Re: Connes & Marcolli paper on renormalization

From: Arnold Neumaier (Arnold.Neumaier_at_univie.ac.at)
Date: 11/12/04 

Date: Fri, 12 Nov 2004 19:08:34 +0000 (UTC)

Hendrik van Hees wrote:
> Eugene Stefanovich wrote:
>
>>Could you please show me the Hamiltonian you are using to
>>describe the time evolution? (reference is OK) How do you prove
>>that the vacuum vector remains unchanged during time evolution?
>
> The QED-Lagrangian (and thus also the Hamiltonian) can be found in any
> textbook on quantum field theory.

Butthe textbook Hamiltonian does _not_ preserve the textbook vacuum!

>>Power counting applies to Hamiltonians (or Lagrangians) written as
>>products of fields. My Hamiltonian cannot be written in terms of
>>fields. It can be only written in terms of creation and annihilation
>>operators of particles.
>
> Then you should check, whether the whole thing is really physically
> meaningful, especially whether the S-matrix is Lorentz invariant,
> unitary, and fulfills the Cluster-Decomposition property. See
> Weinberg's Quantum Theory of fields, Vol I, Cam. Uni. Press for
> details, why local QFTs guarantee these important properties of the S
> matrix.

Actually, creation and annihilation operators _are_ field operators,
and one can reconstruct the standard psi(x) as a linear combination
of the a(x) and a^*(x). Thus Stefanovich constructs a field theory,
but his peculiar philosophy blinds him to this fact.

>>The power counting scheme does not apply.
>>Moreover, I have a proof that the S-matrix of my approach is exactly
>>the same as the S-matrix of renormalized QED. My approach is
>>not just renormalizable. It is finite. Both Hamiltonian and the
>>S-matrix are finite.
>
> Then your theory should be identical with QED in some way.

It is. The representation is related to the usual one by a
perturbatively constructed Bogoliubov transformation which turns the
bare vacuum into the physical vacuum and the bare single-particle states
into the physical single-particle states. The resulting theory is
manifestly finite in perturbation theory.

Arnold Neumaier

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