Quantum Feild Theory Ryder# Chapter 2 Single-particle relativistic wave equations# Relativistic notation# Correspondense of S U ( 2 ) SU(2) S U ( 2 ) S O ( 3 ) SO(3) SO ( 3 ) # U = e i σ ⋅ θ / 2 = cos θ / 2 + i σ ⋅ n sin θ / 2 ↔ R = e i ⋅ J θ U=\mathrm{e}^{\mathrm{i} \boldsymbol{\sigma} \cdot \boldsymbol{\theta} / 2}=\cos \theta / 2+\mathrm{i} \boldsymbol{\sigma} \cdot \mathbf{n} \sin \theta / 2 \leftrightarrow R=\mathrm{e}^{\mathrm{i} \cdot \boldsymbol{J} \boldsymbol{\theta}} U = e i σ ⋅ θ /2 = cos θ /2 + i σ ⋅ n sin θ /2 ↔ R = e i ⋅ J θ Generator of Lorentz Group# x 0 ′ = γ ( x 0 + β x 1 ) , x 1 ′ = γ ( β x 0 + x 1 ) , x 2 ′ = x 2 , x 3 ′ = x 3
x^{0 \prime}=\gamma\left(x^{0}+\beta x^{1}\right), \quad x^{1 \prime}=\gamma\left(\beta x^{0}+x^{1}\right), \quad x^{2 \prime}=x^{2}, \quad x^{3 \prime}=x^{3}
x 0′ = γ ( x 0 + β x 1 ) , x 1′ = γ ( β x 0 + x 1 ) , x 2′ = x 2 , x 3′ = x 3 γ = cosh ϕ , γ β = sinh ϕ ,
\gamma=\cosh \phi, \quad \gamma \beta=\sinh \phi,
γ = cosh ϕ , γ β = sinh ϕ , ( x 0 ′ x 1 ′ x 2 ′ x 3 ′ ) = ( cosh ϕ sinh ϕ 0 0 sinh ϕ cosh ϕ 0 0 0 0 1 0 0 0 0 1 ) ( x 0 x 1 x 2 x 3 )
\left(\begin{array}{l}
x^{0 \prime} \\
x^{1 \prime} \\
x^{2 \prime} \\
x^{3 \prime}
\end{array}\right)=\left(\begin{array}{cccc}
\cosh \phi & \sinh \phi & 0 & 0 \\
\sinh \phi & \cosh \phi & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{array}\right)\left(\begin{array}{l}
x^{0} \\
x^{1} \\
x^{2} \\
x^{3}
\end{array}\right)
x 0′ x 1′ x 2′ x 3′ = cosh ϕ sinh ϕ 0 0 sinh ϕ cosh ϕ 0 0 0 0 1 0 0 0 0 1 x 0 x 1 x 2 x 3 K x = 1 ∂ B i ∂ ϕ ∣ ϕ = 0 = − i ( 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 )
K_{x}=\left.\frac{1 \partial B}{\mathrm{i} \partial \phi}\right|_{\phi=0}=-\mathrm{i}\left(\begin{array}{cccc}
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right)
K x = i ∂ ϕ 1 ∂ B ϕ = 0 = − i 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 K y = − i ( 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 ) , K z = − i ( 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 )
K_{y}=-\mathrm{i}\left(\begin{array}{cccc}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right), \quad K_{z}=-\mathrm{i}\left(\begin{array}{cccc}
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
1 & 0 & 0 & 0
\end{array}\right)
K y = − i 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 , K z = − i 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 Rotation Matrix in SO(3)# R z ( θ ) = ( cos θ sin θ 0 − sin θ cos θ 0 0 0 1 )
R_{z}(\theta)=\left(\begin{array}{ccc}
\cos \theta & \sin \theta & 0 \\
-\sin \theta & \cos \theta & 0 \\
0 & 0 & 1
\end{array}\right)
R z ( θ ) = cos θ − sin θ 0 sin θ cos θ 0 0 0 1 R x ( ϕ ) = ( 1 0 0 0 cos ϕ sin ϕ 0 − sin ϕ cos ϕ ) R y ( ψ ) = ( cos ψ 0 − sin ψ 0 1 0 sin ψ 0 cos ψ )
\begin{aligned}
&R_{x}(\phi)=\left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & \cos \phi & \sin \phi \\
0 & -\sin \phi & \cos \phi
\end{array}\right) \\
&R_{y}(\psi)=\left(\begin{array}{ccc}
\cos \psi & 0 & -\sin \psi \\
0 & 1 & 0 \\
\sin \psi & 0 & \cos \psi
\end{array}\right)
\end{aligned}
R x ( ϕ ) = 1 0 0 0 cos ϕ − sin ϕ 0 sin ϕ cos ϕ R y ( ψ ) = cos ψ 0 sin ψ 0 1 0 − sin ψ 0 cos ψ J z = 1 i d R z ( θ ) d θ ∣ θ = 0 = ( 0 − i 0 i 0 0 0 0 0 ) J x = 1 i d R x ( ϕ ) d ϕ ∣ ϕ = 0 = ( 0 0 0 0 0 − i 0 i 0 ) J y = 1 i d R y ( ψ ) d ψ ∣ ψ = 0 = ( 0 0 i 0 0 0 − i 0 0 )
\begin{aligned}
&J_{z}=\left.\frac{1}{\mathrm{i}} \frac{\mathrm{d} R_{z}(\theta)}{\mathrm{d} \theta}\right|_{\theta=0}=\left(\begin{array}{rrr}
0 & -\mathrm{i} & 0 \\
\mathrm{i} & 0 & 0 \\
0 & 0 & 0
\end{array}\right) \\
&J_{x}=\left.\frac{1}{\mathrm{i}} \frac{\mathrm{d} R_{x}(\phi)}{\mathrm{d} \phi}\right|_{\phi=0}=\left(\begin{array}{rrr}
0 & 0 & 0 \\
0 & 0 & -\mathrm{i} \\
0 & \mathrm{i} & 0
\end{array}\right) \\
&J_{y}=\left.\frac{1}{\mathrm{i}} \frac{\mathrm{d} R_{y}(\psi)}{\mathrm{d} \psi}\right|_{\psi=0}=\left(\begin{array}{rrr}
0 & 0 & \mathrm{i} \\
0 & 0 & 0 \\
-\mathrm{i} & 0 & 0
\end{array}\right)
\end{aligned}
J z = i 1 d θ d R z ( θ ) θ = 0 = 0 i 0 − i 0 0 0 0 0 J x = i 1 d ϕ d R x ( ϕ ) ϕ = 0 = 0 0 0 0 0 i 0 − i 0 J y = i 1 d ψ d R y ( ψ ) ψ = 0 = 0 0 − i 0 0 0 i 0 0 in 4 × 4 4\times 4 4 × 4
J x = − i ( 0 0 0 0 0 0 0 0 0 0 0 1 0 0 − 1 0 ) , J y = − i ( 0 0 0 0 0 0 0 − 1 0 0 0 0 0 1 0 0 ) , J z = − i ( 0 0 0 0 0 0 1 0 0 − 1 0 0 0 0 0 0 ) .
\begin{gathered}
J_{x}=-\mathrm{i}\left(\begin{array}{rrrr}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & -1 & 0
\end{array}\right), \\ J_{y}=-\mathrm{i}\left(\begin{array}{rrrr}
0 & 0 & 0 & 0 \\
0 & 0 & 0 & -1 \\
0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0
\end{array}\right), \\
J_{z}=-\mathrm{i}\left(\begin{array}{rrrr}
0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & 0 & 0
\end{array}\right) .
\end{gathered}
J x = − i 0 0 0 0 0 0 0 0 0 0 0 − 1 0 0 1 0 , J y = − i 0 0 0 0 0 0 0 1 0 0 0 0 0 − 1 0 0 , J z = − i 0 0 0 0 0 0 − 1 0 0 1 0 0 0 0 0 0 . commutation relations# [ K x , K y ] = − i J z \left[K_{x}, K_{y}\right]=-\mathrm{i} J_{z} [ K x , K y ] = − i J z [ J x , K x ] = 0 \left[J_{x}, K_{x}\right]=0 [ J x , K x ] = 0 [ J x , K y ] = i K z \left[J_{x}, K_{y}\right]=\mathrm{i} K_{z} [ J x , K y ] = i K z ] ] ]
A = 1 2 ( J + i K ) B = 1 2 ( J − i K ) . }
\left.\begin{array}{l}
\mathbf{A}=\frac{1}{2}(\mathbf{J}+\mathrm{iK}) \\
\mathbf{B}=\frac{1}{2}(\mathbf{J}-\mathrm{iK}) .
