Mathematical Principles of Theoretical Physics

(Rick Simeone) #1

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),

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