Re: Connes & Marcolli paper on renormalization

From: Eugene Stefanovich (eugenev_at_synopsys.com)
Date: 12/04/04


Date: Sat, 4 Dec 2004 21:14:00 +0000 (UTC)

Arnold Neumaier wrote:
> Eugene Stefanovich wrote:

>
>>>>I do not agree. Any quantum theory should have a classical limit.
>>>
>>>Well, what is the classical limit of QED, in your opinion?
>>>How do you cope with particle creation and destruction in you classical=
>>
>
>>>limit? You fail to discuss this in your book.
>>
>>=20
>>In the classical limit of the "dressed particle" QED, there are
>>particles interacting with each other via (instantaneous) Coulomb and
>>magnetic forces. They are described by the terms of the type
>>a^{\dag}a^{\dag}aa. In addition, acceleration of charges causes emissio=
>
> n
>
>>of photons (bremsstrahlung). In the dressed particle Hamiltonian, this
>>effect is represented by the terms
>>a^{\dag}a^{\dag}c^{\dag}aa. With these simplest contributions, the
>>classical limit of RQD should be very similar (though not identical)
>>to the Maxwell's theory.
>>=20
>>I agree that I should have written more about the classical limit of my=
>
>
>>approach.
>
>
> Since you write in the subjunctive, I suspect that you don't have studied=
>
> the classical limit. Thus you should not demand more from your competitio=
> n.
> It is trivial to write a classical limit in NRQED. But no one has found a=
>
> working classical limit of QED that is covariant. I doubt that you have.

No, I haven't. However, I see clearly how to do it if light emission by
accelerated charges is neglected: I have ready expressions for
interparticle instantaneous potentials which can be used directly in
classical equations of motion.

The situation with light emission and
absorption is trickier. In my approach, light is a flow of particles -
photons, however for photons quantum effects (diffraction,
interference, etc.) are apparent even in the macroscopic classical
world. So, description of photons as classical point particles is not
good. Even in the classical limit, photons should be treated in the
wave picture.

I need to think more about that, but I feel that some mixed
particle-wave "classical" limit of the "dressed particle" theory
is possible. This limit should be close to Maxwell's theory, but
not identical. Some of the differences are:
the photon field has no direct connection with interparticle
Coulomb and magnetic potentials, magnetic forces are not described
by the Biot-Savart law but rather by the Darwin potential,
the interaction between charged particles is instantaneous...

>
>
>
>>>>In this limit, trajectories are well defined.=20
>>>
>>>Why? It depends on the theory, and there is very little.
>>
>>=20
>>See section 7.3 of the book
>
>
> I haven't seen there any covariant dynamical equation for classical
> particles. There is a brief remark on top of p.167 that could count for i=
> t,
> but lacking details nothing can be said. You'd have to give an explicit
> classical Hamiltonian on phase space and prove that the classical replace=
> ment you get
> is compatible with the Poincare group and satisfies some sort of cluster
> separability.

I have quantum Hamiltonian satisfying relativistic invariance.
I started to write it down in subsection 12.2.3.
If we assume that I can continue to write the terms of higher orders,
  and that the series converge, then the classical Hamiltonian
can be obtained from its quantum counterpart
by simply neglecting the non-commutativity of r and p.

As I said above, the tricky part is how to interpret
terms responsible for light emission and absorption (bremsstrahlung) in
  the classical
limit. Photons cannot be described accurately as classical particles
  moving along trajectories.

>
>
>
>>>>If the manifest covariance
>>>>is true, then these trajectories must transform by Lorentz (or, as yo=
>>>
> u
>
>>>>say, they must be observer independent).
>>>
>>>No. There must only be a well-defined recipe how the set of trajectorie=
>>
> s
>
>>>of one observer translates into that of another. And this is what covar=
>>
> iance
>
>>>is about.
>>
>>=20
>>The whole point of the "manifest covariance" is the this well-defined
>>recipe must be nothing else but linear Lorentz formulas. If you allow
>>other recipes (as I do), then you deviate from the strict special
>>relativity.
>
>
> Manifest or not is irrelevant. What counts is only that there is a
> compatible unitary representation of the Poincare group. This _is_
> special relativity.

Great! That's what I was saying all along. Poincare invariance - yes;
manifest covariance - no. However, to be fair, I wouldn't call "special
relativity" a theory which does not respect manifest covariance. It is
relativity, but neither "special" nor "general". Universal Lorentz
transformations, Minkowski spacetime, and manifest covariance are
ingredients of the special relativity, which are not present in my
(still relativistic) approach.