\end{array}\right\}
A = 2 1 ( J + iK ) B = 2 1 ( J − iK ) . } [ A x , A y ] = i A z \left[A_{x}, A_{y}\right]=\mathrm{i} A_{z} [ A x , A y ] = i A z [ B x , B y ] = i B z \left[B_{x}, B_{y}\right]=\mathrm{i} B_{z} [ B x , B y ] = i B z [ A i , B j ] = 0 ( i , j = x , y , z ) . ] \left.\left[A_{i}, B_{j}\right]=0(i, j=x, y, z) .\right] [ A i , B j ] = 0 ( i , j = x , y , z ) . ]
two types of vector# ( j , 0 ) → J ( j ) = i K j ( B = 0 ) , ( 0 , j ) → J ( j ) = − i K ( j ) ( A = 0 ) , }
\left.\begin{array}{ll}
(j, 0) \rightarrow \mathbf{J}^{(j)}=\mathrm{i} \mathbf{K}^{j} & (\mathbf{B}=0), \\
(0, j) \rightarrow \mathbf{J}^{(j)}=-\mathrm{i} \mathbf{K}^{(j)} & (\mathbf{A}=0),
\end{array}\right\}
( j , 0 ) → J ( j ) = i K j ( 0 , j ) → J ( j ) = − i K ( j ) ( B = 0 ) , ( A = 0 ) , } and this in fact corresponds to the two possibilities in ( 2.69 ) (2.69) ( 2.69 )
ξ → exp ( i σ 2 ⋅ θ + σ 2 ⋅ ϕ ) ξ = exp [ i σ 2 ⋅ ( θ − i ϕ ) ] ξ ≡ M ξ
\begin{aligned}
\xi & \rightarrow \exp \left(\mathrm{i} \frac{\sigma}{2} \cdot \boldsymbol{\theta}+\frac{\boldsymbol{\sigma}}{2} \cdot \boldsymbol{\phi}\right) \xi \\
&=\exp \left[\mathrm{i} \frac{\boldsymbol{\sigma}}{2} \cdot(\boldsymbol{\theta}-\mathrm{i} \boldsymbol{\phi})\right] \xi \equiv M \xi
\end{aligned}
ξ → exp ( i 2 σ ⋅ θ + 2 σ ⋅ ϕ ) ξ = exp [ i 2 σ ⋅ ( θ − i ϕ ) ] ξ ≡ M ξ η → exp [ i σ 2 ⋅ ( θ + i ϕ ) ] η ≡ N η
\eta \rightarrow \exp \left[\mathrm{i} \frac{\sigma}{2} \cdot(\boldsymbol{\theta}+\mathrm{i} \phi)\right] \eta \equiv N \eta
η → exp [ i 2 σ ⋅ ( θ + i ϕ ) ] η ≡ N η Relationship betwen M M M N N N M M M # M = exp [ i σ 2 ⋅ ( θ − i ϕ ) ] N = exp [ i σ 2 ⋅ ( θ + i ϕ ) ]
\begin{aligned}
M=\exp \left[\mathrm{i} \frac{\boldsymbol{\sigma}}{2} \cdot(\boldsymbol{\theta}-\mathrm{i} \boldsymbol{\phi})\right]\\
N=\exp \left[\mathrm{i} \frac{\sigma}{2} \cdot(\boldsymbol{\theta}+\mathrm{i} \phi)\right]
\end{aligned}
M = exp [ i 2 σ ⋅ ( θ − i ϕ ) ] N = exp [ i 2 σ ⋅ ( θ + i ϕ ) ] N = ζ M ∗ ζ − 1 with ζ = − i σ 2
N=\zeta M^{*} \zeta^{-1} \quad \text { with } \zeta=-\mathrm{i} \sigma_{2}
N = ζ M ∗ ζ − 1 with ζ = − i σ 2 σ 2 σ ∗ σ 2 = − σ 2 2 σ = − σ 1 ζ M ∗ ζ − 1 = σ 2 exp [ − i 2 σ ∗ ⋅ ( θ + i ϕ ) ] σ 2 = σ 2 2 exp [ i 2 σ ⋅ ( θ + i φ ) ] = N .
\begin{gathered}
\sigma_{2} \sigma^{*} \sigma_{2}=-\sigma_{2}^{2} \sigma=-\sigma_{1} \\
\zeta M^{*} \zeta^{-1}=\sigma_{2} \exp \left[-\frac{\mathrm{i}}{2} \sigma^{*} \cdot(\boldsymbol{\theta}+\mathrm{i} \boldsymbol{\phi})\right] \sigma_{2} \\
=\sigma_{2}^{2} \exp \left[\frac{\mathrm{i}}{2} \boldsymbol{\sigma} \cdot(\boldsymbol{\theta}+\mathrm{i} \boldsymbol{\varphi})\right] \\
=N .
\end{gathered}
σ 2 σ ∗ σ 2 = − σ 2 2 σ = − σ 1 ζ M ∗ ζ − 1 = σ 2 exp [ − 2 i σ ∗ ⋅ ( θ + i ϕ ) ] σ 2 = σ 2 2 exp [ 2 i σ ⋅ ( θ + i φ ) ] = N . Parity# v → − v \mathbf{v} \rightarrow-\mathbf{v} v → − v
K → − K \mathbf{K} \rightarrow-\mathbf{K} K → − K
J → + J \mathbf{J} \rightarrow+\mathbf{J} J → + J
( j , 0 ) ↔ ( 0 , j ) , (j, 0) \leftrightarrow(0, j), \quad ( j , 0 ) ↔ ( 0 , j ) ,
ξ ↔ η . \xi \leftrightarrow \eta . ξ ↔ η .
Irreducible representation of Lorentz Group# ( ξ η ) → ( e 1 / 2 σ ⋅ ( θ − i ϕ ) 0 0 e 1 / 2 σ ⋅ ( θ + i ϕ ) ) ( ξ η ) = ( D ( Λ ) 0 0 D ˉ ( Λ ) ) ( ξ η )
\begin{aligned}
\left(\begin{array}{l}
\xi \\
\eta
\end{array}\right) & \rightarrow\left(\begin{array}{cc}
\mathrm{e}^{1 / 2 \sigma \cdot(\theta-\mathrm{i} \phi)} & 0 \\
0 & \mathrm{e}^{1 / 2 \sigma \cdot(\theta+\mathrm{i} \phi)}
\end{array}\right)\left(\begin{array}{l}
\xi \\
\eta
\end{array}\right) \\
&=\left(\begin{array}{cc}
D(\Lambda) & 0 \\
0 & \bar{D}(\Lambda)
\end{array}\right)\left(\begin{array}{l}
\xi \\
\eta
\end{array}\right)
\end{aligned}
( ξ η ) → ( e 1/2 σ ⋅ ( θ − i ϕ ) 0 0 e 1/2 σ ⋅ ( θ + i ϕ ) ) ( ξ η ) = ( D ( Λ ) 0 0 D ˉ ( Λ ) ) ( ξ η ) D ˉ ( Λ ) = ζ D ∗ ( Λ ) ζ − 1
\bar{D}(\Lambda)=\zeta D^{*}(\Lambda) \zeta^{-1}
D ˉ ( Λ ) = ζ D ∗ ( Λ ) ζ − 1 ζ = − i σ 2 \zeta = -i\sigma^2 ζ = − i σ 2
not Unitary, and not compact
Derive of Dirac equation# θ = 0 \theta=0 θ = 0
ξ → ϕ R , η → ϕ L
\xi \rightarrow \phi_{\mathrm{R}}, \quad \eta \rightarrow \phi_{\mathrm{L}}
ξ → ϕ R , η → ϕ L ϕ R → e 1 / 2 σ ⋅ φ ϕ R = [ cosh ( ϕ / 2 ) + σ ⋅ n sinh ( ϕ / 2 ) ] ϕ R
\begin{aligned}
\phi_{\mathrm{R}} & \rightarrow \mathrm{e}^{1 / 2 \boldsymbol{\sigma} \cdot \boldsymbol{\varphi}} \phi_{\mathrm{R}} \\
&=[\cosh (\phi / 2)+\boldsymbol{\sigma} \cdot \mathbf{n} \sinh (\phi / 2)] \phi_{\mathrm{R}}
\end{aligned}
ϕ R → e 1/2 σ ⋅ φ ϕ R = [ cosh ( ϕ /2 ) + σ ⋅ n sinh ( ϕ /2 )] ϕ R cosh ( ϕ / 2 ) = [ ( γ + 1 ) / 2 ] 1 / 2 sinh ( ϕ / 2 ) = [ ( γ − 1 ) / 2 ] 1 / 2
\begin{aligned}
\cosh(\phi/2)=[(\gamma+1)/2]^{1/2}\\
\sinh(\phi/2)=[(\gamma-1)/2]^{1/2}
\end{aligned}
