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Progress of Theoretical Physics

Relativistic Random Motion Parametrized by Observer's Time

Teruaki Nakagomi

Published May 1, 1989• 3 citations
Publisher

Abstract

A relativistic space-time stochastic process of light-like random motion was constructed in a previous paper by taking proper time as parameter. The same random motion is reformulated by a spatial stochastic process with observer's time as parameter. A crucial point in deriving the latter process from the former is that the two time parameters are stochastically related to each other. Properties of the process are studied, including the Markov property, time evolution equations, transformation properties under the Lorentz group, and an expression for probability over arbitrary space-like hypersurfaces. A formula with close analogy to the Dirac Hamiltonian is also presented. The original model in proper-time formulation is reviewed briefly.

DOI
10.1143/ptp.81.916

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A relativistic space-time stochastic process of light-like random motion was constructed in a previous paper by taking proper time as parameter. The same random motion is reformulated by a spatial stochastic process with observer's time as parameter. A crucial point in deriving the latter process from the former is that the two time parameters are stochastically related to each other. Properties of the process are studied, including the Markov property, time evolution equations, transformation properties under the Lorentz group, and an expression for probability over arbitrary space-like hypersurfaces. A formula with close analogy to the Dirac Hamiltonian is also presented. The original model in proper-time formulation is reviewed briefly.
Given a positive energy solution of the Klein–Gordon equation, the motion of the free, spinless, relativistic particle is described in a fixed Lorentz frame by a Markov diffusion process with nonconstant diffusion coefficient. Proper time is an increasing stochastic process and we derive a probabilistic generalization of the equation (dτ)2=−(1/c2)dXν dXν. A random time-change transformation provides the bridge between the t and the τ domain. In the τ domain, we obtain an M4-valued Markov process with singular and constant diffusion coefficient. The square modulus of the Klein–Gordon solution is an invariant, nonintegrable density for this Markov process. It satisfies a relativistically covariant continuity equation.
The compatibility of special relativity and Quantum Mechanics has been questioned by several authors. The origin of this tension can be traced back mainly to the introduction of the measurement processes and the corresponding wave function reduction, which play a crucial role in Quantum Mechanics. We approach this problem with the help of a recent proposal for a model of Quantum Mechanics, where the measurement is explicitly described as a specific stochastic process. This implements ordinary Quantum Mechanics, where measurement and reduction are treated as phenomenological events of unknown origin without any physical justification. To state clearly the question in general, we first discuss and establish the effect of a Lorentz transformation on a generic wave function in space-time. Alongside the analysis we formulate the relativistic version of the model. We then consider few thought experiments in order to analyze to what extent Quantum Mechanics follows relativistic invariance and find the specific critical points where non invariance possibly occurs. The analysis can shade light to the interpretation of the existing experimental observations.
In this paper, we develop an action formulation of stochastic dynamics in the Hilbert space. In this formulation, the quantum theory of random unitary evolution is easily reconciled with special relativity. We generalize the Wiener process into 1+31+31+3-dimensional spacetime, and then define a scalar random field which keeps invariant under Lorentz transformations. By adding to the action of quantum field theory a coupling term between random and quantum fields, we obtain a random-number action which has the statistical spacetime translation and Lorentz symmetries. The canonical quantization of the theory results in a Lorentz-invariant equation of motion for the state vector or density matrix. We derive the path integral formula of SSS matrix, based on which we develop a diagrammatic technique for doing the calculation. We find the diagrammatic rules for both the stochastic free field theory and stochastic ${\ensuremath{\phi}}^{4}$ theory. The Lorentz invariance of the random SSS matrix is strictly proved by using the diagrammatic technique. We then develop a diagrammatic technique for calculating the density matrix of final quantum states after scattering. In the absence of interaction, we obtain the exact expressions of both SSS matrix and density matrix. In the presence of interaction, we prove a simple relation between the density matrices of stochastic and conventional ${\ensuremath{\phi}}^{4}$ theory. Our formalism leads to an ultraviolet divergence that has a similar origin as that in quantum field theory. The divergence can be canceled by renormalizing the coupling strength to random field. We prove that the stochastic quantum field theory is renormalizable even in the presence of interaction. In the models with a linear coupling between random and quantum fields, the random field excites particles out of the vacuum, driving the Universe towards an infinite-temperature state. The number of excited particles follows the Poisson distribution. The collision between particles is not affected by the random field. But the signals of colliding particles are gradually covered by the background excitations caused by random field.
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}+m2{\mathit{m}}^{2}m2 -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.
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There are two stochastic descriptions of relativistic quantum spinless particles. In the first one, suggested by Feynman's path integral approach to Klein-Gordon propagator, the main ingredients are diffusions in Minkowski space parametrized by some kind of proper time. The second description, on the contrary, is based upon jumb Markov processes giving the space position of the particle in the sense of Newton and Wigner with the time as parameter. In this paper we bridge the gap between these different probabilistic scenarios by constructing the space jump processes by means of space-time diffusions. We give some aplications to the nonrelativistic limit of the hamiltonian semigroup t→exp−(t/¬h)H where H=√−c 2 ¬h 2 Δ+(M 2 c 4 /¬h 2 )−Mc 2 +V(.) Il y a deux descriptions stochastiques de la mecanique quantique relativiste pour une particule sans spin. Dans la premiere, suggere par la theorie de Feynman, les ingredients principaux sont des diffusions dans l'espace de Minkowski parametrise par une sorte de temps propre. La deuxieme description, par contre, emploie des processus de Markov a sauts donnant la position de la particule dans l'espace au sens de Newton et Wigner avec le temps comme parametre. Notre papier va relier les deux prescriptions probabilistes car nous construisons les processus a sauts dans l'espace par les diffusions dans l'espace-temps. Nous donnons aussi des applications a l'etude de la limite non relativiste du semi-groupe hamiltonien t→exp−(t/¬h)H ou H=√−c 2 ¬h 2 Δ+(M 2 c 4 /¬h 2 )−Mc 2 +V(.)