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A version of QED without ultraviolet divergences

A version of QED without ultraviolet divergences - Physics Forum

A version of QED without ultraviolet divergences - Physics Forum. Discuss and ask physics questions, kinematics and other physics problems.


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  #21  
Old 08-04-2004, 04:53 PM
Creighton Hogg
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Default A version of QED without ultraviolet divergences



On Wed, 4 Aug 2004, Bilge wrote:

<>

Hi Bilge,
I thought Fock spaces were the hilbert spaces built from creation and
annihilation operators. Like harmonic oscillator or a free klein-gordon
field. Wouldn't they necessarily *not* include any kindof interaction
just by construction?

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  #22  
Old 08-04-2004, 06:28 PM
Eugene
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Default A version of QED without ultraviolet divergences



Bill Hobba wrote:


I find it hard to believe. Weinberg says that Wilson's approach is
equivalent to the conventional approach. The latter does not solve the
problem of infinities, just hides them in the Hamiltonian.
Therefore I conclude that the former also does not solve this problem.
Though I admit that I may be wrong here since I do not understand
Wilson's approach.

Eugene.

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  #23  
Old 08-04-2004, 06:47 PM
Eugene
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Default A version of QED without ultraviolet divergences



Bilge wrote:


Probably I wasn't clear enough. I do not doubt the ability of QED to
calculate
finite S-matrix (and all related properties, like magnetic moments or
Lamb's shifts) in excellent agreement with experiment. I admire QED for
that. My point is that this great success is achieved in QED at a great
price. The logical consistency of the theory is destroyed. This perfect
S-matrix cannot be obtained from a sensible Hamiltonian by normal rules
of quantum mechanics. There are two ways to calculate the S-matrix in
QED.

1. You can keep the original (finite) Hamiltonian of QED, use standard
perturbation formula for the S-matrix and carefully "subtract"
divergences from the calculated matrix elements in each perturbation
order. Dirac said about this:

"This is just not sensible mathematics. Sensible mathematics involves
neglecting a quantity when it turns out to be small! - not neglecting
it because it is infinitely large and you do not want it!"

2. You can move all divergences into the Hamiltonian in the form of
infinite counterterms (I prefer this approach). Then you can use
the standard QM formula for the S-matrix, and obtain the correct
(finite, accurate) result without any tricks. The "only" problem you
have is infinite Hamiltonian. QFT does not care about this little
annoyance, because
it does not care about time evolution (this is probably why you keep
saying that the problem of infinities has been solved). I do care about
time evolution, so for me the problem is not solved by QED. It is not
solved by electroweak theory and QCD either.

Eugene.


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  #24  
Old 08-04-2004, 07:04 PM
Eugene
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Default A version of QED without ultraviolet divergences



Bilge wrote:



You are talking about Haag's theorem. This theorem does not apply to my
approach. This theorem basically says that "interacting field", i.e.,
the field evolving in time according to the full interacting Hamiltonian
cannot have Lorentz transformation law with respect to boosts.
(note the similarity with the Currie-Jordan-Sudarshan "no go" theorem).

In my approach I don't use "interacting fields" at all. I use free
fields to construct the interacting Hamiltonian and the bost operator.
After that, I discard fields and work exclusively with particle
creation/annihilation operators. The relativistic invariance of
my approach is proven explicitly.



That's probably an unfortunate choice of terminology. I would rather
call them "operators of type B". Actually, there was a rationale why
I called them "unphysical". This is because operators of this type
cannot be present in the S-operator. In my approach, operators of this
type cannot be present in the Hamiltonian as well.

The main problem of the QED Hamiltonian, and why it should be corrected,
is the presence of these "unphysical" or "type B" terms. Trilinear
interactions belong to this type. They are responsible for all these
"unphysical" effect like non-trivial vacuum and "coats" of virtual
particles around real particles. The main idea of my approach is to
apply a unitary transformation which eliminates these "type B"
terms from the interaction and make them "type A", or "physical".

Eugene.



