Re: Finite time calculation in QED

On 2005-04-12, Eugene Stefanovich <eugenev@xxxxxxxxxxxx> wrote:

> I was trying to avoid the calculations you suggested because, in my
> opinion, they have no relationship to the topic of this thread.
> I know pretty well how the Hamiltonian of QED is derived. This is
> described very clearly in sections 8.1 - 8.4 of Weinberg's "The quantum
> theory of fields" vol. 1. The resulting Hamiltonian is written
> explicitly
> in eq. (8.4.22) - (8.4.25) of the book. I think that nobody doubts
> that this is the *correct* Hamiltonian of QED. My suggestion was
> to start discussion directly from this Hamiltonian (or equivalent
> Hamiltonian of your scalar particle theory). In my opinion, deriving
> this Hamiltonian by repeating calculations in sections 8.1 - 8.4 on this
> newsgroup would be a waste of time. I hope that you agree that the
> result of this derivation would be not different from (8.4.22) - (8.4.25).

Allow me to disagree. The Hamiltonian will be different. Different
potential, different Hamiltonian. That's exactly why I wanted to derive
it and at the same time do some work that will turn out to be useful
later (namely propagators aka classical Green functions). And without
the external localizing potential, it is very hard to prepare
localized *multiparticle* states at finite times. At least I don't know
how.

> My point was that this Hamiltonian and even its version with
> counterterms H^c cannot be considered as generator of time evolution,
> because it creates particles out of vacuum and one-particle states.
> I would agree with you that this problem can be solved by redefinition
> of particles. You haven't specified how exactly you are going to perform
> such transition to physical particles.

A free state at t=-oo gets evolved into an eigenstate state of the full
Hamiltonian at t=0. That is how the transition will be performed. What
we haven't gotten to is how this statement is translated into
mathematical language. This is the part where you have to write down and
solve the equations of motion for operators and states in the
interaction picture.

> However, no matter how you do
> that, your theory must satisfy at least 3 axioms:
>
> 1. relativistic invariance,

What is required is relativistic covariance. However, there is now a
preferred frame due to the external potential. Calculations or
experiments in different frames need not be the same, there simply needs
to be a way to translate between them. The motion of the walls of the
external potential will tell you which frame you are in and how to
translate your results into the frame where it is stationary. Consider
the walls of the confining potential to be the walls of a space ship.
The scalar field is like a ball bouncing inside. If in your frame the
space ship is moving, while in mine it is stationary, we don't see the
same motion of the ball. But we can agree on our observations
nonetheless by doing an explicit Lorentz transformation or by
calculating invariant quantities such as the ball's proper time between
bounces.

> 2. trivial time evolution of the "physical" vacuum and 1-particle
> states,

In other words, 1-particle states are stationary of the full
Hamiltonian. Sure they are, we just won't calculate what they are. Of
course, such states exist only if the Hamiltonian itself is invariant
under time translations. However, we'll be dealing with a time-dependent
perturbation potential. Hence the Hamiltonian will have no stationary
states except the physical vacuum, which we also won't calculate
explicitly.

> 3. the S-matrix in you theory must be exactly the same as that
> calculated with the Hamiltonian H^c.

No. Different potential, different Hamiltonian. Different Hamiltonian,
different S-matrix. We won't calculate the S-matrix.

> If you do not accept the above 3 axioms, then you theory disagrees
> with experiment, and has no physical relevance.

On the contrary. The model I proposed is based on a simple and
justifiable physical assumption. The experimental apparatus is not
described by the theory. Without similar assumptions, most of the
examples and problems from any text on quantum mechanics have to be
thrown out the window as well. If you wish to model the apparatus
dynamically within your theory, good luck. At least I don't know how to
do it. However, I'm in good company with all the people working on the
quantum measurement problem.

Igor

.

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