Mathematical Principles of Theoretical Physics

6.2. FOUNDATIONS OF QUANTUM PHYSICS 337

In fact, under the reflection transformation (6.2.17), the Dirac spinorψtransforms as

(6.2.18) ψ ̃=γ^0 ψ (orψ=γ^0 ψ ̃).

Thus (6.2.16) becomes

iγμ ̃∂μψ ̃−

mc

̄h

ψ ̃=γ^0 (iγ^0 ∂ 0 ψ−iγ^0 γkγ^0 ∂kψ−

mc

̄h

ψ) =γ^0 (iγμ∂μ−

mc

h ̄

)ψ= 0.

Here we used the identities

γ^0 γ^0 =I, γ^0 γk=−γk for 1≤k≤ 3.

Remark 6.8.Because the Weyl spinor has two-components, under the reflection transforma-
tion (6.2.17), it is invariant:

x→ −x⇒ψ→ψ(ψthe Weyl spinor).

Hence the Weyl equation is not invariant under the space reflection (6.2.17), which leads to
violation of parity conservation for decays and scatterings involving neutrinos. The violation
of parity conservation was discovered by (Lee and Yang, 1956 ).

Quantum dynamics based on PLD

Due to PLD, as the Lagrangian action of a quantum systemψis given by

L=

∫

L(ψ,Dψ)dxμ,

then its dynamic equation is derived by

(6.2.19)
δ
δ ψ

L= 0.

Hence, we only need to give the Lagrangian action for each of (6.2.9)

1.Schrodinger systems ̈. For the Schr ̈odinger equation (6.2.10), its action takes the form

(6.2.20) Ls=

1

2

i ̄h

(

∂ ψ

∂t

ψ∗−

∂ ψ∗

∂t

ψ

)

−

1

2

(

h ̄^2

2 m

|∇ψ|^2 +V|ψ|^2

)

.

2.Klein-Gordon systems.The action for the Klein-Gordon equation (6.2.11) is as follows

(6.2.21) LKG=

1

2

∂μψ∗∂μψ+

1

2

(mc

̄h

) 2

|ψ|.

3)Weyl systems.The action for the Weyl equation (6.2.13) reads

(6.2.22) Lw=ψ†σμ∂μψ,

whereσ^0 =−I,(σ^1 ,σ^2 ,σ^3 ) =~σ.

4.Dirac systems.The action for the Dirac equation (6.2.16) is

(6.2.23) LD=ψ

(

iγμ∂μ−

mc

h ̄

)

ψ.