The Klein Gordon equation (1926)

Scalar field (J=0) : (x)

2

2

2 2 2

tE m m

2 2p

2( ) 0m

0 0( , ), ( , )A A A B B B

4 vector notation

0 0( , ), ( , )A A A B B B

contravariant

covariant

0 0. .A B A B A B A B A B

A g A A g A

( , )ct x x

4 vectors

( , ) ( , )c t c t

( , )Ep p

c

p i 22 2p p E p

Field theory of

2( ) 0m

Classical electrodynamics, motion of charge –e in EM potential0A (A , ) A

is obtained by the substitution : p p eA

Quantum mechanics : i i eA

The Klein Gordon equation becomes:

2 2 2( ) ( )m V where V ie A A e A

2 14 137e

em The smallness of the EM coupling, , means that it is sensible to

Make a “perturbation” expansion of V in powers of em

Scalar particle – satisfies KG equation

( )V ie A A

Physical interpretation of Quantum Mechanical equations

Schrödinger equation (S.E.)21

2 0t mi

2=

* *( . .) ( . .)i S E i S E continuity 0 eq. .t j

“probability current”

* *2 ( )im j

“probability density”

Physical interpretation of Quantum Mechanics

Schrödinger equation (S.E.)21

2 0t mi

2=

* *( . .) ( . .)i S E i S E continuity 0 eq. .t j

“probability current”

* *2 ( )im j

“probability density”

Klein Gordon equation

**( )t ti

* *( )

( , )

i

j

j

j

2

2

2

tm

2

Pauli and Weisskopf ( , ) EMj e j j

. 0 =t j

j

22E N

Negative energy?

iEtNe

Negative probability??

Want to solve :

Solution :

where

2 4( ) ( ' ) ( ' )Fm x x x x

Feynman propagator Dirac Delta function

2( )m V

4( ) ( ) ' ( ' ) ( ') ( ')Fx x d x x x V x x 2( ) 0m

and

44 ' ( ' ) ( ') ( )d x x x f x f x

Want to solve :

Solution :

where

2 4( ) ( ' ) ( ' )Fm x x x x

Feynman propagator Dirac Delta function

Simplest to solve for propagator in momentum space by taking Fourier transform

2 2

.( ' ) 2 4 .( ' ) 4 41 12 2

( ) ( ' ) ( ' ) ( ' ) ( ' )ip x x ip x xFe m x x d x x e x x d x x

2

2 2 12

( ) ( )Fp m p

2 2 2 4 2 2

.41 1 1 1(2 ) (2 )

( ) , ( )ip x

F Fp m i p m ip x d p e

2( )m V

4( ) ( ) ' ( ' ) ( ') ( ')Fx x d x x x V x x

2

1

(2 ) 2

1

(2 )

2

1

(2 )

2( ) 0m

and

4( ) ( ) ' ( ' ) ( ') ( ')Fx x d x x x V x x

The Born series

Since V(x) is small can solve this equation iteratively :

Interpretation :

'x

x

''x'''x

( ' )F x x

( ')V x

( ) ( )x x 4 ' ( ' ) ( ') ( ')Fd x x x V x x 4 4'' '' ' ( '' ) ( '') ( ''' '') ( ''') ( ''')F Fd x d x x x V x x x V x x ...

but what about negative energy eigenvalues

2 1/ 2( ) ???E m 2p

Feynman – Stuckelberg interpretation

( 0)E ( 0)E

iEte ( )( )i E te

time

space

Two different time orderings giving same observable event :

't t 't t

t

't

4 2 2

.( ' )41 1(2 )

( ' )ip x x

F p m ix x d p e

time

space

't t 't t

3

3

'

(2 ) 2

p

p

i t t id pi e

p.(x'-x)

(p0 integral most conveniently evaluated using contour integration via Cauchy’s theorem )

1/ 22 2p p m

4 2 2

.41 1(2 )

( )ip x

F p m ix d p e

1/ 22 22 2 2 2 20o pp m i p p m i p p m i i

0

4

( ' ).( ' )31

0(2 )0 0

( ' )ip t t

i p x x

F

p p

ex x d p e dp

p i p i

0 If '- 0, choose contour such that ( +ve)I It t p ip p 0 ( ' ) ( ' )Iip t t p t te e

}I

( ' ) 'pi t t

p

iI e t t

0 If '- 0, choose contour such that ( +ve)I It t p ip p 0 ( ' ) ( ' )Iip t t p t te e

( ' ) 'pi t t

p

iI e t t

3

4 2 2 3

.( ' ) ' .( ' )41 1(2 ) (2 ) 2

( ' ) p

p

ip x x i t t ip x xd pF p m ix x d p e i e

time

space

1 2t t 1 2t t

0

. 1

2

ip xp p Vf e

where are positive and negative energy solutions to free KG equation

3 * 3 *( ') ( ') ( ) ( ' ) ( ') ( ) ( ')F p pp px x i d p f x f x t t i d p f x f x t t

Scattering in Quantum Mechanics

Prepare state at t | ( ) |in t i

Time evolution (possibly scattering) | ( ) | ( )in int S t

Observe resulting system in state | ( ) |out t f

QM : probability amplitude :

( ) | ( ) ( ) | | ( )

| |out in out in

fi

t t t S t

f S i S

fi fi fiS iT

Theory confronts experiment - Cross sections and decay rates

S matrix for Klein Gordon scattering

3 *' 0lim ( )pt

d x f i x

' , ( ) | ( )p p out inS t t

*3 ' 4' , ( ) ' ( ') ( ') ( ') ...p p pp

S p p i d x f x V x f x

Relativistic probability density

4( ) ( ) ' ( ' ) ( ') ( ') ...Fx x d x x x V x x 3 * 3 *( ' ) ( ') ( ) ( ' ) ( ') ( ) ( ')F p pp p

x x i d p f x f x t t i d p f x f x t t

fiiT

4 4* *( ) ( )( () ) )() (fi pp p piT i d y i dV y ie A Af y f y f yyf y

4 fiji d y A

* *( ) ( ) ( ) ( )p p pf

pij ie f y f y f y f y

0

. 1

2

ip xp p Vf e

( ).( ) f ii p p x

f ie p p e

Feynman rules

( ' )ie p p

p 'p

,k

Feynman rule associated with Feynman diagram