CHINESE JOURNAL OF PHYSICS VOL. 38, NO. 6 DECEMBER 2000

Stochastic Mechanics in Complex Space

M. S. Wang

Department of Physics and Center for Complex Systems, National Central University,

Chungli, Taiwan 320, R.O.C.
(Received June 22, 2000)

We show that a backward stochastic system with an imaginary diffusion coefficient in complex space is equivalent to a real space quantum system if it obeys a Newtonian-like dynamical law and its final spatial distribution is constrained to be in real space. The ensemble averages of the position, momentum, angular momentum and energy of the stochastic processes are the quantum mechanical expectation values of the corresponding operators. The product of the mean-square deviations of the position and momentum of the stochastic process agreees with the corresponding uncertainty relation of quantum mechanics. The equation of motion is invariant under the combined transformations of changing the time $t\to -t$ and taking the complex conjugate of the equation. The stochastic motion reduces to the classical motion in the limit where the de Broglie wavelengh of the particle vanishes.

PACS. 03.65.Bz $\;$- Foundations, theory of measurement, miscellaneous theories.
PACS. 02.50.Ga $\;$- Markov processes.
PACS. 02.50.Cw - Probability theory.

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