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Physical Review A

Quantum-invariant processes in phase space

Alfredo Luis

Published Nov 30, 2004• 3 citations
Publisher

Abstract

We show the formal equivalence between the phase-space representations of transformations and quantum states. We study invariant quantum input-output transformations in phase space and in Hilbert space. We show that all invariant processes are linear transformations while the converse is not true. Some relevant examples of application of these ideas are examined.

DOI
10.1103/physreva.70.052118

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We show the formal equivalence between the phase-space representations of transformations and quantum states. We study invariant quantum input-output transformations in phase space and in Hilbert space. We show that all invariant processes are linear transformations while the converse is not true. Some relevant examples of application of these ideas are examined.
A phase space representation is defined for quantum mechanical state functions. General requirements of quantum mechanics and some additional, very reasonable restrictions lead to a representation that is uniquely determined apart from a scaling. With one choice of this scaling, the representation is equivalent to that of Bargmann, although different aspects are emphasized here. Equivalence between position and momentum representations leads to a phase space representation that is invariant to, or at most rescaled by, Fourier transform. Phase space equivalents of some common functions are presented including Cartesian Gaussian basis functions and the hydrogen atom ground state function.
The algebra of generalized linear quantum canonical transformations is examined in the prespective of Schwinger's unitary-canonical basis. Formulation of the quantum phase problem within the theory of quantum canonical transformations and in particular with the generalized quantum action-angle phase space formalism is established and it is shown that the conceptual foundation of the quantum phase problem lies within the algebraic properties of the quantum canonical transformations in the quantum phase space. The representations of the Wigner function in the generalized action-angle unitary operator pair for certain Hamiltonian systems with the dynamical symmetry are examined. This generalized canonical formalism is applied to the quantum harmonic oscillator to examine the properties of the unitary quantum phase operator as well as the action-angle Wigner function.
The algebra of generalized linear quantum canonical transformations is examined in the perspective of Schwinger's unitary-canonical operator basis. Formulation of the quantum phase problem within the theory of quantum canonical transformations and in particular with the generalized quantum action-angle phase space formalism is established and it is shown that the conceptual foundation of the quantum phase problem lies within the algebraic properties of the canonical transformations in the quantum phase space. The representations of the Wigner function in the generalized action-angle unitary operator pair for certain Hamiltonian systems with dynamical symmetry is examined. This generalized canonical formalism is applied to the quantum harmonic oscillator to examine the properties of the unitary quantum phase operator as well as the action-angle Wigner function.
Complex numbers appear in the Hilbert space formulation of quantum mechanics, but not in the formulation in phase space. Quantum symmetries are described by complex, unitary or antiunitary operators defining ray representations in Hilbert space, whereas in phase space they are described by real, true representations. Equivalence of the formulations requires that the former representations can be obtained from the latter and vice versa. Examples are given. Equivalence of the two formulations also requires that complex superpositions of state vectors can be described in the phase space formulation, and it is shown that this leads to a nonlinear superposition principle for orthogonal, pure-state Wigner functions. It is concluded that the use of complex numbers in quantum mechanics can be regarded as a computational device to simplify calculations, as in all other applications of mathematics to physical phenomena.