The Zeno’s paradox in quantum theory

TLDR

Recording the track of an unstable particle approximates continuous observation, leading to a paradox we term Zeno’s paradox in quantum theory. The study seeks a quantum‑theoretic expression for the decay probability of an unstable particle over a given interval and discusses its relation to prior work and possible resolutions. The authors formulate a structure theorem concerning semigroups that mathematically transcribes the conclusion about continuous observation. They argue that probabilities for continuously monitored decay have operational meaning, show that continuous observation prevents decay, present a new general structure theorem for semigroups, and subsume the known no‑go theorem as a corollary.

Abstract

We seek a quantum-theoretic expression for the probability that an unstable particle prepared initially in a well defined state ρ 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 conclusion that an unstable particle which is continuously observed to see 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) approximately realizes such continuous observations, the above conclusion seems to pose a paradox which we call Zeno’s paradox in quantum theory. The relation of this result to that of some previous works and 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. We also note that the known ’’no-go’’ theorem concerning the semigroup law for the reduced evolution of any physical system (including decaying systems) is subsumed under our theorem as a direct corollary.