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Particles and Fields
Eur. Phys. J. C 28, 279-295 (2003)
DOI: 10.1140/epjc/s2003-01147-y

Relativistic resonances as non-orthogonal states in Hilbert space

W. Blum and H. Saller

Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, München, Germany


(Received: 28 June 2002 / Revised version: 13 November 2002 / Published online: 14 March 2003)

Abstract
We analyze the energy-momentum properties of relativistic short-lived particles with the result that they are characterized by two 4-vectors: in addition to the familiar energy-momentum vector (timelike) there is an energy-momentum `spread vector' (spacelike). The wave functions in space and time for unstable particles are constructed. For the relativistic properties of unstable states we refer to Wigner's method of Poincaré group representations that are induced by representations of the space-time translation and rotation groups. If stable particles, unstable particles and resonances are treated as elementary objects that are not fundamentally different one has to take into account that they will not generally be orthogonal to each other in their state space. The scalar product between a stable and an unstable state with otherwise identical properties is calculated in a particular Lorentz frame. The spin of an un stable particle is not infinitely sharp but has a `spin spread' giving rise to `spin neighbors'. This opens the possibility of a non-zero scalar product between states with unequal spin. - A first practical application of non-orthogonal states is seen in diffraction dissociation reactions whose large cross-sec tions are attributed to interference of states that are `partially identical'.



© Società Italiana di Fisica, Springer-Verlag 2003