What's up with those Feynman diagrams? [1cm]an Introduction to Quantum Field Theories ·...

What’s up with those Feynman diagrams?

an Introduction to Quantum Field Theories

Martin Nagel

University of Colorado

February 3, 2010

Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 1 / 31

Outline

What you should take away from this lecture:

What do those Feynman diagrams mean?

Where do they come from?

W−

d

b

d

s

s

su, c, t

1

Some appreciation for quantum gauge theories

[Xλ, Xµ] = cνλµXν U(ε(x)) = exp(−igε(x) · X)“

X(R)λ

”

= −cνλµ δWµ = −∂µε(x) − g [ε(x) × Wµ]

�

�

��

�


K 0

φ

B0

Relativistiv Quantum Mechanics

i~∂

∂tψ =

~2

2m∇

2ψ + Vψ

Schrodinger equation

not Lorentz-invariant

can’t handle massless particles

spin must be inserted ’by hand’


Klein-Gordon

Ansatz: Use relativistic energy-momentum relation: E 2 = p2 + m2

⇒ Klein-Gordon equation:

(2 + m2)φ(x, t) = 0 , 2 ≡ ∂µ∂µ =

∂2

∂t2− ∇

2

Problems:

Negative energy solutions E = ±(p2 + m2)1/2

Negative probability density

describes only spinless particles


Klein-Gordon

Ansatz: Use relativistic energy-momentum relation: E 2 = p2 + m2

⇒ Klein-Gordon equation:

(2 + m2)φ(x, t) = 0 , 2 ≡ ∂µ∂µ =

∂2

∂t2− ∇

2

Problems:

Negative energy solutions E = ±(p2 + m2)1/2

Negative probability density

can both be fixed by Feynman interpretation:negative 4-momentum corresponds to antiparticles

describes only spinless particles

particle-antiparticle interpretation arises naturally from a complex Klein-Gordon field φ:correct descripton of bosons


Dirac

Dirac:linear in ∂/∂t and in ∇

Ansatz:

i∂ψ(x, t)

∂t= (−iα · ∇ + βm)ψ(x, t)

demand a KG-type condition:

−∂2ψ/∂t2 = (−∇2 + m2)ψ

which yields the following anti-commutation relations:

{αi , β} = 0 {αi , αj} = 2δij1 β2 = 1


Dirac


Ansatz:

i∂ψ(x, t)

∂t= (−iα · ∇ + βm)ψ(x, t)


−∂2ψ/∂t2 = (−∇2 + m2)ψ


{αi , β} = 0 {αi , αj} = 2δij1 β2 = 1

αi and β are Hermitian (H = α · p + βm)

Trαi = Trβ = 0

Eigenvalues of αi and β are ±1

dimensionality n of αi and β is even

n = 2: {σi} and 1 form a complete set


Dirac


Ansatz:

i∂ψ(x, t)

∂t= (−iα · ∇ + βm)ψ(x, t)


−∂2ψ/∂t2 = (−∇2 + m2)ψ


{αi , β} = 0 {αi , αj} = 2δij1 β2 = 1

αi and β are Hermitian (H = α · p + βm)

Trαi = Trβ = 0

Eigenvalues of αi and β are ±1

dimensionality n of αi and β is even

n = 2: {σi} and 1 form a complete set

⇒ 4 x 4 matrices


9

>>>>>>=

>>>>>>;

Dirac equation I

conventional representation:

αi =

„0 σi

σi 0

«

β =

„1 0

0 −1

«

define γ matrices:γµ = (γ0 = β, γ = βα)

Dirac equation:

(i /∂ − m)ψ = 0 , /∂ ≡ γµ∂µ


Dirac equation I

conventional representation:

αi =

„0 σi

σi 0

«

β =

„1 0

0 −1

«

define γ matrices:γµ = (γ0 = β, γ = βα)

Dirac equation:

(i /∂ − m)ψ = 0 , /∂ ≡ γµ∂µ

Free-particle solutions, with E = ±(p2 + m2)1/2:

ψ =

„φχ

«

e−ip·x =

(

us (p)e−ip+·x , positive 4-momentum pµ+ = (+E , p)

v s (p)e+ip+·x , negative 4-momentum −p+

4-component Dirac spinors:

us(p) =√

E + m

„φs

σ·p

E+mφs

«

v s(p) =√

E + m

„σ·p

E+mχs

χs

«

, s = 1, 2


Dirac equation II

two-component spinors φs=1,2 and χs=1,2 can be distinguished by helicity operator:

h(p) =

σ·p

|p|0

0 σ·p

|p|

!

