Advanced  Quantum  Field  Theory  –  the  fundamental  forces      

QED  • Scalar  electrodynamics  Spinor  electrodynamics  

Nonabelian  gauge  theory  • QCD  Electroweak  

Books   Quantum  Field  Theory  –  Mark  Srednicki  IntroducEon  to  Quantum  Field  Theory  –  Peskin  &  Schroeder  

The  Quantum  Field  Theory  of  Fields  –  Weinberg  

Field  Theory:  A  Modern  Primer  –  Pierre  Ramond  

Quantum  Field  Theory  in  a  nutshell  –  Zee  

}  The  Standard  Model  

Quantum  Field  Theory  and  the  Standard  Model  -‐  Schwartz    

QED  • Scalar  electrodynamics  Spinor  electrodynamics  

Nonabelian  gauge  theory  • QCD  Electroweak  

Books   Quantum  Field  Theory  –  Mark  Srednicki  hOp://www.physics.ucsb.edu/~mark/qT.html  

Web  

Advanced  Quantum  Field  Theory  –  the  fundamental  forces      

hOp://www2.physics.ox.ac.uk/contacts/people/ross  

Outline!!

• QED!Introduction• Photon!propagator• Scalar!QED• Radiative(corrections• Fermions• Fermion(path!integral• Feynman'rules'"!Yukawa&theory• Spinor!QED• Sample&calculations• Beta$functions• Nonabelian)gauge)theory• NAGT"Feynman"rules• BRST!symmetry• Gauge&fixing• Spontaneous*symmetry*breaking• Anomalies• The$Standard$Model

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Quantum  Electrodynamics  (QED)  

Construction of a relativistic field theory

Lagrangian L T V= −

Action

2

1

t

t

S L dt= ∫

Classical path … minimises action

Quantum mechanics … sum over all paths with amplitude /iSe∝ !

•

•

Lagrangian invariant under all the symmetries of nature

(Nonrelativistic mechanics)

-makes it easy to construct viable theories

L = L∫ d 3x, L lagrangian density - D=4

Klein Gordon field φ(x)

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

Manifestly Lorentz invariant

Lagrangian formulation of the Klein Gordon equation

Dφ = 1

complex  scalar  field    

L = L∫ d 3x, L lagrangian density

Klein Gordon field ( )xφ

L = ∂µφ(x)( )† ∂µφ(x) − m2φ(x)†φ(x)

δS = 0 ⇒ ∂L

∂φ− ∂µ

∂L∂(∂µφ)

= 0

Manifestly Lorentz invariant

Euler Lagrange equation

Lagrangian formulation of the Klein Gordon equation

Classical path :

(∂µ∂

µ + m2 )φ = 0 Klein Gordon equation

δL = ∂L

∂φδφ + ∂L

∂(∂µφ)δ (∂µφ)

=

∂L∂φ

− ∂µ∂L

∂(∂µφ)⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎢

⎤

⎦⎥⎥δφ + ∂µ ∂L

∂(∂µφ)δφ

⎛⎝⎜

⎞⎠⎟

surface integral in S..vanishes

δSδφ

= 0

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 φ(x)→ eiαφ(x) …an Abelian (U(1)) gauge symmetry

†0 ( ) ( )( )

µµδ δφ δ φ φ φφ φ

∂ ∂= + ∂ + ↔∂ ∂ ∂L L

L=

iαφ i µα φ∂

†( )( ) ( )

i iµ µµ µα φ α φ φ φφ φ φ⎡ ⎤⎛ ⎞ ⎛ ⎞∂ ∂ ∂= −∂ + ∂ − ↔⎢ ⎥⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦

L L L

0 (Euler lagrange eqs.)

∂µ jµ = 0, jµ = iα

∂L∂(∂µφ)

φ − ∂L∂(∂µφ† )

φ†⎛⎝⎜

⎞⎠⎟

Noether current

True  locally  

The Klein Gordon current

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

Is invariant under …an Abelian (U(1)) gauge symmetry

∂µ jµ = 0, jµ = iα

∂L∂(∂µφ)

φ − ∂L∂(∂µφ† )

φ†⎛⎝⎜

⎞⎠⎟

jµ

KG = −iα φ* ∂µφ −φ ∂µφ*( )

φ(x)→ eiαφ(x)

Classical electrodynamics: motion of particle of charge in EM potential Aµ = ( A0 ,A)

given  by     pµ → pµ +QAµ , i∂µ→ i∂µ +QAµ

⇒ jµ

KG ≡ jµEM

−Q

The Klein Gordon current

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

Is invariant under …an Abelian (U(1)) gauge symmetry

∂µ jµ = 0, jµ = iα

∂L∂(∂µφ)

