dorsal/arxiv
View SchemaThe Schroedinger Problem, Levy Processes Noise in Relativistic Quantum Mechanics
| Authors | P. Garbaczewski, J. R. Klauder, R. Olkiewicz |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9505003 |
| URL | https://arxiv.org/abs/quant-ph/9505003 |
| DOI | 10.1103/PhysRevE.51.4114 |
| Journal | Phys.Rev. E51 (1995) 4114-4131 |
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 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
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"abstract": "The main purpose of the paper is an essentially probabilistic analysis of\nrelativistic quantum mechanics. It is based on the assumption that whenever\nprobability distributions arise, there exists a stochastic process that is\neither responsible for temporal evolution of a given measure or preserves the\nmeasure in the stationary case. Our departure point is the so-called\nSchr\\\"{o}dinger problem of probabilistic evolution, which provides for a unique\nMarkov stochastic interpolation between any given pair of boundary probability\ndensities for a process covering a fixed, finite duration of time, provided we\nhave decided a priori what kind of primordial dynamical semigroup transition\nmechanism is involved. In the nonrelativistic theory, including quantum\nmechanics, Feyman-Kac-like kernels are the building blocks for suitable\ntransition probability densities of the process. In the standard \"free\" case\n(Feynman-Kac potential equal to zero) the familiar Wiener noise is recovered.\nIn the framework of the Schr\\\"{o}dinger problem, the \"free noise\" can also be\nextended to any infinitely divisible probability law, as covered by the\nL\\\u0027{e}vy-Khintchine formula. Since the relativistic Hamiltonians $|\\nabla |$\nand $\\sqrt {-\\triangle +m^2}-m$ are known to generate such laws, we focus on\nthem for the analysis of probabilistic phenomena, which are shown to be\nassociated with the relativistic wave (D\u0027Alembert) and matter-wave\n(Klein-Gordon) equations, respectively. We show that such stochastic processes\nexist and are spatial jump processes. In general, in the presence of external\npotentials, they do not share the Markov property, except for stationary\nsituations. A concrete example of the pseudodifferential Cauchy-Schr\\\"{o}dinger\nevolution is analyzed in detail. The relativistic covariance of related wave",
"arxiv_id": "quant-ph/9505003",
"authors": [
"P. Garbaczewski",
"J. R. Klauder",
"R. Olkiewicz"
],
"categories": [
"quant-ph",
"chem-ph",
"gr-qc",
"hep-th",
"math.PR"
],
"doi": "10.1103/PhysRevE.51.4114",
"journal_ref": "Phys.Rev. E51 (1995) 4114-4131",
"title": "The Schroedinger Problem, Levy Processes Noise in Relativistic Quantum Mechanics",
"url": "https://arxiv.org/abs/quant-ph/9505003"
},
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