Physical Principles and Properties of Unstable States

TL;DR

This paper explores the nature of unstable states in quantum mechanics and quantum field theory, discussing their mathematical representation, physical properties, and the implications for unitarity and Poincaré symmetry.

Contribution

It clarifies how unstable states can be represented as generalized eigenvectors in rigged Hilbert space and discusses their role in quantum field theory without contradicting unitarity.

Findings

Unstable states are not asymptotic states in quantum field theory.

Unstable states can be represented as complex eigenvectors in rigged Hilbert space.

The eigenvalues' real and imaginary parts relate to mass and decay width.

Abstract

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.