Energy-Momentum Properties and the Spin of Short-Lived Particles.

The space–time translation property of a stable particle is characterized by a time-like Lorentz vector (E, <MATH>\vec{k}</MATH>). We show in this contribution that unstable particles are, in addition, characterized by a space-like Lorentz 4-vector of uncertainties, or spreads, (Δ E, <MATH>\hskip2pt\smash{\raisebox{6.5pt}{_{\rightarrow}}\hspace*{-10pt}{\Delta k}}</MATH>). This is true for unstable states created in formation-, in production-, and in decay-experiments. The space-like nature of the spread vector causes a nonzero momentum spread to be present in all Lorentz frames so that there is no Lorentz frame in which the unstable particle is entirely at rest. With the space-like spreadvector (Δ E, <MATH>\hspace*{3pt}\smash{\raisebox{6.5pt}{_{\rightarrow}}\hspace*{-10pt}{\Delta k}}</MATH>) in addition to the time-like (E, <MATH>\smash{\vec{k}}</MATH>), also the rotation property of an unstable particle is affected, and unstable states have an uncertainty in their spin. This means neighboring spin states are occupied in addition to the original spin. Experiments are discussed that show a principal limitation of the accuracy of spin measurement from finite lifetimes. Wave functions for unstable particles are discussed, and we show in the example of a short-lived spin-0 state that the appearance of a spin neighbor in the amplitude is proportional to the inverse lifetime. [ABSTRACT FROM AUTHOR]