spin degree of freedom arises naturally

two-component spinors φ and χ are interchanged under parity⇒ need 4 components


Dirac equation II

two-component spinors φs=1,2 and χs=1,2 can be distinguished by helicity operator:

h(p) =

σ·p

|p|0

0 σ·p

|p|

!

spin degree of freedom arises naturally

two-component spinors φ and χ are interchanged under parity⇒ need 4 components

in chiral representation, the (split-up) Dirac equation becomes:

Eφ = σ · pφ+ mχ

Eχ = −σ · pχ+ mφ

φ and χ become helicity eigenstates for m → 0’mass couples states of different helicity’

Dirac 4-current:

jµ = ψγµψ with ψ = ψ†γ0

ρ = j0 is positive-definite

still negative energy solutions


Interpretation of negative energy solutions

6 qqq

qqq

+m

−m

Epositive-energycontinuumE ≥ +m

negative-energycontinuumE ≤ −m



6 qqq

qqq

+m

−m



Dirac

positive-energy particle can’cascade-down’ through negative-energylevels, without limit

postulate ’vacuum’ ≡ negative-energystates filled with electrons: Dirac ’sea’

stable through Pauli exclusion principle

infinite charge and energy, but onlyrelative quantities observable

missing negative-energy spin-↑ electron= spin-↓ ’hole’ with positive energy andcharge ≡ positron

not a ’single-particle’ theory: e+e− paircreation through excitation → QFT



6 qqq

qqq

+m

−m



Dirac







Feynman

Dirac interpretation fails for bosons(not subject to Pauli exclusion principle)

particles with negative 4-momentumcorrespond to anti-particleswith positive 4-momentum

incoming state ↔ outgoing state

The emission (absorption) of ananti-particle of 4-momentum pµ isphysically equivalent to the absorption(emission) of a particle of 4-momentum−pµ



6 qqq

qqq

+m

−m



Dirac







Feynman

Dirac interpretation fails for bosons(not subject to Pauli exclusion principle)

particles with negative 4-momentumcorrespond to anti-particleswith positive 4-momentum

incoming state ↔ outgoing state

The emission (absorption) of ananti-particle of 4-momentum pµ isphysically equivalent to the absorption(emission) of a particle of 4-momentum−pµ

e−(pi) e−(pf )

V

1

↓e+(−pi) e+(−pf )

V

1


Quantum field theories

Field theories

continuous system (field) with infinite degrees of freedom, localized interactions

{qr (t) , r = 1 . . .N} N→∞−→ φ(x, t)

Lagrange-Hamilton formalism:define Lagrangian, principle of least action ⇒ Euler-Lagrange equations of motion

canonically conjugate field momentum π = ∂L/∂φfield expansion in terms of appropriate normal modes,total energy = sum of individual mode energies

φ(x, t) =

Z ∞

−∞

d3k

(2π)3√

2ω[a(k)ei(kx−ωt) + a∗(k)e−i(kx−ωt)]



Field theories


{qr (t) , r = 1 . . .N} N→∞−→ φ(x, t)



φ(x, t) =

Z ∞

−∞

d3k

(2π)3√

2ω[a(k)ei(kx−ωt) + a†(k)e−i(kx−ωt)]

Quantization

promote fields to operators: φ→ φ



Field theories


{qr (t) , r = 1 . . .N} N→∞−→ φ(x, t)



φ(x, t) =

Z ∞

−∞

d3k

(2π)3√

2ω[a(k)ei(kx−ωt) + a†(k)e−i(kx−ωt)]

Quantization

promote fields to operators: φ→ φ

postulate appropriate commutation relations:

[φ(x, t), π(y, t)] = iδ3(x − y)

[a(k), a†(k′)] = (2π)3δ3(k − k′) [a(k), a(k′)] = [a†(k), a†(k′)] = 0


Gauge theories

Gauge theories are characterized by close interrelation between symmetries, conservationlaws, and dynamics.

Gauge theories = dynamical theories based on local invariance principles

Classical electrodynamics: A and V not uniquely defined for given E and B.

