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The formulism for QED is very similar to the nucleon-pion system in Eqs.(12) and (13). While Eq.(13) for fermion is readily applicable (with appropriate value for k

where

By virtue of the anti-symmetric property of Eq.(21), the continuity equation for the charge-current density is automatically satisfied, i.e.,

In curved space-time Eq.(23b) remains unchanged, while Eq.(23a) becomes

Another form of the Maxwell equations is in terms of the four-vector potentials

---------- (24) |

By the way, Eq.(23b) is automatically satisfied by the relationship between the anti-symmetric tensor fields and the four-vector potentials in Eq.(24).

The 3-vector potential

It can be shown that and A

---------- (32) |

Note that the long range instantaneous Coulomb interaction does not imply a force travelling faster than the speed of light. Although the intereaction is instantaneous, it can be considered as the interaction between two overlapping Coulomb tails (clouds of virtual photons) of the two charged particles, so there is no need for the interaction to travel from one point to another in zero time.

See a slightly different treatment for derivation of the free field equations in "Quantization and Field Equations".

QED concerns mainly with the transverse components, which account for the electromagnetic radiation of accelerating charged particles. The transverse electromagnetic fields provide a simple and physically transparent description of a variety of processes in which real photons are emitted, absorbed, or scattered. The three basic equations for the free-field case are:

where

Eq.(33b) is in a form very similar to the Klein-Gordon Equation Eq.(1l) or Eq.(12) except that the mass term vanishes (because the photon has no rest mass) and the field is a vector (instead of scalar) with two transverse components (polarization) perpendicular to each other. Thus

The quantization rules for the electromagnetic field is very similar to that in Eq.(4):

---------- (36) |

---------- (37) |

Interaction between photon and fermion, e.g., electron can be introduced by demanding local gauge invariance for the formulism. With this constraint on the quantum field theory, the ordinary derivative in Eq.(13) becomes the covariant derivative:

and the interaction takes the form:

where e is the coupling constant. (See appendix on "Abelion/non-Abelion Groups and U(1), SU(2), SU(3)" for a discussion about the concept of gauge or phase transformation.)

- The photon-electron processes are described by substituting Eq.(38b) to H
- The first order graphs -

where the electron field has been decomposed to:

and the symbol**X**may stand for an interaction with an external classical potential, the emission of a photon, or the absorption of a photon. The realization of a particular process depends entirely on the initial and final states. - The second order graphs (tree diagrams) -

- Two-photon annihilation - Pair of an electron and positron into two gamma rays.
- Compton scattering - It occurs when an electron and a photon collide and scatter elastically.
- Moller scattering - It is the scattering of two relativistic electrons.
- Bremsstrahlung - It is the process by which radiation is emitted from an electron as it moves past a nucleus.
- Second order graphs (loop diagrams) -

- Vacuum polarization - It produces a correction to the coupling constant (electric charge) and contributes to the Lamb shift, which is a small splitting of hydrogen atom energy levels caused by the interaction with the virtual pair (of electron and positron).
- Vertex correction - A correction to the electron vertex function, which contributes to the anomalous electron magnetic moment.
- Self-energy - The interaction of a charged particle with its own field and it usually gives rise to infinite self-energy and infinite mass.
- Mass renormalization - The observed mass is generated by combining the bare mass and the (calculated) divergent mass.

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