cosh ( ϕ /2 ) = [( γ + 1 ) /2 ] 1/2 sinh ( ϕ /2 ) = [( γ − 1 ) /2 ] 1/2 ϕ R ( p ) = [ ( γ + 1 2 ) 1 / 2 + σ ⋅ p ^ ( γ − 1 2 ) 1 / 2 ] ϕ R ( 0 )
\phi_{\mathrm{R}}(\mathbf{p})=\left[\left(\frac{\gamma+1}{2}\right)^{1 / 2}+\sigma \cdot \hat{\mathbf{p}}\left(\frac{\gamma-1}{2}\right)^{1 / 2}\right] \phi_{\mathrm{R}}(0)
ϕ R ( p ) = [ ( 2 γ + 1 ) 1/2 + σ ⋅ p ^ ( 2 γ − 1 ) 1/2 ] ϕ R ( 0 ) γ = E / m \gamma=E / m γ = E / m
ϕ R ( p ) = E + m + σ ⋅ p [ 2 m ( E + m ) ] 1 / 2 ϕ R ( 0 )
\phi_{\mathrm{R}}(\mathbf{p})=\frac{E+m+\sigma \cdot \mathbf{p}}{[2 m(E+m)]^{1 / 2}} \phi_{\mathrm{R}}(0)
ϕ R ( p ) = [ 2 m ( E + m ) ] 1/2 E + m + σ ⋅ p ϕ R ( 0 ) ϕ L ( p ) = E + m − φ ⋅ p [ 2 m ( E + m ) ] 1 / 2 ϕ L ( 0 )
\phi_{\mathrm{L}}(\mathbf{p})=\frac{E+m-\varphi \cdot \mathbf{p}}{[2 m(E+m)]^{1 / 2}} \phi_{\mathrm{L}}(0)
ϕ L ( p ) = [ 2 m ( E + m ) ] 1/2 E + m − φ ⋅ p ϕ L ( 0 ) ϕ R ( p ) = E + σ ⋅ p m ϕ L ( p )
\phi_{\mathrm{R}}(\mathbf{p})=\frac{E+\sigma \cdot \mathbf{p}}{m} \phi_{\mathrm{L}}(\mathbf{p})
ϕ R ( p ) = m E + σ ⋅ p ϕ L ( p ) ϕ L ( p ) = E − σ ⋅ p m ϕ R ( p )
\phi_{\mathrm{L}}(\mathbf{p})=\frac{E-\sigma \cdot \mathbf{p}}{m} \phi_{\mathrm{R}}(\mathbf{p})
ϕ L ( p ) = m E − σ ⋅ p ϕ R ( p ) − m ϕ R ( p ) + ( p 0 + σ ⋅ p ) ϕ L ( p ) = 0 ( p 0 − σ ⋅ p ) ϕ R ( p ) − m ϕ L ( p ) = 0 }
\left.\begin{array}{r}
-m \phi_{\mathrm{R}}(\mathbf{p})+\left(p_{0}+\boldsymbol{\sigma} \cdot \mathbf{p}\right) \phi_{\mathrm{L}}(\mathbf{p})=0 \\
\left(p_{0}-\boldsymbol{\sigma} \cdot \mathbf{p}\right) \phi_{\mathrm{R}}(\mathbf{p})-m \phi_{\mathrm{L}}(\mathbf{p})=0
\end{array}\right\}
− m ϕ R ( p ) + ( p 0 + σ ⋅ p ) ϕ L ( p ) = 0 ( p 0 − σ ⋅ p ) ϕ R ( p ) − m ϕ L ( p ) = 0 } ( − m p 0 + σ ⋅ p p 0 − σ ⋅ p − m ) ( ϕ R ( p ) ϕ L ( p ) ) = 0
\left(\begin{array}{cc}
-m & p_{0}+\boldsymbol{\sigma} \cdot \mathbf{p} \\
p_{0}-\boldsymbol{\sigma} \cdot \mathbf{p} & -m
\end{array}\right)\left(\begin{array}{l}
\phi_{\mathrm{R}}(\mathbf{p}) \\
\phi_{\mathrm{L}}(\mathbf{p})
\end{array}\right)=0
( − m p 0 − σ ⋅ p p 0 + σ ⋅ p − m ) ( ϕ R ( p ) ϕ L ( p ) ) = 0 ψ ( p ) = ( ϕ R ( p ) ϕ L ( p ) )
\psi(p)=\left(\begin{array}{l}
\phi_{\mathrm{R}}(\mathbf{p}) \\
\phi_{\mathrm{L}}(\mathbf{p})
\end{array}\right)
ψ ( p ) = ( ϕ R ( p ) ϕ L ( p ) ) γ 0 = ( 0 1 1 0 ) , γ i = ( 0 − σ i σ i 0 )
\color{red}\gamma^{0}=\left(\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right), \quad \gamma^{i}=\left(\begin{array}{cc}
0 & -\sigma^{i} \\
\sigma^{i} & 0
\end{array}\right)
γ 0 = ( 0 1 1 0 ) , γ i = ( 0 σ i − σ i 0 ) ( γ 0 p 0 + γ i p i − m ) ψ ( p ) = 0
\left(\gamma^{0} p_{0}+\gamma^{i} p_{i}-m\right) \psi(p)=0
( γ 0 p 0 + γ i p i − m ) ψ ( p ) = 0 ( γ μ p μ − m ) ψ ( p ) = 0
\left(\gamma^{\mu} p_{\mu}-m\right) \psi(p)=0
( γ μ p μ − m ) ψ ( p ) = 0 helicity# m = 0 m=0 m = 0
( p 0 + σ ⋅ p ) ϕ L ( p ) = 0 ( p 0 − σ ⋅ p ) ϕ R ( p ) = 0
\begin{aligned}
&\left(p_{0}+\boldsymbol{\sigma} \cdot \mathbf{p}\right) \phi_{\mathrm{L}}(\mathbf{p})=0 \\
&\left(p_{0}-\boldsymbol{\sigma} \cdot \mathbf{p}\right) \phi_{\mathrm{R}}(\mathbf{p})=0
\end{aligned}
( p 0 + σ ⋅ p ) ϕ L ( p ) = 0 ( p 0 − σ ⋅ p ) ϕ R ( p ) = 0 σ ⋅ p ^ ϕ L = − ϕ L , σ ⋅ p ^ ϕ R = ϕ R
\boldsymbol{\sigma} \cdot \hat{\mathbf{p}} \phi_{\mathrm{L}}=-\phi_{\mathrm{L}}, \quad \boldsymbol{\sigma} \cdot \hat{\mathbf{p}} \phi_{\mathrm{R}}=\phi_{\mathrm{R}}
σ ⋅ p ^ ϕ L = − ϕ L , σ ⋅ p ^ ϕ R = ϕ R 2.4 Prediction of antiparticles# Algebra of γ \gamma γ # ( i γ μ ∂ μ − m ) ψ = 0
\color{red}\left(\mathrm{i} \gamma^{\mu} \partial_{\mu}-m\right) \psi=0
( i γ μ ∂ μ − m ) ψ = 0 [ − ( γ μ ∂ μ ) ( γ v ∂ v ) − i ( γ μ ∂ μ ) m ] ψ = 0 ( γ μ γ ′ ∂ μ ∂ v + m 2 ) ψ = 0
\begin{array}{r}
{\left[-\left(\gamma^{\mu} \partial_{\mu}\right)\left(\gamma^{v} \partial_{v}\right)-\mathrm{i}\left(\gamma^{\mu} \partial_{\mu}\right) m\right] \psi=0} \\
\left(\gamma^{\mu} \gamma^{\prime} \partial_{\mu} \partial_{v}+m^{2}\right) \psi=0
\end{array}
[ − ( γ μ ∂ μ ) ( γ v ∂ v ) − i ( γ μ ∂ μ ) m ] ψ = 0 ( γ μ γ ′ ∂ μ ∂ v + m 2 ) ψ = 0 1 2 ( γ μ γ v + γ v γ μ ) ≡ 1 2 { γ μ , γ v }
\frac{1}{2}\left(\gamma^{\mu} \gamma^{v}+\gamma^{v} \gamma^{\mu}\right) \equiv \frac{1}{2}\left\{\gamma^{\mu}, \gamma^{v}\right\}
2 1 ( γ μ γ v + γ v γ μ ) ≡ 2 1 { γ μ , γ v } 1 2 { γ μ , γ v } ∂ μ ∂ v ψ + m 2 ψ = 0
\frac{1}{2}\left\{\gamma^{\mu}, \gamma^{v}\right\} \partial_{\mu} \partial_{v} \psi+m^{2} \psi=0
2 1 { γ μ , γ v } ∂ μ ∂ v ψ + m 2 ψ = 0 ( □ + m 2 ) ψ ( x ) = 0
\left(\square+m^{2}\right) \psi(x)=0
( □ + m 2 ) ψ ( x ) = 0 { γ μ , γ v } = 2 g μ v .
\color{red}\left\{\gamma^{\mu}, \gamma^{v}\right\}=2 g^{\mu v} .
{ γ μ , γ v } = 2 g μv . ( γ 0 ) 2 = 1 , ( γ i ) 2 = − 1 , γ μ γ v = − γ v γ μ ( v ≠ μ ) .
\left(\gamma^{0}\right)^{2}=1, \quad\left(\gamma^{i}\right)^{2}=-1, \quad \gamma^{\mu} \gamma^{v}=-\gamma^{v} \gamma^{\mu} \quad(v \neq \mu) .