Gauge invariance: gauge transformation (1) leaves field-strength tensor (2) and thereforeMaxwell’s equations (3) unchanged

Aµ → A′µ = Aµ − ∂µχ (1)

Fµν ≡ ∂µAν − ∂νAµ Fµν → F ′µν = Fµν (2)

∂µFµν = jν (3)

Remember: Maxwell modified Ampere’s law by introducing the displacement current to’save’ the continuity equation ∂µj

µ = 0, which is a statement of local chargeconservation.

Gauge invariance ⇔ charge conservation?


Gauge invariance in quantum mechanics

Schrodinger equation in electromagnetic field:

i∂ψ(x, t)

∂t= [

1

2m(−i∇ − qA)2 + qV ]ψ(x, t) (4)

But solution to (4) is no longer solution after (V ,A) is replaced by (V ′,A′)Schrodinger equation is not gauge-invariant (but ψ itself is not an observable).




i∂ψ(x, t)

∂t= [

1

2m(−i∇ − qA)2 + qV ]ψ(x, t) (4)


to make Schrodinger equation gauge-covariant, gauge transformation must be

accompanied by a transformation of ψ

Aµ → A′µ = Aµ − ∂µχ

ψ → ψ′ = exp(iqχ)ψ




i∂ψ(x, t)

∂t= [

1

2m(−i∇ − qA)2 + qV ]ψ(x, t) (4)



accompanied by a transformation of ψ and a replacement of ∂µ:



∂µ → Dµ ≡ ∂µ + iqAµ

⇒ probability density |ψ|2 and current density ψ∗(Dψ) − (Dψ)∗ψ remain invariant




i∂ψ(x, t)

∂t= [

1

2m(−i∇ − qA)2 + qV ]ψ(x, t) (4)



accompanied by a transformation of ψ and a replacement of ∂µ:



∂µ → Dµ ≡ ∂µ + iqAµ

⇒ probability density |ψ|2 and current density ψ∗(Dψ) − (Dψ)∗ψ remain invariant

Insistence on (local) gauge invariance gives rise to an interaction term

@I


covariant derivative

The argument reversed: the gauge principle

Start by demanding covariance under space-time dependent phase transformations:

ψ(x, t) → ψ′(x, t) = exp[iqχ(x, t)]ψ(x, t)

not possible for any free-particle relativistic wave equation

introduces a new field A (to cancel the ’kinetic’ part of χ)

which itself undergoes an exactly prescribed transformation when ψ → ψ′

that interacts with matter field ψ in an exactly prescribed way

A vector field such as Aµ, necessary to guarantee local phase invariance, is called a

gauge field


Global gauge symmetries

Noether’s theorem:

If the Lagrangian L of a system is invariant under a continuous transformation, thereexists an associated conserved symmetry current ∂µj

µ = 0.

Volume integral of µ = 0 component of a symmetry current is a symmetry operator.





µ = 0.

Volume integral of µ = 0 component of a symmetry current is a symmetry operator.Dirac Lagrangian

LD = ψ(iγµ∂µ − m)ψ

is invariant under global U(1) transformation ψ → ψ′ = e−iαψ

Symmetry current Nµψ = ψγµψ = jµ

Symmetry operator and Hamiltonian:

Nψ =

Z

ψ†ψd3x =

Zd

3p

(2π)3

X

s=1,2

[as†p as

p + bspb

s†p ]

HD =

Zd

3p

(2π)3

X

s=1,2

Ep[as†p as

p − bspb

s†p ]





µ = 0.

Volume integral of µ = 0 component of a symmetry current is a symmetry operator.Dirac Lagrangian


is invariant under global U(1) transformation ψ → ψ′ = e−iαψ

Symmetry current Nµψ = ψγµψ = jµ

Symmetry operator and Hamiltonian:

Nψ =

Z

ψ†ψd3x =

Zd

3p

(2π)3

X

s=1,2

[as†p as

p + bspb

s†p ]

HD =

Zd

3p

(2π)3

X

s=1,2

Ep[as†p as

p − bspb

s†p ]

⇒ postulate anti-commutation relations:

{arp, a

s†q } = {br

p, bs†q } = (2π)3δ3(p − q)δrs or {ψα(x, t), ψ†

β(y, t)} = δ3(x − y)δαβ

i.e. quantum gauge theory enforces the spin-statistics connection!

