Rigged Hilbert space

Preparation-registration processes have been discussed by Amo Bohm and coworkers (see for instance [27, 28]). The issues developed in reference [27] are left out in the present analysis. We mention these references to indicate the existence of subtle and complex mathematical issues related to rigged Hilbert spaces. Here we follow the quantum scattering approach to an extent required to discuss specifically chemical features. 

The I-frame localizes a projected quantum state this frame is a classical physics element, whereas configuration base states x) define a rigged Hilbert space basis [1]. 

The merger is between Special Theory of Relativity and abstract quantum states through rigged Hilbert spaces the inertial frame is used to set up an abstract configuration space in laboratory space. At the abstract level, both formalisms are required. 

The relationship of the Rigged-Hilbert space formulation of quantum mechanics with the formalism of resonant states discussed here requires to be clarified. For example, it is not clear that the complex poles seated on the third quadrant of the complex k plane and their corresponding resonant states U-n play a role in that formalism for times f > 0. As we have shown, these poles are essential to obtain the long time 1/P behavior of decay. In facf the time evolution of decay in terms of fhe resonant state formalism coincides exactly with the numerical solution to the time-dependent Schrbdinger equation of the problem. At a more fundamental level, the above issue is related to the understanding of irreversibility at the quantum level, where the formation and decay of transient states play a fundamental role [93,99]. Further work is required on this issue. 

H. Sailer and W. Blum, On the relation between time representations and inner product spaces, Proceedings from the Jaca Workshop on Rigged Hilbert Spaces and Time Asymmetric Quanmm Theory (eds A. Bohm, M. Gadella and P. Kielanowski),... 

Robert C. Bishop [2004] recently published an independent analysis of the development of Prigogine s research program, with emphasis on work since the mid-1980s. He reports how aspects of work of the BAG did not fit comfortably into the mathematical framework of Hilbert space (HS) in which it had been cast. Eventually it was realized that a different framework — the rigged Hilbert space (RHS) that had been developed earlier for other purposes — was more appropriate for dealing with dynamic systems of the type that interested the BAG. 

The column vector is indicated by square brackets, a row vector by round brackets. The quantum numbers may be determined by the complete set of her-mitian operators commuting with the generator of time evolution. Invariance of the quantum state to frame rotation, origin displacement, parity and other symmetry operations determine quantum numbers for the corresponding irreducible representations. Frame related symmetry operations translate into unitary operator acting on Hilbert space (rigged), e.g. Ta. 

The box-Hilbert space is now rigged with the asymptotic states that can be probed at the laboratory as if they were space separate quantum objects. 

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