1 fermions s 21 fermions s 2 ... i,
TRANSCRIPT
The Lorentz group i i J Rotations B osts Ko
i
i
i
[ , ]
[ , ]
[ , ]
j ijk k
j ijk k
j ijk k
J J i JJ K i KK K i J
εεε
=
=
= −} Generate the group SO(3,1)
102( ( ) )i ijk jk i ix x
M i x x J M K Mσ ρρσ ρ σ ε∂ ∂∂ ∂
= − = =
To construct representa;ons a more convenient (non-‐Hermi;an) basis is
12 ( )i i iN J iK= +
i
† † †i j k
†i j
[ , ]
[ , ]
[ , ] 0
j ijk k
ijk
N N i N
N N i N
N N
ε
ε
=
=
=} (2) (2)SU SU⊗ ( , )representa on ni mt
The Lorentz group i i J Rotations B osts Ko
i
i
i
[ , ]
[ , ]
[ , ]
j ijk k
j ijk k
j ijk k
J J i JJ K i KK K i J
εεε
=
=
= −} Generate the group SO(3,1)
102( ( ) )i ijk jk i ix x
M i x x J M K Mσ ρρσ ρ σ ε∂ ∂∂ ∂
= − = =
To construct representa;ons a more convenient (non-‐Hermi;an) basis is
12 ( )i i iN J iK= +
i
† † †i j k
†i j
[ , ]
[ , ]
[ , ] 0
j ijk k
ijk
N N i N
N N i N
N N
ε
ε
=
=
=
†i i iJ N N= +Representa;ons
( , )n m J n m= +
(0,0) scalar J=0( 1
2 ,0), (0, 12 ) LH and RH "spinors" J= 1
2
( 12 , 1
2 ) vector J=1, etc SU (2)L × SU (2)R
Weyl spinors 1 12 2
R
( ,0) (0, ) Lψ ψ
2-‐component spinors of SU(2)
Rota;ons and Boosts ( ) ( ) ( )L R L R L RSψ ψ→
SL( R) = eiα2
.σ : Rotations
SL( R) = e∓ν2
.σ : Boosts
σ 1 =
0 11 0
⎛
⎝⎜⎞
⎠⎟, σ 2 =
0 −ii 0
⎛
⎝⎜⎞
⎠⎟, σ 3 =
1 00 −1
⎛
⎝⎜⎞
⎠⎟
Weyl spinors 1 12 2
R
( ,0) (0, ) Lψ ψ
2-‐component spinors of SU(2)
( ) ( ) ( )L R L R L RSψ ψ→
SL( R) = eiα2
.σ : Rotations
SL( R) = e∓ν2
.σ : Boosts
†
Dirac spinor
Can combine R, Lψ ψ to form a 4-‐component “Dirac” spinor L
R
=ψ
ψψ⎡ ⎤⎢ ⎥⎣ ⎦
2, ,i ie ωσ µν µνµν µ νψ ψ ωσ ω σ γ γ ω⎡ ⎤→ = = ⎣ ⎦Lorentz transforma;ons
Weyl basis 0 , , 1, 2,3i ijboosts rotations i jω ω→ → =
†
where
γ 0 =0 II 0
⎡
⎣⎢
⎤
⎦⎥ , γ i =
0 −σ i
σ i o
⎡
⎣⎢⎢
⎤
⎦⎥⎥, γ 5 = iγ 0γ 1γ 2γ 3 = − I 0
0 I
⎡
⎣⎢
⎤
⎦⎥
Rota;ons and Boosts
Weyl spinors 1 12 2
R
( ,0) (0, ) Lψ ψ
2-‐component spinors of SU(2)
Rota;ons and Boosts ( ) ( ) ( )L R L R L RSψ ψ→
SL( R) = eiα2
.σ : Rotations
