Background

Quantum Electrodynamics (QED) is the theory of the interaction of quantized electromagnetic fields. It turns out that the interaction of two electromagnetic fields involves the exchange of photons. It was the first successful quantum field theory, but it took quite a bit of doing to make it a success.

Classically an electromagnetic field is given by its electric field E and its magnetic field B. In a system with a charge distribution ρ and a current density distribution J the dynamics of the systen is given by Maxwell's equations:

∇·E = ρ
∇·B = 0
∇×E + (∂B/∂t) = 0
∇×B − (∂E/∂t) = J

Matters are greatly simplified if the field is derived from a vector potential A and a scalar potential φ Then

E = −∇φ − (∂A/∂t)
B = ∇×A

The vector potential A was initially introduced as a mathematical simplification but the Aharonov-Bohm effect shows that it has a physical existence.

Another representation of the electromagnetic field is in terms of the matrix

|  0	−Ex	−Ey	−Ez |
| Ex	0	−Bz	−Bz |
| Ey	Bz	0	−By |
| Ez	−By	B x	0   |

This matrix given over all of space is known as the electromagnetic field tensor F. Its components can be expressed as

F^μν = ∂^μA^ν − ∂^νA^μ

The Lagrangian for the electromagnetic field given by the tensor F can be expressed as

L = −¼F_μνF^μν − J^μA_μ

The first term on the RHS is the kinetic energy of the EM field.

An Electromagnetic (EM) Field,
an Electron and their Interaction

The Lagrangian for this system is the sum of three Lagrangians; i.e.,

L = L_EM + L_elect + L_inter

The Lagrangian for the EM is the kinetic energy term given in the previous section; i.e.,

L_EM = −¼F_μνF^μν

For the other two Lagrangians let ψ be the wave function field for the electron from Schrödinger equation and ψ* is its conjugate adjoint. Then

L_elect = iψ*γ^μ∂_μψ −mψ*ψ

where i is the imaginary unit, γ^μ for μ=0 to 3 are the Dirac matrices and m is the mass of the electron.

For the interaction

L_inter = −qψ*γ^μA_μ

where q is the electrostatic charge of the electron.

The Lagrangian is a function of the vector potential and its time derivative. The time derivative of the vector potential is represented by ∂₀A_μ. This means that there is a momentum vector field M(X) conjugate to the time derivative; i.e.,

M^μ(X) = (∂L(X)/∂(∂₀A_μ)

The Polarization Vector

(To be continued.)

Accuracy

QED is considered to be the most precise of physical theories. The prime example of this precision is the magnetic moment of an electron, This quantity can be calculated using a method based on QED and measured experimentally. The two figures agree out to the tenth decimal place. Richard Feynman characterized this as being equivalent to having two meaurements of the distance from Los Angeles to New York which differ only by the width of a human hair.