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 the temporal evolution of a given measure or preserves the measure in the stationary case. Our departure point is the so-called Schr\"odinger 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, Feynman-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\"odinger problem, the ``free noise'' can also be extended to any infinitely divisible probability law, as covered by the L\'evy-Khintchine formula. Since the relativistic Hamiltonians \ensuremath{\Vert}\ensuremath{\nabla}\ensuremath{\Vert} and \ensuremath{\surd}-\ensuremath{\Delta}+ ${\mathit{m}}^{2}$ -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\"odinger evolution is analyzed in detail. The relativistic covariance of related wave equations is exploited to demonstrate how the associated stochastic jump processes comply with the principles of special relativity.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 the temporal evolution of a given measure or preserves the measure in the stationary case. Our departure point is the so-called Schr\"odinger 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, Feynman-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\"odinger problem, the ``free noise'' can also be extended to any infinitely divisible probability law, as covered by the L\'evy-Khintchine formula. Since the relativistic Hamiltonians \ensuremath{\Vert}\ensuremath{\nabla}\ensuremath{\Vert} and \ensuremath{\surd}-\ensuremath{\Delta}+${\mathit{m}}^{2}$ -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\"odinger evolution is analyzed in detail. The relativistic covariance of related wave equations is exploited to demonstrate how the associated stochastic jump processes comply with the principles of special relativity.

arXiv:quant-ph/9505003Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsMay 1, 199541 citations

Why quantum dynamics can be formulated as a Markov process

Piotr Garbaczewski, Robert Olkiewicz

We invert the well-developed strategy of studying dynamics in terms of probability densities and investigate the problem of the most likely microscopic propagation scenario, which is consistent with the given priori (possibly phenomenological) input-output statistics data for the process taking place in a finite-time interval. A solution of this so-called Schr\"odinger problem is known to provide an adequate probabilistic framework for the measure preserving dynamics which is Markovian. We pay particular attention to the subclass of nonstationary solutions, determined by unitary-wave-packet evolution (Schr\"odinger wave mechanics). The existence of the pertinent Markovian diffusion is known on general grounds, but no explicit demonstration (through detailed computational arguments) until now was available even in the simplest cases. We give a definitive probabilistic description of the free quantum dynamics as a stochastic process solving Schr\"odinger's interpolation problem. The Markov diffusion arised as a particular case singled out by a suitable Feynman-Kac semigroup.We invert the well-developed strategy of studying dynamics in terms of probability densities and investigate the problem of the most likely microscopic propagation scenario, which is consistent with the given priori (possibly phenomenological) input-output statistics data for the process taking place in a finite-time interval. A solution of this so-called Schr\"odinger problem is known to provide an adequate probabilistic framework for the measure preserving dynamics which is Markovian. We pay particular attention to the subclass of nonstationary solutions, determined by unitary-wave-packet evolution (Schr\"odinger wave mechanics). The existence of the pertinent Markovian diffusion is known on general grounds, but no explicit demonstration (through detailed computational arguments) until now was available even in the simplest cases. We give a definitive probabilistic description of the free quantum dynamics as a stochastic process solving Schr\"odinger's interpolation problem. The Markov diffusion arised as a particular case singled out by a suitable Feynman-Kac semigroup.

Physical Review AMay 1, 199519 citations

Stochastic Theory of Quantum Mechanics

M. Godart

The stochastic theory of non-relativistic quantum mechanics presented here relies heavily upon the theory of stochastic processes, with its definitions, theorems and specific vocabulary as well. Its main hypothesis states indeed that the classical trajectories of the particles are identical to the sample functions of a diffusion Markov process, whose conditional probability density is proposed as a substitute for the wave function. The Schroedinger equation and the so-called Nelson equations are used to determine the diffusion tensor and the drift vectors characteristic of such a process. It is then possible to write down the forward and backward Kolmogorov equations that are used to determine the conditional probability density, as well as the Fokker-Planck equation that is used to determine the normal probability density. This method is applied to several simple cases and the results obtained are compared with those of the orthodox theory. Among the most important differences let us mention that any system evolving freely from any original state returns spontaneously to its ground state, that the definitions of the particles velocities and momenta are impossible because the sample functions of a diffusion Markov process nowhere possess a derivative with respect to time and that the so-called collapse of the wave function is a mere mirage explained by the updating choice between new and older initial conditions in the resolution of partial differential equations. We finally attempt to extend the proposed theory to the domain of the relativistic quantum mechanics. It is promising but is clearly unfinished because it has not been possible up to now to solve the Nelson and Kolmogorov equations, except in the very simple case of a free particle.The stochastic theory of non-relativistic quantum mechanics presented here relies heavily upon the theory of stochastic processes, with its definitions, theorems and specific vocabulary as well. Its main hypothesis states indeed that the classical trajectories of the particles are identical to the sample functions of a diffusion Markov process, whose conditional probability density is proposed as a substitute for the wave function. The Schroedinger equation and the so-called Nelson equations are used to determine the diffusion tensor and the drift vectors characteristic of such a process. It is then possible to write down the forward and backward Kolmogorov equations that are used to determine the conditional probability density, as well as the Fokker-Planck equation that is used to determine the normal probability density. This method is applied to several simple cases and the results obtained are compared with those of the orthodox theory. Among the most important differences let us mention that any system evolving freely from any original state returns spontaneously to its ground state, that the definitions of the particles velocities and momenta are impossible because the sample functions of a diffusion Markov process nowhere possess a derivative with respect to time and that the so-called collapse of the wave function is a mere mirage explained by the updating choice between new and older initial conditions in the resolution of partial differential equations. We finally attempt to extend the proposed theory to the domain of the relativistic quantum mechanics. It is promising but is clearly unfinished because it has not been possible up to now to solve the Nelson and Kolmogorov equations, except in the very simple case of a free particle.