SL( R) = e∓ν2
.σ : BoostsDirac spinor
Can combine R, Lψ ψ to form a 4-‐component “Dirac” spinor L
R
=ψ
ψψ⎡ ⎤⎢ ⎥⎣ ⎦
2, ,i ie ωσ µν µνµν µ νψ ψ ωσ ω σ γ γ ω⎡ ⎤→ = = ⎣ ⎦Lorentz transforma;ons
†
(Dirac gamma matrices …new 4-‐vector ) µγ
{ }, 2gµ ν µ ν ν µ µνγ γ γ γ γ γ= + = Clifford Algebra
where
γ 0 =0 II 0
⎡
⎣⎢
⎤
⎦⎥ , γ i =
0 −σ i
σ i o
⎡
⎣⎢⎢
⎤
⎦⎥⎥, γ 5 = iγ 0γ 1γ 2γ 3 = − I 0
0 I
⎡
⎣⎢
⎤
⎦⎥
Weyl spinors 1 12 2
R
( ,0) (0, ) Lψ ψ
2-‐component spinors of SU(2)
Rota;ons and Boosts ( ) ( ) ( )L R L R L RSψ ψ→
SL( R) = eiα2
.σ : Rotations
SL( R) = e∓ν2
.σ : BoostsDirac spinor
Can combine R, Lψ ψ to form a 4-‐component “Dirac” spinor L
R
=ψ
ψψ⎡ ⎤⎢ ⎥⎣ ⎦
2, ,i ie ωσ µν µνµν µ νψ ψ ωσ ω σ γ γ ω⎡ ⎤→ = = ⎣ ⎦Lorentz transforma;ons
where
γ 0 =0 II 0
⎡
⎣⎢
⎤
⎦⎥ , γ i =
0 −σ i
σ i o
⎡
⎣⎢⎢
⎤
⎦⎥⎥, γ 5 = iγ 0γ 1γ 2γ 3 = − I 0
0 I
⎡
⎣⎢
⎤
⎦⎥
†
Note : ψ L( R) =
12 (1∓ γ 5)ψ{ }, 2gµ ν µ ν ν µ µνγ γ γ γ γ γ= + =
(Dirac gamma matrices …new 4-‐vector ) µγ
The Dirac equa;on Fermions described by 4-‐cpt Dirac spinors
ψ†γ 0ψ ≡ψψ =ψ L
†ψ R +ψ R†ψ L Lorentz invariant •
SL( R) = eiα2
.σ : Rotations
SL( R) = e∓ν2
.σ : Boosts
L
R
=ψ
ψψ⎡ ⎤⎢ ⎥⎣ ⎦
2× 2 = 3+1
The Dirac equa;on Fermions described by 4-‐cpt Dirac spinors
ψ†γ 0ψ ≡ψψ =ψ L
†ψ R +ψ R†ψ L Lorentz invariant •
µγNew 4-‐vector •
L
R
=ψ
ψψ⎡ ⎤⎢ ⎥⎣ ⎦
The Dirac equa;on Fermions described by 4-‐cpt Dirac spinors
ψ†γ 0ψ ≡ψψ =ψ L
†ψ R +ψ R†ψ L Lorentz invariant •
L = -iψ γ µ ∂
µ ψ − mψ ψ
µγNew 4-‐vector
Lorentz invariant Lagrangian
•
gµν = Diag(1,−1,−1,−1) (= −gµνSrednicki !)
L
R
=ψ
ψψ⎡ ⎤⎢ ⎥⎣ ⎦
⇒
2× 2 = 3+1
The Dirac equa;on Fermions described by 4-‐cpt Dirac spinors
ψ†γ 0ψ ≡ψψ =ψ L
†ψ R +ψ R†ψ L Lorentz invariant •
L = -iψ γ µ ∂
µ ψ − mψ ψ
µγNew 4-‐vector
From Euler Lagrange equa;on obtain the Dirac equa;on
(−iγ µ ∂
µ− m)ψ = 0
Lorentz invariant Lagrangian
•
δS = 0 ⇒
∂L∂φ
− ∂µ ∂L∂(∂µφ)
= 0 ⎛⎝⎜
⎞⎠⎟
Euler Lagrange equa;on
gµν = Diag(1,−1,−1,−1) (= −gµνSrednicki !)
