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International Journal of Modern Physics A

Stochastic processes in the space of random joint events and equations for transition probabilities of quantum systems

Alexander Alexandrovich Biryukov

Published Jul 10, 2022• 2 citations
Publisher

Abstract

A model of the space of random joint events is constructed. In the space, along with the existence of a symmetric difference of joint events, a new postulate about the existence of a symmetric sum of random joint events is introduced. In the generated space, a stochastic equation of the system motion and an equation for the probabilities of the system transition between two events are constructed. The probability of transition depends on the probabilities of compatibility of two, three, etc. events. The equation is equivalent to the equation of quantum theory if the events are compatible only in pairs. The proposed equation for incompatible events passes into the equation of the Markov chain.

DOI
10.1142/s0217751x22430060

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A model of the space of random joint events is constructed. In the space, along with the existence of a symmetric difference of joint events, a new postulate about the existence of a symmetric sum of random joint events is introduced. In the generated space, a stochastic equation of the system motion and an equation for the probabilities of the system transition between two events are constructed. The probability of transition depends on the probabilities of compatibility of two, three, etc. events. The equation is equivalent to the equation of quantum theory if the events are compatible only in pairs. The proposed equation for incompatible events passes into the equation of the Markov chain.
Abstract The evolution of the system is described as a stochastic process in the space of random joint events. A symmetric difference and a symmetric sum of random joint events are introduced in space. The probability of quantum transitions of the system between certain random events is represented by the sum of the probabilities of joint events (two, three, … ). It is noteworthy that the obtained expression for the transition probabilities coincides with the transition probability in quantum theory, if only pairwise joint random events are taken into account in the model.
The dynamics of the system in the space of random joint events is considered. The symmetric difference of events is introduced in space based on the Kolmogorov axioms. To describe quantum effects in the dynamics of the system, an additional axiom is introduced for random joint events: “the symmetric sum of random events.” In the generated space of random joint events, an equation is constructed for the probability of a system transition between two events. It is shown that for pairwise joint events it is equivalent to the equation of quantum mechanics.
Abstract The evolution of the system is described as a stochastic process in the space of random joint events, in which both a symmetric difference and a symmetric sum of events are introduced. The probability of a system transition between states is represented by a series of double, triple, etc., integrals of real functionals of joint event trajectories. The expression coincides with the transition probability in quantum theory if only pairwise joint random trajectories are taken into account in the model.
The evolution of the system is described as a stochastic process in the space of random joint events. A symmetric difference and a symmetric sum of random joint events are introduced in the space. The probability of quantum transitions of the system between certain random events is represented by the sum of the probabilities of joint events (two, three...). It is noteworthy that the obtained expression for the transition probabilities coincides with that for the transition probability in quantum theory, if only pairwise joint random events are taken into account in the model.
A formulation of quantum statistical ensembles in terms of probability distributions on a projective Hilbert space is developed. The combination of statistically independent systems and the reduction of a system to one of its subsystems are described by means of a tensor product of probability distributions and a reduction formula for the reduced probability distribution. Within this framework the dynamics of open quantum systems is investigated starting form a microscopic system-plus-reservoir model. Employing the Markov approximation of the classical theory of stochastic processes the short time behavior of the conditional transition probability is derived and shown to yield a differential Chapman-Kolmogorov equation. The latter has the form of a Liouville master equation for the reduced probability distribution of the open quantum system. Thus it is shown that the time-dependent wave function of an open quantum system represents a well-defined and unique stochastic process in the space of rays of the underlying Hilbert space. This stochastic process consists of a continuous time evolution generated by a nonlinear Schr\"odinger equation and a discontinuous jump process; the realizations of the processes correspond to those of the Monte Carlo wave function simulation method and to those of the jump-type evolution of the quantum trajectory method. The quantum master equation for the statistical operator is derived as the equation of motion for the two-point correlation function of the stochastic process.
A model for quantum dynamics is presented in terms of a generalization of Markov chains. We first consider measurements and events on an amplitude space. Markov amplitude chains on an amplitude space are then studied. Quantum chains are defined and characterized by a Markov and weak stationarity property. We then consider random phase transformations and study the changes that result in an amplitude space and in the quantum chains due to such transformations. A perturbation expansion involving a potential perturbing the ‘‘free’’ motion is proved. Quantum processes are defined and their relation to quantum chains is discussed. The existence of an arbitrary quantum chain is proved using path chains.