Re: 10 questions on QM postulates



Arnold Neumaier wrote:


However (returning to my simpler notation),
the states |x_1:N> = |x_1,...,x_N> lie in the top
part of the right Gelfand triple = rigged Hilbert space.
This is the name for a triple H in Hbar in H^* of vector spaces,
where Hbar is a Hilbert space, H a dense 'nuclear' subspace
(containing very smooth states with very good behavior at infintity)
and H^* its dual space (containing among others very singular states
and states with very poor behavior at infintity). Observables (in the
weak sense) are bilinear forms, or, which is the same, linear mappings
from H to H^*. The adjoint of such a linear mapping is again an
observable in the weak sense. Annihilation operators a(x) (and their
adjoints a^*(x)) are observables in this weak sense, although they are
not Hermitian (and a fortiori not self-adjoint).


Most physicists take it lightly since the times of Dirac.
They don't bother about self-adjointness or any other functional
analytic concept, unless ignoring it brings them into trouble.

Almost everything they do in the nonrelativistic regime
can be made rigorous in the rigged Hilbert space, so they fare right
even when they imagine wrongly that they work in a Hilbert
space. Thus they get away with their bad practices.
What they call 'Hilbert space' _is_ in fact always a
rigged Hilbert space; they just don't know and don't care.


Why should they care? Is there a specific example where the rigged
Hilbert space picture affects theoretical predictions that can be
measured in experiment?

Eugene Stefanovich


.

Relevant Pages

  • Re: Dynamical Systems and Expansion-Contraction
    ... Irreversibility and Causality, Springer: ... which Antoine defines as in my notation: ... RHS = A triplet where H is a Hilbert space, A is dense ... The Rigged Hilbert Space formalism is able to deal with continuous ...
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  • Re: 10 questions on QM postulates
    ... >>rigged Hilbert space; they just don't know and don't care. ... A working physicists can therefore ignore that part of QFT. ... Arnold Neumaier ...
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  • Re: 10 questions on QM postulates
    ... > Almost everything they do in the nonrelativistic regime ... > can be made rigorous in the rigged Hilbert space, ... > rigged Hilbert space; they just don't know and don't care. ... of a rigorous treatment of interacting relativististic QFTs? ...
    (sci.physics.research)
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