The Schroedinger Problem, Levy Processes Noise in Relativistic Quantum Mechanics

Abstract

The main purpose of the paper is an essentially probabilistic analysis of relativistic quantum mechanics. It is based on the assumption that whenever probability distributions arise, there exists a stochastic process that is either responsible for temporal evolution of a given measure or preserves the measure in the stationary case. Our departure point is the so-called Schrödinger problem of probabilistic evolution, which provides for a unique Markov stochastic interpolation between any given pair of boundary probability densities for a process covering a fixed, finite duration of time, provided we have decided a priori what kind of primordial dynamical semigroup transition mechanism is involved. In the nonrelativistic theory, including quantum mechanics, Feyman-Kac-like kernels are the building blocks for suitable transition probability densities of the process. In the standard "free" case (Feynman-Kac potential equal to zero) the familiar Wiener noise is recovered. In the framework of the Schrödinger problem, the "free noise" can also be extended to any infinitely divisible probability law, as covered by the Lévy-Khintchine formula. Since the relativistic Hamiltonians $|\nabla |$ and $\sqrt{-△+m^2}-m$ are known to generate such laws, we focus on them for the analysis of probabilistic phenomena, which are shown to be associated with the relativistic wave (D'Alembert) and matter-wave (Klein-Gordon) equations, respectively. We show that such stochastic processes exist and are spatial jump processes. In general, in the presence of external potentials, they do not share the Markov property, except for stationary situations. A concrete example of the pseudodifferential Cauchy-Schrödinger evolution is analyzed in detail. The relativistic covariance of related wave

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References in corpus (2)

• Lévy flights in random environments
• Natural boundaries for the Smoluchowski equation and affiliated diffusion processes

Cited by in corpus (21)

• Relativistic Brownian Motion
• Ornstein-Uhlenbeck-Cauchy Process
• Spectral analysis of subordinate Brownian motions in half-line
• Levy flights and nonlocal quantum dynamics
• Feynman-Kac Kernels in Markovian Representations of the Schroedinger Interpolating Dynamics
• Levy flights in confining potentials
• Lévy flights in inhomogeneous environments
• Stochastic Emergent Quantum Gravity
• Burgers' Flows as Markovian Diffusion Processes
• One-dimensional quasi-relativistic particle in the box
• Stochastic Quantization on Lorentzian Manifolds
• Cauchy Noise and Affiliated Stochastic Processes
• Burgers velocity fields and dynamical transport processes
• Nonnegative Feynman-Kac Kernels in Schrödinger's Interpolation Problem
• Lévy-Schrödinger wave packets
• Analytic Continuation of Stochastic Mechanics
• Schroedinger's Interpolating Dynamics and Burgers' Flows
• Comment on "Why quantum mechanics cannot be formulated as a Markov process"
• Markov processes and generalized Schroedinger equations
• (Nonequilibrium) dynamics of diffusion processes with non-conservative drifts
• Stochastic Quantization of Relativistic Theories