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arXiv:quant-ph/0312178

Physical Principles and Properties of Unstable States

Piotr Kielanowski

Published Dec 21, 2003• 5 citations
arXivHTMLPDF

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 Poincaré transformations into the forward light cone.

DOI
10.48550/arxiv.quant-ph/0312178

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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.
Within the rigged Hilbert space formulation of quantum mechanics idealized resonances (without background) are described by generalized eigenvectors of an essentially self-adjoint Hamiltonian with complex eigenvalue and a Breit–Wigner energy distribution. This establishes the link between the S matrix description of resonances by a pole and the usual description of states by vectors, overcomes theoretical problems connected with the deviation from exponential law and simplifies the calculation of the decay rate formula.
In this talk we discuss how to combine aspects of the theory of stable relativistic particles and of the theory of non-relativistic decaying particles into a relativistic theory of unstable particles and resonances. For the theory of stable relativistic particles we use Wigner’s description by unitary irreducible representations (UIR) of the Poincaré group [ 1 ] which has been the foundation for relativistic quantum mechanics and field theory [ 2 ]. For the non-relativistic decaying particles we use a time asymmetric quantum mechanics in which the Hilbert space axiom, that allows only symmetric (unitary group) time evolution, is replaced by time asymmetric boundary conditions which lead to an irreversible , semigroup time evolution [ 3 ]. In the limit in which certain contributions to the energy continuum are neglected, this new quantum mechanics leads to the Weisskopf-Wigner approximate methods [ 4 ] for decay, and to the Breit-Wigner energy distribution [ 5 ] for the resonance lineshape. We thus generalize and combine two fundamental contributions of Wigner into a description of relativistic resonances and decaying states. The central concept of this description is the relativistic Gamow vector, which is a generalized eigenvector (ket) of the self-adjoint invariant square mass operator https://www.w3.org/1998/Math/MathML" display="inline"> M 2 = P μ P μ P μ = P 1 μ + P 2 μ in resonance scattering 1±2→ R →3±4) with a complex eigenvalue, a relativistic Breit-Wigner energy distribution, and an exponential time evolution governed by a semigroup of continuous operators in the space of kets.
The Hilbert-Space Description By the static description of a physical system we mean the rules that assign specified mathematical objects to the states and to the physical quantities of the system, and the prescriptions for calculating the probability distribution of the possible values of every physical quantity when the state of the system is given. In the usual Hilbert-space formulation of quantum mechanics, to each physical system is attached a separable Hilbert space (generally infinite-dimensional) over the complex field. To every physical quantity is associated a linear, self-adjoint, not necessarily bounded operator on. If one deals with a strictly quantum system, then the converse is also generally assumed: every self-adjoint operator on represents some physical quan- tity. The restriction “strictly quantum” is necessary: if the system retains some nonquantum (i.e. classical) feature, so that it requires the algorithm of so-called “superselection rules” (see Chapter 5), then there are self-adjoint operators on that do not represent physical quantities. It should also be stressed that even in the strictly quantum case, most of the self-adjoint operators actually do not represent “interesting” physical quantities: only a few of them represent physical quantities that are useful and meaningful for the description of the physical system (e.g., energy, momentum, position, angular momentum). Therefore, by asserting that every self-adjoint operator on corresponds to some physical quantity we mean there is no a priori impossibility of devising such a correspondence, but we do not claim that this correspondence is present in the real work of experimental physics.
Abstract These Gamow kets span an irreducible representation space for Poincaré transformations which, similar to the Wigner representations for stable particles, are characterized by spin (angular momentum of the partial wave amplitude) and complex mass (position of the resonance pole). The Poincaré transformations of the Gamow kets, as well as of the Lippmann‐Schwinger plane wave scattering states, form only a semigroup of Poincaré transformations into the forward light cone. Their transformation properties are derived. From these one obtains an unambiguous definition of resonance mass and width for relativistic resonances. The physical interpretation of these transformations for the Born probabilities and the problem of causality in relativistic quantum physics is discussed.
We review some results in the theory of non-relativistic quantum unstable systems. We account for the most important definitions of quantum resonances that we identify with unstable quantum systems. Then, we recall the properties and construction of Gamow states as vectors in some extensions of Hilbert spaces, called Rigged Hilbert Spaces. Gamow states account for the purely exponential decaying part of a resonance; the experimental exponential decay for long periods of time physically characterizes a resonance. We briefly discuss one of the most usual models for resonances: the Friedrichs model. Using an algebraic formalism for states and observables, we show that Gamow states cannot be pure states or mixtures from a standard view point. We discuss some additional properties of Gamow states, such as the possibility of obtaining mean values of certain observables on Gamow states. A modification of the time evolution law for the linear space spanned by Gamow shows that some non-commuting observables on this space become commuting for large values of time. We apply Gamow states for a possible explanation of the Loschmidt echo.