arXiv:1206.2917Jun 13, 20123 citations

Schrödinger's interpolation problem and its probabilistic solutions

Piotr Garbaczewski

Abstract: Probablistic solutions of the so called Schrödinger boundary data problem provide for a unique Markovian interpolation between any two strictly positive probability densities designed to form the input-output statistics data for a certain dynamical process taking place in a finite-time interval. The key problem is to select the jointly continuous in all variables positive semigroup kernel, appropriate for the phenomenological (physical) situation. The issue of deriving a microscopic dynamics from the input-output statistics data (analyzed in terms of densities) was addressed, as the Schrödinger problem of a probabilistic interpolation, in a number of publications [1] – [6]. We shall consider Markovian propagation scenarios so remaining within the well established framework, where for any two Borel sets A, B ⊂ R on which the strictly positive boundary densities ρ(x, 0) and ρ(x, T) are defined, the transition probability m(A, B) from the set A to the set B in the time interval T> 0 has a bi-variate density given in a specific factorized form: m(x, y) = f(x)k(x, 0, y, T)g(y), with marginals: ∫Abstract: Probablistic solutions of the so called Schrödinger boundary data problem provide for a unique Markovian interpolation between any two strictly positive probability densities designed to form the input-output statistics data for a certain dynamical process taking place in a finite-time interval. The key problem is to select the jointly continuous in all variables positive semigroup kernel, appropriate for the phenomenological (physical) situation. The issue of deriving a microscopic dynamics from the input-output statistics data (analyzed in terms of densities) was addressed, as the Schrödinger problem of a probabilistic interpolation, in a number of publications [1] – [6]. We shall consider Markovian propagation scenarios so remaining within the well established framework, where for any two Borel sets A, B ⊂ R on which the strictly positive boundary densities ρ(x, 0) and ρ(x, T) are defined, the transition probability m(A, B) from the set A to the set B in the time interval T> 0 has a bi-variate density given in a specific factorized form: m(x, y) = f(x)k(x, 0, y, T)g(y), with marginals: ∫

Technical PhysicsJan 1, 19970 citations

Stochastic Theory of Relativistic Quantum Mechanics

M. Godart

The stochastic theory of relativistic quantum mechanics presented here is modelled on the one that has been proposed previously and that was claimed to be a promising substitute to the orthodox theory in the non-relativistic domain. So it also relies heavily upon the theory of stochastic processes, with its definitions, theorems and specific vocabulary as well. Its main hypothesis states indeed that the classical trajectories of the particles are identical to the sample functions of a diffusion Markov process, whose conditional probability density is proposed as a substitute for the wave function. Along the way, we introduce ad hoc hypotheses for the sole reason that they facilitate and even make possible the further development of the theory. These and the so-called Nelson equations are used to determine the drift vectors and the diffusion tensor characteristic of such a process. It is then possible to write down the Fokker-Planck and Kolmogorov equations that can be used to determine the normal and conditional probability densities. This logical sequence of steps leads to the solution of specific problems and we apply it to the special case of the free particle.The stochastic theory of relativistic quantum mechanics presented here is modelled on the one that has been proposed previously and that was claimed to be a promising substitute to the orthodox theory in the non-relativistic domain. So it also relies heavily upon the theory of stochastic processes, with its definitions, theorems and specific vocabulary as well. Its main hypothesis states indeed that the classical trajectories of the particles are identical to the sample functions of a diffusion Markov process, whose conditional probability density is proposed as a substitute for the wave function. Along the way, we introduce ad hoc hypotheses for the sole reason that they facilitate and even make possible the further development of the theory. These and the so-called Nelson equations are used to determine the drift vectors and the diffusion tensor characteristic of such a process. It is then possible to write down the Fokker-Planck and Kolmogorov equations that can be used to determine the normal and conditional probability densities. This logical sequence of steps leads to the solution of specific problems and we apply it to the special case of the free particle.

arXiv:1412.1508Dec 3, 20140 citations

Lévy–Schrödinger wave packets

Nicola Cufaro Petroni

We analyze the time-dependent solutions of the pseudo-differential Lévy–Schrödinger wave equation in the free case, and compare them with the associated Lévy processes. We list the principal laws used to describe the time evolutions of both the Lévy process densities and the Lévy–Schrödinger wave packets. To have self-adjoint generators and unitary evolutions we will consider only absolutely continuous, infinitely divisible Lévy noises with laws symmetric under change of sign of the independent variable. We then show several examples of the characteristic behavior of the Lévy–Schrödinger wave packets, and in particular of the multimodality arising in their evolutions: a feature at variance with the typical diffusive unimodality of both the corresponding Lévy process densities and usual Schrödinger wavefunctions.We analyze the time-dependent solutions of the pseudo-differential Lévy–Schrödinger wave equation in the free case, and compare them with the associated Lévy processes. We list the principal laws used to describe the time evolutions of both the Lévy process densities and the Lévy–Schrödinger wave packets. To have self-adjoint generators and unitary evolutions we will consider only absolutely continuous, infinitely divisible Lévy noises with laws symmetric under change of sign of the independent variable. We then show several examples of the characteristic behavior of the Lévy–Schrödinger wave packets, and in particular of the multimodality arising in their evolutions: a feature at variance with the typical diffusive unimodality of both the corresponding Lévy process densities and usual Schrödinger wavefunctions.

arXiv:1006.1833Journal of Physics A Mathematical and TheoreticalMar 28, 20118 citations

|\nabla |

{\mathit{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\"odinger evolution is analyzed in detail. The relativistic covariance of related wave equations is exploited to demonstrate how the associated stochastic jump processes comply with the principles of special relativity.