Description of unstable particles in quantum field theory

The consequences of a pole on the second sheet of the S matrix are investigated under the assumption that a certain Green's function has the same pole. It is shown that corresponding to each such pole is an eigenstate of the Hamiltonian with a complex energy. These eigenstates lie in a natural extension of the physical Hilbert space. Because it is the vector space that is modified and not the Hamiltonian, unstable particle states transform covariantly. They have complex energy and momentum but real integer or half-integer spin. Scattering amplitudes involving unstable particles are expressed as residues of poles in a reduction formula of the Lehmann-Symanzik-Zimmermann type. An example from potential theory is worked out in detail.