Construction of the phase operator using logarithm of the annihilation operator

TL;DR

This paper proposes a method to construct a phase operator in quantum mechanics by defining an analytic function of the annihilation operator and using its logarithm, overcoming previous limitations.

Contribution

It introduces a novel approach to define the phase operator via the logarithm of the annihilation operator, avoiding a known mathematical lemma.

Findings

Successfully constructs the phase operator using the logarithm of the annihilation operator.

Provides a condition to circumvent the lemma that excludes the phase operator.

Demonstrates the mathematical consistency of the constructed phase operator.

Abstract

We investigate a lema that excludes existence of the phase operator and present a condition to avoid the lema. A method for construction of an analytic function 'f' of the annihilation operator 'a' is given. f(z) is analytic on some compact domain that does not separate the complex plane. Using these results we obtain ln a. Since [N ,-i ln a]=i, we can use ln a to construct an operator $\Phi$, which satisfies the definition of the phase operator.