A Possible Generalization of Dirac’s Phase Operator and its Relation to Paul’s Phase Operator

Abstract

In this paper we generalize Dirac’s phase operator \(\frac {1}{\sqrt {N}}a\) by defining a new phase operator \(\sqrt {\frac {a+\tilde {a}^{\dagger }}{a^{\dagger }+\tilde {a}}}\) in doubled Fock space, where \(\tilde {a}\) is a fictitious mode which annihilates the fictitious vacuum state \(\left \vert \tilde {0}\right \rangle \). It turns out that \(\sqrt {\frac {a+\tilde {a}^{\dagger }}{a^{\dagger }+\tilde {a}}}\) corresponds to a classical phase in the entangled state representation and is unitary. Remarkably, \(\left \langle \tilde {0}\right \vert \sqrt {\frac {a+\tilde {a}^{\dagger }}{a^{\dagger }+\tilde {a}}}\left \vert \tilde {0}\right \rangle \) is just the Paul’s phase operator whose antinormally ordered form is \({\vdots } \frac {1}{\sqrt {N}}a\vdots \). We also employ the method of integration within ordered product of operators to obtain the Fock representation of Paul’s phase operator, from which one can see how it deffers from Susskind-Glogower’s phase operator.