Re: Finite time calculation in QED

On 2005-04-01, Eugene Stefanovich <eugenev@xxxxxxxxxxxx> wrote:
> Igor Khavkine wrote:
>>
>> I strongly recommend starting this calculation and I offer to clarify
>> whatever steps cause problems. It will be up to you, however, to relate
>> to compare with your own work.
>
> OK, let's do it your way.
> I remind you the model you suggested:
>
> > The model itself is not QED but should have all the important features.
> > Consider a complex scalar field phi. It is minimally coupled to gauge
> > field A. The Lagrangian density is
>
> > L_1 = -|Dphi|^2 - m2 |phi|^2 - V|phi|^2 - (1/4) tr(F2).
>
> > Here D = @ - ieA, the covariant derivative. F is the field strength for
> > A. V is the external potential. There is a preferred frame in which the
> > potential V = V(x) is stationary and is two infinite square wells,
> > one [-L-w,w]x[0,L]x[0,L] and the other [w,w+L]x[0,L]x[0,L].
>
> > The only reason I think a scalar field is easier to handle is because in
> > the case of Dirac fields, the boundary conditions for the infinite
> > square well are not so simple.
>
> > In the infinite past, a particle is placed in each well. At any finite
> > time t the properties of this tate can be examined, for example the
> > expectation vale for the position of the particle in each well. At t=0 a
> > time dependent perturbation can be added to V which sets one of the
> > particles into motion. The effects on the other particle can be
> > calculated.
>
> I have a few comments about the formulation of this problem.
> First, introducing internal potentials looks rather artificial.
> I would prefer to stay closer to reality as much as possible and
> consider just two interacting particle without any extra stuff.
> The decision about retarded/instantaneous nature of interaction can be
> made by examining the dependence of the interparticle force on the
> particles' observables. If you throw away the wells you'll also
> save yourself the aggravation with boundary conditions.
> Then, there is no any advantage in using the scalar field, we can work
> in QED just as well. Moreover, the Hamiltonian in QED is already
> available.

I do not trust conclusions about the speed of the interaction just by
looking at the form of what appears to the force or the potential. I
want to calculate something that is concievably measurable. That is why
I chose the expectation value of the position operator. My choice of
external potential mimics the design of a controlled experiment for such
a measurement.

A priori, we know neither what the vacuum, the 1-particle, or the
2-particle states of the interacting theory look like. We only know that
in the distant past they can be approximated by corresponding free
states. If we do not introduce a localising potential wells, any state
we start with in the distant past will result in scattering by the time
we get to t=0. In other words, if we start with a two particle state in
the distant past, we have no idea what kind of state we'll get at t=0.
With the potential wells we do.

> Now you write
> > Write the Hamiltonian as H = H_0 + H_1, where H_0 is the free (bilinear
> > time independent) part and H_1 the interaction part (the rest).
>
> Done.
>
> > Switch
> > to the interaction picture.
>
> Would you allow me to stay in the Schroedinger picture?
> I would like to see how some simple states, like vacuum or one-particle
> states develop in time. Before attempting to solve a 2-particle problem
> we should make sure that 0-particle and 1-particle systems make sense.

The separation H = H_0 + H_1 into free and interaction parts is
arbitrary. If you choose H_0 = 0 and H = H_1 you recover the
Schroedinger picrure. The difference will appear in the expression for
the propagators and the iteraction vertices. If you choose H_0 = 0, then
all your bare propagators will have value 1. And in addition to the
interaction vertices you expect, you'll also see vertices like
----o---- representing the mass term and ----x---- representing the
kinetic term. I will give all instructions in the interaction picture
where H_0 is taken to be the free scalar field and EM field
Hamiltonians. If you wish you can always compare with the Schroedinger
picture, the map between them is known.

> > Write down the equations of motion for the
> > field operators and the states.
>
> The interaction part H_1 contains trilinear terms like a^*c^*a and a^*ac
> When I evaluate the time evolution of a one-electron state I obtain
>
> exp(iHt) a^* |0> = exp[i(H_0 + H_1)t] a^* |0>
> = exp[i(H_0 + a^*c^*a + a^*ac + ...)t] a^* |0>
> = (1 + iH_0t + ia^*c^*a t+ ia^*ac t+ ...) a^* |0>
> = a^* |0> + it a^*c^* |0> + ...
>
> I see that my electron "created" a photon out of nothing even in the
> lowest (1st) perturbation order. The higher orders are even worse.
> I beg for your help here! The problem is the same in the scalar field
> model that you suggested, because its Hamiltonian also contains unphys
> terms that I consider inappropriate.

First, you did not write down the equations of motion in the interaction
picture as I asked. Second, your solution is not correct, the
Hamiltonian will have a time dependent potential that will kick one of
the particles into motion. In the same vein, in the interaction picture
all particle operators will have time dependence given to them by H_0.
Third, as has been explained several times, the expression you wrote
down has zero physical meaning. You are still far from anything that can
be directly physically interpreted.

You still have the following steps to complete:
* Write down the equations of motion of the field operators and the
states.
* Diagonalize H_0.
* Solve the equations of motion for the field operators phi and A.

Working knowledge of quantum mechanics as well as patience are
prerequisites for this calculation.

Igor

.

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