klein gordon field

3 lagrangian densi, tyL d x=∫L L

Klein Gordon field ( )xφ

( )† 2 †( ) ( ) ( ) ( )x x m x xμμφ φ φ φ∂ ∂ −L =

)0 0S μ

μδφ φ

∂ ∂= −∂ =

∂ ∂ ∂⇒

L L(

} }T V

Manifestly Lorentz invariant

Euler Lagrange equation

Lagrangian formulation of the Klein Gordon equation

Classical path :

(∂μ∂

μ + m2 )φ =0 Klein Gordon equation

New symmetries

( )† 2 †( ) ( ) ( ) ( )x x m x xμμφ φ φ φ∂ ∂ −L =

New symmetries

( )† 2 †( ) ( ) ( ) ( )x x m x xμμφ φ φ φ∂ ∂ −L =

Is invariant under …an Abelian (U(1)) gauge symmetry( ) ( )ix e xαφ φ→

New symmetries

L = ∂μφ(x)( )

†∂μφ(x) −m2φ(x)†φ(x) −m'2(φ2 +φ*2 )

Is not invariant under ( ) ( )ix e xαφ φ→

New symmetries

( )† 2 †( ) ( ) ( ) ( )x x m x xμμφ φ φ φ∂ ∂ −L =

Is invariant under …an Abelian (U(1)) gauge symmetry( ) ( )ix e xαφ φ→

New symmetries

( )† 2 †( ) ( ) ( ) ( )x x m x xμμφ φ φ φ∂ ∂ −L =

Is invariant under

A symmetry implies a conserved current and charge.

Translation Momentum conservation

Rotation Angular momentum conservation

e.g.

…an Abelian (U(1)) gauge symmetry( ) ( )ix e xαφ φ→

New symmetries

( )† 2 †( ) ( ) ( ) ( )x x m x xμμφ φ φ φ∂ ∂ −L =

Is invariant under

A symmetry implies a conserved current and charge.

Translation Momentum conservation

Rotation Angular momentum conservation

e.g.

What conservation law does the U(1) invariance imply?

…an Abelian (U(1)) gauge symmetry( ) ( )ix e xαφ φ→

Noether current

( )† 2 †( ) ( ) ( ) ( )x x m x xμμφ φ φ φ∂ ∂ −L =

Is invariant under ( ) ( )ix e xαφ φ→ …an Abelian (U(1)) gauge symmetry

†0 ( ) ( )( )

μμδ δφ δ φ φ φ

φ φ∂ ∂

= + ∂ + ↔∂ ∂ ∂L L

L =

Noether current

( )† 2 †( ) ( ) ( ) ( )x x m x xμμφ φ φ φ∂ ∂ −L =


†0 ( ) ( )( )


φ φ∂ ∂

= + ∂ + ↔∂ ∂ ∂L L

L =

iαφ

Noether current

( )† 2 †( ) ( ) ( ) ( )x x m x xμμφ φ φ φ∂ ∂ −L =


†0 ( ) ( )( )


φ φ∂ ∂

= + ∂ + ↔∂ ∂ ∂L L

L =

iαφ i μα φ∂

Noether current

( )† 2 †( ) ( ) ( ) ( )x x m x xμμφ φ φ φ∂ ∂ −L =


†0 ( ) ( )( )


φ φ∂ ∂

= + ∂ + ↔∂ ∂ ∂L L

L =

iαφ i μα φ∂

†( )( ) ( )

i iμ μμ μα φ α φ φ φ

φ φ φ⎡ ⎤⎛ ⎞ ⎛ ⎞∂ ∂ ∂

= −∂ + ∂ − ↔⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦

L L L

Noether current

( )† 2 †( ) ( ) ( ) ( )x x m x xμμφ φ φ φ∂ ∂ −L =


†0 ( ) ( )( )


φ φ∂ ∂

= + ∂ + ↔∂ ∂ ∂L L

L =

iαφ i μα φ∂

†( )( ) ( )


φ φ φ⎡ ⎤⎛ ⎞ ⎛ ⎞∂ ∂ ∂

= −∂ + ∂ − ↔⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦

L L L

0 (Euler lagrange eqs.)

Noether current

( )† 2 †( ) ( ) ( ) ( )x x m x xμμφ φ φ φ∂ ∂ −L =


†0 ( ) ( )( )


φ φ∂ ∂

= + ∂ + ↔∂ ∂ ∂L L

L =

iαφ i μα φ∂

†( )( ) ( )


φ φ φ⎡ ⎤⎛ ⎞ ⎛ ⎞∂ ∂ ∂

= −∂ + ∂ − ↔⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦

L L L

0 (Euler lagrange eqs.)

