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Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento B

Generalized phase-space descriptions as linear representations of operators

Manfred Schmutz

Published Jan 1, 1975• 8 citations
Publisher
DOI
10.1007/bf02737684

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This work is a continuation of our previous works concerning linear canonical transformations and phase space representation of quantum theory. It is mainly focused on the description of an approach which allows to establish spinorial representation of linear canonical transformations. This description is started with the presentation of a suitable parameterization of linear canonical transformations which permits to represent them with special pseudo-orthogonal transformations in an operator space. Then the establishment of the spinorial representation is deduced using the well-known relation existing between special pseudo-orthogonal and spin groups. The cases of one dimension and general multidimensional theory are both studied.
In this paper I consider the general mathematical problem of the polar decomposition of an operator in a linear space. Extending the space makes it possible to define a unitary operator related to the original nonhermitean one. By Stone's theorem this guarantees the existence of a phase operator in the extended space. The connection with supersymmetry is pointed out. Applying the general results to harmonic oscillator creation and annihilation operators we regain a phase description originally introduced by Newton. Projecting the phase operator from the extended space to the original one, we find a phase representation for the Boson operators. Introducing the conjugate rotation operator, one can describe the oscillator dynamics in the phase representation. The connection with the Barnett–Pegg phase operator is pointed out.
Representations of quantum state vectors by complex phase space amplitudes, complementing the description of the density operator by the Wigner function, have been defined by applying the Weyl–Wigner transform to dyadic operators, linear in the state vector and anti-linear in a fixed 'window state vector'. Here aspects of this construction are explored, and a connection is established with Gabor's 'windowed Fourier transform'. The amplitudes that arise for simple quantum states from various choices of windows are presented as illustrations. Generalized Bargmann representations of the state vector appear as special cases, associated with Gaussian windows. For every choice of window, amplitudes lie in a corresponding linear subspace of square-integrable functions on phase space. A generalized Born interpretation of amplitudes is described, with both the Wigner function and a generalized Husimi function appearing as quantities linear in an amplitude and anti-linear in its complex conjugate. Schrödinger's time-dependent and time-independent equations are represented on phase space amplitudes, and their solutions described in simple cases.