A Derivation of the Quantized Electromagnetic Field Using Complex Dirac Delta Functions

Abstract

It is shown a complex function $\Phi$ defined to be the product of a real Gaussian function and a complex Dirac delta function satisfies the Cauchy-Riemann equations. It is also shown these harmonic $\Phi$-functions can be included in the solution of the classical electromagnetic field equations to generate the quantum field as a many-particle solution such that the $\Phi$-functions represent the particle states. Creation and destruction operators are defined as usual to add or subtract photons from the particle states. The orbital angular momentum of the $\Phi$-states is interpreted as spin since it emerges from a point source that must be circularly polarized as a requirement of the gauge condition.