Abstract
We review the mathematical tools that are suitable for a formulation of time asymmetry in quantum mechanics. In particular, Hardy functions on a half plane and rigged Hilbert spaces constructed with a subclass of Hardy functions. This time asymmetry often appears in quantum scattering and, in particular, in resonance scattering. We review the construction of Gamow vectors, often considered Gamow states for resonances. A brief summary of the fundamental ideas of time asymmetric quantum mechanics is presented in a last section.
Highlights
In the framework of standard non-relativistic quantum mechanics, the time evolution governed by a self adjoint Hamiltonian H is given by a group of unitary operators on a Hilbert space, depending on the parameter time t
The particle enters into an interaction region, where is subject of some forces that we shall assume that come from the existence of a potential V
Inside the interaction region, time evolution of its state is determined by a Hamiltonian H = H0 + V, which will transform this state
Summary
In the framework of standard non-relativistic quantum mechanics, the time evolution governed by a self adjoint Hamiltonian H is given by a group of unitary operators on a Hilbert space, depending on the parameter time t. Its quantum state is prepared as free This means that its time evolution is given by some sort of free Hamiltonian H0. The particle will abandon the interaction region and evolve again with H0 to be detected in the far future In this case, time evolution is given by a Hamiltonian pair {H0, H}. A interesting situation will occur when the particle spends in the interaction a time which is much larger than the time it would stay if the interaction would be switched off In this case, we say that a metastable state or quantum resonance has been produced. Capture (or preparation of the metastable state) and decay are not mutually symmetric and they are not time reversal of each other This type of asymmetry can be observed in other quantum scattering processes. For other descriptions, which do not exclude Hardy functions, see [9, 10]
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