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- Title
DECAY LAW AND TIME DILATATION.
- Authors
GIACOSA, FRANCESCO
- Abstract
We study the decay law for a moving unstable particle. The usual time-dilatation formula states that the decay width for an unstable state moving with a momentum p and mass M is ~ Γp = ΓM/√p²M² with Γ being the decay width in the rest frame. In agreement with previous studies, we show that in the context of QM as well as QFT, this equation is not correct provided that the quantum measurement is performed in a reference frame in which the unstable particle has momentum p (note, a momentum eigenstate is not a velocity eigenstate in QM). We then give, to our knowledge for the first time, an analytic expression of an improved formula p and we show that the deviation from Γp has a maximum for p/M = √2/3; but is typically very small. Then, the result can be easily generalized to a momentum wave packet and also to an arbitrary initial state. Here, we give a very general expression of the non-decay probability. As a next step, we show that care is needed when one makes a boost of an unstable state with zero momentum/velocity: namely, the resulting state has zero overlap with the elements of the basis of unstable states (it is already decayed!). However, when considering a spread in velocity, one finds again that Γp is typically a very good approximation. The study of a velocity wave packet represents an interesting subject which constitutes one of the main outcomes of the present manuscript. In the end, it should be stressed that there is no whatsoever breaking of special relativity, but as usual in QM, one should specify which kind of measurement on which kind of state is performed.
- Subjects
TIME dilation; RADIOACTIVE decay; QUANTUM measurement; SPECIAL relativity (Physics); WAVE packets; MOMENTUM wave function
- Publication
Acta Physica Polonica B, 2016, Vol 47, Issue 9, p2135
- ISSN
0587-4254
- Publication type
Article
- DOI
10.5506/APhysPolB.47.2135