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Electromagnetic Current

The Lagrangian density in electromagnetic interactions (EM) can be written as:


where is the 4-vector potential. The interpretation of as current follows from the fact that satisfies the current conservation requirement. For illustration, consider an electron-proton scattering. In one photon-exchange approximation, the scattering amplitude is:


where the electron current takes the form:


and the proton current is written as:


is the fine structure constant; k, are the initial and final 4-momenta of the electron; p, the corresponding 4-momenta of the proton and q is the momentum transferred:


is constructed from the proton 4-momenta and the matrices. Due to Lorentz invariance and parity conservation of EM interactions can only be vectors. Therefore, the hadronic current is:


The quantities are functions of only because can be expressed as a function of and the masses involved. Using the following identities




the terms involving can be incorporated into other factors and reduces to the following:


where is constant and M is the proton mass. The factor is inserted to make coincide with the conventional definition of the EM form factor. The current conservation requirement leads to for elastic scattering (). The electric and magnetic form factors and can be shown to be linear combinations of and . In the non-relativistic limit, and are interpreted as the Fourier transforms of the charge and the magnetic moment distributions inside the nucleus.

Bernward Krause
Mon Jan 15 14:57:06 MET 1996