( γ 0 ) 2 = 1 , ( γ i ) 2 = − 1 , γ μ γ v = − γ v γ μ ( v = μ ) . probability current j μ j^\mu j μ # ( i γ μ ∂ μ − m ) ψ = 0
\left(i \gamma^{\mu} \partial_{\mu}-m\right) \psi=0
( i γ μ ∂ μ − m ) ψ = 0 ψ † ( − i γ 0 ∂ ← 0 + i γ i ∂ i − m ) = 0
\psi^{\dagger}\left(-\mathrm{i} \gamma^{0} \stackrel{\leftarrow}{\partial}_{0}+\mathrm{i} \gamma^{i} \partial_{i}-m\right)=0
ψ † ( − i γ 0 ∂ ← 0 + i γ i ∂ i − m ) = 0 From γ i γ 0 = − γ 0 γ i \gamma^i\gamma^0=-\gamma^0\gamma^i γ i γ 0 = − γ 0 γ i
ψ ˉ ( i γ μ ∂ μ + m ) = 0
\bar{\psi}\left(\mathrm{i} \gamma^{\mu}{\partial}_{\mu}+m\right)=0
ψ ˉ ( i γ μ ∂ μ + m ) = 0 ψ ˉ = ψ † γ 0
\color{red}\bar{\psi}=\psi^{\dagger} \gamma^{0}
ψ ˉ = ψ † γ 0 j ˙ μ = ψ ˉ γ μ ψ
\dot{j}^{\mu}=\bar{\psi} \gamma^{\mu} \psi
j ˙ μ = ψ ˉ γ μ ψ ∂ μ j μ = ( ∂ μ ψ ˉ ) γ μ ψ + ψ ˉ γ μ ( ∂ μ ψ ) = ( i m ψ ˉ ) ψ + ψ ˉ ( − i m ψ ) = 0
\begin{aligned}
\partial_{\mu} j^{\mu} &=\left(\partial_{\mu} \bar{\psi}\right) \gamma^{\mu} \psi+\bar{\psi} \gamma^{\mu}\left(\partial_{\mu} \psi\right) \\
&=(i m \bar{\psi}) \psi+\bar{\psi}(-i m \psi)=0
\end{aligned}
∂ μ j μ = ( ∂ μ ψ ˉ ) γ μ ψ + ψ ˉ γ μ ( ∂ μ ψ ) = ( im ψ ˉ ) ψ + ψ ˉ ( − im ψ ) = 0 j 0 = ψ ˉ γ 0 ψ = ψ † ψ = ∣ ψ 1 ∣ 2 + ∣ ψ 2 ∣ 2 + ∣ ψ 3 ∣ 2 + ∣ ψ 4 ∣ 2
j^{0}=\bar{\psi} \gamma^{0} \psi=\psi^{\dagger} \psi=\left|\psi_{1}\right|^{2}+\left|\psi_{2}\right|^{2}+\left|\psi_{3}\right|^{2}+\left|\psi_{4}\right|^{2}
j 0 = ψ ˉ γ 0 ψ = ψ † ψ = ∣ ψ 1 ∣ 2 + ∣ ψ 2 ∣ 2 + ∣ ψ 3 ∣ 2 + ∣ ψ 4 ∣ 2 2.5 Construction of Dirac spinors: algebra of γ \gamma γ # Prove that ψ ˉ ψ \bar\psi \psi ψ ˉ ψ ψ ˉ γ 5 ψ \bar\psi \gamma^5\psi ψ ˉ γ 5 ψ # ψ = ( ϕ R ϕ L )
\psi=\left(\begin{array}{l}
\phi_{\mathrm{R}} \\
\phi_{\mathrm{L}}
\end{array}\right)
ψ = ( ϕ R ϕ L ) ϕ R → exp [ i 2 σ ⋅ ( θ − i ϕ ) ] ϕ R , ϕ L → exp [ i 2 σ ⋅ ( θ + i ϕ ) ] ϕ L
\phi_{\mathrm{R}} \rightarrow \exp \left[\frac{\mathrm{i}}{2} \sigma \cdot(\theta-\mathrm{i} \phi)\right] \phi_{\mathrm{R}}, \quad \phi_{\mathrm{L}} \rightarrow \exp \left[\frac{\mathrm{i}}{2} \sigma \cdot(\theta+\mathrm{i} \phi)\right] \phi_{\mathrm{L}}
ϕ R → exp [ 2 i σ ⋅ ( θ − i ϕ ) ] ϕ R , ϕ L → exp [ 2 i σ ⋅ ( θ + i ϕ ) ] ϕ L ϕ R † → ϕ R † exp [ − i 2 σ ⋅ ( θ + i ϕ ) ] , ϕ L † → ϕ L † exp [ − i 2 σ ⋅ ( θ − i ϕ ) ]
\phi_{\mathrm{R}}^{\dagger} \rightarrow \phi_{\mathrm{R}}^{\dagger} \exp \left[\frac{-\mathrm{i}}{2} \boldsymbol{\sigma} \cdot(\boldsymbol{\theta}+\mathrm{i} \phi)\right], \quad \phi_{\mathrm{L}}^{\dagger} \rightarrow \phi_{\mathrm{L}}^{\dagger} \exp \left[\frac{-\mathrm{i}}{2} \boldsymbol{\sigma} \cdot(\theta-\mathrm{i} \phi)\right]
ϕ R † → ϕ R † exp [ 2 − i σ ⋅ ( θ + i ϕ ) ] , ϕ L † → ϕ L † exp [ 2 − i σ ⋅ ( θ − i ϕ ) ] ψ † ψ = ϕ R † ϕ R + ϕ L † ϕ L
\psi^{\dagger} \psi=\phi_{\mathrm{R}}^{\dagger} \phi_{\mathrm{R}}+\phi_{\mathrm{L}}^{\dagger} \phi_{\mathrm{L}}
ψ † ψ = ϕ R † ϕ R + ϕ L † ϕ L is not invariant. However, the adjoint spinor ψ ˉ defined in (2.104) has compo- nents ψ ˉ = ψ † γ 0 = ( ϕ R † ϕ L † ) ( 0 1 1 0 ) = ( ϕ L † ϕ R † )
\begin{aligned}
&\text { is not invariant. However, the adjoint spinor } \bar{\psi} \text { defined in (2.104) has compo- } \\
&\text { nents } \\
&\qquad \bar{\psi}=\psi^{\dagger} \gamma^{0}=\left(\phi_{\mathrm{R}}^{\dagger} \phi_{\mathrm{L}}^{\dagger}\right)\left(\begin{array}{ll}
0 & 1 \\
1 & 0
\end{array}\right)=\left(\phi_{\mathrm{L}}^{\dagger} \phi_{\mathrm{R}}^{\dagger}\right)
\end{aligned}
is not invariant. However, the adjoint spinor ψ ˉ defined in (2.104) has compo- nents ψ ˉ = ψ † γ 0 = ( ϕ R † ϕ L † ) ( 0 1 1 0 ) = ( ϕ L † ϕ R † ) ψ ˉ ψ = ϕ L † ϕ R + ϕ R † ˙ ϕ L
\bar{\psi} \psi=\phi_{\mathrm{L}}^{\dagger} \phi_{\mathrm{R}}+\phi_{\mathrm{R}}^{\dot{\dagger}} \phi_{\mathrm{L}}
ψ ˉ ψ = ϕ L † ϕ R + ϕ R † ˙ ϕ L ϕ R ↔ ϕ L ,
\phi_{\mathrm{R}} \leftrightarrow \phi_{\mathrm{L}},
ϕ R ↔ ϕ L , so ψ ˉ ψ → ψ ˉ ψ \bar{\psi} \psi \rightarrow \bar{\psi} \psi ψ ˉ ψ → ψ ˉ ψ
γ 5 = ( 1 0 0 − 1 )
\gamma^{5}=\left(\begin{array}{cc}
1 & 0 \\
0 & -1
\end{array}\right)
γ 5 = ( 1 0 0 − 1 ) γ 5 = i γ 0 γ 1 γ γ γ 3 = γ 5 .
\gamma^{5}=\mathrm{i} \gamma^{0} \gamma^{1} \gamma^{\gamma} \gamma^{3}=\gamma_{5} .
γ 5 = i γ 0 γ 1 γ γ γ 3 = γ 5 . ψ ˉ γ 5 ψ = ( ϕ L † ϕ R ) ( 1 0 0 − 1 ) ( ϕ R ϕ L ) = ϕ L † ϕ R − ϕ R + ϕ L .
\begin{aligned}
\bar{\psi} \gamma^{5} \psi &=\left(\phi_{\mathrm{L}}^{\dagger} \phi_{\mathrm{R}}\right)\left(\begin{array}{rr}
1 & 0 \\
0 & -1
\end{array}\right)\left(\begin{array}{l}
\phi_{\mathrm{R}} \\
\phi_{\mathrm{L}}
\end{array}\right) \\
&=\phi_{\mathrm{L}}^{\dagger} \phi_{\mathrm{R}}-\phi_{\mathrm{R}}^{+} \phi_{\mathrm{L}} .
\end{aligned}
ψ ˉ γ 5 ψ = ( ϕ L † ϕ R ) ( 1 0 0 − 1 ) ( ϕ R ϕ L ) = ϕ L † ϕ R − ϕ R + ϕ L . ψ ˉ ψ \bar{\psi} \psi ψ ˉ ψ
ψ ˉ γ 5 ψ \bar{\psi} \gamma_{5} \psi ψ ˉ γ 5 ψ
ψ ˉ γ μ ψ \bar{\psi} \gamma^{\mu} \psi ψ ˉ γ μ ψ
ψ ˉ γ μ γ 5 ψ \bar{\psi} \gamma^{\mu} \gamma^{5} \psi ψ ˉ γ μ γ 5 ψ
ψ ˉ ( γ μ γ v − γ v γ μ ) ψ \bar{\psi}\left(\gamma^{\mu} \gamma^{v}-\gamma^{v} \gamma^{\mu}\right) \psi ψ ˉ ( γ μ γ v − γ v γ μ ) ψ
Solutions of Dirac equation# ψ ( x ) = u ( 0 ) e − i m t positive energy, ψ ( x ) = v ( 0 ) e i m t negative energy. }
\left.\begin{array}{l}
\psi(x)=u(0) \mathrm{e}^{-\mathrm{i} m t} \text { positive energy, } \\
\psi(x)=v(0) \mathrm{e}^{\mathrm{i} m t} \text { negative energy. }
\end{array}\right\}
ψ ( x ) = u ( 0 ) e − i m t positive energy, ψ ( x ) = v ( 0 ) e i m t negative energy. } u ( 1 ) ( 0 ) = ( 1 0 0 0 ) , u ( 2 ) ( 0 ) = ( 0 1 0 0 ) , v ( 1 ) ( 0 ) = ( 0 0 1 0 ) , v ( 2 ) ( 0 ) = ( 0 0 0 1 )
u^{(1)}(0)=\left(\begin{array}{l}
1 \\
0 \\
0 \\
0
\end{array}\right), \quad u^{(2)}(0)=\left(\begin{array}{l}
0 \\
1 \\
0 \\
0
\end{array}\right), \quad v^{(1)}(0)=\left(\begin{array}{l}
0 \\
0 \\
1 \\
0
\end{array}\right), \quad v^{(2)}(0)=\left(\begin{array}{l}
0 \\
0 \\
0 \\
1
\end{array}\right)