φ − ∂L∂(∂µφ† )

φ†⎛⎝⎜

⎞⎠⎟

jµ

KG = −iα φ* ∂µφ −φ ∂µφ*( )

φ(x)→ eiαφ(x)

∂∂t

j0(x)+∇.J(x) = 0

3 0Q d x j= ∫ is the associated conserved charge (Gauss’ law)

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

1,21,2 1,2( ) ( )

i Qx e xαφ φ→

( )( )

† 2 †1 1 1 1

† 2 †2 2 2 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

x x m x x

x x m x x

µµ

µµ

φ φ φ φ

φ φ φ φ

∂ ∂ −

+ ∂ ∂ −

L=

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

( )† 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 x Q i x Qe e iQe xα α αµ µ µ µφ φ φ φ α∂ → ∂ = ∂ + ∂

To obtain invariant Lagrangian look for a modified derivative transforming covariantly

Dµφ → e

iα ( x )Q Dµφ

U(1)  local  gauge  invariance  and  QED    

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

( )( ) ( )i x Qx e xαφ φ→

∂µφ − iQAµφ → e

iα ( x )Q (∂µφ − iQAµφ)+ iQeiα ( x )Qφ ∂µα (x)− iQe

iα ( x )Qφ ∂µα (x)

To obtain invariant Lagrangian look for a modified derivative transforming covariantly

Dµφ → e

iα ( x )Q Dµφ

D iQAµ µ µ= ∂ −

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

U(1)  local  gauge  invariance  and  QED    

Aµ ≡ (V ,A)( )

= ∂µφ( )† ∂µφ − m2φ†φ − iα Aµ (φ ∂µφ* −φ* ∂µφ)+α 2 Aµ Aµ

J µ ≡ (ρ,J)( )

( )( ) ( )iQ xx e xαφ φ→

Dµφ → e

iα ( x )Q Dµφ

A Aµ µ µα→ + ∂

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

Dµ → eiαQ Dµe

− iαQ

jµ = iα

∂L∂(∂µφ)

φ − ∂L∂(∂µφ† )

φ†⎛⎝⎜

⎞⎠⎟

= iα (φ ∂µφ

* −φ* ∂µφ)− 2α2 Aµφ

*φNoether  current  in  free  theory  

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

universal coupling of electromagnetism follows from local gauge invariance

i.e. L = LKG = ∂µφ(x)( )† ∂µφ(x)− m2φ(x)†φ(x)− jµKG Aµ +O(Q2 )

J µ ≡ (ρ,J)( )

( )( ) ( )iQ xx e xαφ φ→

Dµφ → e

iα ( x )Q Dµφ

A Aµ µ µα→ + ∂

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

Dµ → eiαQ Dµe

− iαQ

( )( ) ( )iQ xx e xαφ φ→

Dµφ → e

iα ( x )Q Dµφ

A Aµ µ µα→ + ∂

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

Note : D iQAµ µ µ µ∂ → = ∂ − is equivalent to pµ → pµ +QAµ

universal coupling of electromagnetism follows from local gauge invariance

(∂µ∂

µ + m2 )φ = −Vφ where V = −iQ(∂µ Aµ + Aµ ∂µ )−Q

2 A2

The Euler lagrange equation give the KG equation:

Dµ → eiαQ Dµe

− iαQ

The electromagnetic Lagrangian

F A Aµν µ ν ν µ= ∂ − ∂

, F F A Aµν µν µ µ µα→ → + ∂

− 14 Fµν F

µν

1 2 3

1 3 2

2 3 1

3 2 1

00

00

E E EE B BE B BE B B

− − −⎛ ⎞⎜ ⎟−⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟−⎝ ⎠

2M A Aµ µ Forbidden by gauge invariance

F µν = i

QDµ , Dν⎡⎣ ⎤⎦

⎛⎝⎜

⎞⎠⎟

Invariant:  

Scalar  QED  Lagrangian  

L = (Dµφ)† Dµφ − m

2φ†φ − 14

F µν Fµν

Euler  Lagrange  equaEon  

DµD

µφ − m2φ = 0

Noether  current  

jµ = −iQ[φ†Dµφ − (Dµφ)†φ]

