Re: nonlinearities in QFT
From: Arnold Neumaier (Arnold.Neumaier_at_univie.ac.at)
Date: 10/26/04
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Date: Tue, 26 Oct 2004 17:55:26 +0000 (UTC)
Patrick Van Esch wrote:
> Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote in message news:<4157E1B9.5030302@univie.ac.at>...
>
>>The standard theory gives an S-matrix (or rather an asymptotic series
>>for it) but not a dynamics at finite times. Clearly, measurements happen
>>in finite time, hence cannot be described at present in a fundamental way
>>(i.e., beyond the nonrelativistic QM approximation).
>
> Isn't this a too pessimistic view ? After all, the framework formally
> allows us to have the finite time evolution, it is just that it is
> plagued by infinities and inconsistencies (Haag ?).
Yes. There is no consistent interaction picture in the sense we have it
in QM. Nobody found a consistent notion in QFT of a state at finite times.
Without such a notion there cannot be a dynamical law.
(However, one can compute - nonrigorously, in perturbation theory -
some time-dependent things, namely via the Schwinger-Keldysh
formalism; see, e.g., http://theory.gsi.de/~vanhees/publ/green.pdf )
> But as a "picture
> in the mind" cannot quantum fields be considered as having an evolving
> quantum state in Hilbert space (even if it screws up each time you
> want to write it out explicitly) ?
As pictures in the mind one can have anything one finds helpful.
Probably people working in QFT imagine something like a state evolution
in Hilbert space underlying their formalism. After all, this is how
one justifies that the functional integral works.
The difficulties begin when one tries to draw conclusions that
have quantitative experimental consequences. This is a very efficient
test for distinguishing powerful intuition from feeble ghosts in one's
mind.
This does not mean that there is no dynamical reality underlying
relativistic QFT. It only means that no one has been able to find
a working framework for it.
Arnold Neumaier
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