u ( 1 ) ( 0 ) = 1 0 0 0 , u ( 2 ) ( 0 ) = 0 1 0 0 , v ( 1 ) ( 0 ) = 0 0 1 0 , v ( 2 ) ( 0 ) = 0 0 0 1 γ 0 \gamma^0 γ 0
γ 0 = ( 1 0 0 0 0 1 0 0 0 0 − 1 0 0 0 0 − 1 )
\gamma^{0}=\left(\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1
\end{array}\right)
γ 0 = 1 0 0 0 0 1 0 0 0 0 − 1 0 0 0 0 − 1 obtaioned from Chiral Representation
γ S R 0 = S γ C R 0 S − 1 S = 1 2 ( 1 1 1 − 1 )
\begin{aligned}
\gamma_{\mathrm{SR}}^{0} &=S \gamma_{\mathrm{CR}}^{0} S^{-1} \\
S &=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}
1 & 1 \\
1 & -1
\end{array}\right)
\end{aligned}
γ SR 0 S = S γ CR 0 S − 1 = 2 1 ( 1 1 1 − 1 ) ψ \psi ψ
ψ = S ( ϕ R ϕ L ) = 1 2 ( ϕ R + ϕ L ϕ R − ϕ L )
\psi=S\left(\begin{array}{l}
\phi_{\mathrm{R}} \\
\phi_{\mathrm{L}}
\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}{l}
\phi_{\mathrm{R}}+\phi_{\mathrm{L}} \\
\phi_{\mathrm{R}}-\phi_{\mathrm{L}}
\end{array}\right)
ψ = S ( ϕ R ϕ L ) = 2 1 ( ϕ R + ϕ L ϕ R − ϕ L ) ( ϕ R ϕ L ) → ( ϕ R ′ ϕ L ′ ) = ( e 1 / 2 σ ⋅ ϕ 0 0 e − 1 / 2 σ ⋅ φ ) ( ϕ R ϕ L ) = M ( ϕ R ϕ L )
\left(\begin{array}{l}
\phi_{\mathrm{R}} \\
\phi_{\mathrm{L}}
\end{array}\right) \rightarrow\left(\begin{array}{l}
\phi_{\mathrm{R}}^{\prime} \\
\phi_{\mathrm{L}}^{\prime}
\end{array}\right)=\left(\begin{array}{cc}
\mathrm{e}^{1 / 2 \sigma \cdot \phi} & 0 \\
0 & \mathrm{e}^{-1 / 2 \sigma \cdot \varphi}
\end{array}\right)\left(\begin{array}{l}
\phi_{\mathrm{R}} \\
\phi_{\mathrm{L}}
\end{array}\right)=M\left(\begin{array}{l}
\phi_{\mathrm{R}} \\
\phi_{\mathrm{L}}
\end{array}\right)
( ϕ R ϕ L ) → ( ϕ R ′ ϕ L ′ ) = ( e 1/2 σ ⋅ ϕ 0 0 e − 1/2 σ ⋅ φ ) ( ϕ R ϕ L ) = M ( ϕ R ϕ L ) M S R = S M C R S − 1 = ( cosh ( ϕ / 2 ) σ ⋅ n sinh ( ϕ / 2 ) σ ⋅ n sinh ( ϕ / 2 ) cosh ( ϕ / 2 ) )
M_{\mathrm{SR}}=S M_{\mathrm{CR}} S^{-1}=\left(\begin{array}{cc}
\cosh (\phi / 2) & \sigma \cdot \mathrm{n} \sinh (\phi / 2) \\
\sigma \cdot \mathrm{n} \sinh (\phi / 2) & \cosh (\phi / 2)
\end{array}\right)
M SR = S M CR S − 1 = ( cosh ( ϕ /2 ) σ ⋅ n sinh ( ϕ /2 ) σ ⋅ n sinh ( ϕ /2 ) cosh ( ϕ /2 ) ) cosh ( ϕ / 2 ) = ( E + m 2 m ) 1 / 2 , sinh ( ϕ / 2 ) = ( E − m 2 m ) 1 / 2 , tanh ( ϕ / 2 ) = p E + m
\begin{aligned}
\cosh (\phi / 2) &=\left(\frac{E+m}{2 m}\right)^{1 / 2}, \quad \sinh (\phi / 2)=\left(\frac{E-m}{2 m}\right)^{1 / 2}, \\
\tanh (\phi / 2) &=\frac{p}{E+m}
\end{aligned}
cosh ( ϕ /2 ) tanh ( ϕ /2 ) = ( 2 m E + m ) 1/2 , sinh ( ϕ /2 ) = ( 2 m E − m ) 1/2 , = E + m p M S R = ( E + m 2 m ) 1 / 2 ( 1 0 p z E + m p x − i p y E + m 0 1 p x + i p y E + m − p z E + m p z E + m p x − i p y E + m 1 0 p x + i p y E + m − p z E + m 0 1 )
M_{\mathrm{SR}}=\left(\frac{E+m}{2 m}\right)^{1 / 2}\left(\begin{array}{cccc}
1 & 0 & \frac{p_{z}}{E+m} & \frac{p_{x}-\mathrm{i} p_{y}}{E+m} \\
0 & 1 & \frac{p_{x}+\mathrm{i} p_{y}}{E+m} & \frac{-p_{z}}{E+m} \\
\frac{p_{z}}{E+m} & \frac{p_{x}-\mathrm{i} p_{y}}{E+m} & 1 & 0 \\
\frac{p_{x}+\mathrm{i} p_{y}}{E+m} & \frac{-p_{z}}{E+m} & 0 & 1
\end{array}\right)
M SR = ( 2 m E + m ) 1/2 1 0 E + m p z E + m p x + i p y 0 1 E + m p x − i p y E + m − p z E + m p z E + m p x + i p y 1 0 E + m p x − i p y E + m − p z 0 1 u ( 1 ) = ( E + m 2 m ) 1 / 2 ( 1 p z E + m p + E + m ) , u ( 2 ) = ( E + m 2 m ) 1 / 2 ( 0 p − E + m − p z E + m ) v ( 1 ) = ( E + m 2 m ) 1 / 2 ( p z E + m p + E + m 1 0 ) , v ( 2 ) = ( E + m 2 m ) 1 / 2 ( p − E + m − p z E + m 0 1 )
\begin{aligned}
&u^{(1)}=\left(\frac{E+m}{2 m}\right)^{1 / 2}\left(\begin{array}{c}
1 \\
\frac{p_{z}}{E+m} \\
\frac{p_{+}}{E+m}
\end{array}\right), \quad u^{(2)}=\left(\frac{E+m}{2 m}\right)^{1 / 2}\left(\begin{array}{c}
0 \\
\frac{p_{-}}{E+m} \\
\frac{-p_{z}}{E+m}
\end{array}\right)\\
&v^{(1)}=\left(\frac{E+m}{2 m}\right)^{1 / 2}\left(\begin{array}{c}
\frac{p_{z}}{E+m} \\
\frac{p_{+}}{E+m} \\
1 \\
0
\end{array}\right), \quad v^{(2)}=\left(\frac{E+m}{2 m}\right)^{1 / 2}\left(\begin{array}{c}
\frac{p_{-}}{E+m} \\
\frac{-p_{z}}{E+m} \\
0 \\
1
\end{array}\right)
\end{aligned}
u ( 1 ) = ( 2 m E + m ) 1/2 1 E + m p z E + m p + , u ( 2 ) = ( 2 m E + m ) 1/2 0 E + m p − E + m − p z v ( 1 ) = ( 2 m E + m ) 1/2 E + m p z E + m p + 1 0 , v ( 2 ) = ( 2 m E + m ) 1/2 E + m p − E + m − p z 0 1 where p ± = p x ± i p y . p_{\pm}=p_{x} \pm \mathrm{i} p_{y} . p ± = p x ± i p y . u u u
u ˉ ( 1 ) u ( 1 ) = ( E + m 2 m ) ( 1 0 p z E + m p − E + m ) ( 1 1 − 1 − 1 ) ( 1 p z E + m p + E + m ) = 1 \bar{u}^{(1)} u^{(1)}= \left(\frac{E+m}{2 m}\right)\left(\begin{array}{llll}1 & 0 & \frac{p_{z}}{E+m} & \frac{p_{-}}{E+m}\end{array}\right)\left(\begin{array}{cccc}1 & & & \\ & 1 & & \\ & & -1 & \\ & & & -1\end{array}\right)\left(\begin{array}{c}1 \\ \frac{p_{z}}{E+m} \\ \frac{p_{+}}{E+m}\end{array}\right)=1 u ˉ ( 1 ) u ( 1 ) = ( 2 m E + m ) ( 1 0 E + m p z E + m p − ) 1 1 − 1 − 1 1 E + m p z E + m p + = 1 ( E + m 2 m ) ( 1 0 p z E + m p − E + m ) ( 1 1 − 1 − 1 ) ( 1 0 E z p + m p + E + m ) = 1
\left(\frac{E+m}{2 m}\right)\left(\begin{array}{llll}
1 & 0 & \frac{p_{z}}{E+m} & \frac{p_{-}}{E+m}
\end{array}\right)\left(\begin{array}{cccc}
1 & & & \\
& 1 & & \\
& & -1 & \\
& & & -1
\end{array}\right)\left(\begin{array}{c}
1 \\
\frac{0}{E_{z}} \\
\frac{p+m}{\frac{p_{+}}{E+m}}
\end{array}\right)=1
( 2 m E + m ) ( 1 0 E + m p z E + m p − ) 1 1 − 1 − 1 1 E z 0 E + m p + p + m = 1 u ˉ ( α ) ( p ) u ( α ′ ) ( p ) = δ α α ′ v ˉ ( α ) ( p ) v ( α ′ ) ( p ) = − δ α α ′ u ˉ ( α ) ( p ) v ( α ′ ) ( p ) = 0 u ( α ) + ( p ) u ( α ′ ) ( p ) = v ( α ) + ( p ) v ( α ′ ) ( p ) = E m δ α α ′ .
\begin{aligned}
\bar{u}^{(\alpha)}(p) u^{\left(\alpha^{\prime}\right)}(p) &=\delta_{\alpha \alpha^{\prime}} \\
\bar{v}^{(\alpha)}(p) v^{\left(\alpha^{\prime}\right)}(p) &=-\delta_{\alpha \alpha^{\prime}} \\
\bar{u}^{(\alpha)}(p) v^{\left(\alpha^{\prime}\right)}(p) &=0 \\
u^{(\alpha)+}(p) u^{\left(\alpha^{\prime}\right)}(p) &=v^{(\alpha)+}(p) v^{\left(\alpha^{\prime}\right)}(p)=\frac{E}{m} \delta_{\alpha \alpha^{\prime}} .
