Reader for the course Quantum Field Theory W.J.P. Beenakker

Reader for the course Quantum Field Theory
W.J.P. Beenakker
2016 – 2017
Contents of the lecture course:
1) The Klein-Gordon field
2) Interacting scalar fields and Feynman diagrams
3) The Dirac field
4) Interacting Dirac fields and Feynman diagrams
5) Quantum Electrodynamics (QED)
This reader should be used in combination with the textbook
“An introduction to Quantum Field Theory” (Westview Press, 1995)
by Michael E. Peskin and Daniel V. Schroeder.
Contents
1 The Klein-Gordon field
1.1 Arguments in favour of Quantum Field Theory . . . . . . .
1.2 Lagrangian and Hamiltonian formalism (§ 2.2 in the book)
1.2.1 Noether’s theorem for continuous symmetries . . .
1.3 The free Klein-Gordon theory (real case, § 2.3 in the book)
1.4 Switching on the time dependence (§ 2.4 in the book) . . .
1.5 Quantization of the free complex Klein-Gordon theory . .
1.6 Inversion of the Klein-Gordon equation (§ 2.4 in the book)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
2 Interacting scalar fields and Feynman diagrams
2.1 When are interaction terms small? (§ 4.1 in the book) . . . . . . . . . . . .
2.2 Renormalizable versus non-renormalizable theories . . . . . . . . . . . . . .
2.3 Perturbation theory (§ 4.2 in the book) . . . . . . . . . . . . . . . . . . . .
2.4 Wick’s theorem (§ 4.3 in the book) . . . . . . . . . . . . . . . . . . . . . .
2.5 Diagrammatic notation: Feynman diagrams (§ 4.4 in the book) . . . . . . .
2.6 Scattering amplitudes (§ 4.6 in the book) . . . . . . . . . . . . . . . . . . .
2.7 Non-relativistic limit: forces between particles . . . . . . . . . . . . . . . .
2.8 Translation into probabilities (§ 4.5 in the book) . . . . . . . . . . . . . . .
2.8.1 Decay widths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8.2 Cross sections for scattering reactions . . . . . . . . . . . . . . . . .
2.8.3 CM kinematics and Mandelstam variables for 2 → 2 reactions . .
2.9 Dealing with states in the interacting theory (§7.1 in the book) . . . . . . .
2.9.1 Källén–Lehmann spectral representation . . . . . . . . . . . . . . .
2.9.2 2-point Green’s functions in momentum space (§ 6.3 and 7.1 in the
book) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9.3 Deriving n-particle matrix elements from n-point Green’s functions
2.9.4 The optical theorem (§ 7.3 in the book) . . . . . . . . . . . . . . . .
2.10 The concept of renormalization (chapter 10 in the book) . . . . . . . . . .
2.10.1 Physical perturbation theory (a.k.a. renormalized perturbation theory)
2.10.2 What has happened? . . . . . . . . . . . . . . . . . . . . . . . . . .
2.10.3 Superficial degree of divergence and renormalizability . . . . . . . .
3 The
3.1
3.2
3.3
3.4
Dirac field
Towards the Dirac equation (§ 3.2 and 3.4 in the book) .
Solutions of the free Dirac equation (§ 3.3 in the book) .
Quantization of the free Dirac theory (§ 3.5 in the book)
Discrete symmetries . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
1
3
7
11
17
19
21
26
26
27
31
34
38
43
50
53
53
54
57
59
61
65
70
74
76
81
82
83
85
89
94
97
104
4 Interacting Dirac fields and Feynman diagrams
4.1 Wick’s theorem for fermions . . . . . . . . . . . . . . . . . . .
4.2 Feynman rules for the Yukawa theory . . . . . . . . . . . . . .
4.2.1 Implications of Fermi statistics . . . . . . . . . . . . .
4.2.2 Drawing convention . . . . . . . . . . . . . . . . . . . .
4.2.3 Källén–Lehmann spectral representation for fermions .
4.2.4 Momentum-space Feynman rules for the Yukawa theory
4.3 How to calculate squared amplitudes . . . . . . . . . . . . . .
5 Quantum Electrodynamics (QED)
5.1 Electromagnetism . . . . . . . . . . . . . . . . . . . . . . .
5.2 QED and the gauge principle . . . . . . . . . . . . . . . .
5.2.1 Quantization of the free electromagnetic theory . .
5.3 Feynman rules for QED (§ 4.8 in the book) . . . . . . . . .
5.4 Full fermion propagator (§ 7.1 in the book) . . . . . . . . .
5.5 The Ward –Takahashi identity in QED (§ 7.4 in the book)
5.6 The photon propagator (§ 7.5 in the book) . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
107
107
108
108
110
111
112
113
.
