The last condition is crucial and will be interpreted in the next section in terms of the instantaneous mass at rest of the moving unstable particle. The square modulus of asymptotic expression (13) approximates the survival probability over long times:

For vanishing value of the linear momentum, p = 0, or, equivalently, in the rest reference frame of the unstable particle, the survival probability is approximated over long times as follows:

This interpretation holds if the asymptotic value of the instantaneous mass is considered as the effective mass of the moving unstable particle over long times.

Consequently, the instantaneous decay rate at rest remains approximately unchanged over long times in the reference frame where the unstable particle moves with linear momentum p.

In Section 2.1 how the survival probability at rest, [P.sub.0](t), transforms, due to the relativistic time dilation, in the reference frame where the unstable particle moves with constant linear momentum p is reported.

It is worth noticing that the scaling factor [[chi].sub.p] coincides with the ratio of the asymptotic form of the instantaneous mass [M.sub.p](t) and the asymptotic expression of the instantaneous mass at rest [M.sub.0](t) of the moving unstable particle:

In the latter case, the state vector of the moving unstable particle lies entirely in the subspace [H.sub.[alpha]], and there is no "decay caused by boost."

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.

Here we discussed the dynamical effect of boosts on unstable particles (44).

The time dilation experiments with unstable particles [1-4] are exceptional, because they study systems that are under the action of interaction during sufficiently long time interval.

We applied Poincare-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.

Khalfin, Quantum Theory of Unstable Particles and Relativity, 1997, preprint of Steklov Mathematical Institute, St.