Unified discrete mechanics II: The space and time symmetric hyperincursive discrete Klein-Gordon equation bifurcates to the 4 incursive discrete Majorana real 4-spinors equations

Abstract

Abstract This paper begins with the formalization of the second order hyperincursive discrete Klein-Gordon equation. The temporal second order hyperincursive discrete Klein-Gordon equation is similar to the time-symmetric hyperincursive discrete harmonic oscillator and so bifurcates into a group of 4 incursive discrete real equations of first order. In this group, two equations are the discrete time reverse of the two other equations, giving an oscillator and an anti-oscillator. Firstly, the discrete Klein-Gordon equation, with one space dimension (1D), bifurcates to 4 first order incursive discrete equations that we called the Dubois-Ord-Mann real 4-spinors equations because Ord and Mann obtained the same equations from a stochastic method. Secondly, we generalize to three spatial dimensions (3D) these discrete Dubois-Ord-Mann equations. These 4 discrete equations are then transformed to real partial differential equations which can be written under the generic form of the Dirac quantum 4-spinors equation. Thirdly, we consider a change in the order of space variables and a change of indexes of the functions of the Dubois-Ord-Mann equations. With these changes we obtain the original real 4-Spinors Majorana partial differential equations. Also we obtain the 4 incursive discrete Majorana real equations.