.
.
.
.
.
.
115
115
117
120
124
125
126
129
1
The Klein-Gordon field
The first four lectures cover Chapter 2 of the textbook by Peskin & Schroeder. The relevant
conventions are listed on pages xix–xxi in the book, involving the use of so-called natural
units (~ = c = µ0 = ǫ0 = 1) by absorbing these constants in the relevant fields and
quantities. As a result, a single scale remains: mass. Please familiarize yourself with these
conventions and treat Chapter 1 as reading material, as recommended by the authors.
Throughout this reader you will encounter circled numbers. These numbers match the
markers listed in the outline of the course.
1.1
Arguments in favour of Quantum Field Theory
From particle–wave duality we know that the properties of e.g. electrons and photons are
similar: both objects give rise to diffraction phenomena and carry a particle-like punch.
Historically electromagnetism was first perceived as a field theory and its particle interpretation (photons) was observed later through the photo-electric effect. The other way
around, electrons were first perceived as elementary particles and the field aspects emerged
only once relativistic energies were considered.
1 Question: what is more fundamental, the fields (with particles being derived
quantities resulting from quantization) or the particles (with the fields being
derived quantities resulting from collective many-particle behaviour)?
There are four observations that support the former point of view.
1. Classical physics : as supported by experiment there should be no “action at a distance”, i.e. there should be no forces that are felt everywhere instantaneously. As
a result, the instantaneous laws of Newton and Coulomb had to be replaced by the
local laws of nature of Einstein and Maxwell, based on field theories! . . . However,
strictly speaking a locally defined particle approach is still possible.
2. Relativistic quantum mechanics: as supported by any high-energy collision experiment a relativistic one-particle quantum theory is not feasible. The number of
particles is not conserved, i.e. particles are not indestructible. This differs strongly
from non-relativistic quantum mechanics as formulated by Schrödinger, where massive particles are around forever and can thus be perceived as fundamental. Photons
are massless and are therefore always to be treated relativistically, so we have no
photon conservation.
Let’s recall what happened when we were trying to construct a relativistic quantum mechanical theory for a free particle in flat (Minkowskian) spacetime. The ingredients for the
construction were:
1
• A wave equation that keeps its form under Lorentz-transformations, as required by
the relativity principle.
• A correct quantum mechanical probability interpretation.
p
• The relativistic relation E = p~ 2 + m2 should be built in, in order to ensure that
particle–wave duality is properly incorporated.
The following problems were encountered:
• Negative-energy solutions, leading to an energy spectrum that is unbounded from
below. Dirac solved this for fermionic theories by demanding that the sea of negativeenergy states (Dirac sea) is occupied. Unwanted transitions are then forbidden provided that the exclusion principle applies, which is the case for fermions. However,
that means that the resulting one-particle theory has in fact an infinite number of
particles.
• At energies of the order of the particle mass, extra particles can be liberated from the
Dirac sea. In Dirac’s theory this is called particle–hole creation, which corresponds
to particle–antiparticle pair creation in quantum field theory.
In order to see at what length scales the breakdown of one-particle quantum mechanics
occurs we use the old units for a moment and consider a particle with mass m in a box with
size L. According to Heisenberg’s uncertainty relation the momentum of the particle then
has an uncertainty ∆p = O(~/L). This in turn leads to an uncertainty in the relativistic
p
energy E = p2 c2 + m2 c4 ≥ pc of roughly c∆p = O(~c/L). If this energy uncertainty
exceeds the energy threshold 2mc2 then pair creation may occur. This happens at length
scales L ≤ O(λc ), with λc = ~/(mc) the Compton wavelength. At these length scales
we cannot say anymore that we are dealing with a single particle, since it is accompanied
by a swarm of particle–antiparticle pairs, and a description with an unspecified number
of particles is required! Note that the Compton wavelength is smaller than the de Broglie
wavelength λb = h/p, which is the length scale where the wave-like nature of particles
becomes apparent.
1 The Compton wavelength is the length scale where even the concept of a
single point-like particle breaks down.
