  The main subject of the paper is the description of unstable states in
quantum mechanics and quantum field theory. Unstable states in quantum field
theory can only be introduced as the intermediate states and not as asymptotic
states. The absence of the intermediate unstable states from the asymptotic
states is compatible with unitarity. Thus the concept of an unstable state is
not introduced in quantum field theory despite the fact that an unstable state
has well defined linear momentum, angular momentum and other intrinsic quantum
numbers. In the rigged Hilbert space quantum mechanics one can define vectors
that correspond to the unstable states. These vectors are the generalized
eigenvectors (kets in the rigged Hilbert space) with complex eigenvalues of the
self-adjoint Hamiltonian. The real part of the eigenvalue corresponds to the
mass of an unstable state and the imaginary part is one half of the total
width. Such vectors form the minimally complex semigroup representation of the
Poincar\'e transformations into the forward light cone.
