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International Journal of Modern Physics A

EQUIVALENCE OF STOCHASTIC AND HYDRODYNAMICAL QUANTIZATION

K. Hajra

Published Aug 10, 1989• 1 citation
Publisher

Abstract

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.

DOI
10.1142/s0217751x8900128x

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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.
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.
Markov diffusions corresponding to solutions of the Klein-Gordon equation are introduced in a similar way as in Nelson's stochastic mechanics. A relativistic version of the Nelson-Newton law for Markov diffusions is established. Stochastic generalizations of the proper time and of the relativistic action functional are proposed and a stochastic version of the relativistic principle of least action is stated. The formalism can be regarded as a quantization procedure leading from classical special relativity to the Klein-Gordon equation.
The new scheme of stochastic quantization is proposed. This quantization procedure is equivalent to the deformation of an algebra of observables in the manner of deformation quantization with an imaginary deformation parameter (the Planck constant). We apply this method to the models of nonrelativistic and relativistic particles interacting with an electromagnetic field. In the first case we establish the equivalence of such a quantization to the Fokker-Planck equation with a special force. The application of the proposed quantization procedure to the model of a relativistic particle results in a relativistic generalization of the Fokker-Planck equation in the coordinate space, which in the absence of the electromagnetic field reduces to the relativistic diffusion (heat) equation. The stationary probability distribution functions for a stochastically quantized particle diffusing under a barrier and a particle in the potential of a harmonic oscillator are derived.
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.