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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Dyon spin and statistics: A fiber-bundle theory of interacting magnetic and electric charges

John L. Friedman, Rafael D. Sorkin

Published Nov 15, 1979• 25 citations
Publisher

Abstract

The quantum mechanics of a system of nonrelativistic electrically and magnetically charged particles is formulated by introducing as the classical configuration space a certain U(1) bundle over the set of physical particle configurations. In this way one obtains a well-defined and single-valued classical action; and in the corresponding quantum theory, the origin of half-integral spin and of asymptotic Fermi-Dirac statistics in the description of composite systems of bosons finds its expression in terms of the bundle's structure.

DOI
10.1103/physrevd.20.2511

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The quantum mechanics of a system of nonrelativistic electrically and magnetically charged particles is formulated by introducing as the classical configuration space a certain U(1) bundle over the set of physical particle configurations. In this way one obtains a well-defined and single-valued classical action; and in the corresponding quantum theory, the origin of half-integral spin and of asymptotic Fermi-Dirac statistics in the description of composite systems of bosons finds its expression in terms of the bundle's structure.
The theory of fiber bundles provides a natural setting for the description of macroscopic quantum systems, wherein their classical and quantum features are represented by actions on the base manifolds and the fibers, respectively, of the relevant bundles. We provide realizations of this picture in the description of (a) quasiparticle excitations of many-body systems, especially those in superfluid helium, (b) the interplay between the microscopic and macroscopic dynamics in certain irreversible processes, such as that of a laser, and (c) local thermodynamic equilibrium. In particular, (b) involves the treatment of a dynamical system which is defined on a vector bundle.
In a purely nonrelativistic formulation, a Hamiltonian is obtained for the description of states of a charged particle with spin, interacting with an external electromagnetic field. For the specific cases of spin-1/2 and spin-1, the simplest choices for the Hamiltonian functions lead to a value for the (lowest order) intrinsic magnetic moment equal to the Bohr magneton in both cases. This is the same as is obtained in treatments that start from a relativistic formulation, as in the Dirac theory. The nonrelativistic and relativistic formulations both employ the ‘‘minimal coupling principle,’’ but the resulting Hamiltonians with couplings are, for very general considerations, not completely determined. However, the corresponding ambiguity that occurs in the nonrelativistic theory is also present in the relativistic formulation. The simpler nonrelativistic theory is more transparent in exhibiting these features of the problem and dispels the notion that spin (and the associated magnetic moment) require relativistic formulations for a proper understanding.
We consider configuration spaces of nonidentical pointlike particles. The physically motivated assumption that any two particles cannot be located at the same point in space–time leads to nontrivial topological structure of the configuration space. For a quantum mechanical description of such a system, we classify complex vector bundles over the configuration space and obtain potentials of topological origin, similar to those that occur in the fiber bundle approach to Dirac’s magnetic monopole or in Yang–Mills theory.
By formulating a path integral in the SU(2) fiber bundle space, a nonrelativistic spinor particle moving in external electromagnetic fields is studied. This formulation is a direct extension of Schulman’s [Phys. Rev. 176, 1588 (1968)] continuous treatment for spin; it unifies all spin states to a spherical top model. This theory enjoys a definite classical theory with spin–orbital interactions and the corresponding semiclassical approximation for the path integral. This method is also applicable to a particle moving in a more general Yang–Mills field.
A treatment of the spin-statistics relation in nonrelativistic quantum mechanics due to Berry and Robbins [Proc. R. Soc. London Ser. A 453, 1771–1790 (1997)] is generalized within a group-theoretical framework. The construction of Berry and Robbins is reformulated in terms of certain locally flat vector bundles over n-particle configuration space. It is shown how families of such bundles can be constructed from irreducible representations of the group SU(2n). The construction of Berry and Robbins, which leads to a definite connection between spin and statistics (the physically correct connection), is shown to correspond to the completely symmetric representations. The spin-statistics connection is typically broken for general SU(2n) representations, which may admit, for a given value of spin, both Bose and Fermi statistics, as well as parastatistics. The determination of the allowed values of the spin and statistics reduces to the decomposition of certain zero-weight representations of a (generalized) Weyl group of SU(2n). A formula for this decomposition is obtained using the Littlewood–Richardson theorem for the decomposition of representations of U(m+n) into representations of U(m)×U(n).
In this second paper in a series, we show that the the general statistical approach to nonrelativistic quantum mechanics developed in the first paper yields a representation of quantum spin and magnetic moments based on classical nonrelativistic spinning top models, using Euler angle coordinates. The models allow half-odd-integer spin and predict supraluminal speeds only for electrons and other leptons, which must be treated relativistically. The spin operators in the space-fixed frame satisfy the usual commutation rules, while those in the rotating body-fixed frame satisfy "left-handed" rules. The commutation rules are independent of the structure of the top, so all nonrelativistic rigidly rotating objects must have integer or odd-half-integer spin. Physical boundary conditions restrict all mixed spin states to involve only half-odd-integer or only integer spin eigenstates. For spin 1/2, the theory automatically yields a modified Pauli-Schrödinger equation. The Hamiltonian operator in this equation contains a rigid rotator term and a term involving the square of the magmetic field, as well as an interaction term having the usual form in spherically symmetric and some cylindrically symmetric models, valid for any magnetogyric ratio.