\end{aligned}
u ˉ ( α ) ( p ) u ( α ′ ) ( p ) v ˉ ( α ) ( p ) v ( α ′ ) ( p ) u ˉ ( α ) ( p ) v ( α ′ ) ( p ) u ( α ) + ( p ) u ( α ′ ) ( p ) = δ α α ′ = − δ α α ′ = 0 = v ( α ) + ( p ) v ( α ′ ) ( p ) = m E δ α α ′ . In addition, from (2.96) and (2.136), u u u v v v
u ˉ ( p ) ( γ ⋅ p − m ) = 0 v ˉ ( p ) ( γ ⋅ p + m ) = 0
\begin{array}{r}\bar{u}(p)(\gamma \cdot p-m)=0 \\ \bar{v}(p)(\gamma \cdot p+m)=0\end{array}
u ˉ ( p ) ( γ ⋅ p − m ) = 0 v ˉ ( p ) ( γ ⋅ p + m ) = 0 Projection Operator# The operator m ( γ ) ( γ ⋅ p + m ) = 0. ) m(\gamma)(\gamma \cdot p+m)=0 .) m ( γ ) ( γ ⋅ p + m ) = 0. )
P + = ∑ α u ( α ) ( p ) u ˉ ( α ) ( p )
P_{+}=\sum_{\alpha} u^{(\alpha)}(p) \bar{u}^{(\alpha)}(p)
P + = α ∑ u ( α ) ( p ) u ˉ ( α ) ( p ) is important in many applications. It is a projection operator, since, in view of (2.139),
P + 2 = ∑ α . β u ( α ) ( p ) u ˉ ( α ) ( p ) u ( β ) ( p ) u ˉ ( β ) = ∑ α u ( α ) ( p ) u ˉ ( α ) ( p ) = P +
P_{+}^{2}=\sum_{\alpha . \beta} u^{(\alpha)}(p) \bar{u}^{(\alpha)}(p) u^{(\beta)}(p) \bar{u}^{(\beta)}=\sum_{\alpha} u^{(\alpha)}(p) \bar{u}^{(\alpha)}(p)=P_{+}
P + 2 = α . β ∑ u ( α ) ( p ) u ˉ ( α ) ( p ) u ( β ) ( p ) u ˉ ( β ) = α ∑ u ( α ) ( p ) u ˉ ( α ) ( p ) = P + ( γ ⋅ p − m ) P + = 0
(\gamma \cdot p-m) P_{+}=0
( γ ⋅ p − m ) P + = 0 γ ⋅ p m P + = P + .
\frac{\gamma \cdot p}{m} P_{+}=P_{+} .
m γ ⋅ p P + = P + . P + = ∑ α u ( α ) ( p ) u ˉ ( α ) ( p ) = γ ⋅ p + m 2 m .
P_{+}=\sum_{\alpha} u^{(\alpha)}(p) \bar{u}^{(\alpha)}(p)=\frac{\gamma \cdot p+m}{2 m} .
P + = α ∑ u ( α ) ( p ) u ˉ ( α ) ( p ) = 2 m γ ⋅ p + m . P − = − ∑ α v ( α ) ( p ) v ˉ ( α ) ( p ) = − γ ⋅ p + m 2 m .
P_{-}=-\sum_{\alpha} v^{(\alpha)}(p) \bar{v}^{(\alpha)}(p)=\frac{-\gamma \cdot p+m}{2 m} .
P − = − α ∑ v ( α ) ( p ) v ˉ ( α ) ( p ) = 2 m − γ ⋅ p + m . As expected, P + + P − = 1 P_{+}+P_{-}=1 P + + P − = 1
Tr ( γ ⋅ α ) ( γ ⋅ b ) = Tr ( γ ⋅ b ) ( γ ⋅ a ) = 1 2 Tr a μ b v { γ μ , γ v } = a ⋅ b Tr 1 = 4 a ⋅ b
\begin{aligned}
\operatorname{Tr}(\gamma \cdot \alpha)(\gamma \cdot b) &=\operatorname{Tr}(\gamma \cdot b)(\gamma \cdot a) \\
&=\frac{1}{2} \operatorname{Tr} a_{\mu} b_{v}\left\{\gamma^{\mu}, \gamma^{v}\right\} \\
&=a \cdot b \operatorname{Tr} 1=4 a \cdot b
\end{aligned}
Tr ( γ ⋅ α ) ( γ ⋅ b ) = Tr ( γ ⋅ b ) ( γ ⋅ a ) = 2 1 Tr a μ b v { γ μ , γ v } = a ⋅ b Tr 1 = 4 a ⋅ b ( γ 5 ) 2 = 1 , { γ 5 , γ μ } = 0
\left(\gamma^{5}\right)^{2}=1, \quad\left\{\gamma^{5}, \gamma^{\mu}\right\}=0
( γ 5 ) 2 = 1 , { γ 5 , γ μ } = 0 Tr d 1 … d n = 0 , n odd
\operatorname{Tr} d_{1} \ldots d_{n}=0, \quad n \text { odd }
Tr d 1 … d n = 0 , n odd proof by γ 5 \gamma^5 γ 5
Tr ( γ ⋅ a ) ( γ ⋅ b ) ( γ ⋅ c ) ( γ ⋅ d ) = − Tr ( γ ⋅ b ) ( γ ⋅ a ) ( γ ⋅ c ) ( γ ⋅ d ) + 2 a ⋅ b Tr ( γ ⋅ c ) ( γ ⋅ d )
\begin{aligned}
\operatorname{Tr}(\gamma \cdot a)(\gamma \cdot b)(\gamma \cdot c)(\gamma \cdot d)=&-\operatorname{Tr}(\gamma \cdot b)(\gamma \cdot a)(\gamma \cdot c)(\gamma \cdot d) \\
&+2 a \cdot b \operatorname{Tr}(\gamma \cdot c)(\gamma \cdot d)
\end{aligned}
Tr ( γ ⋅ a ) ( γ ⋅ b ) ( γ ⋅ c ) ( γ ⋅ d ) = − Tr ( γ ⋅ b ) ( γ ⋅ a ) ( γ ⋅ c ) ( γ ⋅ d ) + 2 a ⋅ b Tr ( γ ⋅ c ) ( γ ⋅ d ) Non-relativistic limit and the electron magnetic moment# p μ → p μ − e A μ
p^{\mu} \rightarrow p^{\mu}-e A^{\mu}
p μ → p μ − e A μ E → E − e ϕ , p → p − e A .
E \rightarrow E-e \phi, \quad \mathbf{p} \rightarrow \mathbf{p}-e \mathbf{A} .
E → E − e ϕ , p → p − e A . γ 0 ( E − e ϕ ) ψ − γ ⋅ ( p − e A ) ψ = m ψ .
\gamma^{0}(E-e \phi) \psi-\gamma \cdot(\mathbf{p}-e \mathbf{A}) \psi=m \psi .
γ 0 ( E − e ϕ ) ψ − γ ⋅ ( p − e A ) ψ = m ψ . γ 0 = ( 1 0 0 − 1 ) , γ = ( 0 σ − σ 0 ) , ψ = ( u v )
\gamma^{0}=\left(\begin{array}{rr}
1 & 0 \\
0 & -1
\end{array}\right), \quad \gamma=\left(\begin{array}{rr}
0 & \sigma \\
-\sigma & 0
\end{array}\right), \quad \psi=\left(\begin{array}{l}
u \\
v
\end{array}\right)
γ 0 = ( 1 0 0 − 1 ) , γ = ( 0 − σ σ 0 ) , ψ = ( u v ) ( E − e ϕ ) ( u − v ) − ( p − e A ) ⋅ ( 0 σ − σ 0 ) ( u v ) = m ( u v )
(E-e \phi)\left(\begin{array}{r}
u \\
-v
\end{array}\right)-(\mathbf{p}-e \mathbf{A}) \cdot\left(\begin{array}{rr}
0 & \sigma \\
-\sigma & 0
\end{array}\right)\left(\begin{array}{l}
u \\
v
\end{array}\right)=m\left(\begin{array}{l}
u \\
v
\end{array}\right)
( E − e ϕ ) ( u − v ) − ( p − e A ) ⋅ ( 0 − σ σ 0 ) ( u v ) = m ( u v ) ( E − e ϕ ) u − σ ⋅ ( p − e A ) v = m u ,
(E-e \phi) u-\sigma \cdot(\mathbf{p}-e \mathbf{A}) v=m u,
( E − e ϕ ) u − σ ⋅ ( p − e A ) v = m u , − ( E − e ϕ ) v + σ ⋅ ( p − e A ) u = m v
-(E-e \phi) v+\sigma \cdot(\mathbf{p}-e \mathbf{A}) u=m v
− ( E − e ϕ ) v + σ ⋅ ( p − e A ) u = m v v = ( E + m − e ϕ ) − 1 σ ⋅ ( p − e A ) u .
v=(E+m-e \phi)^{-1} \sigma \cdot(\mathbf{p}-e \mathbf{A}) u .
v = ( E + m − e ϕ ) − 1 σ ⋅ ( p − e A ) u . v ≈ 1 2 m σ ⋅ ( p − e A ) u = 0 ( v c ) u
v \approx \frac{1}{2 m} \boldsymbol{\sigma} \cdot(\mathbf{p}-\mathrm{eA}) u=0\left(\frac{v}{c}\right) u
v ≈ 2 m 1 σ ⋅ ( p − eA ) u = 0 ( c v ) u E u = σ ⋅ π σ ⋅ π 2 m u + m u + e ϕ u
E u=\frac{\boldsymbol{\sigma} \cdot \pi \boldsymbol{\sigma} \cdot \pi}{2 m} u+m u+e \phi u
E u = 2 m σ ⋅ π σ ⋅ π u + m u + e ϕ u W u = [ 1 2 m ( σ ⋅ π ) ( σ ⋅ π ) + e ϕ ] u .
