The quantum absolute phase observable

Abstract

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

12 pages, REVTEX