Time Asymmetric Quantum Mechanics and Relativistic Resonances

Arno Böhma, Hani Kaldass, S. Wickramasekara

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.

Arno Böhm

Arno Böhm, Fatih Erman, Haydar Uncu

This article is a review of time asymmetric quantum theory and its consequences applied to the resonance and decay phenomena. We first give some phenomenological results about resonances and decaying states to support the popular idea that resonances characterized by a width \Gamma and decaying states characterized by a lifetime \tau are different appearances of the same physical entity. Based on Weisskopf-Wigner (WW) methods, one obtains approximately \frac{\hbar}{\tau}\approx \Gamma. However, using standard axioms of quantum physics it is not possible to establish a rigorous theory to which the various WW methods can be considered as approximations. In standard quantum theory, the set of states and the set of observables are mathematically identified and described by the same Hilbert space H. Modifying this Hilbert space axiom to a Hardy space axiom one distinguishes the prepared (in) states and detected (out) observables. This leads to semi-group time evolution and to beginnings of time for individual microsystems. As a consequence of this time asymmetric theory one derives \frac{\hbar}{\tau} = \Gamma as an exact relation, and this unifies resonances and decaying states. Finally, we show that this unification can also be extended to the relativistic regime.

H. D. Zeh

T. E. Phipps

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.