Unified discrete mechanics III: the hyperincursive discrete Klein-Gordon equation bifurcates to the 4 incursive discrete Majorana and Dirac equations and to the 16 Proca equations

Abstract

This paper presents the second order hyperincursive discrete Klein-Gordon equation in three spatial dimensions. This discrete Klein-Gordon equation bifurcates to 4 incursive discrete equations. We present the 4 incursive discrete Dubois-Ord-Mann real equations and the corresponding first order partial differential equations. In three spatial dimensions, the two oscillators given by these equations are now entangled with one spatial dimension, so the 4 equations form a whole. Then, we deduce the 4 incursive discrete Dubois-Majorana real equations and the corresponding first order partial differential equations. Next, these Dubois-Majorana real equations are presented in the generic form of the Dirac 4-spinors equation. In making a change in the indexes of the 4 functions in the Dubois-Majorana equations, the 4 first order partial differential equations become identical to the Majorana real 4-spinors equations. Then, we demonstrate that the Majorana equations bifurcate to the 8 real Dirac first order partial differential equations that are transformed to the original Dirac 4-spinors equations. Then, we give the 4 incursive discrete Dirac 4-spinors equations. Finally, we show that there are 16 discrete functions associated to the space and time symmetric discrete Klein-Gordon equation. This is in agreement with the Proca thesis on the 16 components of the Dirac wave function in 4 groups of 4 equations. In this paper, we restricted our derivation of the Majorana and Dirac equations to the first group of 4 equations depending on 4 functions.