††

0, 2 ( ) ( )

iej jμμ μ μ μφ φ

φ φ⎛ ⎞∂ ∂

∂ = = −⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

L L Noether current

The Klein Gordon current

( )† 2 †( ) ( ) ( ) ( )x x m x xμμφ φ φ φ∂ ∂ −L =

Is invariant under ( ) ( )ix e xαψ ψ→ …an Abelian (U(1)) gauge symmetry

††

0, 2 ( ) ( )


φ φ⎛ ⎞∂ ∂

∂ = = −⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

L L


( )† 2 †( ) ( ) ( ) ( )x x m x xμμφ φ φ φ∂ ∂ −L =


††

0, 2 ( ) ( )


φ φ⎛ ⎞∂ ∂

∂ = = −⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

L L

( )* *KGj ieμ μ μφ φ φ φ= − ∂ − ∂


( )† 2 †( ) ( ) ( ) ( )x x m x xμμφ φ φ φ∂ ∂ −L =


††

0, 2 ( ) ( )


φ φ⎛ ⎞∂ ∂

∂ = = −⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

L L

( )* *KGj ieμ μ μφ φ φ φ= − ∂ − ∂

This is of the form of the electromagnetic current we used for the KG field


( )† 2 †( ) ( ) ( ) ( )x x m x xμμφ φ φ φ∂ ∂ −L =


††

0, 2 ( ) ( )


φ φ⎛ ⎞∂ ∂

∂ = = −⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

L L

( )* *KGj ieμ μ μφ φ φ φ= − ∂ − ∂

This is of the form of the electromagnetic current we used for the KG field

3 0Q d x j=∫ is the associated conserved charge

Suppose we have two fields with different U(1) charges :

1,2

1,2 1,2( ) ( )i Qx e xαφ φ→

..no cross terms possible (corresponding to charge conservation)

L = ∂μφ(x)( )

†∂μφ(x) −m2φ(x)†φ(x) + λ φ

4+

λ '

M 2φ

6+ ...

Additional terms

L = ∂μφ(x)( )

†∂μφ(x) −m2φ(x)†φ(x) + λ φ

4+

λ '

M 2φ

6+ ...

Additional terms

}

Renormalisable D ≤4

L = ∂μφ(x)( )

†∂μφ(x) −m2φ(x)†φ(x) + λ φ

4+

λ '

M 2φ

6+ ...

Additional terms

}

Renormalisable D ≤4

If M ? 103GeV , "Effective" Field theory approximately renormalisable

U(1) local gauge invariance and QED


( )( ) ( )i x Qx e xαφ φ→


( )† 2 †( ) ( ) ( ) ( )x x m x xμμφ φ φ φ∂ ∂ −L = not invariant due to derivatives

( )( ) ( )i x Qx e xαφ φ→

( ) ( ) )( ()( ) (( ) )i x Q i i xx QQ i x Qe e iQe xiQA iQA iQe xαμ μ

α α αμ μ μ μμφ φ φ φ φαφ φ α∂ → ∂ = ∂− − − ∂+ ∂



( )( ) ( )i x Qx e xαφ φ→



To obtain invariant Lagrangian look for a modified derivative transforming covariantly

( )i xD e Dαμ μφ φ→



( )( ) ( )i x Qx e xαφ φ→



To obtain invariant Lagrangian look for a modified derivative transforming covariantly


D iQAμ μ μ=∂ −

A Aμ μ μα→ +∂Need to introduce a new vector field

( )( ) ( )iQ xx e xαφ φ→


A Aμ μ μα→ +∂

( )† 2 †( ) ( ) ( ) ( )D x D x m x xμμφ φ φ φ−L = is invariant under local U(1)

( )( ) ( )iQ xx e xαφ φ→




Note : D iQAμ μ μ μ∂ → =∂ − is equivalent to p p eAμ μ μ→ +

universal coupling of electromagnetism follows from local gauge invariance

( )( ) ( )iQ xx e xαφ φ→






2 2 2( ) ( )m V where V ie A A e Aμ μ μμ μ μψ ψ∂ ∂ + =− =− ∂ + ∂ −

The Euler lagrange equation give the KG equation:

( )( ) ( )iQ xx e xαφ φ→






( )†KG 2 † 2( ) ( ) ( ) ( ) ( )KGx x m x x j A O eμ μμ μφ φ φ φ∂ ∂ − − +i.e. L =L =

