On Dirac equation on a time scale

Abstract

We consider the non-autonomous linear Dirac equation on a time scale containing important discrete, continuous, and quantum time scales. A representation of the solutions is established via an approximate solutions in terms of unknown phase functions with the error estimates. JWKB and other asymptotic representations are discussed. The adiabatic invariants of the Dirac equation are described by using a small parameter method. We also calculate the transition probabilities for the Dirac equation. Using the asymptotic solutions we show that the electron-positron transition probability during a long period of time is about 1/3. Since this probability is high, there is a simple explanation of the stability of the revolution of an electron about the proton only by the electromagnetic field. Indeed when the electron is far from the proton, it is attracted by the electromagnetic field of the proton. When the electron approaches closer to the proton, it turns to the positron which is repelling from the proton by the same electromagnetic field.