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arXiv:hep-th/9309158 · Modern Physics Letters A

TRANSITION AMPLITUDES WITHIN THE STOCHASTIC QUANTIZATION SCHEME

H. HÜFFEL, H. NAKAZATO

Published Oct 20, 1994• 4 citations
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Abstract

Quantum mechanical transition amplitudes are calculated within the stochastic quantization scheme for the free nonrelativistic particle, the Harmonic oscillator and the nonrelativistic particle in a constant magnetic field; we conclude with free Grassmann quantum mechanics.
DOI
10.1142/s0217732394002793

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As an alternative formulation of quantum mechanics, path integral is based on the notion of transition amplitude which gives the wave function of a quantum system at a time t f by acting on the wave function at an earlier time ti.We show that for a general quadratic form for the Lagrangian of the system, transition amplitude has the form f (t f -ti)e i h S class., where S class. is the classical action.We then present an algebraic method to evaluate the function f (t f -ti) without refereing to the path integral calculations.We examine the presented method to the cases of free particle and harmonic oscillator and obtain their propagators.
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.