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AIP conference proceedings

Wigner quantization problem for external forces

E. Kapuścik, A. Horzela

Published Jan 1, 1998• 2 citations
Publisher

Abstract

The Wigner problem of finding most general quantum mechanical commutation relations compatible with equations of motion is discussed for particles influenced by quantized external forces. The problem is solved here under an assumption that Wigner commutation relations determine minimal Lie algebras with initial observables as generators.
DOI
10.1063/1.57098

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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.
In quantum mechanics, physical observables are represented by operators on a certain Hilbert space. The question of how such operators commute, has been a matter of discussion. In the standard perspective, the operators corresponding to the position and momentum of a system are assumed to satisfy the canonical commutation relations. It is known that these relations imply that the Hamilton and Heisenberg equations of motion are compatible as operator equations. However, Wigner showed that the inverse statement is not true. Therefore, it is a much weaker constraint to impose the compatibility of the equations of motion. For any physical system, this results in a set of compatibility conditions, which form the core of Wigner quantization. The key to finding operators that are subject to the compatibility conditions is provided by Lie superalgebras and their representations. Lie superalgebras can be defined as algebras generated by odd elements satisfying particular superbracket relations. Using these defining relations, Lie superalgebra generators can be found that obey the compatibility conditions. The Lie superalgebra elements act as operators on a vector space if we consider Lie superalgebra representations. Various physical systems are investigated in this thesis in the context of Wigner quantization. For each of these systems solutions are found in terms of Lie superalgebra generators, after which specific representations are examined. In such representations, the focus lies on determining particular physical properties of the system. Most of the studied systems are harmonic oscillator models. First, we examine a set-up of coupled harmonic oscillators, for which the interaction is represented by an interaction matrix. Then we focus on the angular momentum content of a 3N3N3N-dimensional harmonic oscillator. Finally, our attention goes to two one-dimensional systems, namely the free particle and the Berry-Keating-Connes Hamiltonian. The latter of these Hamiltonians is notorious for its possible connection with the Riemann hypothesis. All of the aforementioned Hamiltonians have been extensively investigated in the context of canonical quantization, so that our results can be compared to the well-known canonical case.
The problem of whether or not the equations of motion of a quantum system determine the commutation relations was posed by E.P.Wigner in 1950. A similar problem (known as "The Inverse Problem in the Calculus of Variations") was posed in a classical setting as back as in 1887 by H.Helmoltz and has received great attention also in recent times. The aim of this paper is to discuss how these two apparently unrelated problems can actually be discussed in a somewhat unified framework. After reviewing briefly the Inverse Problem and the existence of alternative structures for classical systems, we discuss the geometric structures that are intrinsically present in Quantum Mechanics, starting from finite-level systems and then moving to a more general setting by using the Weyl-Wigner approach, showing how this approach can accomodate in an almost natural way the existence of alternative structures in Quantum Mechanics as well.
The relation between canonical commutation rules and a shape of acting force is analysed. It is shown that canonical commutation relations and Newtonian equations of motion for a single particle dynamics do imply the force to be of the Lorentz type but the inverse is not true. The example of a single particle motion in a constant magnetic field shows that equations of motion allow an alternative to canonical commutation relations to exist and it is a Lie algebra. The algebra found is of the type of algebras found in studies leading towards noncommutative geometry approaches to physical problems.
In a system of coupled harmonic oscillators, the interaction can be represented by a real, symmetric and positive definite interaction matrix.The quantization of a Hamiltonian describing such a system has been done in the canonical case.In this paper, we take a more general approach and look at the system as a Wigner quantum system.Hereby, one does not assume the canonical commutation relations, but instead one just requires the compatibility between the Hamilton and Heisenberg equations.Solutions of this problem are related to the Lie superalgebras gl(1|n) and osp(1|2n).We determine the spectrum of the considered Hamiltonian in specific representations of these Lie superalgebras and discuss the results in detail.We also make the connection with the well-known canonical case.
It has been shown by Wigner that the equations of motion of a quantum-mechanical system with a hamiltonian H=12p2+V(q)H=\frac{1}{2}{p}^{2}+V(q)H=21​p2+V(q) need not imply the commutation relation i[p,q]=1i[p,q]=1i[p,q]=1. Particular examples are furnished by the free particle and the harmonic oscillator. On the other hand, Wigner points out that the above commutation rule does follow from the equations of motion if V(q)V(q)V(q) is of the form V=aq+bV=aq+bV=aq+b or V=aq3V=a{q}^{3}V=aq3. In the present note, this latter result is generalized to the case of an arbitrary potential V=aqn+bV=a{q}^{n}+bV=aqn+b, where nnn is an odd integer; the last-named potentials are, in fact, particular cases of those satisfying a general criterion contained in the article.