Advances in High Energy Physics

Volume 2015, Article ID 461987, 6 pages

http://dx.doi.org/10.1155/2015/461987

## On the Velocity of Moving Relativistic Unstable Quantum Systems

Institute of Physics, University of Zielona Góra, Ulica Prof. Z. Szafrana 4a, 65-516 Zielona Góra, Poland

Received 10 October 2015; Revised 13 December 2015; Accepted 17 December 2015

Academic Editor: Shi-Hai Dong

Copyright © 2015 K. Urbanowski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We study properties of moving relativistic quantum unstable systems. We show that in contrast to the properties of classical particles and quantum stable objects the velocity of freely moving relativistic quantum unstable systems cannot be constant in time. We show that this new quantum effect results from the fundamental principles of the quantum theory and physics: it is a consequence of the principle of conservation of energy and of the fact that the mass of the quantum unstable system is not defined. This effect can affect the form of the decay law of moving relativistic quantum unstable systems.

#### 1. Introduction

Physicists studying the decay processes of unstable quantum systems moving with the velocity relative to the rest reference frame of an observer and trying to derive theoretically the decay law of such systems are confronted with the following problem: Which of the two possible assumptions, or perhaps (where is the momentum of the moving unstable system), will get decay law correctly describing the real properties of such system. When one considers classical physics decay processes, the mentioned assumptions both lead to the decay law of the same form. Namely, from the standard, text book considerations, one finds that if the decay law of the unstable particle in rest has the exponential form , then the decay law of the moving particle with momentum is , where denotes time, is the decay rate (time and are measured in the rest reference frame of the particle), is the relativistic Lorentz factor, , , and is the decay law of the particle moving with the constant velocity (we use units, and thus ). It is almost common belief that this equality is valid also for any in the case of quantum decay processes and does not depend on the model of the unstable system considered. The cases and both were studied in the literature. The assumption was used in [1, 2] to derive the survival probability . From these studies, it follows that in the case of moving quantum unstable systems the relation is valid to a sufficient accuracy only for not more than a few lifetimes and that for times much longer than a few lifetimes there is (see [2, 3]). The assumption was used, for example, in [4], to derive the decay law of moving quantum unstable systems. Unfortunately, the result obtained in [4] is similar to the case : only for no more than a few lifetimes. What is more, it appears that the assumption may lead to the relation , that is, to the result never observed in experiments [5, 6].

Unfortunately, the experiments did not give any decisive answer for the problem which is the correct assumption: or ? It is because all known tests of the relation were performed for times (where is the lifetime) (see, e.g., [7, 8]). Note that the same relation obtained in [1, 2, 4] is approximately valid for the same times (see also discussion in [9–13]). The problem seems to be extremely important in accelerator physics where the correct interpretation of the obtained results depends on knowledge of the properly calculated decay law of the moving unstable particles created in the collisions observed. Similarly, the proper interpretation of results of observations of astrophysical processes in which a huge numbers of elementary particles (including unstable one) are produced is impossible without knowing the correct form of the decay law of unstable particles created in these processes. So the further theoretical studies of the above-described problem are necessary and seem to be important.

In this paper, we analyze general properties of unstable quantum system from the point of view of fundamental principles of physics and quantum theory. Here, we show that the principle of the conservation of the energy does not allow any moving quantum unstable system to move with the velocity constant in time.

#### 2. Quantum Unstable Systems

The main information about properties of quantum unstable systems is contained in their decay law, that is, in their survival probability. Let the reference frame be the common inertial rest frame for the observer and for the unstable system. Then, if one knows that the system in the rest frame is in the initial unstable state , which was prepared at the initial instant , one can calculate its survival probability, , which equals , where is the survival amplitude, , , is the total self-adjoint Hamiltonian of the system under considerations, , and is the Hilbert space of states of the considered system. So in order to calculate the amplitude , one should know the state . Within the standard approach, the unstable state is modeled as the following wave packet [14–18]:where is the lower bound of the continuous part of the spectrum of and vectors solve the following:Eigenvectors are normalized as follows: We require the state to be normalized; so it has to be . Thus, which allows one to represent the amplitude as the Fourier transform of the mass (energy) distribution function, :where and , for [14–23] (see also [1–6]). From the last relation and from the Riemann-Lebesque lemma, it follows that as . It is because, from the normalization condition , it follows that is an absolutely integrable function (note that this approach is also applicable in Quantum Field Theory models [24, 25]). The typical form of the survival probability is presented in Figure 1, where the calculations were performed for having the Breit-Wigner form:assuming for simplicity that . Here, is a step function: consider for and for .