The lagrangian density is
L = ψ-bar (iγ_μ (∂_μ + ieA_μ + ieB_μ) - m) ψ
- F_μν F^μν.
Wikipedia says that A is the EM field of "the electron itself" and B is the external field.
The first question is how many fields there are in the lagrangian? It looks like each electron has its own field ψ, and also its own EM field A.
Apparently, there is also a free EM field present as well as an external field whose origin may be charges far away.
The basic idea of a lagrangian density is to find a local extreme value in a spacetime integral of that density.
Suppose that we have just a slow-moving single electron in a static external electric field B.
The lagrangian above treats the energy of the combined electric fields A + B as potential energy. The mass of the electron is potential energy, and is constant.
But the potential of the electron in the field A + B is strange. If B is the static potential of other electrons, then the potential in B should have a minus sign in the lagrangian. That is correct in the above formula.
And what is the potential energy of the electron in its own electric field? If the potential energy is negative, then the sign in the above formula is wrong.
The above formula seems to mention the total energy of an electron three times as potential energy:
1. potential energy in its own field,
2. mass,
3. -2 × energy of the electric field of the electron, in the last term.
A drum skin QED lagrangian
How could we write a lagrangian which makes sense?
We had the analogy where a drum skin is pressed with a finger. The static electric field of the electron is the depression in the skin.
The depression is like a negative electric field and the finger is a positive charge sitting at the bottom of the depression.
The variables are the skin depression and the finger position.
L = eA - F^2,
where A is the skin depression as a positive value, F^2 is the skin deformation energy and e is the downward force on the finger as a positive value.
What if we add another pressing finger? We have to introduce another field A_2 for it. The other finger stays in its own depression.
Note that the pressing force for each finger i only releases energy for its own depression A_i and ignores other depressions. We need separate terms F_i^2 for each A_i.
We can implement a repulsive force between the fingers by adding the energy of the total field F^2. If the fingers move closer, then F^2 grows.
There is a problem, though. Each A_i should be independent of the other A_j. But the term F^2 couples them. What to do? We may multiply the energy associated with each i with a large number M. The private field of each i is then "rigid".
To accommodate drum skin vibrations as well as an external field, we need to introduce yet another field B which is not visible for any of the electrons. Drum skin vibrations live in the global field B.
The total QED lagrangian is
L = Σ_i M (e A_i(x_i) - F_i^2) - F^2,
where x_i is the position of the i'th electron and F^2 is the positive energy of the total field, which is the sum of all A_i and B. M is the large positive number which makes the static field of each electron rigid.
The mass of an electron is
M (F_i^2 - e A_i(x_i) + F_i^2.
We should still add kinetic energies to the lagrangian.
The classical lagrangian
The lagrangian is
L = L_field + L_interaction
= -1/(4μ) F^αβ F_αβ - A J,
where Wikipedia says A J means "many terms expressing the electric currents of other charged fields in terms of their variables".
If there are no pointlike charges but a finite charge density, then the lagrangian above might be similar to the lagrangian of an attractive force, and it would calculate correctly the field inside and around a uniform charge density ball.
But if we have two such charge balls, the lagrangian then attracts them together, which is wrong.
We need to check the literature. Has anyone got the EM lagrangian right?
If there are no pointlike charges but a finite charge density, then the lagrangian above might be similar to the lagrangian of an attractive force, and it would calculate correctly the field inside and around a uniform charge density ball.
But if we have two such charge balls, the lagrangian then attracts them together, which is wrong.
We need to check the literature. Has anyone got the EM lagrangian right?
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