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International Journal of Geometric Methods in Modern Physics

Quantum mechanics without quantization: Representing a universal kinematical group

Gerald A. Goldin

Published Mar 27, 2025• 0 citations
Publisher

Abstract

A certain infinite-dimensional Lie group, obtained originally by regularizing and then exponentiating a singular local current algebra constructed from canonical fields, has been shown over several decades to describe a wide variety of quantum systems, via its continuous unitary representations. The group is a semidirect product of Schwartz space functions under pointwise addition with a diffeomorphism group under composition. Previously unsuspected possibilities were predicted this way, including anyons and nonabelian anyons in two-dimensional space. Last year David Sharp and I offered a much simpler, direct construction. We proposed fundamental reasons why this group serves as a universal kinematical group for quantum systems with mass in an arbitrary physical space. This paper beyond summarizing that construction, discusses more extensively what it means to obtain quantum mechanics without actually quantizing classical observables in phase space, and answers some fundamental questions. Directions are suggested for obtaining still greater generality of kinematical and dynamical possibilities by modifying the initial assumptions.

DOI
10.1142/s0219887825400419

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A certain infinite-dimensional Lie group, obtained originally by regularizing and then exponentiating a singular local current algebra constructed from canonical fields, has been shown over several decades to describe a wide variety of quantum systems, via its continuous unitary representations. The group is a semidirect product of Schwartz space functions under pointwise addition with a diffeomorphism group under composition. Previously unsuspected possibilities were predicted this way, including anyons and nonabelian anyons in two-dimensional space. Last year David Sharp and I offered a much simpler, direct construction. We proposed fundamental reasons why this group serves as a universal kinematical group for quantum systems with mass in an arbitrary physical space. This paper beyond summarizing that construction, discusses more extensively what it means to obtain quantum mechanics without actually quantizing classical observables in phase space, and answers some fundamental questions. Directions are suggested for obtaining still greater generality of kinematical and dynamical possibilities by modifying the initial assumptions.
About 15 years ago, we (Heinz-Dietrich Doebner and I) proposed a special type of nonlinear modification of the usual Schrödinger time-evolution equation in quantum mechanics. Our equation was motivated by certain unitary representations of the group of diffeomorphisms of physical space, in the framework of either nonrelativistic local current algebra or quantum Borel kinematics. Subsequently, we developed this and related approaches to nonlinearity in quantum mechanics considerably further, to incorporate theories of measurement, groups of nonlinear gauge transformations, symmetry and invariance properties, unification of a large family of nonlinear perturbations, and possible physical contexts for quantum nonlinearity. Some of our results and highlights of some open questions are summarized.
The ill-posed problem of quantizing space-time is replaced by a more determined and well-posed problem of regularizing quantum dynamics. \nThe problem is then to eliminate the Heisenberg singularity from quantum mechanics as economically as possible. \n \nThe concepts of regular and singular groups are explained and the Heisenberg singularity defined. This singularity infests not only the theory of space-time, \nbut also the Bose-Einstein statistics and the theory of the gauge fields and interactions. It is responsible for most of the infinities of present quantum field theory. \n \nThe key new conceptual step is to turn attention from observables to "dynamicals", the observable-valued-functions of time which actually enters into the Heisenberg dynamical equations. The dynamicals have separate algebras from the algebra and Lie algebra of the observables. This reconception allows for the possibility of clock-system entanglement that is missing from the usual singular dynamics, and implied by the concept of quantum space-time. \n\nThe dynamical Lie algebra and the resulting Lie group are regularized for an example system, the time-dependent isotropic harmonic oscillator of arbitrary finite dimension. The result is a quantize space-time, but also momentum-energy and every other dynamical variable in the theory. This method is readily extended to general dynamic quantum systems.
It is possible that a complete physical theory can be written entirely in terms of operators such as current densities, rather than in terms of field operators. The current densities in these models correspond in general to distributions of unbounded operators. Such a theory is reviewed for the case of nonrelativistic quantum mechanics. It is found that one can exponentiate the current algebra to obtain a group, which may then be represented by unitary (hence bounded) operators in Hilbert space. This procedure is analogous to exponentiating the canonical commutation relations and obtaining the Weyl group. For nonrelativistic quantum mechanics without spin, the group is the semidirect product J∧K Schwartz space J(R3) with a group K of certain C∞ diffeomorphisms from R3 onto itself. For f ∈ J and φ ∈ K, the composition mapping (f,φ)→f∘φ defines the semidirect product law. The Gel'fand-Vilenkin formalism for ``nuclear Lie groups'' is suitable for the representation theory of such a group. Almost all of the physical information is contained in a cylindrical measure μ on J′, the dual of the nuclear space J. The Fourier transform of μ can be interpreted as an expectation functional with respect to the state of lowest energy. The nonrelativistic Fock representation (including the theory of n particles in a box) is examined in this formalism. Conditions on μ are systematically developed which suffice to recover all of the infinitesimal generators on a common, dense, invariant domain in the Hilbert space. For the Weyl group slightly weaker conditions than for the semidirect product case would suffice. Gaussian measures in J′, as well as the measures defining n-particle representations of J ∧K, satisfy these conditions.
A formalism of quantum kinematics which rests on the regular representation of a Lie group is proposed. This representation leads to explicit canonical commutation relations for non-Abelian dynamical variables which, together with the Lie algebra, define the kinematic algebra. Generalised equations of motion for the group-parameter-dependent operators, as well as generalised wave equations, are introduced over the group manifold considered as a background arena. The formalism proposed affords a method of geometric quantisation stemming directly from the observed symmetries of a system.
Constructing a classical mechanical system associated with a given quantum-mechanical one entails construction of a classical phase space and a corresponding Hamiltonian function from the available quantum structures and a notion of coarser observations. The Hilbert space of any quantum-mechanical system naturally has the structure of an infinite-dimensional symplectic manifold ('quantum phase space'). There is also a systematic, quotienting procedure which imparts a bundle structure to the quantum phase space and extracts a classical phase space as the base space. This works straightforwardly when the Hilbert space carries weakly continuous representation of the Heisenberg group and one recovers the linear classical phase space . We report on how the procedure also allows extraction of nonlinear classical phase spaces and illustrate it for Hilbert spaces being finite dimensional (spin- j systems), infinite dimensional but separable (particle on a circle) and infinite dimensional but non-separable (polymer quantization). To construct a corresponding classical dynamics, one needs to choose a suitable section and identify an effective Hamiltonian. The effective dynamics mirrors the quantum dynamics provided the section satisfies conditions of semiclassicality and tangentiality.