>
>
>
>>>>When the unitary equivalence of different forms of dynamics is proved,=
>>>
>
>>>>the states are not subject to transformation.
>>>
>>>Of course the states must transform, too. You can see this already
>>>in the nonrelativistic case when you go from the Schroedinger represent=
>>
> ation
>
>>>to the interaction representation. Same physics, different states and
>>>observables, related by a unitary transformation.
>>
>>=20
>>This is not correct. The connection between the instant and point forms=
>
> =20
>
>>of dynamics is not of the type as connection between the Schroedinger,
>>Heisenberg, and interaction representation. The definition of states
>>does not change when a unitary transformation between different
>>forms is made(see below).
>
>
> The definition of states is independent of the representation but the
> states themselves _must_ be transformed unitarily. And they are, in spite=
> of
> what you say below.
>
>
>
>>>>If you unitarily transform
>>>>both states and operators, then, of course, the physical content of
>>>>the theory does not change. That's rather trivial.
>>>
>>>This is also what happens in the equivalence of Shirikov's version
>>>with yours. You keep the vacuum, hee keeps the Hamiltonian; to relate t=
>>
> he
>
>>>two you must transform both.
>>
>>=20
>>That's why I said that my approach and Shirokov's approach are triviall=
>
> y
>
>>identical. This is not the case with instant and point form dynamics.
>>The unitary transformations between them does not touch the states.
>>Otherwise the proof of the Sokolov's theorem could be done in just one
>>sentence.=20
>
>
> The only nontrivial thing in the equivalence is to exhibit the right
> unitary transform to do that. I haven't read Sokolov's proof,
> but the classical version of this transform was exhibited by Bakamjian in=
>
> Phys. Rev. 121, 1849=961851 (1961),
> and the quantum version is similar.

The Sokolov's proof is not much different from the Bakamjian's version.
The significant addition made by Sokolov is demonstration that the
unitary operator connecting the generators in the two forms satisfies
Coester's condition which is necessary for the preservation of the
S-matrix.

You can find Sokolov's arguments in

S.N.Sokolov, "Physical equivalence of the point and instantaneous
forms of relativistic dynamics", Theor. Math. Phys. 24 (1975), 799

S.N.Sokolov, A.N.Shatnii, "Physical equivalence of the three forms
of relativistic dynamics and addition of interactions in the front and
instant forms", Theor. Math. Phys. 37 (1978), 1029.

You may notice that neither Bakamjian nor Sokolov apply the
unitary transformation to the observables of individual particles.
The definitions of particles and their free states do not change
by this transformation. Only 10 generators of the Poincare group
are transformed.

By rereading Sokolov's papers I found a couple of sentences which
explain how he
understood the "physical equivalence" and vindicate my point of view.
These are the first two sentences in the 1978 paper:

"If one adopts the point of view, first expressed by Heisenberg, that
  all experimental information about the physical world is ultimately
deduced from scattering experiments and reduces to knowledge of
certain elements of the scattering matrix (or the analogous classical
quantity), then different dynamical theories which lead to the same S
matrix must be regarded as physically equivalent. It is a simple
matter to construct both nonrelativistic and relativistic examples of
different Hamiltonians that give the same S matrix."

I do not adopt Heisenberg's point of view. For me, different
Hamiltonians and forms of dynamics are physically distinguishable,
even though they give the same S matrix, and even though
experimental observations
of time dynamics are extremely difficult.

[...]
>
>
>>4. The agreement with experiment is a great plus, but any consistent
>>theory should be also free of internal contradictions.
>>This is not the case with SR and GR.
>
>
> There are no internal contradictions between quantum field theory
> and special relativity.

If you define special relativity as a theory requiring the manifest
covariance of all physical laws and if you cast QFT in the form
of Hamiltonian dynamical theory (as I did) the contradiction
becomes apparent - this is the CJS theorem which says that
trajectories (in the classical limit, of course) do not transform
covariantly.

However, if you drop the requirement of manifest covariance and
require only Poincare invariance (as I did), then the contradiction
disappears. That's the main point of my book.

The contradiction does not show up also if you keep the manifest
covariance and pay attention only to the S matrix in QFT. That's
the way QFT is currently presented (I leave aside the time dependent
approaches you mentioned in previous posts. I haven't understood
these approaches yet.) In this formulation, QFT is not a complete
theory.

> And it is not even clear that there are any
> with general relativity. See the topic
> 'Is quantum mechanics compatible with general relativity?'
> in my theoretical physics FAQ at
> http://www.mat.univie.ac.at/~neum/physics-faq.txt

I think the main contradiction is the following: General relativity
(just as the special relativity) implies that position and time
are equivalent quantities (different components of the same 4-vector).
On the other hand,
in quantum mechanics, position and time play very different roles.
Position is an observable (Hermitian operator), and time is just a
classical
parameter labeling different reference frames. This controversy was
discussed from different points of view by Isham in gr-qc/9210011.
Personally, I don't see any prospect of resolving this problem unless
the GR approach to gravity (the curvature of the 4D spacetime) is
abandoned.

Eugene Stefanovich.

>
>
> Arnold Neumaier
>



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