The quantum phase problem: steps toward a resolution

Abstract

Defining the observable $\varphi$ canonically conjugate to the number observable $\mathbf{N}$ has long been an open problem in quantum theory. The problem stems from the fact that $\mathbf{N}$ is bounded from below. In a previous work we have shown how to define the absolute phase observable $\mathbf{\Phi }\equiv |\varphi |$ by suitably restricting the Hilbert space of $x$ and $p$ like variables. Here we show that also from the classical point of view, there is no rigorous definition for the phase even though it's absolute value is well defined.

21 pages, REVTEX