L
R
=ψ
ψψ⎡ ⎤⎢ ⎥⎣ ⎦
⇒ 2× 2 = 3+1
The Dirac equa;on Fermions described by 4-‐cpt Dirac spinors ψ
ψ†γ 0ψ ≡ψψ =ψ L
†ψ R +ψ R†ψ L Lorentz invariant •
L = -iψ γ µ ∂
µ ψ − mψ ψ
µγNew 4-‐vector
From Euler Lagrange equa;on obtain the Dirac equa;on
(−iγ µ ∂
µ− m)ψ ≡ (−i ∂ − m)ψ = 0
The Lagrangian
•
gµν = Diag(1,−1,−1,−1)( )
pµ = (m,0,0,0), γ 0 −1( )ψ = −1 1
1 −1⎛
⎝⎜⎞
⎠⎟ψ = 0 2-‐components projected out
The Dirac equa;on Fermions described by 4-‐cpt Dirac spinors ψ
ψ†γ 0ψ ≡ψψ =ψ L
†ψ R +ψ R†ψ L Lorentz invariant •
L = -iψ γ µ ∂
µ ψ − mψ ψ
µγNew 4-‐vector
From Euler Lagrange equa;on obtain the Dirac equa;on
(−iγ µ ∂
µ− m)ψ ≡ (i ∂ − m)ψ = 0
The Lagrangian
•
Momentum components off shell Projected out
gµν = Diag(1,−1,−1,−1)( )
−iγ µ ∂µ + m( ) −iγ µ ∂
µ − m( )ψ = 0
⇒ ∂2+ m2( )ψ = 0
pµ = (m,0,0,0), γ 0 −1( )ψ = −1 1
1 −1⎛
⎝⎜⎞
⎠⎟ψ = 0 2-‐components projected out
The Dirac equa;on
L = -iψ γ µ ∂
µ ψ − mψ ψ
µγ
Fermions described by 4-‐cpt Dirac spinors ψ
New 4-‐vector
The Lagrangian
•
ψ γ µ ∂µ ψ =ψ †γ 0(γ 0 ∂
0−γ i ∂i )ψ =ψ † I 0
0 I⎛
⎝⎜⎞
⎠⎟∂0− 0 I
I 0⎛
⎝⎜⎞
⎠⎟0 −σ i
σ i 0
⎛
⎝⎜⎜
⎞
⎠⎟⎟∂i
⎛
⎝⎜⎜
⎞
⎠⎟⎟ψ
=ψ † I 00 I
⎛
⎝⎜⎞
⎠⎟∂0−
σ i 0
0 −σ i
⎛
⎝⎜⎜
⎞
⎠⎟⎟∂i
⎛
⎝⎜⎜
⎞
⎠⎟⎟ψ ≡ψ L
†σµ∂µψ L +ψ R
†σ µ ∂µψ R
σ µ = 1,σ i( ), σµ= 1,−σ i( )
ψ†γ 0ψ ≡ψψ =ψ L
†ψ R +ψ R†ψ L Lorentz invariant •
ψ L ≡σ 2ψ R
* ∼ (12
,0)
Can construct LH spinors out of RH an;spinors and vice-‐versa
Proof : ψ L → eσ!"
2.νψ L ?
ψ R → e−σ!"
2.νψ R
σ 2ψ R* →σ 2e
−σ!"*
2.ν"
ψ R* = σ 2e
−σ!"*
2.ν"
σ 2σ 2ψ R* = e
σ!"
