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 Poincaré transformations into the forward light cone.
>An unstable particle capable of undergoing real decay - N + theta is considered in the nonrelativistic second quantization theory (Lee model). Perturbation theory and the expansion of an arbitrary state in eigen states are extended to the case of an unstable particle. The quantity which plays the role of the state norm of such a particle is determined. A new method is proposed for determining the fraction of time a stable particle V capable of undergoing the virtual transformation into N + theta stays in the state V and in the state N + theta . If the method is applied to an unstable particle V, one can find the time it is in each of the two states, but the answer is expressed in complex numbers. (auth)