Wave Particle Duality is codified in Euler's equation where e^ix=cosx+isinx. Indeed, Euler's equation is the ultimate wave equation. e is a continuous function. Waves are an example of continuous functions. i is discreet; discontinuous. A quantum of energy is an example of a discontinuous function. Number theory has it that the reals, measured along the x axis or along the cosine, are considered as being discreet and discontinuous . The imaginary complex numbers measured along the y axis or sine is considered as being continuous. If this is true note what Euler's equation does. It measures a continuous, object , e , discontinuously along the cosine, x. Conversely it measures a discontinuous object, i, continuously along the sine, y. This is the basis of wave particle duality. It has been established that an electron , as a particle, has wave characteristics (double slit experiment) and that a photon, as a wave, has particle characteristics (photoelectric effect). The explanation is simple. e is a photon with continuous (reals ) characteristics. i is an electron with discontinuous (complex) characteristics. i is literally the quaternion i. e corresponds to the quaternion j, and the neutrino is the quaternion k. Again wave particle duality exists because Euler's equation says it must exist.

We must also account for the scalar version of these particles. The scalar version is simply our standard classical description of the photon. As a scalar it has no definable position thus it has no energy of position. The scalar photon is all kinetic energy and travels at the speed of light for all observers. Quaternion algebra says that a scalar is a sphere and that a vector is a point in or upon that sphere. As a scalar the light travels along the surface of a sphere thus it is in truth an angular velocity. As long as all observers are within or upon this sphere then all observers will measure the same angular velocity. On a spinning disk all points on the disk will measure the same angular velocity or the same rpm. The measurement in or upon the sphere is similar.

We must also account for the scalar version of these particles. The scalar version is simply our standard classical description of the photon. As a scalar it has no definable position thus it has no energy of position. The scalar photon is all kinetic energy and travels at the speed of light for all observers. Quaternion algebra says that a scalar is a sphere and that a vector is a point in or upon that sphere. As a scalar the light travels along the surface of a sphere thus it is in truth an angular velocity. As long as all observers are within or upon this sphere then all observers will measure the same angular velocity. On a spinning disk all points on the disk will measure the same angular velocity or the same rpm. The measurement in or upon the sphere is similar.