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On the Stochastic Quantization Method: Characteristics and Applications to Singular Systems

Masahiko Kanenaga, Mikio Namiki

Published Jan 1, 1996• 0 citations
OpenAlex

Abstract

Introducing the generalized Langevin equation, we extend the stochastic quantization method so as to deal with singular dynamical systems beyond the ordinary territory of quantum mechanics. We also show how the uncertainty relation is built up to the quantum mechanical limit with respect to fictitious time, irrespective of its initial value, within the framework of the usual stochastic quantization method.

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Introducing the generalized Langevin equation, we extend the stochastic quantization method so as to deal with singular dynamical systems beyond the ordinary territory of quantum mechanics. We also show how the uncertainty relation is built up to the quantum mechanical limit with respect to fictitious time, irrespective of its initial value, within the framework of the usual stochastic quantization method.
The ill-posed problem of quantizing space-time is replaced by a more determined and well-posed problem of regularizing quantum dynamics. \nThe problem is then to eliminate the Heisenberg singularity from quantum mechanics as economically as possible. \n \nThe concepts of regular and singular groups are explained and the Heisenberg singularity defined. This singularity infests not only the theory of space-time, \nbut also the Bose-Einstein statistics and the theory of the gauge fields and interactions. It is responsible for most of the infinities of present quantum field theory. \n \nThe key new conceptual step is to turn attention from observables to "dynamicals", the observable-valued-functions of time which actually enters into the Heisenberg dynamical equations. The dynamicals have separate algebras from the algebra and Lie algebra of the observables. This reconception allows for the possibility of clock-system entanglement that is missing from the usual singular dynamics, and implied by the concept of quantum space-time. \n\nThe dynamical Lie algebra and the resulting Lie group are regularized for an example system, the time-dependent isotropic harmonic oscillator of arbitrary finite dimension. The result is a quantize space-time, but also momentum-energy and every other dynamical variable in the theory. This method is readily extended to general dynamic quantum systems.
The stochastic quantization method is developed in a possible “Heisenberg” operator theory of stochastic processes so designed as to keep a formal analogy to quantum mechanics. The general theory is first formulated for stochastic processes of the Wiener type and then its application to quantization of boson fields is presented. It seems that the operator formalism is convenient to examine implications of the stochastic quantization method. Within the theoretical framework we develop the “Feynman-Dyson” approach to field propagators in perturbation theory. This approach enables us to simplify and systematize practical calculations in comparison with the Langevin equation method. In the Appendix we formulate the Green function approach to field propagators using the characteristic functional.
Langevin methods have proved useful in simulating lattice field theories of many kinds. These methods are essentially a practical realization of the stochastic quantization approach to quantum mechanical systems. The authors wish to show that the Langevin method is effective in calculations of non-relativistic quantum mechanical systems. For a one-dimensional system a more conventional approach through the diagonalization of the Hamiltonian matrix using the method of Sturm sequencing is more efficient. However, in two and higher dimensions where the Hamiltonian is no longer tridiagonal the Langevin scheme is more efficient and accurate. It follows therefore that Langevin methods may actually have a great deal to offer in quantum mechanical problems in high dimensions especially in cases where the potential does not have spherical symmetry.
The stochastic representation of quantum mechanics is used to give a purely stochastic treatment of the quantum Langevin equation. A general coupling involving positions and velocities of the particle and harmonic modes, with arbitrary dependence on the particle's position, is dealt with. Summing over the modes in the path integral and using stochastic integrals, the author obtains explicitly the reduced measure, that is, the stochastic process representing the time evolution of the small system in the presence of noise. Contact is made with the operatorial representation, and a relation is established between the correlations of the noise in both representations. A general condition on the coupling constants is given for recovering time locality in the continuum limit, and the effects of renormalisation are re-analysed within this purely stochastic approach.
This is the fourth paper of a series devoted to mathematically derive the Schrödinger equation using stochastic constructs. We then show that Quantum Mechanics has a stochastic support by following two paths: we first present a stochastic derivation already known in the literature and show that it is equivalent to the characteristic function derivation, made by us in the first paper of this series. However, this approach does not furnish the true stochastic dynamics of the quantum mechanical system, but only an averaged one. We then present a new derivation of the Schrödinger equation based on Langevin equations that present all features of a true stochastic system. These two approaches then improve our understanding of Quantum Mechanics to encompass stochastic behavior.