The Spin Spectrum of an Unstable Particle

Abstract

A state vector (vector-valued distribution) of an unstable particle can be a 3-momentum eigenvector in at most one inertial frame as a consequence of the necessary width of the energy spectrum. We investigate this frame dependence of momentum eigenvectors of unstable particles and demonstrate that: (i) It is compatible with the principle of relativity. (ii) It leads to a distinction between two ways of defining the spin of the particle which are equivalent for stable particles. (iii) One definition, called kinematical spin, yields a precise value, and is determined by considerations of detailed balancing and other means of counting the degeneracy of the momentum eigenvectors. (iv) The second definition, called dynamical spin, need not yield a sharp value, is determined by the dynamics of the decay process, and is measured by observing angular distributions of decay products. It is shown that an unstable particle with vanishing kinematical spin (pion) may have a small admixture of non-isotropic angular distribution of unpolarized decay products in the rest frame of the decay products. The order of magnitude of the effect is estimated and the coupling between the mass and spin spectrum is studied in the case of local interactions without derivative coupling.