Spatial Aspects of a 2x2 Matrix Model of the Relativistic Momentum Energy Equation and Applications to Light

In a set of notes [2]-[6], it was seen that a 2x2 matrix formulation of E2=p2 + m2 (1) naturally existed. This formulation used quantum mechanical type linear algebra and included a time evolution operator exp(-iHt). The model produced quantum mechanical zitterbewegung, but was also applicable to a classical spring. A momentum operator also existed with eigenvalues which had the same modulus value as the energy. In this note, we suggest a form for a spatial evolution operator and apply it to X, a position vector which only exists in the p=0 case. Here [H,X]=V. No such matrix exists for p>0. We find X(t) and X(x) and show that they both have the same time and spatial frequencies suggesting wavelike motion. We then apply the model to a photon moving in the y direction in a frame moving in the x direction. We suggest that the model predicts that a photon has wavelike motion with frequency proportional to p and a wavelength proportional to 1/p. Finally, we apply the 2x2 matrix model to the electromagnetic energy density equation to argue that E (electric field) and B(magnetic field) must be periodic in time with the same frequency.