Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 086, 13 pages      arXiv:1109.0598      https://doi.org/10.3842/SIGMA.2011.086
Contribution to the Proceedings of the Workshop “Supersymmetric Quantum Mechanics and Spectral Design”

Time Asymmetric Quantum Mechanics

Arno R. Bohm a, Manuel Gadella b and Piotr Kielanowski c
a) Department of Physics, University of Texas at Austin, Austin, TX 78712, USA
b) Departamento de FTAO, Universidad de Valladolid, 47071 Valladolid, Spain
c) Cinvestav, Dept Fis, Mexico City 07000, DF Mexico

Received January 30, 2011, in final form August 22, 2011; Published online September 03, 2011

Abstract
The meaning of time asymmetry in quantum physics is discussed. On the basis of a mathematical theorem, the Stone-von Neumann theorem, the solutions of the dynamical equations, the Schrödinger equation (1) for states or the Heisenberg equation (6a) for observables are given by a unitary group. Dirac kets require the concept of a RHS (rigged Hilbert space) of Schwartz functions; for this kind of RHS a mathematical theorem also leads to time symmetric group evolution. Scattering theory suggests to distinguish mathematically between states (defined by a preparation apparatus) and observables (defined by a registration apparatus (detector)). If one requires that scattering resonances of width Γ and exponentially decaying states of lifetime τ=h/Γ should be the same physical entities (for which there is sufficient evidence) one is led to a pair of RHS's of Hardy functions and connected with it, to a semigroup time evolution t0≤t<∞, with the puzzling result that there is a quantum mechanical beginning of time, just like the big bang time for the universe, when it was a quantum system. The decay of quasi-stable particles is used to illustrate this quantum mechanical time asymmetry. From the analysis of these processes, we show that the properties of rigged Hilbert spaces of Hardy functions are suitable for a formulation of time asymmetry in quantum mechanics.

Key words: resonances; arrow of time; Hardy spaces.

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