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.

The time plot is explicitly evaluated for $\pi$ decay. Insofar as the distribution of mass values for the $\pi$ meson starts with the $\mu$ mass (assumed stable), the time plot is not merely the conventional decay exponential $e^{-\frac{\tau }{{\tau }_0}}$. 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.

• Received 16 March 1959

DOI:https://doi.org/10.1103/PhysRev.115.1079