The electromagnetic Lagrangian


F A Aμν μ ν ν μ=∂ −∂

1 2 3

1 3 2

2 3 1

3 2 1

0

0

0

0

E E E

E B B

E B B

E B B

− − −⎛ ⎞⎜ ⎟−⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟−⎝ ⎠



, F F A Aμν μν μ μ μα→ → +∂

1 2 3

1 3 2

2 3 1

3 2 1

0

0

0

0

E E E

E B B

E B B

E B B

− − −⎛ ⎞⎜ ⎟−⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟−⎝ ⎠




14

EM j AF F μνμν

μμ=− −L

1 2 3

1 3 2

2 3 1

3 2 1

0

0

0

0

E E E

E B B

E B B

E B B

− − −⎛ ⎞⎜ ⎟−⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟−⎝ ⎠




14

EM j AF F μνμν

μμ=− −L

1 2 3

1 3 2

2 3 1

3 2 1

0

0

0

0

E E E

E B B

E B B

E B B

− − −⎛ ⎞⎜ ⎟−⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟−⎝ ⎠

2M A Aμμ Forbidden by gauge invariance




14

EM j AF F μνμν

μμ=− −L

The Euler-Lagrange equations give Maxwell equations !

F jμν νμ∂ =

1 2 3

1 3 2

2 3 1

3 2 1

0

0

0

0

E E E

E B B

E B B

E B B

− − −⎛ ⎞⎜ ⎟−⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟−⎝ ⎠

0)A A

μν μ ν

∂ ∂−∂ =

∂ ∂ ∂L L

(





14

EM j AF F μνμν

μμ=− −L


F jμν νμ∂ =

. , 0

. 0,

tE

t

ρ ∂∇ = ∇× + =

∂∂

∇ = ∇× − =∂

BE E

B B j≡

1 2 3

1 3 2

2 3 1

3 2 1

0

0

0

0

E E E

E B B

E B B

E B B

− − −⎛ ⎞⎜ ⎟−⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟−⎝ ⎠

0)A A

μν μ ν

∂ ∂−∂ =

∂ ∂ ∂L L

(





14

EM j AF F μνμν

μμ=− −L


F jμν νμ∂ =

. , 0

. 0,

tE

t

ρ ∂∇ = ∇× + =

∂∂

∇ = ∇× − =∂

BE E

B B j≡

1 2 3

1 3 2

2 3 1

3 2 1

0

0

0

0

E E E

E B B

E B B

E B B

− − −⎛ ⎞⎜ ⎟−⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟−⎝ ⎠

0)A A

μν μ ν

∂ ∂−∂ =

∂ ∂ ∂L L

(


EM dynamicsfollows from a local gauge symmetry!!

The photon propagator

( )F A A jμν μ ν ν μ νμ μ μ∂ =∂ ∂ −∂ ∂ =

• The propagators determined by terms quadratic in the fields, using the Euler Lagrange equations.

2 2 2( ) ( )m V where V ie A A e Aμ μ μμ μ μψ ψ∂ ∂ + =− =− ∂ + ∂ −

The Klein Gordon propagator (reminder)

2 4( ) ( ' ) ( ' )Fm x x x xμμ δ∂ ∂ + Δ − = −

In momentum space:

Δ∞F (p) = i

−p2 +m2 ±iε

With normalisation convention used in Feynman rules = inverse of momentum space operator multiplied by -i


( )F A A jμν μ ν ν μ νμ μ μ∂ =∂ ∂ −∂ ∂ =



( )F A A jμν μ ν ν μ νμ μ μ∂ =∂ ∂ −∂ ∂ =

Gauge ambiguity

A Aμ μ μα→ +∂ 2A Aμ μμ μ α∂ → ∂ +∂



( )F A A jμν μ ν ν μ νμ μ μ∂ =∂ ∂ −∂ ∂ =

Gauge ambiguity

A Aμ μ μα→ +∂ 2A Aμ μμ μ α∂ → ∂ +∂

Choose as

1

(gauge fixing)

Aμμξ

− ∂



( )F A A jμν μ ν ν μ νμ μ μ∂ =∂ ∂ −∂ ∂ =

Gauge ambiguity

A Aμ μ μα→ +∂ 2A Aμ μμ μ α∂ → ∂ +∂

i.e. with suitable “gauge” choice of α (“ξ” gauge) want to solve

21 1(1 ) ( ) ( (1 ) )A A g A jμ ν ν μ νμ ν μ ν

μ μ μξ ξ∂ ∂ − − ∂ ∂ ≡ ∂ − − ∂ ∂ =

Choose as

1

(gauge fixing)