W u=\left[\frac{1}{2 m}(\sigma \cdot \pi)(\sigma \cdot \pi)+e \phi\right] u .
W u = [ 2 m 1 ( σ ⋅ π ) ( σ ⋅ π ) + e ϕ ] u . ( σ ⋅ A ) ( σ ⋅ B ) = A ⋅ B + i σ ⋅ ( A × B )
(\boldsymbol{\sigma} \cdot \mathbf{A})(\boldsymbol{\sigma} \cdot \mathbf{B})=\mathbf{A} \cdot \mathbf{B}+\mathrm{i} \boldsymbol{\sigma} \cdot(\mathbf{A} \times \mathbf{B})
( σ ⋅ A ) ( σ ⋅ B ) = A ⋅ B + i σ ⋅ ( A × B ) ( σ ⋅ π ) 2 = π ⋅ π + i σ ⋅ ( π × π ) = ( p − e A ) 2 + i σ ⋅ ( p − e A ) × ( p − e A )
\begin{aligned}
(\boldsymbol{\sigma} \cdot \pi)^{2} &=\pi \cdot \pi+\mathrm{i} \boldsymbol{\sigma} \cdot(\pi \times \pi) \\
&=(\mathbf{p}-e \mathbf{A})^{2}+\mathrm{i} \boldsymbol{\sigma} \cdot(\mathbf{p}-e \mathbf{A}) \times(\mathbf{p}-e \mathbf{A})
\end{aligned}
( σ ⋅ π ) 2 = π ⋅ π + i σ ⋅ ( π × π ) = ( p − e A ) 2 + i σ ⋅ ( p − e A ) × ( p − e A ) p × A + A × p .
\mathbf{p} \times \mathbf{A}+\mathbf{A} \times \mathbf{p} .
p × A + A × p . [ p i , A j ] = − i ℏ ∂ i A j
\left[p_{i}, A_{j}\right]=-\mathrm{i} \hbar \partial_{i} A_{j}
[ p i , A j ] = − i ℏ ∂ i A j ( p i A j − p j A i ) + ( A i p j − A j p i ) = − i ℏ ( ∂ i A j − ∂ j A i ) .
\left(p_{i} A_{j}-p_{j} A_{i}\right)+\left(A_{i} p_{j}-A_{j} p_{i}\right)=-\mathrm{i} \hbar\left(\partial_{i} A_{j}-\partial_{j} A_{i}\right) .
( p i A j − p j A i ) + ( A i p j − A j p i ) = − i ℏ ( ∂ i A j − ∂ j A i ) . p × A + A × p = − i ℏ ∇ × A = − i ℏ B ,
\mathbf{p} \times \mathbf{A}+\mathbf{A} \times \mathbf{p}=-\mathrm{i} \hbar \nabla \times \mathbf{A}=-\mathrm{i} \hbar \mathbf{B},
p × A + A × p = − i ℏ∇ × A = − i ℏ B , H = 1 2 m ( p − e A ) 2 + e ϕ − e ℏ 2 m σ ⋅ B .
H=\frac{1}{2 m}(\mathbf{p}-e \mathbf{A})^{2}+e \phi-\frac{e \hbar}{2 m} \boldsymbol{\sigma} \cdot \mathbf{B} .
H = 2 m 1 ( p − e A ) 2 + e ϕ − 2 m e ℏ σ ⋅ B . 2.7 The relevance of the Poincare group: spin operators and the zero mass limit# Operator expression for angular momentum and boosts# x ′ = x cos θ + y sin θ y ′ = − x sin θ + y cos θ z ′ = z
\begin{aligned}
&x^{\prime}=x \cos \theta+y \sin \theta \\
&y^{\prime}=-x \sin \theta+y \cos \theta \\
&z^{\prime}=z
\end{aligned}
x ′ = x cos θ + y sin θ y ′ = − x sin θ + y cos θ z ′ = z J z f ( x , y , z ) = i lim θ → 0 [ f ( x ′ , y ′ , z ) − f ( x , y , z ) θ ] = i lim θ → 0 [ f ( x + y θ , y − x θ , z ) − f ( x , y , z ) θ ] = i ( y ∂ f ∂ x − x ∂ f ∂ y ) J z = − i ( x ∂ ∂ y − y ∂ ∂ x ) .
\begin{aligned}
J_{z} f(x, y, z) &=\mathrm{i} \lim _{\theta \rightarrow 0}\left[\frac{f\left(x^{\prime}, y^{\prime}, z\right)-f(x, y, z)}{\theta}\right] \\
&=\mathrm{i} \lim _{\theta \rightarrow 0}\left[\frac{f(x+y \theta, y-x \theta, z)-f(x, y, z)}{\theta}\right] \\
&=\mathrm{i}\left(y \frac{\partial f}{\partial x}-x \frac{\partial f}{\partial y}\right) \\
J_{z} &=-\mathrm{i}\left(x \frac{\partial}{\partial y}-y \frac{\partial}{\partial x}\right) .
\end{aligned}
J z f ( x , y , z ) J z = i θ → 0 lim [ θ f ( x ′ , y ′ , z ) − f ( x , y , z ) ] = i θ → 0 lim [ θ f ( x + y θ , y − x θ , z ) − f ( x , y , z ) ] = i ( y ∂ x ∂ f − x ∂ y ∂ f ) = − i ( x ∂ y ∂ − y ∂ x ∂ ) . J x = − i ( y ∂ ∂ z − z ∂ ∂ y ) , J y = − i ( z ∂ ∂ x − x ∂ ∂ z )
J_{x}=-\mathrm{i}\left(y \frac{\partial}{\partial z}-z \frac{\partial}{\partial y}\right), \quad J_{y}=-\mathrm{i}\left(z \frac{\partial}{\partial x}-x \frac{\partial}{\partial z}\right)
J x = − i ( y ∂ z ∂ − z ∂ y ∂ ) , J y = − i ( z ∂ x ∂ − x ∂ z ∂ ) and we can easily prove that
[ J x , J y ] = i J z \left[J_{x}, J_{y}\right]=\mathrm{i} J_{z} \quad [ J x , J y ] = i J z
Definition of generator
X α = i ( ∂ x ′ ∂ a α ∣ a = 0 ∂ ∂ x + ∂ y ′ ∂ a α ∣ a = 0 ∂ ∂ y + ∂ z ′ ∂ a α ∣ a = 0 ∂ ∂ z + ∂ t ′ ∂ a α ∣ a = 0 ∂ ∂ t ) = i ∂ x ′ μ ∂ a α ∂ ∂ x μ ( α = 1 , … , r )
\begin{aligned}
X_{\alpha} &=\mathrm{i}\left(\left.\frac{\partial x^{\prime}}{\partial a^{\alpha}}\right|_{a=0} \frac{\partial}{\partial x}+\left.\frac{\partial y^{\prime}}{\partial a^{\alpha}}\right|_{a=0} \frac{\partial}{\partial y}+\left.\frac{\partial z^{\prime}}{\partial a^{\alpha}}\right|_{a=0} \frac{\partial}{\partial z}+\left.\frac{\partial t^{\prime}}{\partial a^{\alpha}}\right|_{a=0} \frac{\partial}{\partial t}\right) \\
&=\mathrm{i} \frac{\partial x^{\prime \mu}}{\partial a^{\alpha}} \frac{\partial}{\partial x^{\mu}}(\alpha=1, \ldots, r)
\end{aligned}
X α = i ( ∂ a α ∂ x ′ a = 0 ∂ x ∂ + ∂ a α ∂ y ′ a = 0 ∂ y ∂ + ∂ a α ∂ z ′ a = 0 ∂ z ∂ + ∂ a α ∂ t ′ a = 0 ∂ t ∂ ) = i ∂ a α ∂ x ′ μ ∂ x μ ∂ ( α = 1 , … , r ) K x = i ( t ∂ ∂ x + x ∂ ∂ t )
K_{x}=\mathrm{i}\left(t \frac{\partial}{\partial x}+x \frac{\partial}{\partial t}\right)
K x = i ( t ∂ x ∂ + x ∂ t ∂ ) K y = i ( t ∂ ∂ y + y ∂ ∂ t ) , K z = i ( t ∂ ∂ z + z ∂ ∂ t )
K_{y}=\mathrm{i}\left(t \frac{\partial}{\partial y}+y \frac{\partial}{\partial t}\right), \quad K_{z}=i\left(t \frac{\partial}{\partial z}+z \frac{\partial}{\partial t}\right)
K y = i ( t ∂ y ∂ + y ∂ t ∂ ) , K z = i ( t ∂ z ∂ + z ∂ t ∂ ) [ K x , K y ] = − i J z and cyclic perms, [ K x , J y ] = i K z and cyclic perms, [ K x , J x ] = 0 , etc., }
\left.\begin{array}{rl}
{\left[K_{x}, K_{y}\right]=-\mathrm{i} J_{z}} & \text { and cyclic perms, } \\
{\left[K_{x}, J_{y}\right]=\mathrm{i} K_{z}} & \text { and cyclic perms, } \\
{\left[K_{x}, J_{x}\right]=0,} & \text { etc., }
