Re: How real are the "Virtual" partticles?
From: Eugene Stefanovich (eugenev_at_synopsys.com)
Date: 03/26/05
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Date: Sat, 26 Mar 2005 08:42:45 +0000 (UTC)
jsolomon@mail.com wrote:
> Eugene Stefanovich wrote:
>
>>Let me then summarize the main points of our discussion
>>(the way I see it).
>>My major original claim was that in the current QED theory
>>(in the limit of infinite cutoff) there is
>>no Hamilton operator H that can be simultaneously used
>>in both perturbative expansion for the S-matrix and in the
>>formula for the time evolution operator exp(iHt). The
>>latter formula remains ill-defined because
>>a) existing H have infinite coefficients
>>b) existing H have trilinear terms that lead to the "instability"
>>of electrons and vacuum.
>>
>
>
> I would like to share my thoughts on your approach. You apply a
> unitary transformation to the "old" Hamiltonian to obtain a "new"
> Hamiltonian. Therefore your approach is mathematically equivalent to
> the old approach.
I disagree. Two unitarily equivalent Hamiltonians H and H' = UHU^{-1}
are NOT physically equivalent. True, they have the same spectrum of
eigenvalues, and the S-matrices are the same for most U.
However, the eigenfunctions of the two Hamiltonians are different,
and the time evolutions described by them are different too.
It is possible (though difficult) to distinguish experimentally
theories with Hamiltonians H and H'.
> The potential in your approach is that you are
> looking at the problem from a different "point of view".
>
> Some problems become more clear when the "point of view" is changed.
> For example it makes more sense to describe the laws of planetary
> motion from the "point of view" of being at the center of the solar
> system. Of course we could transform our coordinate system to the
> surface of the earth and describe the motion of the planets from that
> "point of view". However in that case the laws of planetary motion
> would be extremely complicated. Although both "points of view" are
> mathematically equivalent we can see that having the correct "point of
> view" is very important in gaining insight into a problem.
>
> Now when your Hamiltonian is used the vacuum state and single particle
> states are very simple. Your vacuum is almost the same as what we use
> to imagine when we thought of a vacuum, i.e., nothing is there. Your
> single particle states are also very uncomplicated. Therefore changing
> the from the "old" Hamiltonian to the "new" Hamiltonian simplifies that
> part of the discussion. However the question remains as to what is
> most fundumental. At this point, you have to start with the old
> Hamiltonian to get anywhere. That is, you can write out the free
> non-interacting part of the Hamiltonian easily enough but you cannot
> write out the intaction terms. To get these you have to go back to the
> original Hamiltonian where the interaction terms are known and apply
> your methods to obtain the new Hamiltonian. Therefore it seems that
> the original Hamiltonian is more fundamental because we always have to
> start with it.
>
> What you really want to do is derive your Hamiltonian from basic
> principles and not start with the old Hamiltonian. So I would like to
> know have you made any progress on this? What approaches have you
> tried?
You are exactly right. I have no idea how to derive the dressed particle
Hamiltonian from first principles. The only way I know is rather
ugly:
field quantization -> renormalization -> dressing.
If such a first principles derivation were known we could delete from
the vocabulary of physics such curse words as regularization and
renormalization. Probably "quantum fields" and "gauge invariance" will
have to go too. There are few principles which, I believe will survive:
1. relativistic invariance
2. instant form of dynamics
3. full dressing (simple vacuum and 1-particle states)
4. cluster separability
There is an approach which attempts to derive the form of interaction
based on these principles only
E. Kazes, Analytic theory of relativistic interactions,
Phys. Rev. D 4 (1971), 999.
(the author does not use the principle 3 above, but it seems to
be rather simple to incorporate this principle in his approach).
The idea is to first fix the topology (the creation-annihilation
operator structure) of the interaction terms in the Hamiltonian and
boost operators, and then use Poincare commutation relations as
a set of differential equations that can be solved to determine
the momentum dependence of the coefficients in interaction terms.
I like this approach very much, but I haven't tried to apply it
to QED. If somebody wants to try, I would be eager to see the results.
Eugene Stefanovich.
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