Pauli exclusion principle: identical fermions are antisymmetric under exchange of state labels

two-fermion state: |p1, s1;p2, s2〉 ∝ as1†p1

as2†p2

|0〉Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 13 / 31

Local gauge symmetries I


Local gauge symmetries II

Demand invariance of Dirac Lagrangian


under local U(1) phase transformation

ψ(x, t) → ψ′(x, t) = e−iqχ(x,t)ψ(x, t)

given the replacement∂µ → Dµ = ∂µ + iqAµ

⇒ The Lagrangian has gained an interaction term

LD∂µ→Dµ

−→ LlocalD = LD + Lint where Lint = −q ψγµψ

| {z }Aµ









LD∂µ→Dµ


| {z }Aµ

Global symmetries

conserved symmetry current, symmetry operator ⇒ conserved quantum number




LD = ψ(iγµ∂µ − m)ψ − 1

4FµνF

µν





LD∂µ→Dµ


| {z }Aµ

Global symmetries

conserved symmetry current, symmetry operator ⇒ conserved quantum number

Local symmetries

requires introduction of compensating gauge fields

symmetry current becomes dynamical current that couples to gauge field

prescribes a unique form of the interaction


Maxwell Lagrangian

= jµ

Perturbation theory

split up Hamiltonian into free and interaction parts: H = H0 + H ′

Heisenbeg picture: states (operators) are time-independent (time-dependent)

Schrodinger picture: states (operators) are time-dependent (time-independent)

Interaction picture:

time dependence of operators generated by H0:dAI (t)

dt= −i[AI (t), H0]

time dependence of states generated by H′: id

dt|ψ(t)〉I = H′

I (t) |ψ(t)〉I


Perturbation theory






dt= −i[AI (t), H0]


dt|ψ(t)〉I = H′

I (t) |ψ(t)〉I

S-matrix = amplitude to find a particular final state |f 〉 in time-evolved initial state |i〉

〈f |ψ(∞)〉I =D

f |S|ψ(−∞)E

I=D

f |S |iE

Dyson expansion:

S =∞X

n=0

(−i)n

n!

Z

. . .

Z

d4x1d

4x2 . . . d4xnT{H′

I (x1)H′I (x2) . . . H′

I (xn)}

with H ′I (t) =

RH′

I (x, t)d3x

and time-ordering symbol

T{H′I (x1)H′

I (x2)} = H′I (x1)H′

I (x2) for t1 > t2

= H′I (x2)H′

I (x1) for t1 < t2


QED

H ′QED = −Lint = qψγµψAµ

fermion field

ψ(x) =

Zd

3p

(2π)3

1p

2Ep

X

s

(aspu

s(p)e−ip·x + bs†p v s (p)eip·x )

ψ(x) =

Zd

3p

(2π)3

1p

2Ep

X

s

(bspv

s(p)e−ip·x + as†p us(p)eip·x )

{arp, a

s†q } = {br

p, bs†q } = (2π)3δ3(p − q)δrs

{arp, a

sq} = {ar†

p , as†q } = {ar(†)

p , bs(†)q } = {br

p, bsq} = {br†

p , bs†q } = 0

one-particle state: |p, s〉 ≡p

2Epas†

p |0〉

electromagnetic field

Aµ(x) =

Zd

3k

(2π)3

1√2ωk

3X

λ=0

(αλk εµλ(k)e−ik·x + αλ†k εµ∗λ (k)eik·x)

[αλk , αλ′†k′ ] = −gλλ

′

(2π)3δ3(k − k′)

indep

enden

toper

ator

sco

mm

ute

:[αλ

(†)

k,a

r(†)

p]=

[αλ

(†)

k,b

r(†)

p]=

0


e+e− scattering: 0th order

0th order term in Dyson expansion: 1

S(0)fi = 〈f | 1 |i〉 =

D

0|as′

k′ar′

p′as†k ar†

p |0E

(16EkEpEk′Ep′)1/2




S(0)fi = 〈f | 1 |i〉 =

D

0|as′

k′ar′

p′as†k ar†

p |0E


AAK

��

vacuum expectation value




S(0)fi = 〈f | 1 |i〉 =

D

0|as′

k′ar′

p′as†k ar†

p |0E


AAK

��


General strategy:

(anti-)commute a†’s to the left

(anti-)commute a’s to the right

’pick up’ δ-functions on the way

and use 〈0| a†i = ai |0〉 = 0




S(0)fi = 〈f | 1 |i〉 =

D

0|as′

k′ar′

p′as†k ar†

p |0E


AAK

��


General strategy:

(anti-)commute a†’s to the left

(anti-)commute a’s to the right

’pick up’ δ-functions on the way

and use 〈0| a†i = ai |0〉 = 0

. . . S(0)fi = 2Ek (2π)4δ3(k − k

′)2Ep(2π)3δ3(p − p′)

e+(k, r)

e−(p, s)

e+(k′, r′)

e−(p′, s′)

1


e+e− scattering: 1st order

S(1)fi = −ie

Z

d4x˙k

′

p′˛˛ ψ(x)γµψ(x)Aµ(x) |kp〉= 0

Note: only 1 electromagnetic field operator Aµ

no free photons in either initial or final states

αλk (αλ†k ) commutes all the way to the right (left)



S(1)fi = −ie

Z

d4x˙k

′





⇒ amplitude vanishes!



S(1)fi = −ie

Z

d4x˙k

′





⇒ amplitude vanishes!

Can generally decompose any free field into positive- and negative-frequency parts:

φ+(x) =R

d3p

(2π)31√2Ep

ape−ip·x φ−(x) =

Rd

3p

(2π)31√2Ep

a†pe

+ip·x

so that φ(x) = φ+(x) + φ−(x) and 〈0| φ−(x) = φ+(x) |0〉 = 0


ψ(x)γµψ(x)Aµ(x)

’three-fields-at-a-point’ interaction

1


Bremsstrahlung

˙kλp

′

s′

˛˛ ψ(x)γµψ(x)Aµ(x) |ps〉


e−(p, s)

e−(p′, s′)

γ(k, λ)

1


Bremsstrahlung

˙kλp

′

s′

˛˛ ψ(x)γµψ(x)Aµ(x) |ps〉= 0


e−(p, s)

e−(p′, s′)

γ(k, λ)

1


since pµ 6= p′µ + kµ

Bremsstrahlung

˙kλp

′

s′



e−(p, s)

e−(p′, s′)

γ(k, λ)

1

isolated diagram, takes place in vacuumfor bremsstrahlung to occur, we need a nucleus to absorb some of the momentumadditional interaction needed ⇒ 2nd order process:



Bremsstrahlung

˙kλp

′

s′



e−(p, s)

e−(p′, s′)

γ(k, λ)

1

isolated diagram, takes place in vacuumfor bremsstrahlung to occur, we need a nucleus to absorb some of the momentumadditional interaction needed ⇒ 2nd order process:

γ

e−(p, s)

e−(p′, s′)

γ(k, λ)

1



e+e− scattering: 2nd order

S(2)fi =

(−ie)2

2!

ZZ

d4x1d

4x2

˙k

′

s′ , p′

r′

˛˛

×T [ψ(x1)γµψ(x1)Aµ(x1)ψ(x2)γ

νψ(x2)Aν(x2)] |ks , pr 〉 (16EkEpEk′Ep′ )1/2

16 operators, 12 out of those inside time-ordering symbol

6 momentum integrals

⇒ we need a more efficient formalism!


Wick’s theorem

Wick’s theorem

T{a1a2 . . . an} = N{a1a2 . . . an + all possible contractions}

T{a1a2a3a4} = N{a1a2a3a4 + a1a2a3a4 + a1a2a3a4 + a1a2a3a4 + a1a2a3a4 +

+a1a2a3a4 + a1a2a3a4 + a1a2a3a4 + a1a2a3a4 + a1a2a3a4}


Wick’s theorem

Wick’s theorem




normal ordering symbol N() places operators in normal order , i.e. all a(k)’s to theright of all a†(k ′)’s, so that 〈0|N(any number of operators) |0〉 = 0


Wick’s theorem

Wick’s theorem




normal ordering symbol N() places operators in normal order , i.e. all a(k)’s to theright of all a†(k ′)’s, so that 〈0|N(any number of operators) |0〉 = 0decompose field φ into positive- and negative-frequency parts:φ+(x) ∝ a(k)e−ikx φ−(x) ∝ a†(k)e+ikx

contraction of two fields:

φ(x)φ(y) ≡ ˆ

φ+(x), φ−(y)˜

= 〈0|φ(x)φ(y) |0〉 for x0 > y 0

ˆφ+(y), φ−(x)

