Representations of the Poincaré group associated with unstable particles

The problem of a relativistically covariant description of unstable particles is reexamined. We follow the approach which associates a unitary reducible representation of the Poincaré group with a larger isolated system, and compare it with the one ascribing a nonunitary irreducible representation to the unstable particle alone. It is shown that the problem originates in the choice of the subspace ${\mathcal{H}}_u$ of the state Hilbert space which could be related to the unstable particle. Translational invariance of ${\mathcal{H}}_u$ is proved to be incompatible with unitarity of the boosts. Further, we propose a concrete choice of ${\mathcal{H}}_u$ and argue that in most cases of the actual experimental arrangements this subspace is effectively one dimensional. A correct slow-down for decay of the moving particles is obtained.