We consider in general terms dynamical systems with finite-dimensional, non-simply connected configuration-spaces. The fundamental group is assumed to be finite. We analyze in full detail those ambiguities in the quantization procedure that arise from the non-simply connectedness of the classical configuration space. We define the quantum theory on the universal cover but restrict the algebra of observables $Ø$ to the commutant of the algebra generated by deck-transformations. We apply standard superselection principles and construct the corresponding sectors. We emphasize the relevance of all sectors and not just the abelian ones.
We consider in general terms dynamical systems with finite-dimensional, non-simply connected configuration-spaces. The fundamental group is assumed to be finite. We analyze in full detail those ambiguities in the quantization procedure that arise from the non-simply connectedness of the classical configuration space. We define the quantum theory on the universal cover but restrict the algebra of observables Ø to the commutant of the algebra generated by deck-transformations. We apply standard superselection principles and construct the corresponding sectors. We emphasize the relevance of all sectors and not just the abelian ones.
This paper is concerned with the relationship between the group G of all symmetries of an unconstrained dynamical system and the space of its quantum states. The chief aim is to establish the validity of the claim that "quantizing a system" means deciding what G is and determining all projective unitary representations of G. The mathematical tool suited to this purpose is the theory of central extensions of an arbitrary group G by the circle group T (and the closely related cohomology group H2 (G, T)). The fundamental structural property is that there is a universal state space [Formula: see text] embedding all states, a unitary representation space of the universal central extension [Formula: see text] of G, on which the Pontryagin dual H2 (G, T)⋆ of H2 (G, T) operates as a group of su-perselection rules, so that the superselection sectors are [Formula: see text] and observables are self-adjoint operators Vh → Vh. This leads to a classification of time-evolution operators respecting the symmetries by H2 (G, T) and a general understanding of "anomalies" as arising from a mismatch of sectors to which symmetry transformations and time-evolution refer. For the special case of a Lie group G of symmetries, [Formula: see text] of G coincides with [Formula: see text] the universal covering group, if G is semisimple. More generally, depending on whether or not G is semisimple and/or simply connected, inequivalent quantizations are of topological, algebraic or "mixed" origin; in the algebraic case, the relationship between anomalous conservation laws and invariant dynamics is explicitly given. Illustrative examples worked out include the Euclidean group E (d) and the (connected) Poincaré group P (d, 1) for arbitrary spatial dimension d and the discrete groups (Z/N)2 and Z2 appropriate for planar square crystals — in particular, the absence of "fractional" angular momentum in the plane is established and the difficulties specific to (2,1)-dimensional quantum field theories precisely formulated. The paper concludes by noting that a number of ambiguities generally supposed to affect traditional procedures of quantization resolve themselves satisfactorily in this approach. Mathematically, the paper is self-contained as far as the concepts and results are concerned; occasionally, proofs of well-known facts are reworked when they serve to illustrate the value of the symmetry point of view. The theory of universal central extensions of groups, including important special cases, is treated in detail in an appendix by M. S. Raghunathan, in view of its significance for quantization and the absence of a similar account in the literature.
Quantization is defined as the act of assigning an appropriate C*-algebra [Formula: see text] to a given configuration space Q, along with a prescription mapping self-adjoint elements of [Formula: see text] into physically interpretable observables. This procedure is adopted to solve the problem of quantizing a particle moving on a homogeneous locally compact configuration space Q=G/H. Here [Formula: see text] is chosen to be the transformation group C*-algebra corresponding to the canonical action of G on Q. The structure of these algebras and their representations are examined in some detail. Inequivalent quantizations are identified with inequivalent irreducible representations of the C*-algebra corresponding to the system, hence with its superselection sectors. Introducing the concept of a pre-Hamiltonian, we construct a large class of G-invariant time-evolutions on these algebras, and find the Hamiltonians implementing these time-evolutions in each irreducible representation of [Formula: see text]. “Topological” terms in the Hamiltonian (or the corresponding action) turn out to be representation-dependent, and are automatically induced by the quantization procedure. Known “topological” charge quantization or periodicity conditions are then identically satisfied as a consequence of the representation theory of [Formula: see text].
