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Zeitschrift für Naturforschung A

Quantum Mechanics as a Generalization of Nambu Dynamics to the Weyl-Wigner Formalism

Iwo Białynicki‐Birula, P. J. Morrison

Published Feb 1, 1997• 0 citations
Publisher

Abstract

Abstract It is shown that Nambu dynamics can be generalized to any number of dimensions by replacing the 0(3) algebra, a prominent feature of Nambu's formulation, by an arbitrary Lie algebra. For the infinite dimensional algebra of rotations in phase space one obtains quantum mechanics in the Weyl-Wigner representation from the generalized Nambu dynamics. Also, this formulation can be cast into a canonical Hamiltonian form by a natural choice of canonically conjugate variables.

DOI
10.1515/zna-1997-1-205

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Abstract It is shown that Nambu dynamics can be generalized to any number of dimensions by replacing the 0(3) algebra, a prominent feature of Nambu's formulation, by an arbitrary Lie algebra. For the infinite dimensional algebra of rotations in phase space one obtains quantum mechanics in the Weyl-Wigner representation from the generalized Nambu dynamics. Also, this formulation can be cast into a canonical Hamiltonian form by a natural choice of canonically conjugate variables.
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The Wigner problem, i.e. the investigation of general quantum mechanical commutation relations consistent with the Heisenberg evolution equations of a given shape, is studied. We follow a recently proposed generalization of this idea within which the classical analogy is postulated only for the shape of the time evolution equations and not for a Hamiltonian itself. This links our investigation to the problem of alternative Hamiltonians of classical mechanics and to canonically inequivalent phase-space descriptions of physical systems governed by the same Newton equations of motion. Warned that the time evolution and the other symmetry generators may be given ambiguously even in the formalism of classical mechanics, we do not a priori assume the shape of their quantum analogues. Instead we only require that the set of basic algebraic relations, which quantum mechanical observables are to obey, has a Lie algebra structure. Such a requirement appears to be sufficient to find solutions for simple oscillator-like dynamics. New algebras of quantum mechanical observables are not constructed as a linear envelope of the Heisenberg algebra, and their representations reflect physical results unexpected in the framework of the canonical approach. We illustrate our approach in detail for the example of the one-dimensional harmonic oscillator using the representation of the generalized coherent states.
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.
General structural features of dynamical theories can be exhibited in relations between classical and quantum mechanics; the essential structure is a Lie algebra of basic dynamical-variable fanctions providing transformationgroup infinitesimal generators. It is shown that the particular representation chosen is not important for the dynamical structure analysis by considering possibilities of transcribing classical and quantum mechanics each into the natural representation of the other. This leads to an outlining of the formal theory of generalized dynamics by constructing a class of Lie algebras including those of classical and quantum mechanics as special cases. (L.N.N.)
Quantum mechanics is formulated on a quantum mechanical phase space. The algebra of observables and states is represented by an algebra of functions on a phase space that fulfils a certain coherence condition, expressing the quantum mechanical superposition principle. The trace operation is an integration over phase space. In the case where the canonical variables independently run from - infinity to + infinity formalism reduces to the representation of quantum mechanics by Wigner distributions. However, the notion of coherent algebra allows one to apply the formalism to spaces for which the Wigner mapping is not known. The quantum mechanics of a particle in a plane in polar coordinates is discussed as an example.
A formalism is presented in which quantum particle dynamics can be developed on its own rather than `quantization' of an underlying classical theory. It is proposed that the unification of probability and dynamics should be considered as the basic feature of quantum theory. Arguments are given to show that when such a unification is attempted at the configuration space level, the wave funtions of Schro¨\ddot{o}o¨dinger theory appear as the natural candidates for the desired unification. A *-algebra AQ\mathcal{A}_{Q}AQ​ of (not necessarily bounded) linear operators acting on an appropriate dense set of these wave functions appears as the arena for quantum kinematics. A simple generalization of an existing formalism in noncommutative geometry is employed to develop the notion of generalized algebraic symplectic structure (GASS) which can accomodate classical and quantum symplectic structures as special cases. Quantum kinematics and dynamics is developed in in the framework of a noncommutative Hamiltonian system employing an appropriate GASS based in $ \mathcal{A}_{Q}$. The Planck constant is introduced at only one place -- in the quantum symplectic form; its appearance at conventional places is then automatic. Unitary Wigner symmetries appear as canonical transformations in the noncommutative Hamiltonian system. A straightforward treatment of quantum - classical correspondence is given in terms of appropriate GASSes.