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International Journal of Modern Physics A

THE GEOMETRIC PHASE AND THE SCHWINGER TERM IN SOME MODELS

Harald Grosse, Edwin Langmann A

Published Aug 20, 1992• 12 citations
Publisher

Abstract

We discuss the quantization of fermions interacting with external fields and observe the occurrence of equivalent as well as inequivalent representations of the canonical anticommutation relations. Implementability of gauge and axial gauge transformations leads to generators which fulfil an algebra of current with a Schwinger term. This term can be written as a cocycle and leads to the boson-fermion correspondence. Transport of a quantum-mechanical system along a closed loop of parameter space may yield a geometric phase. We discuss models for which nonintegrable phase factors are obtained from the adiabatic parallel transport. After second quantization, one obtains, in addition, a Schwinger term. Depending on the type of transformation, a subtle relationship between these two obstructions can occur. We indicate finally how we may transport density matrices along closed loops in parameter space.

DOI
10.1142/s0217751x92002301

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We discuss the quantization of fermions interacting with external fields and observe the occurrence of equivalent as well as inequivalent representations of the canonical anticommutation relations. Implementability of gauge and axial gauge transformations leads to generators which fulfil an algebra of current with a Schwinger term. This term can be written as a cocycle and leads to the boson-fermion correspondence. Transport of a quantum-mechanical system along a closed loop of parameter space may yield a geometric phase. We discuss models for which nonintegrable phase factors are obtained from the adiabatic parallel transport. After second quantization, one obtains, in addition, a Schwinger term. Depending on the type of transformation, a subtle relationship between these two obstructions can occur. We indicate finally how we may transport density matrices along closed loops in parameter space.
We discuss a generalization of the conventional bosonization procedure to the case of current-current interactions which have a natural representation in terms of current instead of fermion-number-density operators. A consistent bosonization procedure requires a geometrical quantization of the Hamiltonian action of ${\mathit{W}}_{\mathrm{\ensuremath{\infty}}}$ on its coadjoint orbits. An integrable example of a nontrivial realization of this symmetry is presented by the Calogero-Sutherland model. For an illustrative nonintegrable example we consider transverse gauge interactions and calculate the fermion Green function.
We present an algebraic method to derive the structure at the basis of the mapping of bosonic algebras of creation and annihilation operators into fermionic algebras, and vice versa, introducing a suitable identification between bosonic and fermionic generators. The algebraic structure thus obtained corresponds to a deformed Grassmann-type algebra, involving anticommuting Grassmann-type variables. The role played by the latter in implementing gauge invariance in second quantization within our procedure is then discussed. This discussion includes the application of the mapping to the case of the bosonic and fermionic harmonic oscillator Hamiltonians.
The exchange phase for two spins is studied here from the point of view of the quantization of a fermion in the framework of Nelson’s stochastic mechanics. This introduces a direction vector attached to a space–time point depicting the spin degrees of freedom. In this formalism, a fermion appears as a scalar particle attached with a magnetic-flux quantum, and a quantum spin can be described in terms of an SU(2) gauge bundle. This helps us to recast the Berry–Robbins formalism where the exchange phase appears as an unfamiliar geometric phase arising out of the ‘exchange rotation’ in a transported spin basis in terms of gauge currents. However, for polarized fermions, the exchange phase is found to be given by the Berry phase.
Even though its classical equations of motion are then left invariant, when an action is redefined by an additive total derivative or divergence term (in time, in the case of a mechanical system) such a transformation induces non-trivial consequences for the system’s canonical phase space formulation. This is even more true and then in more subtle ways for the canonically quantized dynamics, with, in particular, an induced transformation in the unitary configuration space representation of the Heisenberg algebra being used for the quantum system. When coupled to a background gauge field, such considerations become crucial for a proper understanding of the consequences for the system’s quantum dynamics of gauge transformations of that classical external background gauge field, while under such transformations, the system’s degrees of freedom, abstract quantum states and quantum dynamics are certainly strictly invariant. After a detailed analysis of these different points in a general context, these are then illustrated specifically in the case of the quantum Landau problem with its classical external background magnetic vector potential for which the most general possible parametrized gauge choice is implemented herein. The latter discussion aims as well to clarify some perplexing statements in the literature regarding the status of gauge choices to be made for the magnetic vector potential for that quantum system. The role of the global spacetime symmetries of the Landau problem and their gauge invariant Noether charges is then also emphasized.
The exchange phase for two spins is studied here from the point of view of the quantization of a fermion in the framework of Nelson’s stochastic mechanics. This introduces a direction vector attached to a space–time point depicting the spin degrees of freedom. In this formalism, a fermion appears as a scalar particle attached with a magnetic-flux quantum, and a quantum spin can be described in terms of an SU(2) gauge bundle. This helps us to recast the Berry–Robbins formalism where the exchange phase appears as an unfamiliar geometric phase arising out of the ‘exchange rotation’ in a transported spin basis in terms of gauge currents. However, for polarized fermions, the exchange phase is found to be given by the Berry phase.