L = − 14 Fµν F

µν + jµ Aµ + ∂µφ* ∂µφ − m

2φ*φ

The electromagnetic Lagrangian

F A Aµν µ ν ν µ= ∂ − ∂

, F F A Aµν µν µ µ µα→ → + ∂

LEM = − 14 Fµν F

µν + jµ Aµ

1 2 3

1 3 2

2 3 1

3 2 1

00

00

E E EE B BE B BE B B

− − −⎛ ⎞⎜ ⎟−⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟−⎝ ⎠

2M A Aµ µ Forbidden by gauge invariance

F µν = i

QDµ , Dν⎡⎣ ⎤⎦

⎛⎝⎜

⎞⎠⎟

The electromagnetic Lagrangian

F A Aµν µ ν ν µ= ∂ − ∂

, F F A Aµν µν µ µ µα→ → + ∂

LEM = − 14 Fµν F

µν + jµ Aµ

The Euler-Lagrange equations give Maxwell equations !

F jµν νµ∂ =. , 0

. 0,

tEt

ρ ∂∇ = ∇× + =∂∂∇ = ∇× − =∂

BE E

B B j≡

1 2 3

1 3 2

2 3 1

3 2 1

00

00

E E EE B BE B BE B B

− − −⎛ ⎞⎜ ⎟−⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟−⎝ ⎠

0)A A

µν µ ν

∂ ∂− ∂ =∂ ∂ ∂L L

(

2M A Aµ µ Forbidden by gauge invariance

EM dynamics follows from a local gauge symmetry!!

N.B. εµνρσ∂µFρσ = 0( )

Degrees  of  freedom  

Aµ (x) : 4  degrees  of  freedom?  

Lorentz  transformaEon:  can  remove  one  more  degree  of  freedom  

No  Eme  derivaEve  of    A0 …no  conjugate  momentum  and  no  dynamics  

Originates  from  gauge  invariance  …too  many  degrees  of  freedom  

LEM = − 14 Fµν Fµν + jµ Aµ

= − 12 ∂µ Aν ∂µ Aν + 12 ∂

µ Aν ∂ν Aµ + jµ Aµ

(c.f.  Chapter  8  Schwartz)  

No  Eme  derivaEve  of    A0 …no  conjugate  momentum  and  no  dynamics  

Originates  from  gauge  invariance  …too  many  degrees  of  freedom  

Gauge  fixing:  

nµAµ = 0n2 > 0, axial gaugen2 = 0, lightcone gaugen2 < 0, temporal gauge{  •

∂µAµ = 0, Lorentz gauge•

A Aµ µ µα→ + ∂ 2A Aµ µµ µ α∂ → ∂ + ∂ Rξ − gauge: ∂

2α = − 1ξ∂µ Aµ•

Coulomb  (radiaEon,  transverse)  gauge:    

k.A k( ) = 0•

e.g.  Coulomb  gauge   L = − 14 Fµν F

µν

⇒

Gauge  invariance:  

can  choose  α  to  give      

αα  

⇒

 Coulomb  gauge  preserved  by  gauge  transformaEons  with    

i.e.  have  residual  gauge  symmetry  to  set  A0  =  0  

Other  equs  of  moEon  give    

Free  photon  field  

∂2Aµ − ∂µ ∂ν Aν( ) = jµ

For a free field in Coulomb gauge ∂2A(x) = 0In analogy with the scalar case:

aλ (k) and aλ† (k) are annihilation and creation operators after canonical quantisation

See  Srednicki  chapter  3    and  chapters  5,  55  (LSZ  reducEon)  

polarisaEon  vectors  

i.e k.A k( ) = 0

PolarisaEon  vectors  -‐  Coulomb  gauge  

right-‐polarizaEon  

leT-‐polarizaEon  

= (0,1)

4-‐vector  form  

0 | Aµ | p,εj = εµ

je− ip.x

Ward  idenEty  

Sample  basis:  

Lorentz  transformaEon:  

physical  polarisaEon  states  

Unphysical  ?  

e.g.  

Λνµ =

32 1 0 − 121 1 0 −10 0 1 012 1 0 12

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

, Λνµε1

ν = ε1µ + 1

Epµ , Λν

µ pν = pµ

Λ  in  liOle  group,  SO(3)  

Ward  idenEty  

Sample  basis:  

Lorentz  transformaEon:  

QED  matrix  elements:  

M = ε µ Mµ → ε

µ +pµE

⎛

⎝⎜⎞

⎠⎟Mµ '

Unphysical  polarisaEon  must  not  contribute   ⇒ pµ Mµ

' = 0

i.e. pµ Mµ ' = p '

µ Mµ ' = pµ Mµ = 0 Ward  idenEty  

physical  polarisaEon  states  

Unphysical  ?  

(also  required  by  gauge  invariance)