The Zeno’s paradox in quantum theory

A quantum-theoretic expression is sought for the probability that an unstable particle prepared initially in a well-defined state will be found to decay sometime during a given interval. It is argued that probabilities like this which pertain to continuous monitoring possess operational meaning. A simple natural approach to this problem leads to the startling conclusion that an unstable particle which is continuously observed whether it decays will never be found to decay. Since recording the track of an unstable particle (which can be distinguished from its decay products) realizes such continuous observations to a close degree of approximation, the above conclusion poses a paradox which we call Zeno's Paradox in Quantum Theory. Its implications and possible resolutions are briefly discussed. The mathematical transcription of the above-mentioned conclusion is a structure theorem concerning semigroups. Although special cases of this theorem are known, the general formulation and the proof given here are believed to be new. The known ''no-go'' theorem concerning the semigroup law for the reduced evolution of any physical system (including decaying systems) is subsumed under the theorem as a direct corollary.