The quantum absolute phase observable

Abstract

Defining the observable ${\bf ϕ}$ canonically conjugate to the number observable ${\bf N}$ has long been an open problem in quantum theory. Here we show how to define the absolute phase observable ${\bf Φ}\equiv |{\bfϕ}|$ by suitably restricting the Hilbert space of $x$ and $p$ like variables. This ${\bf Φ}$ is actually the absolute value of the phase and has the correct classical limit. A correction to the ``cosine'' ${\bf C}$ and ``sine'' ${\bf S}$ operators of Carruthers and Nieto is obtained.