to be a physical symmetry, it must leave the Hilbert space $\mathcal{H}_θ$ associated with the $θ$-vacuum invariant ($P: \mathcal{H}_θ\rightarrow \mathcal{H}_θ$), 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 $\mathcal{H}_θ$ 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 $\barθ=0$ (mod $π$), and that for complex quark mass matrix $M$, if a generalized parity operator $\mathcal{P}$ is a symmetry, then the value of $θ$ gets determined so that it exactly cancels $Arg Det M$, again giving $\barθ=0$ (mod $π$). These results establish the equivalence of the Hamiltonian and Lagrangian approaches to the strong CP problem.