One calls attention to the fact that the stochastic physical systems are not random completely. They have both random and regular components of their evolution. Dynamic system is considered to be a special case of physical system with vanishing stochastic component of evolution. Mathematical technique for description of regular evolution component of physical systems (stochastic and deterministic) is constructed. The regular component of the system S evolution is described explicitly by the statistical average system , which is a continuous dynamic system. The action for is reduced to the form of the action for a set of interacting identical deterministic systems S_d. The form of interaction of S_d describes implicitly the character of the stochastic component of S evolution. Interplay between the physical system S and the statistical average system in the measurement process is discussed. There are at least two different kind of measurement: single measurements (S-measurements) connected with measurements in S and mass measurements (M-measurements) connected with measurements in . Conventional identification of S-measurement and M-measurement is a source of many misunderstandings and paradoxes.
One calls attention to the fact that the stochastic physical systems are not random completely. They have both random and regular components of their evolution. Dynamic system is considered to be a special case of physical system with vanishing stochastic component of evolution. Mathematical technique for description of regular evolution component of physical systems (stochastic and deterministic) is constructed. The regular component of the system S evolution is described explicitly by the statistical average system , which is a continuous dynamic system. The action for is reduced to the form of the action for a set of interacting identical deterministic systems S_d. The form of interaction of S_d describes implicitly the character of the stochastic component of S evolution. Interplay between the physical system S and the statistical average system in the measurement process is discussed. There are at least two different kind of measurement: single measurements (S-measurements) connected with measurements in S and mass measurements (M-measurements) connected with measurements in . Conventional identification of S-measurement and M-measurement is a source of many misunderstandings and paradoxes.
We develop a statistical model of microscopic stochastic deviation from classical mechanics based on a stochastic process with a transition probability that is assumed to be given by an exponential distribution of infinitesimal stationary action. We apply the statistical model to stochastically modify a classical mechanical model for the measurement of physical quantities reproducing the prediction of quantum mechanics. The system+apparatus always has a definite configuration at all times, as in classical mechanics, fluctuating randomly following a continuous trajectory. On the other hand, the wavefunction and quantum mechanical Hermitian operator corresponding to the physical quantity arise formally as artificial mathematical constructs. During a single measurement, the wavefunction of the whole system+apparatus evolves according to a Schrödinger equation and the configuration of the apparatus acts as the pointer of the measurement so that there is no wavefunction collapse. We will also show that while the outcome of each single measurement event does not reveal the actual value of the physical quantity prior to measurement, its average in an ensemble of identical measurements is equal to the average of the actual value of the physical quantity prior to measurement over the distribution of the configuration of the system.
In the present paper we consider the problem of the description of an arbitrary generalized quantum measurement with outcomes in a measurable space. Analysing the unitary invariants of a separable statistical realization of a quantum instrument, we present the most general form of a possible integral representation of an instrument. We introduce the notion of a stochastic realization of an instrument and establish a one-to-one correspondence between the class of unitarily and phase equivalent separable statistical realizations and the equivalence class of stochastic realizations of an instrument. We further single out the invariant class of unitarily and phase equivalent separable statistical realizations for which the integral representation of an instrument is the same for all statistical realizations from this class and is wholly determined by the invariants of this class. We call the special form of this integral representation the quantum stochastic representation of an instrument. We show that the description of a generalized direct quantum measurement can be considered in the frame of a new general approach based on the notion of a family of quantum stochastic evolution operators satisfying the orthonormality relation. This approach gives not only the complete statistical description of any generalized direct quantum measurement but the complete description in a Hilbert space of the stochastic behaviour of a quantum system under a generalized direct measurement in the sense of specifying the probabilistic transition law governing the change from the initial state to a final one under a single measurement. Under this approach a unitary evolution of an isolated quantum system is included as a special case. In the frame of the proposed approach, which we call quantum stochastic approach, all possible schemes of measurements upon a quantum system can be considered. In the case of repeated or continuous in time measurements the quantum stochastic approach allows to define, in the most general case, the notion of the family of posterior pure state trajectories (quantum trajectories in discrete or continuous time) in the Hilbert space of a quantum system and to give their probabilistic treatment.