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  #25  
Old 08-04-2004, 07:09 PM
Eugene
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Default A version of QED without ultraviolet divergences



Creighton Hogg wrote:


Fock space is simply a direct sum of n-particle Hilbert spaces.
In QFT you need to describe processes in which the number of particles
can change. You cannot work in a Hilbert space with fixed number of
particles. So, you just put together n-particle Hilbert spaces with
all possible values of n, and obtain the Fock space. In the Fock space
you can neglect interaction if you choose the non-interacting
Hamiltonian, or you can take the interaction into account if you
choose the interacting Hamiltonian.

Eugene.

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  #26  
Old 08-04-2004, 07:24 PM
Bilge
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Default A version of QED without ultraviolet divergences

Creighton Hogg:

Right. And it appears to not be possible to add an interaction
other than for a field that's asymptotically free.


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  #27  
Old 08-05-2004, 01:55 AM
Eugene
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Default A version of QED without ultraviolet divergences



Bill Hobba wrote:


Thank you for these references. I briefly looked at them and see that
they offer more than I can swallow. I simply do not understand this
language. It will take me some time to figure out what they are talking
about. If you can show me that EFT or Wilsonian approach can produce
a finite Hamiltonian from which both time evolution and S-matrix
can be calculated using standard quantum-mechanical formulas, I will
shut up. I bet they cannot do that.

I understand much better your third reference


Let me quote from it

" You might still be bothered by many things in the previous section.
When we renormalize physical quantities such as charge and mass, you
might be thinking that these quantities are observable and are not
infinte. So how can you get away with making them divergent and then
ignoring it?! The answer to that question is actually deeper than it
first seems.

First of all, there is a flaw to the skeptic's argument that the
electron is not infinitely massive or carries infinite charge. In fact,
according to QFT, it does! The reason we don't see it is subtle but
beautiful. If the electron has infinite charge, then it has an infinite
amount of energy from the electromagnetic field. This energy manifests
itself by the uncertainty principle which says that the field is allowed
to create and destroy particles in very short times; such particles are
called ``virtual particles''. With this huge amount of energy, the field
is able to produce many particles with charge all around the electron.
But because these virtual particles are charged, they line up with the
field and dampen the strength, analogously to dielectrics in classical
electrodynamics. Hense as you go further away from the electron, its
effective charge becomes weaker due to this dielectric effect, thus
lowering the charge of the electron to the values we measure. "

The infinite "bare" charge is what is present in the mutilated QED
Hamiltonian. The finite "screened" charge is what appears in the
S-matrix as a coefficient in front of scattering amplitudes.
While you are working with S-matrix only you can pretend that everything
is fine and finite. However, when you try to calculate the time
evolution you will face infinite "bare" masses and charges in the
Hamiltonian.

In my approach this infinite Hamiltonian is unitarily transformed
and made finite. This is called "clothing transformation", i.e.,
the transformation from "bare" to "clothed" particles. As a result of
this transformation the "coats" of virtual particles disappear.
These coats are unphysical anyway: nobody have seen these virtual
particles.

Again, I am not questioning successes of QFT in calculating S-matrix
properties. I can easily believe that these successful results can
be achieved in a variety of ways: conventional renormalization,
Wilson's renormalization, EFT, etc. I am claiming that neither of these
approaches is capable of producing a finite Hamiltonian and describe
the time evolution in a consistent fashion.

Eugene.

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  #28  
Old 08-05-2004, 06:58 PM
Bilge
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Default A version of QED without ultraviolet divergences

Eugene:


The logical consistency of the theory is not destroyed and I've
already given you the reasons why at least twice, along with a
reference.


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  #29  
Old 08-05-2004, 07:44 PM
Eugene
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Default A version of QED without ultraviolet divergences



Bilge wrote:

The reference you gave me is for standard QFT dealing with the
S-matrix only. S-matrix (asymptotic limits of the time evolution)
is reproduced extremely well there. I see logical inconsistency
in the fact that this theory cannot describe equally well the
time evolution at intermediate times when particles actually
interact with each other. This is because QFT swept infinities
under the rug. This rug is Hamiltonian. Without finite Hamiltonian
you cannot do time evolution. Without time evolution the theory
is not consistent.

Eugene.


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  #30  
Old 08-06-2004, 01:25 PM
Bilge
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Default A version of QED without ultraviolet divergences

Eugene:



OK, whatever you say, since you keep saying the same thing
despite correction.

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