Quantum Electrodynamics on Null Planes and Applications to Lasers

The conventional formulation of quantum electrodynamics in which the system develops from one space-like hyperplane to the next is here replaced by one in which the development proceeds over null hyperplanes. For detailed study a quantized electromagnetic field $A^{\mu }$ is chosen to interact with a quantized spin-0 particle field $\Phi$ in an unquantized electromagnetic field $A_{\mathrm{ext}}^{}{}_{}{}^{\mu }$ as background. If the latter is chosen to be a laser field, the $\Phi -A_{\mathrm{ext}}^{}{}_{}{}^{\mu }$ interaction permits exact closed-form solutions (Volkov) and allows the construction of wave packets which cannot be done in the usual formulation. The perturbation solution for the $S$ matrix is therefore conveniently based on the Furry picture. The null-plane formulation has various advantages. In particular, the gauge problem which causes difficulties in the usual theory is absent in the null-plane gauge chosen here. Since there are only two dynamically independent components of $A^{\mu }$, the commutation relations, field equations, gauge conditions, and vacuum definition are all mutually consistent. A natural nullplane gauge is used. Similarities and differences between this and the conventional theory are pointed out. As an application the Compton scattering of a charged particle with a laser beam is shown to lead to an intensity-dependent frequency shift. The controversy on this issue is settled here without divergent phase factors, because our wave-packet description permits a clean separation of the particle beam from the laser.