ABSTRACT
Well, in the first place, it leads to great anxiety as to whether it’s going to be correct or not...
P.A.M. Dirac (At age 60, when asked about his feelings on discovering the Dirac equation)
5.1. The Dirac equation and the algebra of gamma matrices
It is well-known that the Dirac equation is obtained by linearizing the energy-momentum relation
p2 +m2 = 0, (5.1.1)
where p is the momentum four-vector, and m is the electron rest mass. In the case of a free electron, the Dirac equation writes
(~α · ~ˆp+ βm− Eˆ)|ψ >= 0, (5.1.2)
where α1, α2, α3 and β are the Dirac matrices:
α1 =
0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0
; α2 = 0 0 0 −i 0 0 i 0 0 −i 0 0 i 0 0 0
;
α3 =
0 0 1 0 0 0 0 −1 1 0 0 0 0 −1 0 0
; β = 1 0 0 0 0 1 0 0 0 0 −1 0 0 0 0 −1
, (5.1.3) while Eˆ is the energy operator. For the sake of simplicity, we again drop the notation ̂, which designates operators, and we shall reserve this sign for other purposes.