˜= 〈0|φ(y)φ(x) |0〉 for x0 < y 0

ff

= 〈0|Tφ(x)φ(y) |0〉

Feynman propagator

= amplitude for (KG, Dirac, ...) particle to propagate from

y to xx to y

ff

= momentum-space Green’s function for (KG, Dirac, ...) differential operatorMartin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 22 / 31


Wick’s theorem for fermions

time- and normal-ordered products and contractions pick up one minus sign for eachfermion interchange:

T (ψ1ψ2ψ3ψ4) = (−1)3ψ3ψ1ψ4ψ2 if x03 > x0

1 > x04 > x0

2

N(apaqa†r ) = (−1)2a†

r apaq = (−1)3a†r aqap

ψ(x)ψ(y) ≡(

{ψ+(x), ψ−(y)} for x0 > y 0

−{ψ+(y), ψ−(x)} for x0 < y 0

ψ(x)ψ(y) = ψ(x)ψ(y) = 0

N(ψ1ψ2ψ3ψ4) = −ψ1ψ3N(ψ2ψ4)

ZZ

d4x1d

4x2

˙k

′

s′ , p′

r′

˛˛T [ψ(x1)γ

µψ(x1)Aµ(x1)ψ(x2)γνψ(x2)Aν(x2)] |ks , pr 〉

external states are not |0〉 ⇒ not-fully-contracted terms do not necessarily vanish!



An uncontracted ψ operator inside N-product has 2 terms: ψ+ on the far right and ψ−

on the far left.One contribution for each way of anti-commuting the a of ψ+ past an initial-state a†,and for each way of anti-commuting the a† of ψ− past a final-state a.


operators fully contracted internally

〈p′, k′ | (ψγµψAµ)x (ψγνψAν)y | p, k〉〈p′, k′ | (ψγµψAµ)x (ψγ

νψAν)y | p, k〉




νψAν)y | p, k〉

γ

y x

1

γ

y x

1

×

e+(k, r)

e−(p, s)

e+(k′, r′)

e−(p′, s′)

1




νψAν)y | p, k〉

γ

y x

1

γ

y x

1

Exponentiation of the disconnected diagrams:disconnected pieces (’vacuum bubbles’) exponentiate to an overall phase

factor giving the shift of the energy of the interacting vacuum state(as opposed to the free-theory vacuum state)

⇒ no sensible contribution to S-matrix element!


external states partially contracted

〈p′, k′ | (ψγµψAµ)x (ψγνψAν)y | p, k〉

γ

e+(k, r)

e−(p, s)

e+(k′, r′)

e−(p′, s′)

y x

1

+ x ↔ y

+ (p, s) ↔ (k, r) , (p′, s ′) ↔ (k ′, r ′)


e+(k, r)

e−(p, s)

e+(k′, r′)

e−(p′, s′)

x

y

1

+ x ↔ y

+ (p, s) ↔ (k, r) , (p′, s ′) ↔ (k ′, r ′)


external states partially contracted


γ

e+(k, r)

e−(p, s)

e+(k′, r′)

e−(p′, s′)

y x

1

+ x ↔ y

+ (p, s) ↔ (k, r) , (p′, s ′) ↔ (k ′, r ′)


e+(k, r)

e−(p, s)

e+(k′, r′)

e−(p′, s′)

x

y

1

+ x ↔ y

+ (p, s) ↔ (k, r) , (p′, s ′) ↔ (k ′, r ′)

diagrams with loops connected to only one external line have nothing to do withthe scattering process itself; they are a property of the external states in theinteracting theory, i.e. they represent corrections to the ’no-scattering’ diagram

⇒ taken care of by Renormalization


external states fully contracted


γ

e+(k, r)

e−(p, s)

e+(k′, r′)

e−(p′, s′)y

x

1

+ x ↔ y


γ

e+(k, r)

e−(p, s)

e+(k′, r′)

e−(p′, s′)

y x

1

+ x ↔ y




γ

e+(k, r)

e−(p, s)

e+(k′, r′)

e−(p′, s′)y

x

1

+ x ↔ y


γ

e+(k, r)

e−(p, s)

e+(k′, r′)

e−(p′, s′)

y x

1

+ x ↔ y

only fully connected diagrams, in which all external lines are connected to each

other, contribute in a non-trivial way to the S-matrix!




