We study the time evolution of decaying particles in the framework of certain renormalizable models of relativistic quantum field theory. Significant differences between the latter and the non-relativistic quantum mechanical case are found —in particular, regarding the so-called quantum Zeno effect. In some cases, the finite time behaviour is well defined after having performed a conventional renormalization. However, in other situations, the latter does not suffice to obtain finite survival probabilities. In all cases, the crucial consistency with Fermi’s golden rule holds. The analysis of the finite time evolution in relativistic quantum field theory (RQFT) and, in particular, the behaviour at short times of the survival probability of unstable particles in RQFT is a relevant matter as far as the foundations of quantum mechanics are concerned and has some peculiarities which make it to be worth studying. But before going to RQFT, we will briefly recapitulate the main features of the survival amplitude (SA) at
The manifestation of measurements, randomly distributed in time, on the evolution of quantum systems are analyzed in detail. The set of randomly distributed measurements (RDM) is modeled within the renewal theory, in which the distribution is characterized by the probability density function (PDF) W(t) of times t between successive events (measurements). The evolution of the quantum system affected by the RDM is shown to be described by the density matrix satisfying the stochastic Liouville equation. This equation is applied to the analysis of the RDM effect on the evolution of a two level systems for different types of RDM statistics, corresponding to different PDFs W(t). Obtained general results are illustrated as applied to the cases of the Poissonian [W(t) ~ e^{-w_r t}] and anomalous [W(t) ~ 1/t^{1+α}, (αis smaller or equal to 1)], RDM statistics. In particular, specific features of the quantum and inverse Zeno effects, resulting from the RDM, are thoroughly discussed.
Our main aim is to prove a more general version of the quantum Zeno effect. Then we discuss some examples of the quantum Zeno effect. Furthermore, we discuss a possibility that based on the quantum Zeno effect and certain experiments one could check whether, from the statistical point of view, a concrete system behaves like a quantum system. The more general version of quantum Zeno effect can be helpful to prove that the brain acts like in a quantum system. The proof of our main result is based on certain formulas describing probability distributions of time series related to quantum measurements.
In two previous papers a causal interpretation of the quantum theory was developed which involved the hypothesis that a quantum-mechanical system contains a precisely defined particle variable x but that, at present, we are restricted to calculating the probability density
P(x,t) that the particle is at the position x. It was shown that the assumption that $P(\mathrm{x}, t)={|\ensuremath{\psi}(\mathrm{x}, t)|}^{2}$ is consistent, in the sense that if it holds initially, the equations of motion of the particles will cause this relation to be maintained for all time. In this paper, we extend the theory by showing that as a result of random collisions, an arbitrary probability density will ultimately decay into one with a density of ${|\ensuremath{\psi}(\mathrm{x}, t)|}^{2}$. Since all quantum-mechanical experiments to date have been concerned with statistical ensembles of systems that have been colliding with other systems for a very long time, it is therefore inevitable that as we draw samples from such ensembles, the probability density of systems with particles at the point x will be equal to ${|\ensuremath{\psi}(\mathrm{x}, t)|}^{2}$.In the previous papers we also pointed out that, within the conceptual framework of the causal interpretation, it was possible to suggest mathematical theories more general than are permitted by the usual interpretation and that these more general theories might be needed in the domain of ${10}^{\ensuremath{-}13}$ cm, where present theories seem to fail. However, if these more general theories should apply at the level of ${10}^{\ensuremath{-}13}$ cm, then there would be a tendency to create discrepancies between
P and ${|\ensuremath{\psi}|}^{2}$, a tendency whose cumulative effects should be felt even at the atomic level, where the more general theory ought to approach the usual theory. However, because those discrepancies have been shown to die out as a result of collisions, we can expect that under normal conditions the difference between
P and ${|\ensuremath{\psi}|}^{2}$ would be negligible. Conditions are suggested, however, in which this difference might be appreciable, and experiments are indicated which might be able to test for the existence of such discrepancies.
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.
In this work we consider a new example of the recently introduced quantum Hamlet effect. We consider an especial, abstract, unstable quantum system whose evolution during a small time interval is interrupted by frequent measurements. Here three different final situations exist. First one corresponds to quantum Zeno effect, second one - to quantum anti-Zeno effect and third one - to so-called quantum Hamlet effect. By quantum Zeno effect final non-decay probability is function of number of the decay measurements variable and dynamical degree parameter equivalent to two. When measurements number tends toward infinity non-decay probability has the one limit, or, it tends analytically toward one and system stands non-decayed. By quantum anti-Zeno effect final non-decay probability is function of number of the decay measurements variable and dynamical degree parameter equivalent to one. When measurements number tends toward infinity non-decay probability has the zero limit, or, it tends analytically toward zero and system becomes decayed. By quantum Hamlet effect, final non-decay probability is function of two variable, number of the decay measurements and dynamical degree. When measurements number tends toward infinity and dynamical degree toward one, final non-decay probability depends not only of final value of given variables, but, also, on the ways on which given variables tends toward their final values. It means that final no-decay probability has not (analytical) limit, or that there is no {\it analytical} prediction on the final no-decay probability. To be decayed or no-decayed that is analytically unsolvable question for given quantum system.