1 fermions s 21 fermions s 2 ... i,

Fermions S = 1

2

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


( ) ( ) ( )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



SL( R) = eiα2

.σ : Rotations

SL( R) = e∓ν2

.σ : BoostsDirac spinor


R

=ψ

ψψ⎡ ⎤⎢ ⎥⎣ ⎦


†

(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



SL( R) = eiα2

.σ : Rotations

SL( R) = e∓ν2

.σ : BoostsDirac spinor


R

=ψ

ψψ⎡ ⎤⎢ ⎥⎣ ⎦


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




µγNew 4-‐vector •

L

R

=ψ

ψψ⎡ ⎤⎢ ⎥⎣ ⎦




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




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



L = -iψ γ µ ∂

µ ψ − mψ ψ

µγNew 4-‐vector


(−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



L = -iψ γ µ ∂

µ ψ − mψ ψ

µγNew 4-‐vector


(−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 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 )


(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


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

Fermion Lagrangian

Dirac fermion:

Dirac equa;on

Majorana fermion:

Free par;cle solu;on to the Dirac equa;on

( ),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

1 fermions s 21 fermions s 2 ... i,

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