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]
�
�
��
�
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 2 / 31
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’
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 3 / 31
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 4 / 31
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 4 / 31
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 5 / 31
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
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 5 / 31
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
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 5 / 31
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
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 5 / 31
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
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 5 / 31
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
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 5 / 31
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
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 5 / 31
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 6 / 31
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 6 / 31
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 7 / 31
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 7 / 31
Interpretation of negative energy solutions
6 qqq
qqq
+m
−m
Epositive-energycontinuumE ≥ +m
negative-energycontinuumE ≤ −m
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 8 / 31
Interpretation of negative energy solutions
6 qqq
qqq
+m
−m
Epositive-energycontinuumE ≥ +m
negative-energycontinuumE ≤ −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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 8 / 31
Interpretation of negative energy solutions
6 qqq
qqq
+m
−m
Epositive-energycontinuumE ≥ +m
negative-energycontinuumE ≤ −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
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 8 / 31
Interpretation of negative energy solutions
6 qqq
qqq
+m
−m
Epositive-energycontinuumE ≥ +m
negative-energycontinuumE ≤ −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
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 8 / 31
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)]
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 9 / 31
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)]
Quantization
promote fields to operators: φ→ φ
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 9 / 31
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)]
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 9 / 31
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?
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 10 / 31
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).
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 11 / 31
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).
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 11 / 31
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).
to make Schrodinger equation gauge-covariant, gauge transformation must be
accompanied by a transformation of ψ and a replacement of ∂µ:
Aµ → A′µ = Aµ − ∂µχ
ψ → ψ′ = exp(iqχ)ψ
∂µ → Dµ ≡ ∂µ + iqAµ
⇒ probability density |ψ|2 and current density ψ∗(Dψ) − (Dψ)∗ψ remain invariant
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 11 / 31
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).
to make Schrodinger equation gauge-covariant, gauge transformation must be
accompanied by a transformation of ψ and a replacement of ∂µ:
Aµ → A′µ = Aµ − ∂µχ
ψ → ψ′ = exp(iqχ)ψ
∂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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 11 / 31
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 12 / 31
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.
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 13 / 31
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.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 ]
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 13 / 31
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.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 ]
⇒ 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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 14 / 31
Local gauge symmetries II
Demand invariance of Dirac Lagrangian
LD = ψ(iγµ∂µ − m)ψ
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 15 / 31
Local gauge symmetries II
Demand invariance of Dirac Lagrangian
LD = ψ(iγµ∂µ − m)ψ
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
Global symmetries
conserved symmetry current, symmetry operator ⇒ conserved quantum number
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 15 / 31
Local gauge symmetries II
Demand invariance of Dirac Lagrangian
LD = ψ(iγµ∂µ − m)ψ − 1
4FµνF
µν
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
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 15 / 31
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 16 / 31
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
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 16 / 31
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
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 16 / 31
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 17 / 31
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 18 / 31
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
AAK
���
vacuum expectation value
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 18 / 31
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
AAK
���
vacuum expectation value
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 18 / 31
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
AAK
���
vacuum expectation value
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 18 / 31
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)
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 19 / 31
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)
⇒ amplitude vanishes!
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 19 / 31
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)
⇒ 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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 19 / 31
ψ(x)γµψ(x)Aµ(x)
’three-fields-at-a-point’ interaction
1
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 20 / 31
Bremsstrahlung
˙kλp
′
s′
˛˛ ψ(x)γµψ(x)Aµ(x) |ps〉
’three-fields-at-a-point’ interaction
e−(p, s)
e−(p′, s′)
γ(k, λ)
1
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 20 / 31
Bremsstrahlung
˙kλp
′
s′
˛˛ ψ(x)γµψ(x)Aµ(x) |ps〉= 0
’three-fields-at-a-point’ interaction
e−(p, s)
e−(p′, s′)
γ(k, λ)
1
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 20 / 31
since pµ 6= p′µ + kµ
Bremsstrahlung
˙kλp
′
s′
˛˛ ψ(x)γµψ(x)Aµ(x) |ps〉= 0
’three-fields-at-a-point’ interaction
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:
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 20 / 31
since pµ 6= p′µ + kµ
Bremsstrahlung
˙kλp
′
s′
˛˛ ψ(x)γµψ(x)Aµ(x) |ps〉= 0
’three-fields-at-a-point’ interaction
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 20 / 31
since pµ 6= p′µ + kµ
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!
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 21 / 31
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}
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 22 / 31
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}
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 22 / 31
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}
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
e+e− scattering: 2nd order
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!
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 23 / 31
e+e− scattering: 2nd order
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.
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 24 / 31
e+e− scattering: 2nd order
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.
