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arXiv:hep-th/0212282 · Fortschritte der Physik

Time asymmetric quantum theory – III. Decaying states and the causal Poincaré semigroup

Arno Böhm, Hani Kaldass, Sujeewa Wickramasekara

Published May 13, 2003• 19 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1002/prop.200310075

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Time asymmetric quantum theory – II. Relativistic resonances from S‐matrix poles

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Abstract Relativistic resonances and decaying states are described by representations of Poincaré transformations, similar to Wigner's definition of stable particles. To associate decaying state vectors to resonance poles of the S‐matrix, the conventional Hilbert space assumption (or asymptotic completeness) is replaced by a new hypothesis that associates different dense Hardy subspaces to the in‐ and out‐scattering states. Then one can separate the scattering amplitude into a background amplitude and one or several “relativistic Breit‐Wigner” amplitudes, which represent the resonances per se. These Breit‐Wigner amplitudes have a precisely defined lineshape and are associated to exponentially decaying Gamow vectors which furnish the irreducible representation spaces of causal Poincaré transformations into the forward light cone.

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Abstract A slight modification of one axiom of quantum theory changes a reversible theory into a time asymmetric theory. Whereas the standard Hilbert space axiom does not distinguish mathematically between the space of states (in‐states of scattering theory) and the space of observables (out‐“states” of scattering theory) the new axiom associates states and observables to two different Hardy subspaces which are dense in the same Hilbert space and analytic in the lower and upper complex energy plane, respectively. As a consequence of this new axiom the dynamical equations (Schrödinger or Heisenberg) integrate to a semigroup evolution. Extending this new Hardy space axiom to a relativistic theory provides a relativistic theory of resonance scattering and decay with Born probablilities that fulfill Einstein causality and the exponential decay law.

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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.
Abstract Relativistic resonances and decaying states are described by representations of Poincaré transformations, similar to Wigner's definition of stable particles. To associate decaying state vectors to resonance poles of the S‐matrix, the conventional Hilbert space assumption (or asymptotic completeness) is replaced by a new hypothesis that associates different dense Hardy subspaces to the in‐ and out‐scattering states. Then one can separate the scattering amplitude into a background amplitude and one or several “relativistic Breit‐Wigner” amplitudes, which represent the resonances per se. These Breit‐Wigner amplitudes have a precisely defined lineshape and are associated to exponentially decaying Gamow vectors which furnish the irreducible representation spaces of causal Poincaré transformations into the forward light cone.
Starting from the phenomenological lineshape, relativistic Gamow vectors are defined. They span an irreducible representation ([ j , s R ]) of the causal Poincaré semigroup. Their transformation properties are presented, from which follow the exponential time evolution of the (relativistic) Gamow states (of spin j and mass s R = ( M − iΓ/2) 2 ). The preparation and analysis of decay data are complicated by the presence of a continuous background integral—always omitted in the Weisskopf–Wigner approximation—in the complex basis vector expansion for the in-state of a resonance scattering experiment. This background integral, which is related to the background term in the scattering amplitude, gives rise to deviations from the exponential time evolution. To what extent a prepared state decays exponentially depends on the experiment and not the resonance per se .
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.
Gamow vectors in non-relativistic quantum mechanics are generalized eigenvectors (kets) of self-adjoint Hamiltonians with complex eigenvalues. Like the Dirac kets, they are mathematically well defined in the Rigged Hilbert Space. Gamow kets are derived from the resonance poles of the S-matrix. They have a Breit-Wigner energy distribution, an exponential decay law, and are members of a basis vector expansion whose truncation gives the finite dimensional effective theories with a complex Hamiltonian matrix. They also have an asymmetric time evolution described by a semigroup generated by the Hamiltonian, which expresses a fundamental quantum mechanical arrow of time. These Gamow kets are generalized to relativistic Gamow vectors by extrapolating from the Galilei group to the Poincare group. This leads to semigroup representations of the Poincare group which are characterized by spin j and complex invariant mass square. In these non-unitary representations the Lorentz subgroup is unitarily represented and the four-momenta are "minimally complex" in the sense that the four-velocity is real. The relativistic Gamow vectors have all the properties listed above for the non-relativistic Gamow vectors and are therefore ideally suited to describe relativistic resonances and quasistable particles.
For a quantum theory that includes exponentially decaying states and Breit–Wigner resonances, which are related to each other by the lifetime–width relation , where τ is the lifetime of the decaying state and Γ is the width of the resonance, one has to go beyond the Hilbert space and beyond the Schwartz–Rigged Hilbert Space of the Dirac formalism. One has to distinguish between prepared states, using a space , and detected observables, using a space , where −(+) refers to analyticity of the energy wavefunction in the lower (upper) complex energy semiplane. This differentiation is also justified by causality: a state needs to be prepared first, before an observable can be measured in it. The axiom that will lead to the lifetime–width relation is that Φ + and Φ − are Hardy spaces of the upper and lower semiplane, respectively. Applying this axiom to the relativistic case for the variable leads to semigroup transformations into the forward light cone (Einstein causality) and a precise definition of resonance mass and width.
Abstract A slight modification of one axiom of quantum theory changes a reversible theory into a time asymmetric theory. Whereas the standard Hilbert space axiom does not distinguish mathematically between the space of states (in‐states of scattering theory) and the space of observables (out‐“states” of scattering theory) the new axiom associates states and observables to two different Hardy subspaces which are dense in the same Hilbert space and analytic in the lower and upper complex energy plane, respectively. As a consequence of this new axiom the dynamical equations (Schrödinger or Heisenberg) integrate to a semigroup evolution. Extending this new Hardy space axiom to a relativistic theory provides a relativistic theory of resonance scattering and decay with Born probablilities that fulfill Einstein causality and the exponential decay law.