Modern Physics Letters A

STOCHASTIC INTERPRETATION OF POSITION-MOMENTUM UNCERTAINTY RELATION

K. Hajra

Published Aug 10, 1989• 3 citations

Abstract

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.

DOI: 10.1142/s0217732389001684

Related papers

EQUIVALENCE OF STOCHASTIC, KLAUDER AND GEOMETRIC QUANTIZATION

K. Hajra, P. Bandyopadhyay

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.

International Journal of Modern Physics AMar 10, 19926 citations

Relativistic Brownian motion and the space–time approach to quantum mechanics

Sisir Roy

An attempt has been made to extend the stochastic quantization procedures introduced by Nelson in the nonrelativistic case to the relativistic case in the four-dimensional Finsler space. The space–time in the microdomain is considered to be quantized and a more general concept of probability is needed to have a consistent and complete theory of quantum mechanics.

Journal of Mathematical PhysicsJan 1, 198013 citations

STOCHASTIC QUANTIZATION AND THE BERRY PHASE

Pratul Bandyopadhyay, K. Hajra

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.

International Journal of Modern Physics AJul 20, 19912 citations

EQUIVALENCE OF STOCHASTIC AND HYDRODYNAMICAL QUANTIZATION

K. Hajra

On the basis of earlier work on relativistic generalization of Nelson’s stochastic quantization procedure introducing an anisotropy in the internal space it is shown here that in the non-relativistic limit the equivalence of stochastic and hydrodynamical quantizations formulated respectively by Nelson and Takabayashi can be achieved. Some difficulties regarding interpretation in both the formalisms may possibly be removed from the geometry of internal space-time.

International Journal of Modern Physics AAug 10, 19891 citation

Stochastic mechanics of a relativistic spinless particle

Gian Fabrizio De Angelis

An extension of Nelson’s stochastic mechanics to the relativistic domain is proposed. To each pure state of a spinless relativistic quantum particle corresponds a Markov process t ]ξt, where the random variable ξt represents, at every time t, the space position of the particle in the sense of Newton and Wigner. The process t ]ξt is not a diffusion but the usual Nelson’s theory is restored in the nonrelativistic limit.

Journal of Mathematical PhysicsJun 1, 199014 citations

Stochastic quantum mechanics and quantum spacetime : a consistent unification of relativity and quantum theory based on stochastic spaces

Eduard Prugovečki

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.

Jan 1, 198410 citations