The description of physical systems obtained in the classical limit of quantum mechanics is not consistent with the one given by classical statistical mechanics. In qantum mechanics the complete description of a physical system is given by specifying the wave function, which evolves completely deterministically. Probability comes into play only when a measurement is made. Classical statistical mechanics, on the contrary, concerns systems whose description is not complete and where probability enters as a consequence of a lack of knowledge. We show that Nelson's stochastic mechanics, whose predictions are identical with those of quantum mechanics, provides nevertheless a unified description of the deterministic and random aspects of quantum mechanics which eliminates the inconsistency outlined above and gives a simple and satisfactory solution to the measurement problem.
The formalism of dynamics in the space-time, where motion of free particles is primordially stochastic, is considered. The conventional dynamic formalism, obtained for the space-time, where the motion of free particles is primordially deterministic, seems to be unsuitable. The statistical ensemble of stochastic (or deterministic) systems is considered to be the main object of dynamics. At such an approach one can describe deterministic, stochastic and quantum particles by means of the uniform technique. The quantum particle is described as a stochastic particle, i.e. without a reference to the quantum principles. Besides, by means of this technique one can describe classical inviscid fluid. There are four different versions of the formalism. One of versions uses equations, which describes evolution of the labelling (Lagrangian coordinates at fixed initial conditions for the density and the momentum density. These fixed initial conditions enter in the evolution equations. A nonuniqueness of irrotational flow and the mismatch between the velocity and the momentum density appear at this description. The mismatch is determined by the fluid vorticity and by some arbitrary parameter. Appearance of mismatch may be associated with the turbulence. The united formalism is purely dynamic. Even describing stochastic systems, it does not refer to probability and probabilistic structures.
The formalism of the particle dynamics in the space-time, where motion of free particles is primordially stochastic, is considered. The conventional dynamic formalism, obtained for the space-time, where the motion of free particles is primordially deterministic, seems to be unsuitable. The statistical ensemble of stochastic (or deterministic) systems is considered to be the main object of dynamics. At such a logical reloading the statistical description becomes a component of dynamics, and capacities of dynamics increase. At such an approach one can describe deterministic, stochastic and quantum particles by means of the uniform technique. The quantum particle is described as a stochastic particle, i.e. without a reference to the quantum principles. Besides, by means of this technique one can describe classical inviscid fluid. There are four different versions of the formalism: (1) description in Euler dynamic variables, (2) description in Lagrange dynamic variables, (3) description in terms of the generalized stream function, (4) description in terms of the wave function. The uniform formalism is purely dynamic. Even describing stochastic systems, it does not refer to probability and probabilistic structures. In relativistic case the uniform formalism can describe pair production and pair annihilation.
Introduction to statistical ensembles The concept of a random or stochastic process or function represents a generalization of the idea of a set of random variables x 1 , x 2 , …, when the set is no longer countable and the variables form a continuum. We therefore introduce a continuous parameter t , such as time, that labels the variates. We call x ( t ) a random process or a random function of t if x does not depend on t in a deterministic way. Random processes are encountered in many fields of science, whenever fluctuations are present. Examples of a real random process x ( t ) are the fluctuating voltage across an electrical resistor, and the coordinates of a particle under-going Brownian motion. We shall see shortly that the optical field generated by any realistic light source must also be treated as a random function of position and time. Of course the parameter t may also stand for some quantity other than time, but for simplicity we shall take it to represent time. In our applications x ( t ) will frequently represent a Cartesian component of the electric or magnetic field vector in a light beam. To begin with we shall take x ( t ) to be real, but complex random processes will also be encountered. The ensemble average As x does not depend on t deterministically, we can only describe its values statistically, by some probability distribution or probability density.