So, if we were to use a particle approach that is defined locally, it cannot be a single-particle
approach since multi-particle objects will unavoidably feature.
3. Many-particle quantum mechanics: the particle interpretation of a quantum mechanical theory can change radically in a different physical environment (cf. particles becoming waves in low-temperature superfluid 4 He, coherent states in a driven oscillator
system, . . . ). That means that the nature of particles can change!
2
4. The observation that all particles of the same type and in the same physical setting
are always the same everywhere . This hints at a description of physics that spans
all of space and time.
1.2
Lagrangian and Hamiltonian formalism (§ 2.2 in the book)
2a In order to set up quantum field theory we first consider classical field
theory in the Lagrangian and Hamiltonian formalism. The philosophy behind
this is that wave equations can be viewed as equations of motion for the wave
functions, i.e. the fields. This is best formulated in terms of Lagrangians for
continuous systems. Such Lagrangians are particularly suitable for discussing
symmetries, the cornerstones of relativistic quantum field theory.
Classical Lagrangian formalism: for a finite number of degrees of freedom the Lagrangian is given by
L {qj (t)}, {q̇j (t) = dqj (t)/dt}, t = T − V ,
where qj are generalized coordinates, T is the kinetic energy and V the potential energy.
Hamilton’s variation principle: classical solutions to the equations of motion (classical
Rt
paths) are obtained by finding the extrema of the action S = t12 dt L under synchronous
variations of the paths while keeping the endpoints fixed.
q
q(t2 )
Variation around the classical path
q(t)
for a free particle
qcl (t)
q(t1 )
t1
t
t2
The condition for a stationary action reads
Z t2
dt L = 0 for qj (t) → qj (t) + δqj (t) such that δqj (t1 ) = δqj (t2 ) = 0 .
δS = δ
t1
From this it follows that
t = t2 X Z t2 X Z t2 ∂L
X ∂L
∂L
d ∂L ∂L
dt
δqj = 0 .
δqj +
δ q̇j =
δqj
−
+
dt
∂q
∂
q̇
∂
q̇
∂q
dt
∂
q̇
j
j
j
j
j
t
t
1
1
t
=
t
1
j
j
j
3
This has to be true for all δqj , so from this the Lagrange equations follow:
∂L
d ∂L =
∀
.
∂qj
j dt ∂ q̇j
These are the equations of motion for a system without boundary conditions.
y
y
(xj , yj )
y(x)
−→
x
x
Figure 1: A classical, non-relativistic example of a continuous system.
Now we switch from a discrete set of particles to a field. A field is a dynamical system with
a continuous, infinite number of degrees of freedom, i.e. at least one degree of freedom for
each point in space. An example is given by the string in figure 1, in which case gradients
in x will enter V as elastic energy (see also Ex. 1). In the field-theory case the discrete
set of generalized coordinates {qj (t)} is replaced by a continuous generalized coordinate
φ(x), where x is a spacetime four-vector. In this way we treat ~x and t on equal footing,
as required for a relativistic approach. The Lagrangian L {qj }, {q̇j }, t is replaced by a
Lagrangian density L(φ(x), ∂µ φ), which depends on the generalized cooordinate φ(x) and
the corresponding four-velocity ∂µ φ(x). The fact that the derivates with respect to time
and space should be combined into a four-velocity ∂µ φ(x) is needed for a proper relativistic treatment, as we will see later on. In practice we only work with Lagrangian densities,
so we usually refer to L in a sloppy way as ‘the Lagrangian’.
Now that we have a Lagrangian, we need to formulate Hamilton’s variation principle for
continuous systems:
Z x2
Z t2 Z
4
δS = δ
dt d~x L(φ, ∂µ φ) ≡ δ
d x L(φ, ∂µ φ) = 0
t1
x1
|~
x|→∞
for φ(x) → φ(x) + δφ(x) such that δφ(x) −−−
−→ 0 and ∀ δφ(t1 , ~x ) = δφ(t2 , ~x ) = 0 .
~x
This means that the system evolves between two field configurations that are kept fixed at
the temporal and spatial boundaries of the four-dimensional integration region. The latter
requirement follows from the fact that we will consider systems with finite properties only.