Aμμξ

− ∂



( )F A A jμν μ ν ν μ νμ μ μ∂ =∂ ∂ −∂ ∂ =

Gauge ambiguity

A Aμ μ μα→ +∂ 2A Aμ μμ μ α∂ → ∂ +∂

i.e. with suitable “gauge” choice of α (“ξ” gauge) want to solve

21 1(1 ) ( ) ( (1 ) )A A g A jμ ν ν μ νμ ν μ ν

μ μ μξ ξ∂ ∂ − − ∂ ∂ ≡ ∂ − − ∂ ∂ =

In momentum space the photon propagator is

1

22 2

1(1 ) (1 )

p pii g p p p g

p pμ νμν μ ν

μν ξξ

−⎛ ⎞⎛ ⎞

− − − = − + −⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

(‘t Hooft Feynman gauge ξ=1)

Choose as

1

(gauge fixing)

Aμμξ

− ∂


(

(3) (2) (1))

The Standard Model

SU SU U⊗ ⊗Extension to non-Abelian symmetry

(

(3) (2) (1))

The Standard Model

SU SU U⊗ ⊗

(2) local gauge invarianceSU

Q → eig2α (x).

σ2Q

1 2 3

0 1 0 1 0

1 0 0 0 1

i

iσ σ σ

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠ ⎝ ⎠

, ,R R

L

uQ u d

d

⎛ ⎞=⎜ ⎟

⎝ ⎠

σ i

2,σ j

2

⎡

⎣⎢⎢

⎤

⎦⎥⎥=iε ijk

σ k

2

⎛

⎝⎜

⎞

⎠⎟

Extension to non-Abelian symmetry

(

(3) (2) (1))

The Standard Model

SU SU U⊗ ⊗


Q → eig2α (x).

σ2Q

1 2 3

0 1 0 1 0

1 0 0 0 1

i

iσ σ σ

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠ ⎝ ⎠

, ,R R

L

uQ u d

d

⎛ ⎞=⎜ ⎟

⎝ ⎠

2D2

iiig Wμ μ μσ

=∂ +

, , 2 ,i i i ijk j kW W g Wμ μ μ μα ε α→ −∂ −

where


DμQ→ e

ig2α (x).σ2 DμQ

(

(3) (2) (1))

The Standard Model

SU SU U⊗ ⊗


Q → eig2α (x).

σ2Q

1 2 3

0 1 0 1 0

1 0 0 0 1

i

iσ σ σ

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟− −⎝ ⎠ ⎝ ⎠ ⎝ ⎠

, ,R R

L

uQ u d

d

⎛ ⎞=⎜ ⎟

⎝ ⎠

2D2

iiig Wμ μ μσ

=∂ +

, , 2 ,i i i ijk j kW W g Wμ μ μ μα ε α→ −∂ −

where


DμQ→ e

ig2α (x).σ2 DμQ

i.e. Need 3 gauge bosons SU(N ) : N 2 −1 gauge bosons (dimension of the adjoint representation)( )

Symmetry :

Local conservation of 2 weak isospin charges

A non-Abelian (SU(2)) local gauge field theory

Gauge boson(J=1)Wa=1..3

Weak coupling, α2

( )2 ( ). aig xa ab

r r r rst s t

e

W W f W

τ

μ μ μ μα α

Ψ → Ψ

→ − ∂ −

á

2( )rrig W μ

μ μλ γΨ ∂ − Ψ

,W Z±

u d

e eν

Neutral currents

, ,L L

R R R

u

d e

u d e

ν⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Weak Interactions

SU(2) local gauge theory

Symmetry :

Local conservation of 3 strong colour charges

QCD : a non-Abelian (SU(3)) local gauge field theory

Ψa → eig3α (x).λ( )

b

aΨa

Gμr → Gμ

r −∂μαr −g3 f rstα sGμ

t

3( )rrig G μ


q

q

q

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

Symmetry :

3 8

1 0 0 1 0 01

0 1 0 , 0 1 03

0 0 0 0 0 2

λ λ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟= − =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠

1 2

0 1 0 0 0

1 0 0 , 0 0 , ...

