Finding EQ of Motion for Systems of Particles

[K.J.Healey]

This is not homework; I'm currently in my first particle physics course (survey) and I had a question.
Suppose you have some Lagrangian describing a system. Let's say two electrons, with an interaction. How would you go about finding say, the "force" on either particle, or the potential. Even in approximation? What method should I even begin to learn about?

I guess my problem is I understand what's in these density equations, but how do we convert that to anything usable, for things as simple as two electrons? So I'm looking for more methods, than description of the system.

 

[pam]

"force" is very little used in particle physics.
The "potential" between two particles is usually found by a circuitous route.
First the scattering amplitude is calculated in Born approximation.
Then the potential is taken to be the Fourier transform of the scattering amplitude.
For instance, the scattering amplitude for two electrons is A~e^2/q^2.
Its Fourier transform is V~e^2/r.

 

[lbrits]

For a good discussion, see the textbook by Zee. In any case, the type of force that you're thinking of is (usually) mediated through by gauge bosons (e.g. photon), and the two particles interact by exchange of such a boson. The tree level Feynman diagram (meaning, no loops) is essentially just the propagator of the photon, and this is, as pam said, $$\sim \tfrac{1}{q^2}$$ where $$q$$ is the momentum.

Also, as pam said, to relate this amplitude to semi-classical mechanics, you need to do some first-quantization. In ordinary quantum mechanics, you can solve scattering problems perturbatively using the Born approximation. There the first term in the amplitude is simply the Fourier transform of the intermediate propagator. If you don't mind some math, here is a quick derivation of the Born approximation: http://www.physics.thetangentbundle.net/wiki/Quantum_mechanics/Born_approximation along with a scattering calculation for a Yukawa $$\tfrac{e^{-\mu r}}{r}$$ type potential http://www.physics.thetangentbundle.net/wiki/Quantum_mechanics/scattering_off_of_a_Yukawa_potential . In the last example you'll see the "photon" propagator pop out at the end, at which point you can make contact with field theory.