On the stochastic mechanics of the free relativistic particle

Details On the stochastic mechanics of the free relativistic particle On the stochastic mechanics of the free relativistic particle

2001-08-31

arxiv.org/abs/quant-ph/0108139v1

Journal of Mathematical Physics, 42 (2001), 4846-4856

Physics

Quantum Physics

Scientific paper

Given a positive energy solution of the Klein-Gordon equation, the motion of the free, spinless, relativistic particle is described in a fixed Lorentz frame by a Markov diffusion process with non-constant diffusion coefficient. Proper time is an increasing stochastic process and we derive a probabilistic generalization of the equation $(d\tau)^2=-\frac{1}{c^2}dX_{\nu}dX_{\nu}$. A random time-change transformation provides the bridge between the $t$ and the $\tau$ domain. In the $\tau$ domain, we obtain an $\M^4$-valued Markov process with singular and constant diffusion coefficient. The square modulus of the Klein-Gordon solution is an invariant, non integrable density for this Markov process. It satisfies a relativistically covariant continuity equation.

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