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International Journal of Modern Physics A

EQUIVALENCE OF STOCHASTIC, KLAUDER AND GEOMETRIC QUANTIZATION

K. Hajra, P. Bandyopadhyay

Published Mar 10, 1992• 6 citations
Publisher

Abstract

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.

DOI
10.1142/s0217751x92000545

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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.
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 geometrical approach to phase-space quantization introduced by Klauder [KQ] is interpreted in terms of a universal magnetic field acting on a free particle moving in a higher dimensional configuration space; quantization corresponds to freezing the particle to its first Landau level. The Geometric Quantization [GQ] scheme appears as the natural technique to define the interaction with the magnetic field for a particle on a general Riemannian manifold. The freedom of redefining the operators' ordering makes it possible to select that particular definition of the Hamiltonian which is adapted to a specific polarization; in this way the first Landau level acquires the expected degeneracy. This unification with GQ makes it clear how algebraic relations between classical observables are or are not preserved under quantization. From this point of view all quantum systems appear as the low energy sector of a generalized theory in which all classical observables have a uniquely assigned quantum counterpart such that Poisson bracket relations are isomorphic to the commutation relations.
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.
I: Stochastic Quantum Mechanics.- 1 / Nonrelativistic Stochastic Quantum Mechanics.- 1.1 Stochastic Values and Stochastic Phase Spaces.- 1.2. Sharp Localizability and Systems of Imprimitivity.- 1.3. Stochastic Localizability and Systems of Co variance.- 1.4. Phase-Space Representations of the Galilei Group.- 1.5. Localizability in Nonrelativistic Stochastic Phase Spaces.- 1.6. Stochastic Phase-Space Systems of Covariance.- 1.7. Resolution Generators and Gauge Freedom in Stochastic Phase Spaces.- 1.8. Stochastic Probability Currents and Their Sharp-Point Limits.- 1.9. Propagators in Nonrelativistic Stochastic Phase Spaces.- 1.10. Quantum Stochastic Processes and Phase-Space Path Integrals.- 1.11. Semiclassical Propagators in Stochastic Phase Spaces.- 1.12. The Physical Nature and the Measurement of Proper Wave Functions and of Stochastic Position and Momentum.- Notes.- 2 / Relativistic Stochastic Quantum Mechanics.- 2.1. The Nonexistence of Relativistic Probability Currents at Sharp Space time Points.- 2.2. The Unfeasibility of Sharp Localization in Relativistic Quantum Mechanics.- 2.3. Phase-Space Representations of the Poincare Group.- 2.4. Resolution Generators and Relativistic Phase-Space Representations.- 2.5. Localizability in Relativistic Stochastic Phase Spaces.- 2.6. Systems of Covariance in Relativistic Stochastic Phase Spaces.- 2.7. Relativistic Canonical Commutation Relations and Gauge Freedom.- 2.8. Relativistically Covariant and Conserved Probability and Charge Currents.- 2.9. Propagators in Relativistic Stochastic Phase Spaces.- 2.10. Relativistic Extended Quantum Particles in External Electromagnetic Fields.- Notes.- 3 / Statistical Mechanics on Stochastic Phase Spaces.- 3.1. Phase-Space Representations of Quantum Statistical Mechanics.- 3.2. Informational Equivalence and Completeness of POV Measures.- 3.3. Generalized Stochastic Phase-Space Systems of Covariance.- 3.4. Classical and Quantum Mechanics in L2 (?N).- 3.5. Classical and Quantum Statistical Mechanics in Master Liouville Space ?(?N).- 3.6. Quantum Observables as Density Distributions on Stochastic Phase Spaces.- 3.7. Unified Formulations of Classical and Quantum Boltzmann Equations and Scattering Cross-Sections.- Notes.- II: Quantum Spacetime.- 4 / Reciprocity Theory and the Geometrization of Stochastic Quantum Mechanics.- 4.1. The Operational Foundations of Special and General Relativity.- 4.2. Coordinate Fluctuation Amplitudes for Coherent Flows of Quantum Test Particles.- 4.3. Quantum Lorentz Frames and Minkowski Quantum Spacetime.- 4.4. General Quantum Spacetimes and Stochastic Geometries.- 4.5. Reciprocity Theory and Born's Quantum Metric Operator.- 4.6. Unitary Internal Symmetries and Generalized Representations of Relativistic Canonical Commutation Relations.- Notes.- 5 / Field Theory on Quantum Spacetime.- 5.1. Free Scalar Fields on Minkowski Quantum Spacetime.- 5.2. LSZ Formalism and Causality on Quantum Spacetime.- 5.3. Lagrangian Field Theory on Quantum Spacetime.- 5.4. Dirac Bispinors and Free Fields on Quantum Spacetime.- 5.5. Photon Localizability and the Electromagnetic Field on Quantum Spacetime.- 5.6. Reciprocally Invariant Quantum Electrodynamics for Extended Fermions.- Notes.- Appendix A / Elements of Measure Theory.- Appendix B / Elements of Operator Theory on Hilbert Spaces.- Notes.- References.
It is shown that in the context of stochastic-phase-space formulations of quantum mechanics a nonlocal type of nonrelativistic as well as relativistic dynamics for a massive spin-zero particle moving in an external electromagnetic field can be formulated in ${L}^{2}(\ensuremath{\Gamma})$ space. In a sharp-point limit $s\ensuremath{\rightarrow}+0$ the nonrelativistic dynamics merges into the conventional local model. The relativistic dynamics is gauge invariant, it can be formulated covariantly in the relativistic phase space Mm{M}_{m}Mm​, it possesses a covariant propagator, it leaves the space of positive-energy solutions of the Klein-Gordon equation invariant, and it gives rise to a covariant and conserved probability current on stochastic phase space. Furthermore, for states describing a particle moving at nonrelativistic speeds in relation to a given inertial frame, the relativistic propagation contracts into its nonrelativistic counterpart. The sharp-point limit $s\ensuremath{\rightarrow}+0$ of the relativistic SSS matrix also exists in each such inertial frame. The necessity of interactions acting at nonsharp configuration points for a consistent formulation of covariant relativistic one-particle dynamics is traced to the question of the nonexistence of covariant position operators in relativistic theories. The mandatory character of this feature is shown to be firmly rooted in the theory of quantum measurement.