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arXiv:quant-ph/0512127

A Possible Generalization of Quantum Mechanics

Yu Peng Tian

Published Dec 16, 2005• 0 citations
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Abstract

A "minimal" generalization of Quantum Mechanics is proposed, where the Lagrangian or the action functional is a mapping from the (classical) states of a system to the Lie algebra of a general compact Lie group, and the wave function takes values in the corresponding group algebra. This formalism admits a probability interpretation and a suitable dynamics, but has no obvious classical correspondence. Allowing the Lagrangian or the action functional to take values in a general Lie algebra instead of only the real number field (actually the u(1) algebra) enlarges the extent of possible physical laws that can describe the real world. The generalized quantum dynamics of a point particle in a background gauge field is given as an example, which realizes the gauge invariance by a Wilson line structure and shows that some Schrodinger-like equation can be deduced within this formalism. Some possible developments of this formalism are also discussed.

DOI
10.48550/arxiv.quant-ph/0512127

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A "minimal" generalization of Quantum Mechanics is proposed, where the Lagrangian or the action functional is a mapping from the (classical) states of a system to the Lie algebra of a general compact Lie group, and the wave function takes values in the corresponding group algebra. This formalism admits a probability interpretation and a suitable dynamics, but has no obvious classical correspondence. Allowing the Lagrangian or the action functional to take values in a general Lie algebra instead of only the real number field (actually the u(1) algebra) enlarges the extent of possible physical laws that can describe the real world. The generalized quantum dynamics of a point particle in a background gauge field is given as an example, which realizes the gauge invariance by a Wilson line structure and shows that some Schrodinger-like equation can be deduced within this formalism. Some possible developments of this formalism are also discussed.
This article presents a mathematical framework for group-theoretical formalism of quantum mechanics and discusses the geometric quantization of the classical systems associated with a Lie group. The classical-quantum correspondence is realized by identifying quantum observable algebra and its classical analogue with the set of distributions with compact supports on a Lie group and on the associated Lie algebra respectively, both having a convolution-type associative algebraic structure. The general mathematical constructs are illustrated by studying systems associated with the Heisenberg-Weyl group. It is shown that the exponential mapping from Heisenberg-Weyl algebra to the corresponding Lie group gives Weyl quantization.
We start by rewriting classical mechanics in a quantum mechanical fashion and point out that the only difference with quantum theory re- sides at one point. There is a classical analogon of the collapse of the wavefunction and an extension of the usual Born rule is proposed which might solve this problem. We work only with algebra’s over the real (com- plex) numbers, general number fields of finite characteristic allowing for finite dimensional representations of the commutation relations are not considered given that such fields are not well ordered and do not give rise to a well defined probability interpretation. Our theory generalizes however to discrete spacetimes and finite dimensional algebra’s. Looking at physics this way, spacetime itself distinguishes itself algebraically by means of well chosen commutation relations and there is further nothing special about it meaning it has also a particle interpretation just like any other dynamical variable. Likewise, there is no reason for the dynamics to be Hamiltonian and therefore we have a nonconservative formulation of quantum physics at hand. The harmonic oscillator (amongst few oth- ers) distinguishes itself because the algebra forms a finite dimensional Lie algebra; the classical and quantum (discrete) harmonic oscillator are stud- ied in a some more generality and examples are given which are neither classical, nor quantum.
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.)
In this paper we consider a general action principle for mechanics written by means of the elements of a Lie algebra. We study the physical reasons why we have to choose precisely a Lie algebra to write the action principle. By means of such an action principle we work out the equations of motion and a technique to evaluate perturbations in a general mechanics that is equivalent to a general interaction picture. Classical or quantum mechanics come out as particular cases when we make realizations of the Lie algebra by derivations into the algebra of products of functions or operators, respectively. Later on we develop in particular the applications of the action principle to classical and quantum mechanics, seeing that in this last case it agrees with Schwinger's action principle. The main contribution of this paper is to introduce a perturbation theory and an interaction picture of classical mechanics on the same footing as in quantum mechanics.
A weakness which has previously seemed unavoidable in particle interpretations of quantum mechanics (such as in the de Broglie-Bohm model) is addressed here and a resolution proposed. The weakness in question is the lack of action and reaction occurring between the model's field (or "pilot wave") and the particle. Although the field acts on the particle, the particle does not act back on the field. It is shown here that this rather artificial feature is, in fact, not necessary and can be fully eliminated while remaining consistent with the usual quantum predictions. Mathematically this amounts to demonstrating that there exists a suitable Lagrangian density function which generates equations coinciding with quantum mechanics yet incorporates the desired action and reaction. As a by-product, an appealing explanation emerges to another long-standing question, namely why the mathematical formalism of quantum mechanics seems only to be describing fields when measurements generally detect localised particles. A further bonus is that the hitherto unrelated concept of a gauge transformation is found to arise naturally as an essential part of the formalism. In particular, the phase S of the gauge transformation is seen to be the action function describing the hidden motion of the particle.
A generalization of quantum mechanics is given in which the central mathematical concept is the analogue of the action in classical mechanics. It is therefore applicable to mechanical systems whose equations of motion cannot be put into Hamiltonian form. It is only required that some form of least action principle be available. It is shown that if the action is the time integral of a function of velocity and position (that is, if a Lagrangian exists), the generalization reduces to the usual form of quantum mechanics. In the classical limit, the quantum equations go over into the corresponding classical ones, with the same action function. As a special problem, because of its application to electrodynamics, and because the results serve as a confirmation of the proposed generalization, the interaction of two systems through the agency of an intermediate harmonic oscillator is discussed in detail. It is shown that in quantum mechanics, just as in classical mechanics, under certain circumstances the oscillator can be completely eliminated, its place being taken by a direct, but, in general, not instantaneous, interaction between the two systems. The work is non-relativistic throughout.