Abstract
Defining the observable φ canonically conjugate to the number observable N has long been an open problem in quantum theory. The problem stems from the fact that N is bounded from below. In a previous work we have shown how to define the absolute phase observable Φ≡|φ| 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.
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Gour, G. The Quantum Phase Problem: Steps Toward a Resolution. Foundations of Physics 32, 907–926 (2002). https://doi.org/10.1023/A:1016059229336
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DOI: https://doi.org/10.1023/A:1016059229336