Advances in High Energy Physics

Volume 2018 (2018), Article ID 4657079, 9 pages

https://doi.org/10.1155/2018/4657079

## Moving Unstable Particles and Special Relativity

1763 Braddock Court, San Jose, CA 95125, USA

Correspondence should be addressed to Eugene V. Stefanovich; moc.sysponys@venegue

Received 5 December 2017; Accepted 12 March 2018; Published 19 April 2018

Academic Editor: Krzysztof Urbanowski

Copyright © 2018 Eugene V. Stefanovich. 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

In Poincaré-Wigner-Dirac theory of relativistic interactions, boosts are dynamical. This means that, just like time translations, boost transformations have a nontrivial effect on internal variables of interacting systems. In this respect, boosts are different from space translations and rotations, whose actions are always universal, trivial, and interaction-independent. Applying this theory to unstable particles viewed from a moving reference frame, we prove that the decay probability cannot be invariant with respect to boosts. Different moving observers may see different internal compositions of the same unstable particle. Unfortunately, this effect is too small to be noticeable in modern experiments.

#### 1. Introduction

Time dilation is one of the most spectacular predictions of special relativity. This theory predicts that any time-dependent process slows down by the universal factor of when viewed from a reference frame moving with the speed (and rapidity ). The textbook example of such a time-dependent process is the decay law of an unstable particle at rest. The function is the probability of finding the unstable particle at time , if it was prepared with 100% certainty at time . Then, according to special relativity, the decay law of a moving particle should be exactly times slower:Indeed, this prediction was confirmed in numerous measurements [1–4]. The best accuracy of 0.1% was achieved in experiments with relativistic muons [5, 6].

However, the exact validity of (1) is still a subject of controversy. One point of view [7–9] is that special-relativistic time dilation was derived in the framework of classical theory and may not be directly applicable to unstable particles, which are fundamentally quantum systems without well-defined masses, velocities, positions, and so on.

However, such a quantum clock as an unstable particle cannot be at rest (i.e., cannot have zero velocity or zero momentum) and simultaneously be at a definite point (due to the quantum uncertainty relation). So, the standard derivation of the moving clock dilation is inapplicable for the quantum clock. The related quantum-mechanical derivation must contain some reservations and corrections. Shirokov [10]

Indeed, detailed quantum-mechanical calculations [8, 10–12] suggest that (1) is not accurate, and that corrections to this formula should be expected, especially at large times, exceeding multiple lifetimes. Although these corrections are too small to be observed in modern experiments, their presence casts doubt on the limits of applicability of Einstein’s special relativity.

Unfortunately, the corrections to (1) were derived in [8, 10–12] under certain assumptions and approximations. So, the question remains whether one can design a relativistic model in which the decay slowdown will acquire exactly the form (1) demanded by special relativity [13, 14]?

In order to answer this question we will analyze the status of interactions in special relativity from a more general point of view. We are going to prove that under no circumstances the decay law transforms with respect to boosts exactly as in (1).

#### 2. Materials and Methods

##### 2.1. Inertial Transformations

The theory of relativity tries to connect views of different inertial observers. The principle of relativity says that all such observers are equivalent; that is, two inertial observers performing the same experiment will obtain the same results.

There are four classes of inertial transformations, space translations, time translations, rotations, and boosts, and their actions on observed systems differ very much (see Table 1). For example, it is easy to describe the results of space translations and rotations. An observer displaced by the 3-vector sees all atoms in the Universe simply shifted in the opposite direction . This shift is absolutely exact and universal. It applies to all systems, however complicated. The same can be said about rotations. One can switch to the point of view of the rotated observer by simply rotating all atoms in the Universe. For example, rotation through the angle about the -axis results in the transformation of coordinateswhich is independent on the composition of the observed system and on its interactions. Due to this exact universality, we can regard space translations and rotations as purely geometrical or* kinematical* transformations.