γ

e+(k, r)

e−(p, s)

e+(k′, r′)

e−(p′, s′)y

x

1

+ x ↔ y


γ

e+(k, r)

e−(p, s)

e+(k′, r′)

e−(p′, s′)

y x

1

+ x ↔ y



Note: vertices are labeled for illustration purposes only!momentum-space propagators carry no implied order of emission/absorption;both time-ordered processes are always included!




γ

e+(k, r)

e−(p, s)

e+(k′, r′)

e−(p′, s′)y

x

1

+ x ↔ y


γ

e+(k, r)

e−(p, s)

e+(k′, r′)

e−(p′, s′)

y x

1

+ x ↔ y



Note: vertices are labeled for illustration purposes only!momentum-space propagators carry no implied order of emission/absorption;both time-ordered processes are always included!


γγ

but those photons are not real, they are virtual!

Virtual particles

space-time integral enforces 4-momentum conservation at each vertex

photon cannot have ’the right energy’: photon is virtual or off-mass-shell, withq2 6= m2

γ = 0

due to ’energy-time uncertainty relation’, a virtual photon can live for a time

∆t .~

∆E


Virtual particles

space-time integral enforces 4-momentum conservation at each vertex

photon cannot have ’the right energy’: photon is virtual or off-mass-shell, withq2 6= m2

γ = 0

due to ’energy-time uncertainty relation’, a virtual photon can live for a time

∆t .~

∆E

In quantum field theories, particles interact by exchanging virtual field quanta, whichmediate the force

a given order of perturbation series corresponds to the same number of verteces,where a field quantum is emitted or absorbed

replacing ∆t ≈ ∆r/c and ∆E ≈ mc2 for massive field quanta gives the effectiverange of the associated force (Yukawa)

∆r .~

mc


Feynman rules for QED

fermion propagator:

photon propagator:

vertex:

incoming photon:

outgoing photon:

incoming fermion:

outgoing fermion:

incoming antifermion:

outgoing antifermion:

p

1

p →

1

µ

1

p →

1

p →

1

p p

1

p p

1

p →

p →

1

p →

p →

1

=i(/p + m)

p2 + m2 + iε

=−igµνp2 + iε

= iqγµ

= εµ(p)

= ε∗µ(p)

= us(p)

= us(p)

= v s (p)

= v s (p)


Non-Abelian gauge theories

QED

Strong and weak forces are built on generalizations of gauge principle



QED = Abelian U(1) gauge theory

invariant under local phase transformations of a single field: ψ → ψ′ = exp(iqχ)ψ

different transformations commute: eiqχ1 · eiqχ2 = e

iqχ2 · eiqχ1

Strong and weak forces are built on generalizations of gauge principle




invariant under local phase transformations of a single field: ψ → ψ′ = exp(iqχ)ψ

different transformations commute: eiqχ1 · eiqχ2 = e

iqχ2 · eiqχ1

Strong and weak forces = non-Abelian gauge theories

local phase transformations involve more than one field ⇒ they do not commute




invariant under local phase transformations of a single field

different transformations commute


∂µ → Dµ = ∂µ + iqAµ


Aµ transforms independent of q ⇒ photon is neutral














gauge fields belong to regular (adjoint) representation of gauge group

coupling constant enters transformation law of gauge fields through non-vanishingcommutator, i.e. gauge fields carry non-Abelian ’charge’ degrees of freedom














the field quanta will necessarily interact with themselves

g

g

g

1

g

g

g

g

1














the field quanta will necessarily interact with themselves

All three dynamical theories in the Standard Model are based on a local gauge principle:

SU(3)c × SU(2)L × U(1)Y| {z }

��


electroweak = (weak isospin × hypercharge)strong

Summary

Feynman diagrams

formalism that allows convenient organization and visualization of QED processes inperturbation series

Feynman diagrams = representations of momentum-space amplitudes

each diagram represents a specific contribution to the scattering matrix, by a precisemathematical correspondance (’Feynman rules’)

contributions to S-matrix elements only from fully connected, amputated diagrams!disconnected diagrams (vacuum bubbles) represent the evolution of the free-theoryvacuum state into the interacting-theory vacuum stateexternal line corrections represent the evolution of the free-theory single-particle stateinto the interacting-theory single-particle state

Quantum gauge theories

gauge principle: demand invariance of Lagrangian under local phase transformationsof matter fields

requires introduction of compensating gauge fields

which couple to matter fields in an exactly determined way


What's up with those Feynman diagrams? [1cm]an Introduction to Quantum Field Theories ·...

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