Re: 10 questions on QM postulates

Hendrik van Hees wrote:

> Arnold Neumaier wrote:
>
>>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.
>
> Yeah, that's right. So in other words, I have to "smear" the usual Fock
> states with wavepackets to get vectors in the Hilbert space.

... and the field operators with test functions to get densely defined
operators on the Hilbert space.


> If one is
> honest, one must do this all the time, I agree with you, especially
> when deriving the LSZ-reduction formula and the like.

Yes. A few people actually do it, for example Scharf's book on QED.


> On the other hand, isn't this what the axiomatic quantum-field theorists
> try to do, and they never could get a definite answer to the question
> of a rigorous treatment of interacting relativististic QFTs?

They can, in 2 and 3 dimensions (superrenormalizable theories).
For 4 dimensions, the known analytic estimates are too weak to
prove the existence of the required limits.


> I don't
> know much about the current status of this endeaver, but most
> physicists, I know and work with, do not think very high of this quite
> mathematical branch of QFT. One once said about it: "Ah, that's very
> deep stuff, but they cannot explain more than the free particle. They
> can not even prove QED to exist." Is this still true?

In 4D, yes.
But some of this stuff will be essential for a final solution
of the problem when it comes. A mathematician can tell structure...
So for me it is just one of the boundary conditions to be met;
the oher boundary condition is what physicists are doing.

A working physicists can therefore ignore that part of QFT.
But someone looking for better foundations of QFT
(which currently is a mess!) cannot.


Arnold Neumaier

.

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