⇒ define contractions of field operators with external states:
ψ(x) |ps 〉 = ψ+(x) |ps〉 = e−ip·xus (p) |0〉
〈ps | ψ(x) = 〈ps | ψ−(x) = 〈0| e+ip·x us(p)
ψ(x) |ks〉 = ψ+(x) |ks 〉 = e−ik·x v s(k) |0〉
〈ks |ψ(x) = 〈ks |ψ−(x) = 〈0| e+ik·xv s (k)
...incoming fermion
...outgoing fermion
...incoming anti-fermion
...outgoing anti-fermion
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 24 / 31
e+e− scattering: 2nd order
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.
⇒ define contractions of field operators with external states:
ψ(x) |ps 〉 = ψ+(x) |ps〉 = e−ip·xus (p) |0〉
〈ps | ψ(x) = 〈ps | ψ−(x) = 〈0| e+ip·x us(p)
ψ(x) |ks〉 = ψ+(x) |ks 〉 = e−ik·x v s(k) |0〉
〈ks |ψ(x) = 〈ks |ψ−(x) = 〈0| e+ik·xv s (k)
...incoming fermion
...outgoing fermion
...incoming anti-fermion
...outgoing anti-fermion
An uncontracted A operator inside N-product vanishes (no photons in external states)
⇒ two A’s must be contracted internally
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 24 / 31
operators fully contracted internally
〈p′, k′ | (ψγµψAµ)x (ψγνψAν)y | p, k〉 〈p′, k′ | (ψγµψAµ)x (ψγ
νψAν)y | p, k〉
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 25 / 31
operators fully contracted internally
〈p′, k′ | (ψγµψAµ)x (ψγνψAν)y | p, k〉 〈p′, k′ | (ψγµψAµ)x (ψγ
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 25 / 31
operators fully contracted internally
〈p′, k′ | (ψγµψAµ)x (ψγνψAν)y | p, k〉 〈p′, k′ | (ψγµψAµ)x (ψγ
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!
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 25 / 31
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 ′)
〈p′, k′ | (ψγµψAµ)x (ψγνψAν)y | p, k〉
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 ′)
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 26 / 31
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 ′)
〈p′, k′ | (ψγµψAµ)x (ψγνψAν)y | p, k〉
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 26 / 31
external states fully 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′, 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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 27 / 31
external states fully 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′, 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
only fully connected diagrams, in which all external lines are connected to each
other, contribute in a non-trivial way to the S-matrix!
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 27 / 31
external states fully 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′, 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
only fully connected diagrams, in which all external lines are connected to each
other, contribute in a non-trivial way to the S-matrix!
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!
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 27 / 31
external states fully 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′, 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
only fully connected diagrams, in which all external lines are connected to each
other, contribute in a non-trivial way to the S-matrix!
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!
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 27 / 31
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 28 / 31
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 28 / 31
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)
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 29 / 31
Non-Abelian gauge theories
QED
Strong and weak forces are built on generalizations of gauge principle
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 30 / 31
Non-Abelian gauge theories
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 30 / 31
Non-Abelian gauge theories
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 = non-Abelian gauge theories
local phase transformations involve more than one field ⇒ they do not commute
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 30 / 31
Non-Abelian gauge theories
QED = Abelian U(1) gauge theory
invariant under local phase transformations of a single field
different transformations commute
ψ → ψ′ = exp(iqχ)ψ
∂µ → Dµ = ∂µ + iqAµ
Aµ → A′µ = Aµ − ∂µχ
Aµ transforms independent of q ⇒ photon is neutral
Strong and weak forces = non-Abelian gauge theories
local phase transformations involve more than one field ⇒ they do not commute
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 30 / 31
Non-Abelian gauge theories
QED = Abelian U(1) gauge theory
invariant under local phase transformations of a single field
different transformations commute
ψ → ψ′ = exp(iqχ)ψ
∂µ → Dµ = ∂µ + iqAµ
Aµ → A′µ = Aµ − ∂µχ
Aµ transforms independent of q ⇒ photon is neutral
Strong and weak forces = non-Abelian gauge theories
local phase transformations involve more than one field ⇒ they do not commute
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 30 / 31
Non-Abelian gauge theories
QED = Abelian U(1) gauge theory
invariant under local phase transformations of a single field
different transformations commute
ψ → ψ′ = exp(iqχ)ψ
∂µ → Dµ = ∂µ + iqAµ
Aµ → A′µ = Aµ − ∂µχ
Aµ transforms independent of q ⇒ photon is neutral
Strong and weak forces = non-Abelian gauge theories
local phase transformations involve more than one field ⇒ they do not commute
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 30 / 31
Non-Abelian gauge theories
QED = Abelian U(1) gauge theory
invariant under local phase transformations of a single field
different transformations commute
ψ → ψ′ = exp(iqχ)ψ
∂µ → Dµ = ∂µ + iqAµ
Aµ → A′µ = Aµ − ∂µχ
Aµ transforms independent of q ⇒ photon is neutral
Strong and weak forces = non-Abelian gauge theories
local phase transformations involve more than one field ⇒ they do not commute
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
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 }
��
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 30 / 31
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
Martin Nagel (CU Boulder) Quantum Field Theories February 3, 2010 31 / 31
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