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CRC Press eBooks

Groups and Semigroups in Scattering Theory - Poincare Transformations and Decaying States

Arno Böhma, Hani Kaldass, S. Wickramasekara

Published Nov 30, 2003• 0 citations
Publisher

Abstract

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.

DOI
10.1201/9781482269079-95

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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.

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Relativistic resonances, semigroup representation of Poincar transformations, the exponential decay law and deviations thereof

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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 .

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Relativistic Gamow Vectors

Arno Böhm, Hani Kaldass, Sujeewa Wickramasekara +1 more

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.

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Time asymmetric quantum theory and its applications in non-relativistic and relativistic quantum systems

Hai Viet Bui

The exact relation τ = ~/Γ between the width Γ of a resonance and the lifetime\nτ for a decay of this resonance could not be obtained in the conventional quantum\ntheory based on the Hilbert space as well as on the Schwartz space in which the spaces\nof the states and of the observables are described by the same space. Furthermore,\nboth dynamical evolution of the states (in Schrodinger picture) and observables (in\nHeisenberg picture) are symmetrically in time, given by an unitary group with time\nextending over −∞ < t < +∞. This time symmetric evolution is a mathematical\nconsequence of Von Neumann theorem for the dynamical differential equations,\nSchrodinger equation for the state or Heisenberg equation for the observable, under\neither the Hilbert space or Schwartz space boundary condition. However, this unitary\ngroup evolution violates causality. In order to get a quantum theory in which the\nexact relation and causality are obtained, one has to replace the Hilbert space or the\nSchwartz space boundary condition by a new boundary condition based on the Hardy\nspace axioms in which the space of the states and the space of the observables are\ndescribed by two distinguishable Hardy spaces. As a consequence of the new Hardy\nspace axiom, one obtains, instead of the time symmetric evolution for the states and\nthe observables, time asymmetrical evolutions for the states and observables which\nare described by two semi-groups. The time asymmetrical evolution predicts an finite\nbeginning of time t0(= 0) for quantum system which can be experimental observed\nby directly measuring the lifetime of the decaying state as the beginning of time of\nthe ensemble in scattering experiments. A resonance obeying the exponential time\nevolution can then be described by a Gamow vector, which is defined as superposition\nof the exact out-plane wave states with exact Breit-Wigner energy distribution.

•Jan 1, 2016•0 citations
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.
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 .
The exact relation τ = ℏ/Γ between the width Γ of a resonance and the lifetime τ for the decay of this resonance could not be obtained in standard quantum theory based on the Hilbert space or Schwartz space axiom in non-relativistic physics as well as in the relativistic regime. In order to obtain the exact relation, one has to modify the Hilbert space axiom or the Schwartz space axiom and choose new boundary conditions based on the Hardy space axioms in which the space of the states and the space of the observables are described by two different Hardy spaces. As consequences of the new Hardy space axioms, one obtains, instead of the symmetric time evolution for the states and the observables, asymmetrical time evolutions for the states and observables which are described by two semi-groups. A relativistic resonance obeying the exponential time evolution can be described by a relativistic Gamow vector, which is defined as superposition of the exact out-plane wave states with a Breit-Wigner energy distribution of the width Γ.
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.
The exact relation τ = ~/Γ between the width Γ of a resonance and the lifetime\nτ for a decay of this resonance could not be obtained in the conventional quantum\ntheory based on the Hilbert space as well as on the Schwartz space in which the spaces\nof the states and of the observables are described by the same space. Furthermore,\nboth dynamical evolution of the states (in Schrodinger picture) and observables (in\nHeisenberg picture) are symmetrically in time, given by an unitary group with time\nextending over −∞ < t < +∞. This time symmetric evolution is a mathematical\nconsequence of Von Neumann theorem for the dynamical differential equations,\nSchrodinger equation for the state or Heisenberg equation for the observable, under\neither the Hilbert space or Schwartz space boundary condition. However, this unitary\ngroup evolution violates causality. In order to get a quantum theory in which the\nexact relation and causality are obtained, one has to replace the Hilbert space or the\nSchwartz space boundary condition by a new boundary condition based on the Hardy\nspace axioms in which the space of the states and the space of the observables are\ndescribed by two distinguishable Hardy spaces. As a consequence of the new Hardy\nspace axiom, one obtains, instead of the time symmetric evolution for the states and\nthe observables, time asymmetrical evolutions for the states and observables which\nare described by two semi-groups. The time asymmetrical evolution predicts an finite\nbeginning of time t0(= 0) for quantum system which can be experimental observed\nby directly measuring the lifetime of the decaying state as the beginning of time of\nthe ensemble in scattering experiments. A resonance obeying the exponential time\nevolution can then be described by a Gamow vector, which is defined as superposition\nof the exact out-plane wave states with exact Breit-Wigner energy distribution.