Formal relation between Pegg-Barnett and Paul quantum phase frameworks

Abstract

The problem of defining a Hermitian quantum phase operator is nearly as old as quantum mechanics itself. Throughout the years, a number of solutions was proposed, ranging from abstract operator formalisms to phase-space methods. In this work, we connect two of the most prominent approaches: Pegg-Barnett and Paul formalisms, by proving that the Paul formalism is equivalent to the Pegg-Barnett formalism applied to an infinitely amplified state. Our findings fill in a conceptual gap in the understanding of the quantum phase problem.