We explain why, in a configuration space that is multiply connected, i.e., whose fundamental group is nontrivial, there are several quantum theories, corresponding to different choices of topological factors. We do this in the context of Bohmian mechanics, a quantum theory without observers from which the quantum formalism can be derived. What we do can be regarded as generalizing the Bohmian dynamics on R3N to arbitrary Riemannian manifolds, and classifying the possible dynamics that arise. This approach provides a new understanding of the topological features of quantum theory, such as the symmetrization postulate for identical particles. For our analysis we employ wave functions on the universal covering space of the configuration space.
We reconsider the (non-relativistic) quantum theory of indistinguishable particles on the basis of Rieffel’s notion of [Formula: see text]-algebraic (“strict”) deformation quantization. Using this formalism, we relate the operator approach of Messiah and Greenberg (1964) to the configuration space approach pioneered by Souriau (1967), Laidlaw and DeWitt-Morette (1971), Leinaas and Myrheim (1977), and others. In dimension [Formula: see text], the former yields bosons, fermions, and paraparticles, whereas the latter seems to leave room for bosons and fermions only, apparently contradicting the operator approach as far as the admissibility of parastatistics is concerned. To resolve this, we first prove that in [Formula: see text] the topologically non-trivial configuration spaces of the second approach are quantized by the algebras of observables of the first. Secondly, we show that the irreducible representations of the latter may be realized by vector bundle constructions, among which the line bundles recover the results of the second approach. Mathematically speaking, representations on higher-dimensional bundles (which define parastatistics) cannot be excluded, which render the configuration space approach incomplete. Physically, however, we show that the corresponding particle states may always be realized in terms of bosons and/or fermions with an unobserved internal degree of freedom (although based on non-relativistic quantum mechanics, this conclusion is analogous to the rigorous results of the Doplicher–Haag–Roberts analysis in algebraic quantum field theory, as well as to the heuristic arguments which led Gell-Mann and others to qcd (i.e. Quantum Chromodynamics)).
In section 6.5 we looked at the problem of evaluating quantum mechanical path integrals in curvilinear co-ordinates. For the model considered there (a point particle in two dimensions) the choice of curvilinear co-ordinates was capricious, but the use of curvilinear co-ordinates is natural in describing constrained systems. For example, suppose the particle in two dimensions was forced to move on a ring of radius R . The problem of path roughness would then go away, since the scalar curvature would be constant along the path. However, we acquire a new problem in that the configuration space for the particle (the circle) is no longer simply connected – a notion that we shall define later. This, in turn, can induce a new term in the action. We stress that this term is in no way a replacement for that due to roughness. For one thing, it is of order ħ , rather than of order ħ 2 . Most importantly it is not unique , leading to an ambiguity in quantisation. This ambiguity is a general property of quantisation on non-simply-connected manifolds. In this chapter we shall sketch how such O( ħ ) terms (topological ‘charges’) arise naturally in the path integral formalism which, being an ‘integral’ over paths in the configuration space of the system, is ideally suited for handling non-trivial configuration spaces.
We reconsider the (non-relativistic) quantum theory of indistinguishable particles on the basis of Rieffel's notion of C*-algebraic (`strict') deformation quantization. Using this formalism, we relate the operator approach of Messiah and Greenberg (1964) to the configuration space approach due to Souriau (1967), Laidlaw and DeWitt (1971), Leinaas and Myrheim (1977), and others. The former allows parastatistics, whereas the latter apparently leaves room for bosons and fermions only. This seems to contradict the operator approach as far as the admissibility of parastatistics is concerned. To resolve this, we first prove that the topologically nontrivial configuration spaces of the second approach are quantized by the algebras of observables of the first. Second, we show that the irreducible representations of the latter may be realized by vector bundle constructions, which include the line bundles of the second approach: representations on higher-dimensional bundles (which define parastatistics) cannot be excluded a priori. However, we show that the corresponding particle states may always be realized in terms of bosons and/or fermions with an unobserved internal degree of freedom. Although based on non-relativistic quantum mechanics, this conclusion is analogous to the rigorous results of the Doplicher-Haag-Roberts analysis in algebraic quantum field theory, as well as to the heuristic arguments which led Gell-Mann and others to QCD.