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International Journal of Modern Physics A

STOCHASTIC QUANTIZATION AND THE BERRY PHASE

Pratul Bandyopadhyay, K. Hajra

Published Jul 20, 1991• 2 citations
Publisher

Abstract

We study here the Berry phase from the point of view of stochastic quantization. The relativistic generalization of Nelson’s stochastic quantization procedure can be achieved when the Brownian motion process is taken into account in the internal space (apart from that in the external space). This effectively considers the particle as one that is stochastically extended, and nonrelativistic quantum mechanics is obtained in the sharp point limit. This can be formulated in terms of a gauge-field-theoretical extension of the particle. This inherent gauge field gives rise to the holonomy in a Hermitian line bundle, which appears as an extra phase factor in the adiabatic limit for a parameter-dependent Hamiltonian, and determines a unique connection. When the Hamiltonian is degenerate, the holonomy is defined in a complex vector bundle. In the case of a fermion, this Berry connection is found to be related to the Wess-Zumino term and can be considered to be of topological origin.

DOI
10.1142/s0217751x91001490

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We study here the Berry phase from the point of view of stochastic quantization. The relativistic generalization of Nelson’s stochastic quantization procedure can be achieved when the Brownian motion process is taken into account in the internal space (apart from that in the external space). This effectively considers the particle as one that is stochastically extended, and nonrelativistic quantum mechanics is obtained in the sharp point limit. This can be formulated in terms of a gauge-field-theoretical extension of the particle. This inherent gauge field gives rise to the holonomy in a Hermitian line bundle, which appears as an extra phase factor in the adiabatic limit for a parameter-dependent Hamiltonian, and determines a unique connection. When the Hamiltonian is degenerate, the holonomy is defined in a complex vector bundle. In the case of a fermion, this Berry connection is found to be related to the Wess-Zumino term and can be considered to be of topological origin.
The relativistic generalization of stochastic quantization helps us to introduce a stochastic-phase-space formulation when a relativistic quantum particle appears as a stochastically extended one. The nonrelativistic quantum mechanics is obtained in the sharp point limit. This also helps us to introduce a gauge-theoretical extension of a relativistic quantum particle when for a fermion the group structure of the gauge field is SU(2). The sharp point limit is obtained when we have a minimal contribution of the residual gauge field retained in the limiting procedure. This is shown to be equivalent to the geometrical approach to the phase-space quantization introduced by Klauder if it is interpreted in terms of a universal magnetic field acting on a free particle moving in a higher-dimensional configuration space when quantization corresponds to freezing the particle to its first Landau level. The geometric quantization then appears as a natural consequence of these two formalisms, since the Hermitian line bundle introduced there finds a physical meaning in terms of the inherent gauge field in stochastic-phase-space formulation or in the interaction with the magnetic field in Klauder quantization.
Heisenberg’s uncertainty relation is derived here from Nelson’s stochastic quantization procedure introducing Brownian motion in the internal space of a particle apart from that in the external space. This helps us to have a relativistic generalisation of Nelson’s formalism when a relativistic quantum particle appears to be a stochastically extended one. The nonrelativistic formalism is obtained in the sharp point limit. Quantum probability is found to be a manifestation of the internal geometry in this sharp point limit.
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.
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.