^{1}

^{3}.

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.

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

However, the exact validity of (

Indeed, detailed quantum-mechanical calculations [

Unfortunately, the corrections to (

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 (

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

Inertial transformations.

Transformation | Type | Parameter | Generator | Meaning of generator |
---|---|---|---|---|

Space translation | Kinematical | Distance | | Total momentum |

Rotation | Kinematical | Angle | | Total angular momentum |

Time translation | Dynamical | Time | | Total energy (Hamiltonian) |

Boost | Dynamical | Rapidity | | Boost operator |

Time translation is also an inertial transformation, because repeating the same experiment at different times will not change the outcome. However, this transformation is by no means kinematical. Time evolutions of interacting systems can be very complicated. Their description requires intimate knowledge of the system’s composition, state, and interactions acting between system’s parts. We will say that time translations are

Now, what about boosts? Are they kinematical or dynamical? In nonrelativistic classical physics boosts are definitely regarded as kinematical, they simply change velocities of all atoms in the Universe. However, things become more complicated in relativistic physics, as we shall see below.

Description of boost transformations is the central subject of special relativity. Einstein based his approach on the already mentioned relativity postulate and on his second postulate about the invariance of the speed of light. It is remarkable how all results of special relativity can be derived from these two simple and undeniable statements.

Consider the light clock shown in Figure

Light clock: (a) at rest and (b) in motion perpendicular to the clock’s axis.

Let us now consider the same clock oriented parallel to its velocity, as in Figure

Light clock: (a) at rest and (b) in motion parallel to the clock’s axis. The time evolution is shown in three frames stacked up vertically.

Formulas (

We can also make a clock, in which, instead of the light pulse, we have a massive steel ball bouncing between the two mirrors. The ball’s speed

As a consistency check, by applying the velocity addition law (

Special relativity explains how all these particular results can be generalized into Lorentz transformations for the times and positions of events. For example, any event having 4-coordinates

However, it is important to note that all the above derivations used model systems without interactions. The second Einstein postulate is formally applicable only to light pulses and events associated with them. So, strictly speaking, we are not allowed to extend results of special relativity beyond corpuscular optics. In addition, one can show [

How can we be confident that the same conclusions apply to interacting systems? For example, what if the steel ball is bouncing between plates of a charged capacitor? Can we be sure that Lorentz formulas (

Here we meet the following fork in the road. On the one hand, we can choose to postulate that the laws of special relativity established above are valid independent of interactions. Then boosts should be rigorously kinematical, just as space translations and rotations. This nonobvious postulate is tacitly assumed in all textbooks. In particular, it was used in numerous attempts [

Alternatively, we can assume that, similar to time translations, boosts are dynamical; that is, they involve interactions, and their actions cannot be expressed by simple universal formulas, like (

We are interested in application of inertial transformations to quantum systems. Properties of such systems are described by objects in the Hilbert space

Important role is played by the so-called “infinitesimal transformations” or generators. They are represented by Hermitian operators in

Commutators of the Hermitian generators are fully determined by the structure of the Poincaré group [

According to Wigner and Weinberg [

Each normalized state vector

Observations of the unstable particle can be also described in the quantum-logical language of yes-no questions, like “Do we see the unstable particle?” and an observable, which can take two values 1 or 0, corresponding to the possible answers “yes” or “no.” Obviously, the Hermitian operator of this observable is the projection

Next we should find out how the quantity

Using available 1-particle irreducible representations

According to Dirac and Weinberg [

A Poincaré-Wigner-Dirac relativistic quantum description of any isolated interacting system is constructed in a similar manner. In the Hilbert space

As we mentioned at the end of Section

In classical relativistic physics, this hypothesis is known as the condition of “invariant trajectories” or “manifest covariance.” The well-known Currie-Jordan-Sudarshan theorem [

One idea was that Hamiltonian dynamics is not suitable for describing relativistic interactions. Instead, various non-Hamiltonian theories were developed [

Another idea is to abandon particles and replace them by (quantum) fields [

We cannot accept this point of view, because it has nothing to say about such interacting time-dependent system as the unstable particle.

Our preferred way to resolve the Currie-Jordan-Sudarshan controversy is to abandon the hypothesis of “invariant trajectories” and admit that boost transformations are dynamical. Actually, even in the original Dirac’s paper [

Therefore, we should have

Our conclusion about the dynamical character of boosts disagrees with the usual special-relativistic “geometrical” view on boosts. In particular, we can no longer claim that

Returning to our example of unstable particle, we can say that when the observer at rest sees the pure unstable particle

Suppose that (

It is important that (

Here we discussed the dynamical effect of boosts on unstable particles (

The time dilation experiments with unstable particles [

Perhaps, the most convincing evidence for the dynamical character of boosts was obtained in the Frascati experiment [

We applied Poincaré-Wigner-Dirac theory of relativistic interactions to unstable particles. In particular, we were interested in how the same particle is seen by different moving observers. We proved that the decay probability cannot be invariant with respect to boosts. Different moving observers may see different internal compositions of the same particle. In spite of being very small, this effect is fundamentally important as it sets the limit of applicability for special relativity.

The author declares no conflicts of interest.