Phase Operator Problem and Macroscopic Extension of Quantum Mechanics

Manuel Gadella

Masanao Ozawa

To find an operator representation of the phase variable of a single-mode electromagnetic field, the Schrodinger representation is extended to a larger Hilbert space including states with infinite excitation by nonstandard analy- sis, so that the self-adjoint phase operator is acting on the extended Hilbert space.It is shown that this phase operator is a Naimark extension of the optimal probability operator-valued measure for the phase parameter already known in the quantum estimation theory, and is naturally considered as the controversial limit of the phase operators on finite dimensional spaces recently proposed by Pegg and Barnett.Eventually, two of recent promising attempts to obtain the correct statistics of the phase variable in quantum mechanics is synthesized in the present framework based on new mathematical ideas concerning the number system including infinite and infinitesimal numbers.

Berthold‐Georg Englert, Krzysztof Wódkiewicz

The concept of intrinsic and operational observables in quantum mechanics is introduced. In any realistic description of a quantum measurement that includes a macroscopic detecting device, it is possible to construct from the statistics of the recorded raw data a set of operational quantities that correspond to the intrinsic quantum-mechanical observable. This general approach is illustrated by the example of an operational measurement of the position and the momentum of a particle, as well as by an analysis of the operational detection of the phase of an optical field. For the latter we identify the intrinsic phase operator and report its explicit form.

Franklin E. Schroeck

Nicolaas P. Landsman

M. V. Lokajíček

Time flow has been embodied in time-dependent Schroedinger equation representing one of the foundations of quantum mechanics. Pauli's criticism (1933) has, however, indicated that the assumptions concerning representation Hilbert space have led to some contradictions. Many authors have tried to solve this discrepancy practically without any actual success. The reason may be seen in that two different problems have been mixed: the problem of Pauli (being more general and more important) and non-unitarity of exponential phase operator of linear harmonic oscillator introduced by Dirac, as demonstrated in 1964. The problem will be discussed in a broad historical context and a solution based on extension of representation Hilbert space will be shown.