\end{array}\right\}
[ K x , K y ] = − i J z [ K x , J y ] = i K z [ K x , J x ] = 0 , and cyclic perms, and cyclic perms, etc., ⎭ ⎬ ⎫ 3.2 The real scalar field: variational principle and Noether’s theorem# Euler-Lagrange Equation# ∂ L ∂ ϕ − ∂ ∂ x μ [ ∂ L ∂ ( ∂ μ ϕ ) ] = 0
\frac{\partial \mathscr{L}}{\partial \phi}-\frac{\partial}{\partial x^{\mu}}\left[\frac{\partial \mathscr{L}}{\partial\left(\partial_{\mu} \phi\right)}\right]=0
∂ ϕ ∂ L − ∂ x μ ∂ [ ∂ ( ∂ μ ϕ ) ∂ L ] = 0 Lagrangian for Klein-Gordon Equation# L = 1 2 g κ λ ( ∂ κ ϕ ) ( ∂ λ ϕ ) − m 2 2 ϕ 2
\mathscr{L}=\frac{1}{2} g^{\kappa \lambda}\left(\partial_{\kappa} \phi\right)\left(\partial_{\lambda} \phi\right)-\frac{m^{2}}{2} \phi^{2}
L = 2 1 g κλ ( ∂ κ ϕ ) ( ∂ λ ϕ ) − 2 m 2 ϕ 2 This equation is written deliberately to give the Klein-Gordon Equation
Symmetry and conserved quantity(Noether’s theorem), There’s another type of conserved quantity which is topological in nature, and whose conservation has nothing to do with Nother’s theorem# Energy-Momentum tensor# 3.3 Complex scalar fields and the electromagnetic field# Conservation of charge from noether’s theorem# ϕ = ( ϕ 1 + i ϕ 2 ) / 2 ϕ ∗ = ( ϕ 1 − i ϕ 2 ) / 2
\begin{aligned}
\phi &=\left(\phi_{1}+\mathrm{i} \phi_{2}\right) / \sqrt{2} \\
\phi^{*} &=\left(\phi_{1}-\mathrm{i} \phi_{2}\right) / \sqrt{2}
\end{aligned}
ϕ ϕ ∗ = ( ϕ 1 + i ϕ 2 ) / 2 = ( ϕ 1 − i ϕ 2 ) / 2 L = ( ∂ μ ϕ ) ( ∂ μ ϕ ∗ ) − m 2 ϕ ∗ ϕ
\mathscr{L}=\left(\partial_{\mu} \phi\right)\left(\partial^{\mu} \phi^{*}\right)-m^{2} \phi^{*} \phi
L = ( ∂ μ ϕ ) ( ∂ μ ϕ ∗ ) − m 2 ϕ ∗ ϕ ( □ + m 2 ) ϕ = 0 ( □ + m 2 ) ϕ ∗ = 0
\begin{gathered}
\left(\square+m^{2}\right) \phi=0 \\
\left(\square+m^{2}\right) \phi^{*}=0
\end{gathered}
( □ + m 2 ) ϕ = 0 ( □ + m 2 ) ϕ ∗ = 0 Symmetry
ϕ → e − i λ ϕ , ϕ ∗ → e i Λ ϕ ∗
\phi \rightarrow \mathrm{e}^{-\mathrm{i} \lambda} \phi, \quad \phi^{*} \rightarrow \mathrm{e}^{\mathrm{i} \Lambda} \phi^{*}
ϕ → e − i λ ϕ , ϕ ∗ → e i Λ ϕ ∗ δ ϕ = − i Λ ϕ , δ ϕ ∗ = i Λ ϕ ∗
\delta \phi=-\mathrm{i} \Lambda \phi, \quad \delta \phi^{*}=\mathrm{i} \Lambda \phi^{*}
δ ϕ = − i Λ ϕ , δ ϕ ∗ = i Λ ϕ ∗ δ ( ∂ μ ϕ ) = − i Λ ∂ μ ϕ , δ ( ∂ μ ϕ ∗ ) = i Λ ∂ μ ϕ ∗
\delta\left(\partial_{\mu} \phi\right)=-\mathrm{i} \Lambda \partial_{\mu} \phi, \quad \delta\left(\partial_{\mu} \phi^{*}\right)=\mathrm{i} \Lambda \partial_{\mu} \phi^{*}
δ ( ∂ μ ϕ ) = − i Λ ∂ μ ϕ , δ ( ∂ μ ϕ ∗ ) = i Λ ∂ μ ϕ ∗ Φ = − i ϕ , Φ ∗ = i ϕ ∗ , X = 0.
\Phi=-\mathrm{i} \phi, \quad \Phi^{*}=\mathrm{i} \phi^{*}, \quad X=0 .
Φ = − i ϕ , Φ ∗ = i ϕ ∗ , X = 0. J μ = ∂ L ∂ ( ∂ μ ϕ ) ( − i ϕ ) + ∂ L ∂ ( ∂ μ ϕ ∗ ) ( i ϕ ∗ )
J^{\mu}=\frac{\partial \mathscr{L}}{\partial\left(\partial_{\mu} \phi\right)}(-\mathrm{i} \phi)+\frac{\partial \mathscr{L}}{\partial\left(\partial_{\mu} \phi^{*}\right)}\left(\mathrm{i} \phi^{*}\right)
J μ = ∂ ( ∂ μ ϕ ) ∂ L ( − i ϕ ) + ∂ ( ∂ μ ϕ ∗ ) ∂ L ( i ϕ ∗ ) J μ = i ( ϕ ∗ ∂ μ ϕ − ϕ ∂ μ ϕ ∗ )
J^{\mu}=\mathrm{i}\left(\phi^{*} \partial^{\mu} \phi-\phi \partial^{\mu} \phi^{*}\right)
J μ = i ( ϕ ∗ ∂ μ ϕ − ϕ ∂ μ ϕ ∗ ) ∂ μ J μ = 0 ,
\partial_{\mu} J^{\mu}=0,
∂ μ J μ = 0 , Q = ∫ J 0 d V = i ∫ ( ϕ ∗ ∂ ϕ ∂ t − ϕ ∂ ϕ ∗ ∂ t ) d V
\begin{aligned}
Q &=\int J^{0} \mathrm{~d} V \\
&=\mathrm{i} \int\left(\phi^{*} \frac{\partial \phi}{\partial t}-\phi \frac{\partial \phi^{*}}{\partial t}\right) \mathrm{d} V
\end{aligned}
Q = ∫ J 0 d V = i ∫ ( ϕ ∗ ∂ t ∂ ϕ − ϕ ∂ t ∂ ϕ ∗ ) d V L = ( ∂ μ ϕ 1 ) ( ∂ μ ϕ 1 ) + ( ∂ μ ϕ 2 ) ( ∂ μ ϕ 2 ) − m 2 ( ϕ 1 2 + ϕ 2 2 )
\mathscr{L}=\left(\partial_{\mu} \phi_{1}\right)\left(\partial^{\mu} \phi_{1}\right)+\left(\partial_{\mu} \phi_{2}\right)\left(\partial^{\mu} \phi_{2}\right)-m^{2}\left(\phi_{1}^{2}+\phi_{2}^{2}\right)
L = ( ∂ μ ϕ 1 ) ( ∂ μ ϕ 1 ) + ( ∂ μ ϕ 2 ) ( ∂ μ ϕ 2 ) − m 2 ( ϕ 1 2 + ϕ 2 2 ) ϕ = i ϕ 1 + j ϕ 2
\phi=\mathrm{i} \phi_{1}+\mathrm{j} \phi_{2}
ϕ = i ϕ 1 + j ϕ 2 ϕ = i ϕ 1 + j ϕ 2
\phi=\mathbf{i} \phi_{1}+\mathbf{j} \phi_{2}
ϕ = i ϕ 1 + j ϕ 2 L = ( ∂ μ ϕ ) ⋅ ( ∂ μ ϕ ) − m 2 ϕ ⋅ ϕ .
\mathscr{L}=\left(\partial_{\mu} \boldsymbol{\phi}\right) \cdot\left(\partial^{\mu} \boldsymbol{\phi}\right)-m^{2} \boldsymbol{\phi} \cdot \boldsymbol{\phi} .
L = ( ∂ μ ϕ ) ⋅ ( ∂ μ ϕ ) − m 2 ϕ ⋅ ϕ . ϕ 1 ′ + i ϕ 2 ′ = e − i Λ ( ϕ 1 + i ϕ 2 ) , ϕ 1 ′ − i ϕ 2 ′ = e i Λ ( ϕ 1 − i ϕ 2 ) ,
\begin{aligned}
&\phi_{1}^{\prime}+\mathrm{i} \phi_{2}^{\prime}=\mathrm{e}^{-\mathrm{i} \Lambda}\left(\phi_{1}+\mathrm{i} \phi_{2}\right), \\
&\phi_{1}^{\prime}-\mathrm{i} \phi_{2}^{\prime}=\mathrm{e}^{\mathrm{i} \Lambda}\left(\phi_{1}-\mathrm{i} \phi_{2}\right),
\end{aligned}
ϕ 1 ′ + i ϕ 2 ′ = e − i Λ ( ϕ 1 + i ϕ 2 ) , ϕ 1 ′ − i ϕ 2 ′ = e i Λ ( ϕ 1 − i ϕ 2 ) , ϕ 1 ′ = ϕ 1 cos Λ + ϕ 2 sin Λ ϕ 2 ′ = − ϕ 1 sin Λ + ϕ 2 cos Λ . }
\left.\begin{array}{l}
\phi_{1}^{\prime}=\phi_{1} \cos \Lambda+\phi_{2} \sin \Lambda \\
\phi_{2}^{\prime}=-\phi_{1} \sin \Lambda+\phi_{2} \cos \Lambda .
\end{array}\right\}
ϕ 1 ′ = ϕ 1 cos Λ + ϕ 2 sin Λ ϕ 2 ′ = − ϕ 1 sin Λ + ϕ 2 cos Λ. } Rotations in two dimensions form the group of S O ( 2 ) SO(2) SO ( 2 )
e i Λ ( e i Λ ) ∗ = 1
e^{i\Lambda}(e^{i\Lambda})^*=1
e i Λ ( e i Λ ) ∗ = 1 Then S O ( 2 ) ≈ U ( 1 ) SO(2)\approx U(1) SO ( 2 ) ≈ U ( 1 )
Since Λ \Lambda Λ Λ ( x μ ) \Lambda(x^\mu) Λ ( x μ )