Relativistic Theory of Unstable Particles. II

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 $\rho \left(m^2\right)$ 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 $\rho \left(m^2\right)$]. The proper-time propagation of this state gives the decay amplitude, and its modulus is ideally the experimentally observed decay-time plot.