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THE PHASE REPRESENTATION OF COVARIANT PHASE OBSERVABLES

Juha-Pekka Pellonpää

Published Jan 1, 2001• 0 citations
OpenAlex

Abstract

Abstract. Covariant phase observables are obtained by defining simple conditions for mappings from the set of phase wave functions (unit vectors of the Hardy space) to the set of phase probability densities. The existence of phase probability density for any phase wave function, the existence of interference effects, and the natural phase shift covariance are those simple conditions. The nonlocalizability of covariant phase observables is proved.

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Abstract. Covariant phase observables are obtained by defining simple conditions for mappings from the set of phase wave functions (unit vectors of the Hardy space) to the set of phase probability densities. The existence of phase probability density for any phase wave function, the existence of interference effects, and the natural phase shift covariance are those simple conditions. The nonlocalizability of covariant phase observables is proved.
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Abstract Covariant phase observables constitute a simple solution of the quantum phase problem of a single‐mode optical field. They share three important properties: their range of values is the phase interval [0,2π), they are covariant under shifts generated by the number operator (which is necessary for coherent state phase measurements), and they have the uniform phase distribution in number states. Moreover, some phase observables have been measured (e.g. the phase observable associated to the Q‐function). The canonical phase observable has some additional properties which distinguish it from other covariant phase observables: it generates number shifts, it is uniquely associated to the polar decomposition of the lowering operator (Dirac's idea [1]), it has a projection valued covariant dilation (Newton's extension [2]), and it has a projection valued discretization (the Pegg‐Barnett formalism [3]). The single‐mode covariant phase theory can easily be extended to the two‐mode theory of covariant phase difference observables. Finally, most of all phase theories have connections to the covariant phase theory.