## Abstract

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 nonconstant diffusion coefficient. Proper time is an increasing stochastic process and we derive a probabilistic generalization of the equation (d τ)^{2} = - (1/c^{2})dX_{v} dX_{v}. A random time-change transformation provides the bridge between the t and the τ domain. In the τ domain, we obtain an double-struck M sign ^{4}-valued Markov process with singular and constant diffusion coefficient. The square modulus of the Klein-Gordon solution is an invariant, nonintegrable density for this Markov process. It satisfies a relativistically covariant continuity equation.

Original language | English (US) |
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Pages (from-to) | 4846-4856 |

Number of pages | 11 |

Journal | Journal of Mathematical Physics |

Volume | 42 |

Issue number | 10 |

DOIs | |

State | Published - Oct 2001 |

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics