Re: Dynamical Systems and Expansion-Contraction

>>From Osher Doctorow

A. Bohm, H. D. Doebner, P. Kielanowski (Eds.) of U. Texas Austin,
Irreversibility and Causality, Springer: Berlin, 1998, is the reference
that I was referring to. J. P. Antoine wrote the first paper in that
volume, "Quantum mechanics beyond Hilber space," which regarded the
change from the usual Hilbert space formalism as the most
revolutionary. I meant to add Rigged Hilbert Spaces (RHS) to the
previous post, which Antoine defines as in my notation:

1) RHS = A triplet (A, H, B) where H is a Hilbert space, A is dense
subspace of H with a locally convex topology with a stronger notion of
convergence than that inherited from H (a finer topology than the norm
topology inherited from H), B is the space of continuous antilinear
functionals on A.

Examples of RHS include the Schwartz distributions spaces over R or
R^N. One usually also requires that A satisfy completeness,
reflexivity, countable intersection (A is intersection of countable
family of Hilbert spaces Hn, n = 1, 2, 3, ..., so that it is a Frechet
space) and nuclearity (for each n, the embedded Hm --> Hn is a
Hilbert-Schmidt operator for some m > n).

The Rigged Hilbert Space formalism is able to deal with continuous
spectra.

The ordinary Hilbert space of QM does not allow a time-asymmetric
formulation even though the physics of our universe is time-asymmetric
according to A. Bohm and N. L. Harshman in "Quantum theory in the
rigged Hilbert space - irreversibility from causality,"
arXiv:quant-ph/9805063 v1 21 May 1998. Time-asymmetric solutions are
selected for the classical equation of general relativity (Big Bang,
Big Crunch, black hole, white hole) and electromagnetism
(retarded-advance) and have to apply as well to the mathematical theory
of Quantum physics according to Bohm and Harshman, and the rigged
Hilbert Space theory allows time asymmetry.

Osher Doctorow

.

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