Relativistic Brownian Motion

We solve the problem of formulating Brownian motion in a relativistically covariant framework in 1 + 1 and 3 + 1 dimensions. We obtain covariant Fokker-Planck equations with (for the isotropic case) a differential operator of invariant d'Alembert form. Treating the spacelike and timelike fluctuations separately, we show that it is essential to take into account the analytic continuation of " unphysical " fluctuations in order to achieve these results. 1 Introduction Nelson 1 , in 1966, constructed the Schrödinger equation from an analysis of Brownian motion by identifying the forward and backward average velocities of a Brownian particle with the real and imaginary parts of a wave function. He pointed out that the basic process involved is defined nonrelativistically, and can be used if relativistic effects can be safely neglected. The development of a relativistically covariant formulation of Brownian motion could therefore provide some insight into the structure of a relativistic quantum theory. In recent years, Brownian motion has been applied as a mechanism for the description of irreversible evolution of quantum mechnical states, for example, collapse of wave functions as result of measurement. One finds that the wave function with a stochastic term added to the Schrödinger equation (represented in the Hilbert space of states 2 or in their projective representation 3) evolves to a mixed state of alternative outcomes of a measurement with the same probability distribution as given by the calculation of a priori (Born) probabilities in the framework of the standard quantum theory 2,3. One of the motivations for the work of Adler and Horwitz 3 was based on the existence of a statistical mechanics, developed by Adler 4 and Adler and Millard 5 for the description of the equilibrium state of a general class of quantum fields; in this theory, the thermal average of commutator expressions take on the values implied by the familiar complex canonical quantum theory and quantum field theory. The Brownian motion corrections to the standard theories may be thought of as arising from the fluctuations around the equilibrium state. The development of a relativistically covariant theory of Brownian motion could make an extension of these ideas to relativistic quantum theory and quantum field theory more accessible. The program of stochastic quantization of Parisi and Wu 6 assumes the existence of relativistic wave equations, and applies a statistical approach closely related to path in-tegrals. The Wiener distribution of nonrelativistic Brownian motion, …