Is the operator $\hat O_{\alpha}$ which is defined in the following a unitary operator?
Operator $\hat O_{\alpha}$ is supposed to act on composite states with two explicit components, such that
$$| \phi, \psi \rangle \mapsto \hat O_{\alpha} | \phi, \psi \rangle := | e^{i~\alpha[ \phi, \psi ]}~\phi, e^{-i~\alpha[ \phi, \psi ]}~\psi \rangle,$$
where $\phi$ and $\psi$ denote the two explicit components, and $\alpha[ \phi, \psi ]$ is a generally non-zero "phase value" as some specific function of the two explicit components.
Is operator $\hat O_{\alpha}$ unitary for any arbitrary "phase function $\alpha$", or perhaps only for some suitable specific cases?
Might there be any other reasons to argue that operator $\hat O_{\alpha}$ is not a valid operator to be considered in quantum-mechanics?
