iT

fi= !i d 4 y" f

p '+

+* ( y) V ( y) fp+

+ ( y) = i d 4 y" fp '+

+* ( y) ie( Aµ#µ+ #

µAµ ) f

p+

+ ( y)

4 fiji d y Aµ

µ= ! "

jµ

fi = !ie fp '+

+* ( y) "µ

fp+

+ ( y)#$

%& ! "

µf

p '+

*+ ( y)#$

%& f

p+

+ ( y)( )

0

. 1

2

ip x

pp V

f e±=

!

= !e( p

f+ p

i)µe

i( pf! p

i).y

Feynman rules

( ' )ie p pµ µ+

p 'p

,k µ

Feynman rule associated with Feynman diagram

Compton scattering of a π meson

,k ! ', 'k !

p 'pp k+ p 'pp k!

,k ! ', 'k !

,k ! ', 'k !

p 'p

Feynman rules 2

2 2( )

( ) V

V ie A A A

m

eµ µ

µ

µ

µ

µ

! !" " +

"

#

= # + #

=

"

Klein Gordon

( ' )ie p p! !+

p 'p

,k !

2 2

i

p m! External photon !"

p

2ie

,k ! ', 'k !

p 'p

!" !"#

Compton scattering of a π meson

,k ! ', 'k !

p 'pp k+ p 'p'p k!

,k ! ', 'k ! ,k ! ', 'k !

p 'p

2

2 2

2 2

( ) [ .(2 ) '.(2 ' ')( )

.(2 ' ) '.(2 ') 2 . ']( ')

fi

iie p k p k

p k m

ip k p k i

p k m

! !

! ! ! !

= " + ++ "

+ " " "" "

iM

3 32 42 4 2

4 6

1 (2 )( )

2 2 (2 ) 2 2

C DC D A B

A B C D

d p d pVd p p p p V

E E V E E

!" #

!= + $ $

Av

M

Compton scattering of a π meson

,k ! ', 'k !

p 'pp k+ p 'pp k!

,k ! ', 'k ! ,k ! ', 'k !

p 'p

2 2

2 2

.(2 ) '.(2 ' ')( )

.(2 ' ') '.(2 ' ') 2 . '( )

fi

ip k p k

p k m

ip k p k i

p k m

! !

! ! ! !

= + ++ "

+ " " "+ "

M

[ ]

2 2

22

( . ')

1 (1 cos )klab m

d

d m

! " # #$

% &=' ()* + + ,

22 2 2

0 2

8| . 8.10 3.10

3total k

dd GeV mb

d m!

" !#" $ $ $

== % = =

%& !

2

/ 1

2|

total k m

mk

!"#

>>!

( . '. 0 gauge)p p! != =

The Dirac equationRELATIVISTIC QUANTUM MECHANICS

Dirac's derivation

The modern view - group representation theory

•

•

The Lorentz groupi i

J Rotations B osts Ko

i

i

i

[ , ]

[ , ]

[ , ]

j ijk k

j ijk k

j ijk k

J J i J

J K i K

K K i J

!

!

!

=

=

= "

} Generate the group SO(3,1)

102

( ( ) )i ijk jk i ix xM i x x J M K M! ""! " ! #$ $

$ $= % = =

To construct representations a more convenient (non-Hermitian) 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

!

!

=

=

=

} (2) (2)SU SU! ( , )representa on ni mt

The Lorentz groupi i

J Rotations B osts Ko

i

i

i

[ , ]

[ , ]

[ , ]

j ijk k

j ijk k

j ijk k

J J i J

J K i K

K K i J

!

!

!

=

=

= "

} Generate the group SO(3,1)

102

( ( ) )i ijk jk i ix xM i x x J M K M! ""! " ! #$ $

$ $= % = =

To construct representations a more convenient (non-Hermitian) 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

!

!

=

=

=

†

i i iJ N N= +

Representations

( , )n m J n m= +

1 1 12 2 2

1 12 2

scalar

LH and RH sp

(0,0) J=0

( ,0), (0 inors

vector

, ) J=

( , ) J=1, etc

Weyl spinors 1 12 2

R

( ,0) (0, )

L

! !

2-component spinors of SU(2)

Rotations and Boosts( ) ( ) ( )L R L R L R

S! !"

SL( R)

= ei!

2."

: Rotations

SL( R)

= e±!

2.#

: Boosts

Dirac spinor

Can combine R, L

! ! to form a 4-component “Dirac” spinor L

R

=!

!!" #$ %& '

2, ,

i ie

!" µ# µ#µ# µ #$ $ !" ! " % % !& '( = = ) *Lorentz transformations

where

!0=

0 " I

" I 0

#

$%

&

'( , !

i=

0 )i

")i

o

#

$%%

&

'((, !

5= i! 0! 1! 2! 3

=I 0

0 " I

#

$%

&

'(

Weyl basis0

, , 1, 2,3i ijboosts rotations i j! !" " =

†

†

Weyl spinors 1 12 2

R

( ,0) (0, )

L

! !

2-component spinors of SU(2)

Rotations and Boosts( ) ( ) ( )L R L R L R

S! !"

SL( R)

= ei!

2."

: Rotations

SL( R)

= e±!

2.#

: Boosts

Dirac spinor

Can combine R, L

! ! to form a 4-component “Dirac” spinor L

R

=!

!!" #$ %& '

2, ,

i ie

!" µ# µ#µ# µ #$ $ !" ! " % % !& '( = = ) *Lorentz transformations

where

!0=

0 " I

" I 0

#

$%

&

'( , !

i=

0 )i

")i

o

#

$%%

&

'((, !

5= i! 0! 1! 2! 3

=I 0

0 " I

#

$%

&

'(

†

Note : 1( ) 52

(1 )L R

! " != !

(Dirac gamma matrices, …new 4-vector )µ!

{ }, 2gµ ! µ ! ! µ µ!" " " " " "= + =

The Dirac equation

µ!

Fermions described by 4-cpt Dirac spinors !

New 4-vector

( ) 0i mµ

µ! "# $ =

. ( )ip xe u p! "

=

( ) ( )( ) ( ) 0p m u p p m u pµ

µ! " # " =/

In momentum space :