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arXiv:hep-th/0004173 · Modern Physics Letters A

BERRY CONNECTIONS AND INDUCED GAUGE FIELDS IN QUANTUM MECHANICS ON SPHERE

Hitoshi Ikemori, S. Kitakado, Hideharu Otsu, Toshiro Sato

Published Jun 14, 2000• 2 citations
arXivHTMLPDF

Abstract

Quantum mechanics on sphere Sn is studied from the viewpoint that the Berry's connection has to appear as a topological term in the effective action. Furthermore we show that this term is the Chern–Simons term of gauge variables that correspond to the extra degrees of freedom of the enlarged space.
DOI
10.1142/s0217732300001365

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In non-adiabatic cases the topological action associated with Berry's phase and the corresponding effective Hamiltonian are obtained by path-integral method. We also give the non-adiabatic transition probability amplitude in the first-order approximation. It is thereby shown that the Berry's phase and the induced gauge structure have universality of existence. As an example, dynamics of induced monople relating to the Bitter-Dubber's experiment is analysed in terms of induced gauge field.
I describe a condensed matter system in terms of interacting quantum fields. In this electronic system, gauge interactions of non electromagnetic origin are generated effectively with a topological term in the effective action and play a leading role in describing the physics. Using the renormalization group and large-N expansion technique, I analyze the low energy sector of a U (1) × U (1) model in three dimensional space-time with a potential of sixth order in the scalar fields coupled to mixed Chern-Simons gauge fields. From the quantum effective potential, I derive various renormalization group functions and the fixed points associated with a quantum phase transition.
The author considers the quantum mechanics coupled to the gauge field which has the Chern-Simons term as the action. The generalized Hamiltonian formalism of the system is constructed and canonical and path-integral quantizations are performed. In solving the Gauss law constraint, the gauge field is represented by a multivalued function and the charge density operator of the particle. The gauge potential is singular and also induces a singular magnetic field through the magnetic flux is finite. By a phase transformation, the singular gauge field is eliminated. Then the transformed wavefunction absorbing the gauge field induces a novel phase factor under 2 pi -rotation in space. Thus the author shows that the transformed wavefunction describes exotic spin state. The extension of these analyses to the relativistic case is also presented.
A hidden gauge theory structure of quantum mechanics which is invisible in its conventional formulation is uncovered. Quantum mechanics is shown to be equivalent to a certain Yang–Mills theory with an infinite-dimensional gauge group and a nondynamical connection. It is defined over an arbitrary symplectic manifold which constitutes the phase space of the system under consideration. The "matter fields" are local generalizations of states and observables; they assume values in a family of local Hilbert spaces (and their tensor products) which are attached to the points of phase space. Under local frame rotations they transform in the spinor representation of the metaplectic group Mp(2N), the double covering of Sp(2N). The rules of canonical quantization are replaced by two independent postulates with a simple group-theoretical and differential-geometrical interpretation. A novel background-quantum split symmetry plays a central role.