Physics – Quantum Physics
Scientific paper
1995-05-09
Phys.Rev. E51 (1995) 4114-4131
Physics
Quantum Physics
Latex file
Scientific paper
10.1103/PhysRevE.51.4114
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 Schr\"{o}dinger 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 Schr\"{o}dinger problem, the "free noise" can also be extended to any infinitely divisible probability law, as covered by the L\'{e}vy-Khintchine formula. Since the relativistic Hamiltonians $|\nabla |$ and $\sqrt {-\triangle +m^2}-m$ 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-Schr\"{o}dinger evolution is analyzed in detail. The relativistic covariance of related wave
Garbaczewski Piotr
Klauder John R.
Olkiewicz Robert
No associations
LandOfFree
The Schroedinger Problem, Levy Processes Noise in Relativistic Quantum Mechanics does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The Schroedinger Problem, Levy Processes Noise in Relativistic Quantum Mechanics, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The Schroedinger Problem, Levy Processes Noise in Relativistic Quantum Mechanics will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-2110