Difficulty with a Kinematic Concept of Unstable Particles: the SZ.-Nagy Extension and the Matthews-Salam-Zwanziger Representation

Abstract. We discuss the possibility of describing unstable systems, or dissipative systems in general, by vectors in a Hilbert space, evolving in time according to some non-unitary group or semigroup of translations. If the states of the unstable or dissipative system are embedded ina larger Hilbert space containing "decay products " as well, so that the time evolution f the system as a whole becomes unitary, we show that the infinitesimal generator necessarily has all energies from minus to plus infinity in its spectrum. This result supplements and extends the well-known fact that a positive energy spectrum is incompatible with a decay law bounded by a decreasing exponential. As an example of both facts, we discuss Zwanziger's irreducible, nonunitary representation f the Poincar6 group; and we find its minimal, unitary extension (the Sz.-Nagy construction). The answer provides a mathematically canonical approach to the Matthews-Salam theory of wave functions for unstable, elementary particles, where the spectrum difficulty was already recognized. We speculate on the possibility that the Matthews-Salam-Zwanziger rep esen-tation might be a strong coupling approximation i the relativistic version of the Wigner-Weisskopf theory, but we have not shown the existence of a physically acceptable mode