2.ν"
σ 2ψ R* using σ 2σ i
*σ 2 = −σ i
ψ R ≡σ 2ψ L
* ∼ (0, 12
)
Majorana fermions
ψ Maj =ψ L
ψ R =σ 2ψ L*
⎡
⎣⎢⎢
⎤
⎦⎥⎥
only 2 independent components
Majorana mass m(ψ L†ψ R +ψ R
†ψ L ) → M ψ Ltσ 2ψ L ≡ M ψ! Lψ L
Parity
i i i iJ J K K→ →−
0 0( , ) ( , )i ix x x x xµ = → −
i i iN J iK= +† †i i i iN N N N→ →
L R0
R L
0=
0I
Iψ ψ
ψ ψ γ ψψ ψ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
→ = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠
i.e. ψ L†σ
µψ L ↔ψ R
†σµψ R
∂0 → ∂0 , ∂i → −∂i σµ = (1,σ i ), σ
µ= (1,−σ i )
i.e. ψ L†σ
µ∂µψ L ↔ψ R
†σ µ ∂µψ R
LKinetic=ψ L†σ
µ∂µψ L +ψ R
†σ µ ∂µψ R preserves parity: (neutrinos violate parity)
Charge conjuga;on
LKinetic=ψ L†σ
µ∂µψ L +ψ R
†σ µ ∂µψ R
invariant under ψ L →σ 2ψ R* , ψ R → −σ 2ψ L
* (σ 2σµσ 2 = σ 2
µT)
LEM =Q ψ L†σ
µAµψ L +ψ R
†σ µAµψ R( )EM and QCD interac;ons preserve charge conjuga;on:
provided QAµ → −QAµ
Weak interac;ons do not preserve charge conjuga;on:
Rela;on to Srednicki nota;on
(2,1) ≡ψ a
gµν = Diag(−1,1,1,1( )
i
† † †i j k
†i j
[ , ]
[ , ]
[ , ] 0
j ijk k
ijk
N N i N
N N i N
N N
ε
ε
=
=
= Hermitian conjugation N ↔ N †
ψ R†ψ L ≡ψ L
Tσ 2ψ L →−iψ aεabψ b
ε12 = −ε 21 = 1 (c. f .gµν )
Rela;on to Srednicki nota;on
(2,1) ≡ψ a , (1,2) = ψ a[ ]† ≡ψ !a†
gµν = Diag(−1,1,1,1( )
i
† † †i j k
†i j
[ , ]
[ , ]
[ , ] 0
j ijk k
ijk
N N i N
N N i N
N N
ε
ε
=
=
= Hermitian conjugation N ↔ N †
Dot denotes SU(2)R index
Conven;on: RH (do^ed) fields always ψ†
ψ R =σ 2ψ L* ≡ψ † !a = ε !a !bψ !b
† ψ L ≡ψ a
Rela;on to Srednicki nota;on
Invariants :
ψ L†χR = −iε !a !bψ !a
†χ !b†
ψ R†χL = σ 2ψ L
*( )† χL =ψ LTσ 2χL = −iε abψ bχa
ψ L
†σµ∂µψ L +ψ R
†σ µ∂µψ R ≡ ξaσ µa !c∂µξ
† !c + χ !a†σ
µ !ac∂µχc
= χ †σµ∂µχ + ξ†σ
µ∂µξ + ∂µ ξσ µξ†( )
ψ =ψ L
ψ R
⎛
⎝⎜
⎞
⎠⎟ ≡
χc
ξ† !c⎛
⎝⎜⎜
⎞
⎠⎟⎟
gµν = Diag(−1,1,1,1( )
ψ Lα ≡ψ a , ψ Rα ≡ψ † !a
ψaχa ≡ψ χ
( ),µγ β βα=
γ 0 =0 II 0
⎡
⎣⎢
⎤
⎦⎥ , γ i =
0 −σ i
σ i o
⎡
⎣⎢⎢
⎤
⎦⎥⎥, γ 5 = iγ 0γ 1γ 2γ 3 = − I 0
0 I
⎡
⎣⎢
⎤
⎦⎥
0
0 00 0i
II
σγ γ
σ⎛ ⎞ ⎛ ⎞
= =⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠
Dirac-‐Pauli basis
Weyl basis
0 00 0
II
σα β
σ⎛ ⎞ ⎛ ⎞
= =⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠
0 1 2 35
00I
iI
γ γ γ γ γ ⎡ ⎤= = ⎢ ⎥
⎣ ⎦
α =
σ 00 −σ
⎛
⎝⎜
⎞
⎠⎟ β = 0 I
I 0⎛
⎝⎜⎞
⎠⎟
{ }, 2gµ ν µ ν ν µ µνγ γ γ γ γ γ= + = gµν = Diag(1,−1,−1,−1)( )
Free par;cle solu;on
−iγ µ ∂
µ− m( )ψ = 0 . ( )ip xe u pψ −=
γ µ pµ − m( )u( p) ≡ p − m( )u( p) = 0
α .P + βm( )u = Eu
.
