We show using the Hamiltonian formalism that if parity is a good symmetry of QCD, then the strong $CP$ phase $\overline{\theta }$ 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 $[P,\mathrm{\Omega }]=0$—where $\mathrm{\Omega }$ is the generator of large gauge transformations—independently enforces $\overline{\theta }=0$ (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 $\overline{\theta }=0(\mathrm{mod}\pi )$. These results establish the equivalence of the Hamiltonian and Lagrangian approaches to the strong $CP$ problem.

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• Extensions of gauge sector
• Quantum chromodynamics
• Discrete symmetries
• Spacetime symmetries