0 0 0 0 0 0

i

iλ λ−⎛ ⎞ ⎛ ⎞

⎜ ⎟ ⎜ ⎟= =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠


The strong interactions

QCD Quantum Chromodynamics Symmetry :

Local conservation of 3 strong colour charges

QCD : a non-Abelian (SU(3)) local gauge field theory

Gauge boson(J=1)“Gluons”

α

α

q

q

Ga=1..8

Strong coupling, α3

Ψa → eiα (x).λ( )b

aΨa

Gμr → Gμ

r − 1g3∂μα

r − f rstα sGμt

3( )rrig G μ


q

q

q

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

Symmetry :

SU(3)

Matter Sector “chiral”

L

u

d

⎛ ⎞⎜ ⎟⎝ ⎠

L

u

d

⎛ ⎞⎜ ⎟⎝ ⎠

L

u

d

⎛ ⎞⎜ ⎟⎝ ⎠

RuRu

Ru

(3)SU

Family Symmetry?

(2)SU

Neutral

Up

Down

(2)SU

Family Symmetry?

e

Le

ν⎛ ⎞⎜ ⎟⎝ ⎠

ReL

μνμ

⎛ ⎞⎜ ⎟⎝ ⎠

RμL

τντ

⎛ ⎞⎜ ⎟⎝ ⎠

Rτ

L

c

s

⎛ ⎞⎜ ⎟⎝ ⎠

L

c

s

⎛ ⎞⎜ ⎟⎝ ⎠

L

c

s

⎛ ⎞⎜ ⎟⎝ ⎠

L

t

b

⎛ ⎞⎜ ⎟⎝ ⎠

L

t

b

⎛ ⎞⎜ ⎟⎝ ⎠

RdRd

RdRc

RcRc

RsRs

Rs

RtRt

Rt Rb

Rb

Rb

Partial Unification

(3) (2) (1)SU SU U⊗ ⊗

L

t

b

⎛ ⎞⎜ ⎟⎝ ⎠


L

u

d

⎛ ⎞⎜ ⎟⎝ ⎠

L

u

d

⎛ ⎞⎜ ⎟⎝ ⎠

L

u

d

⎛ ⎞⎜ ⎟⎝ ⎠

RuRu

Ru

(3)SU

Family Symmetry?

(2)SU

Neutral

Up

Down

(2)SU

Family Symmetry?

e

Le

ν⎛ ⎞⎜ ⎟⎝ ⎠

ReL

μνμ

⎛ ⎞⎜ ⎟⎝ ⎠

RμL

τντ

⎛ ⎞⎜ ⎟⎝ ⎠

Rτ

L

c

s

⎛ ⎞⎜ ⎟⎝ ⎠

L

c

s

⎛ ⎞⎜ ⎟⎝ ⎠

L

c

s

⎛ ⎞⎜ ⎟⎝ ⎠ L

t

b

⎛ ⎞⎜ ⎟⎝ ⎠

L

t

b

⎛ ⎞⎜ ⎟⎝ ⎠

L

t

b

⎛ ⎞⎜ ⎟⎝ ⎠

RdRd

RdRc

RcRc

RsRs

Rs

RtRt

Rt Rb

Rb

Rb

}Wμ±

Partial Unification

(3) (2) (1)SU SU U⊗ ⊗ }Wμ±


L

u

d

⎛ ⎞⎜ ⎟⎝ ⎠

L

u

d

⎛ ⎞⎜ ⎟⎝ ⎠

L

u

d

⎛ ⎞⎜ ⎟⎝ ⎠

RuRu

Ru

(3)SU

Family Symmetry?

(2)SU

Neutral

Up

Down

(2)SU

Family Symmetry?

e

Le

ν⎛ ⎞⎜ ⎟⎝ ⎠

ReL

μνμ

⎛ ⎞⎜ ⎟⎝ ⎠

RμL

τντ

⎛ ⎞⎜ ⎟⎝ ⎠

Rτ

L

c

s

⎛ ⎞⎜ ⎟⎝ ⎠

L

c

s

⎛ ⎞⎜ ⎟⎝ ⎠

L

c

s

⎛ ⎞⎜ ⎟⎝ ⎠ L

t

b

⎛ ⎞⎜ ⎟⎝ ⎠

L

t

b

⎛ ⎞⎜ ⎟⎝ ⎠

L

t

b

⎛ ⎞⎜ ⎟⎝ ⎠

RdRd

RdRc

RcRc

RsRs

Rs

RtRt

Rt Rb

Rb

Rb

}Wμ±

Partial Unification

(3) (2) (1)SU SU U⊗ ⊗ }Wμ±

}1..8aGμ=

klein gordon field

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