In 1968, Dashen and Sharp obtained a certain singular Lie algebra of local densities and currents from canonical commutation relations in nonrelativistic quantum field theory. The corresponding Lie group is infinite dimensional: the natural semidirect product of an additive group of scalar functions with a group of diffeomorphisms. Unitary representations of this group describe a wide variety of quantum systems, and have predicted previously unsuspected possibilities – notably, anyons and nonabelian anyons in two space dimensions. We present here foundational reasons why this semidirect product group serves as a universal kinematical group for quantum mechanics. We obtain thus a unified account of all possible quantum kinematics for systems with mass in an arbitrary physical space, and clarify the role played by topology in quantum mechanics. Our development does not require quantization of classical phase space; rather, the classical limit follows from the quantum mechanics. We also consider the relationship of our development to Heisenberg quantization.
In 1968, Dashen and Sharp obtained a certain singular Lie algebra of local densities and currents from canonical commutation relations in nonrelativistic quantum field theory. The corresponding Lie group is infinite dimensional: the natural semidirect product of an additive group of scalar functions with a group of diffeomorphisms. Unitary representations of this group describe a wide variety of quantum systems, and have predicted previously unsuspected possibilities – notably, anyons and nonabelian anyons in two space dimensions. We present here foundational reasons why this semidirect product group serves as a universal kinematical group for quantum mechanics. We obtain thus a unified account of all possible quantum kinematics for systems with mass in an arbitrary physical space, and clarify the role played by topology in quantum mechanics. Our development does not require quantization of classical phase space; rather, the classical limit follows from the quantum mechanics. We also consider the relationship of our development to Heisenberg quantization.
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.
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.
Unitary representations of diffeomorphism groups predict some unusual possibilities in quantum theory, including non-standard statistics and certain nonlinear effects. Many of the fundamental physical properties of “anyons” were first derived from their study by R. Menikoff, D.H. Sharp, and the author. This paper surveys new applications in two other domains: first (with Menikoff and Sharp) some surprising conclusions about the nature of quantum vortex configurations in ideal, incompressible fluids; second (with H.-D. Doebner) a natural description of dissipative quantum mechanics by means of a nonlinear Schrödinger equation different from the sort usually studied. Our equation follows from including a diffusion current in the equation of continuity.
The work can be considered as an essay on mathematical and conceptual structure of nonrelativistic quantum mechanics (QM) which is related here to some other (more general, but also to more special and “approximative”) theories. QM is here primarily reformulated in an equivalent form of a Poisson system on the phase space consisting of density matrices, where the “observables”, as well as “symmetry generators” are represented by a specific type of real valued (densely defined) functions, namely the usual quantum expectations of corresponding selfadjoint operators. It is shown in this paper that inclusion of additional (“nonlinear”) symmetry generators (i.e. “Hamiltonians”) into this reformulation of (linear) QM leads to a considerable extension of the theory: two kinds of quantum “mixed states” should be distinguished, and operator ‐ valued functions of density matrices should be used in the r ˆ ole of “nonlinear observables”. A general framework for physical theories is obtained in this way: By different choices of the sets of “nonlinear observables” we obtain, as special cases, e.g. classical mechanics on homogeneous spaces of kinematical symmetry groups, standard (linear) QM, or nonlinear extensions of QM; also various “quasiclassical approximations” to QM are all subtheories of the presented extension of QM - a version of the extended quantum mechanics (EQM). A general interpretation scheme of EQM extending the usual statistical interpretation of QM is also proposed. Eventually, EQM is shown to be (included into) a C -algebraic (hence linear) quantum theory. Mathematical formulation of these theories is presented. The presentation includes an analysis of problems connected with differentiation on infinite ‐ dimensional manifolds, as well as a solution of some problems connected with the work with only densely defined unbounded real‐valued functions on the (infinite dimensional) “phase space” which correspond to unbounded operators (generators) and to their nonlinear generalizations. Also “nonlinear deformations” of unitary representations of kinematical symmetry Lie groups are introduced. Possible applications are briefly discussed, and some specific examples are presented. The text contains also brief reviews of Hamiltonian classical mechanics, as well as of QM. Mathematical appendices make the paper nearly selfcontained.