Mathematical Principles of Theoretical Physics

88 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS

whereψ= (ψ 1 ,ψ 2 ,ψ 3 ,ψ 4 )Tis the Dirac spinor, andγμ= (γ^0 ,γ^1 ,γ^2 ,γ^3 )is the Dirac matri-
ces, defined by

(2.5.39) γ^0 =

(

I 0

0 −I

)

, γk=

(

0 σk

−σk 0

)

fork= 1 , 2 , 3 ,

where the Pauli matrices are given by

σ 1 =

(

0 1

1 0

)

, σ 2 =

(

0 −i

i 0

)

, σ 3 =

(

1 0

0 − 1

)

.

The Lagrange action of (2.5.38) is

(2.5.40) L=

∫

M^4

ψ(iγμ∂μ−

mc

h ̄

)ψ

√

−gdx,

whereψ=ψ†γ^0.
The Lagrange density of (2.5.40) is

L=ψ

(

iγμ∂μ−

mc

̄h

)

ψ

is Lorentz invariant. To see this, recall the spinor transformation defined by (2.2.63):

̃xμ=Lνμxν ⇒ψ ̃=Rψ,

which satisfies that (2.2.62):

(2.5.41) (iγμ ̃∂μ−

mc

h ̄

)ψ ̃=R(iγμ∂μ−

mc

h ̄

)ψ,

where ̃∂μ=∂/∂x ̃μ,Ris as in (2.2.67). By (2.2.68) and

(2.5.42) R†γ^0 =γ^0 R†, (γ^0 )†=γ^0.

It follows from (2.5.41) and (2.4.2) that

ψ ̃(iγμ∂ ̃μ−

mc

h ̄

)ψ ̃=ψ†R†γ^0 R

(

iγμ∂μ−

mc

h ̄

)

ψ

=(byψ†γ^0 =ψ)

=ψ(iγμ∂μ−

mc

h ̄

)ψ.

It implies that the Lagrange action (2.5.40) is Lorentz invariant.
It is easy to see that

δL

δ ψ∗

= 0 ⇔ (iγμ∂μ−

mc

̄h

)ψ= 0.