Phase Operator Problem and Macroscopic Extension of Quantum Mechanics

To find the Hermitian phase operator of a single-mode electromagnetic field in quantum mechanics, the Schrödinger representation is extended to a larger Hilbert space augmented by states with infinite excitation by nonstandard analysis. The Hermitian phase operator is shown to exist on the extended Hilbert space. This operator is naturally considered as the controversial limit of the approximate phase operators on finite dimensional spaces proposed by Pegg and Barnett. The spectral measure of this operator is a Naimark extension of the optimal probability operator-valued measure for the phase parameter found by Helstrom. Eventually, the two promising approaches to the statistics of the phase in quantum mechanics is synthesized by means of the Hermitian phase operator in the macroscopic extension of the Schrödinger representation.