The quantum phase problem: steps toward a resolution

Abstract

Defining the observable ${\bf ϕ}$ canonically conjugate to the number observable ${\bf N}$ has long been an open problem in quantum theory. The problem stems from the fact that ${\bf N}$ is bounded from below. In a previous work we have shown how to define the absolute phase observable ${\bf Φ}\equiv |{\bfϕ}|$ 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.