Re: Connes & Marcolli paper on renormalization

From: Arnold Neumaier (Arnold.Neumaier_at_univie.ac.at)
Date: 11/24/04 

Date: Wed, 24 Nov 2004 07:47:07 +0000 (UTC)

Eugene Stefanovich wrote:
> Arnold Neumaier wrote:
>
>>Eugene Stefanovich wrote:
>>
>>>The Hamiltonian of renormalized QED is infinite
>>
>>Not necessarily. It is a limit of well-defined Hamiltonians with
>>perturbatively defined cutoff-dependent coupling coefficients,
>>and this limit might well turn out to exist in the resolvent sense.
>>Indeed, this happens for similar problem in 1D quantum mechanics with
>>singular potentials. You'd at least try to understand the part which is
>>known before you draw your conclusions.
>
> I don't see how Hamiltonian can be finite when parameters e and m
> turn out to be infinite after renormalization. Could you give me
> a reference to similar problems in QM?

I had given you one (delta-potential) quite early in our discussions,
but you chose to ignore it.

>>>and contains self-interaction of particles and vacuum.
>>
>>This does not make it ill-defined. Indeed, this Hamiltonian is exactly
>>the same as that of Shirokov, whose theory you called equivalent with
>>yours.
>
> In Shirokov's approach the dressing transformation redefines states
> (vacuum vector and creation and annihilation operators). So, the
> Hamiltonian is kept (mathematically)
> the same, while definitions of particles change.

These particles are just quasiparticles in the standard sense.
This is quite in line with the standard nonrelativisitc case,
where the vacuum is the ground state and the quasi-particles are the
excitations of the vacuum. Except that no infinities occur in the
nonrelativistic case.

> The redefinition of
> particles in the Shirokov's approach makes it wrong to say that
> he keeps the old QED Hamiltonian with self-interaction. The self-
> interaction effects got absorbed in the new definitions of particles,
> and the physical content of the Hamiltonian changes.

This is the usual dressing defining quasiparticles for any Hamiltonian.
He shows that one just used the standard Hamiltonian and then
looks at excitations of the physical vacuum instead of the bare ones.
Then everything is finite (In the UV, perturbatively).
Thus what you say does not contradict my conclusions.

Arnold Neumaier

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