58 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS
Then we have
(2.2.57) E=cP±mc^2.
Inserting the operatorsEandP 1 in (2.2.45) and (2.2.46) into (2.2.57), and taking the mass
operator as
±mc^2 =mc^2 α 0 , α 0 =(
I 0
0 −I
)
, I=
(
1 0
0 1
)
,
we derive the following Dirac equations:
(2.2.58) ih ̄
∂ ψ
∂t=−ihc ̄ (~α·∇)ψ+mc^2 α 0 ψ,whereψ= (ψ 1 ,ψ 2 ,ψ 3 ,ψ 4 )Tis a four-component Dirac spinor, and~α= (α 1 ,α 2 ,α 3 )is as
defined by (2.2.48).
Usually, we multiply both sides of (2.2.58) by the matrixα 0 , and denote
(2.2.59) γμ= (γ^0 ,γ^1 ,γ^2 ,γ^3 ),
where
γ^0 =α 0 =(
I 0
0 −I
)
, γk=α 0 αk=(
0 σk
−σk 0)
for 1≤k≤ 3.The matricesγμare the Dirac matrices. Then the Dirac equations (2.2.58) are in the form
(2.2.60)
(
iγμ∂μ−mc
̄h)
ψ= 0.When we consider the case under an electromagnetic field, we need to replace∂μin
(2.2.60) byDμgiven by (2.2.54).
Remark 2.25.The reason why the scalar-valued momentum operatorsPare taken in the
form of (2.2.46) is due to the following Einstein energy-momentum formulas:
E^2 =c^2 P 02 =c^2 ~P^2 for massm= 0 ,
E^2 =c^2 (P 1 +mcα 0 )^2 =c^2 ~P^2 +m^2 c^4 form 6 = 0 ,whereP 0 ,P 1 are as in (2.2.46), and~Pis as in (2.2.45).
2.2.7 Dirac spinors
The Dirac matrices (2.2.59) are not invariant under the Lorentz transformations, i.e.γμ=
(γ^0 ,γ^1 ,γ^2 ,γ^3 )is not a 4-D vector operator. Hence, the covariance of the Dirac equations
(2.2.60) requires that under the transformation
(2.2.61) ̃xμ=Lxν, L= (Lμν)as in (2.2.9),