Relativistic Theory of Unstable Particles. II

Abstract

This paper is a direct continuation of an earlier paper (I) where an attempt was made to set up a field-theoretic foundation for the theory of mean mass and lifetime of an unstable particle. It was argued in I that the decay-time plot of a beam of unstable particles is a concept peculiar to a single-particle theory; that from a field-theoretic point of view, mass (the variable conjugate to proper time) rather than time has the primary significance. Here we show that the spectral function $\ensuremath{\rho}({m}^{2})$ appearing in the (field-theoretic) one-particle propagator has a direct significance as the probability of finding in production an unstable particle of mass $m$. This allows us to define a "one-particle" state for the unstable particle as a superposition of its outgoing decay states suitably weighted in mass space [with a factor which is the square-root of $\ensuremath{\rho}({m}^{2})$]. The proper-time propagation of this state gives the decay amplitude, and its modulus is ideally the experimentally observed decay-time plot.The time plot is explicitly evaluated for $\ensuremath{\pi}$ decay. Insofar as the distribution of mass values for the $\ensuremath{\pi}$ meson starts with the $\ensuremath{\mu}$ mass (assumed stable), the time plot is not merely the conventional decay exponential ${e}^{\ensuremath{-}\frac{\ensuremath{\tau}}{{\ensuremath{\tau}}_{0}}}$. There are additional terms which become important about a hundred lifetimes after the particle is created.Finally we compare the time plots for particle and antiparticle decays on the basis of $\mathrm{CTP}$ invariance.