  The quantum mechanical description of the evolution of an unstable system
defined initially as a state in a Hilbert space at a given time does not
provide a semigroup (exponential) decay law. The Wigner-Weisskopf survival
amplitude, describing reversible quantum transitions, may be dominated by
exponential type decay in pole approximation at times not too short or too
long, but, in the two channel case, for example, the pole residues are not
orthogonal, and the evolution does not correspond to a semigroup (experiments
on the decay of the neutral $K$-meson system strongly support the semigroup
evolution postulated by Lee, Oehme and Yang, and Yang and Wu). The scattering
theory of Lax and Phillips, originally developed for classical wave equations,
has been recently extended to the description of the evolution of resonant
states in the framework of quantum theory. The resulting evolution law of the
unstable system is that of a semigroup, and the resonant state is a
well-defined function in the Lax-Phillips Hilbert space. In this paper we apply
this theory to a relativistically covariant quantum field theoretical form of
the (soluble) Lee model. We construct the translation representations with the
help of the wave operators, and show that the resulting Lax-Phillips $S$-matrix
is an inner function (the Lax-Phillips theory is essentially a theory of
translation invariant subspaces). In the special case that the $S$-matrix is a
rational inner function, we obtain the resonant state explicitly and analyze
its particle ($V, N, \theta$) content. If there is an exponential bound, the
general case differs only by a so-called trivial inner factor, which does not
change the complex spectrum, but may affect the wave function of the resonant
state.