.A A
B B
mI p u uE
u up mI
σ
σ⎛ ⎞⎛ ⎞ ⎛ ⎞
=⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟− ⎝ ⎠ ⎝ ⎠⎝ ⎠
. ( )
. ( )B A
A B
p u E m u
p u E m u
σ
σ
= −
= +
0 00 0
II
σα β
σ⎛ ⎞ ⎛ ⎞
= =⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠
Dirac Pauli basis -‐ c.f. Chapter 38 Srednicki for Weyl basis solu;on
( ),µγ β βα=
i.e.
For the 2 E>0 solu;ons , we may take (1) (2)1 0
0 1χ χ⎛ ⎞ ⎛ ⎞
= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
( ) ( ) ,s sAu χ=
( ) ( ).s sB
pu
E mσ
χ=+
Posi;ve energy 4-‐spinor solu;ons of Dirac’s equa;on
( )
( )( ), 0, 1, 2.
s
ss
u N E spE m
χσ
χ
⎛ ⎞⎜ ⎟= > =⎜ ⎟⎜ ⎟+⎝ ⎠
. ( )
. ( )B A
A B
p u E m u
p u E m u
σ
σ
= −
= +
0,If p E m= = +
( ) 0 0B BE m u u+ = ⇒ =
For the 2 E<0 solu;ons , we may take ( ) ( ) ,s sBu χ=
( ) ( ) ( ). .s s sA
p pu
E m E mσ σ
χ χ= = −− +
u(s+2) = N '−
σ .pE + m
χ (s)
χ (s)
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
, E < 0, s = 1,2
Orthonormal states ( )† ( ) 0,r su u r s= ≠
Non-‐rela;vis;c correspondance
( ) ( ) ( ) ( )2 2 2 2/ / / /(1) (2) (3) (4)
1 0 0 00 1 0 0, , ,
0 0 1 00 0 0 1
imc t imc t imc t imc te e e eψ ψ ψ ψ− − + +
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟= = = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
! ! ! !
Nega;ve energy 4-‐spinor solu;ons of Dirac’s equa;on
. ( )
. ( )B A
A B
p u E m u
p u E m u
σ
σ
= −
= +
Spin – for state at rest
( ) ( ) ( ) ( )2 2 2 2/ / / /(1) (2) (3) (4)
1 0 0 00 1 0 0, , ,
0 0 1 00 0 0 1
imc t imc t imc t imc te e e eψ ψ ψ ψ− − + +
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟= = = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
! ! ! !
Since we have a (two-‐fold) degeneracy there must be some operator which commutes with the energy operator and whose eigenvalues label the two states
33
3
1 0 0 00 1 0 000 0 1 000 0 0 1
σσ
⎛ ⎞⎜ ⎟−⎛ ⎞ ⎜ ⎟Σ ≡ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎜ ⎟⎜ ⎟−⎝ ⎠
3 (1,2) (1,2)Eigenvalues 1 ψ ψ± Σ = ±
0p =
00σ
σ⎛ ⎞
Σ ≡ ⎜ ⎟⎝ ⎠
( )2 231 1 12 4 2 2, has eigenvalues IΣ = Σ ±! ! ! !
1 12 2 is spin operator S corresponding to S⇒ Σ =!
Spin – for state NOT at rest 0p ≠
[ ] [ ]1 1 12 2 2 no longer a commuting observable , , . 0H P mα βΣ Σ = Σ + ≠! ! !
Helicity
!!
!!. 01. ,
2 0 .
p pp p
pp
σ
σ
⎛ ⎞⎜ ⎟Σ ≡ =⎜ ⎟⎝ ⎠"
0 00 0
II
σα β
σ⎛ ⎞ ⎛ ⎞
= =⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠
Eigenvalues 12
+ !p
s
12
− !p
s
(More generally, in arbitrary frame, spin given by boos;ng result at rest -‐ (0, ) ' )s s s sµ µ µ νν= ⇒ = Λ