4
From this variation principle it follows that
δS =
Z
x2
Z
x2
4
d x
x1
=
4
∂L
∂L
δφ +
δ(∂µ φ)
∂φ
∂(∂µ φ)
d x ∂µ
x1
Z x2
∂L
∂L
∂L
4
δφ = 0
δφ +
d x
− ∂µ
∂(∂µ φ)
∂φ
∂(∂µ φ)
x1
for all allowed variations δφ. According to Gauss’ divergence theorem, the first integral in
the final expression vanishes since it gives rise to an integral over the boundary of the fourdimensional integration region. The final result is the so-called Euler–Lagrange equation
for a stationary action:
∂L ∂L
=
.
∂µ
∂(∂µ φ)
∂φ
We get the same equation for each extra field occuring in L.
An immediate consequence of the variation principle is that the equation of motion (Euler–
Lagrange equation) does not change if we add a φ-dependent four-divergence to the Lagrangian: L → L + ∂µ Gµ . The reason is that this extra term adds a boundary contribution to S. Such a boundary contribution remains unaffected by a field variation with
fixed boundaries. Note that we have not considered the possibility of having terms in the
Lagrangian that couple φ(t, ~x ) to φ(t, ~y ). This follows from the locality requirement that
we have to impose on viable quantum field theories. As a result, only φ(x) and ∂µ φ(x)
occur.
Just like the Lagrangian, the Hamiltonian H in the discrete case becomes an integral of
the Hamiltonian density H in the continuous case:
Z
Z
X
∂φ
~ π) ≡ d~x π
−L ,
H {qj }, {pj } ≡
pj q̇j − L −→
d~x H(φ, ∇φ,
∂t
j
with the conjugate momenta for both cases defined as
pj ≡
∂L
∂ q̇j
−→
π ≡
∂L
.
∂(∂φ/∂t)
Note the preferred treatment of t with respect to ~x in the definition of H: ∂φ/∂t occurs
in the definition of π . That means that t and ~x are not treated on equal footing in the
Hamiltonian formalism, making the Hamiltonian formalism less suitable for dealing with
relativistic field theories than the Lagrangian formalism. We will need to know H, though,
for performing the quantization of the classical theory.
Example: consider the following Lagrangian containing a set of fields labeled by a ∈ N
L {φa }, {∂µ φa }
=
1 2 1 ~
1
1
1
φ̇a − (∇φa )2 − m2 φ2a = (∂µ φa )(∂ µ φa ) − m2 φ2a .
2
2
2
2
2
5
P 2
A summation convention is implied here, so φ2a =
a φa . Note that according to the
~
~ a ). Using
standard convention in the book ∂µ φa = (∂0 φa , ∇φa ) and ∂ µ φa = (∂0 φa , −∇φ
Einstein’s standard summation convention for repeated Minkowski indices, the Euler–
Lagrange equations then read
∂µ (∂ µ φa ) + m2 φa = ( + m2 )φa = 0 ,
i.e. all fields φa satisfy the familiar Klein-Gordon equation. The conjugate momenta and
the Hamiltonian density are given by
πa =
∂L
= φ̇a
∂ φ̇a
and
H = φ̇a πa − L =
1 2 1 ~
1
πa + (∇φa )2 + m2 φ2a .
2
2
2
The first (kinetic) term in the Hamiltonian density corresponds to the energy cost of “moving” in time, the second (elastic) term to the energy cost of “shearing” in space, and the
third (mass) term is the energy cost of having the field around at all. Note that in deriving
this Hamiltonian we sum over all fields in the term φ̇a πa . This makes sense, since all fields
φa are independent.
2b Question: apart from being local, what requirements do we have to impose
on the Lagrangian density of a relativistic quantum field theory?
Relativity principle: the guiding principle will be the relativity principle, which states
that in each inertial frame the physics should be the same. One option is to use a passive transformation to go from one inertial frame to the other. In that case we have
to find a relativistic wave equation that keeps its form under Lorentz transformations:
Df (x) = 0 ⇒ D ′ f ′ (x′ ) = 0, where D is a differential operator and f a field. The
prime indicates Lorentz-transformed objects. Alternatively, we can physically transform
all fields and demand the relativistic wave equation to be invariant. This is called an active
transformation. To phrase it differently, if a field satisfies the equation of motion, then the
same should hold for the Lorentz-transformed field:
Df (x) = 0
⇒
Df ′ (x) = 0 .
This is automatically guaranteed if the associated Lagrangian density L is a Lorentz scalar
field, since the action S will in that case be Lorentz invariant and therefore an extremum
of the action will indeed yield another extremum upon Lorentz transformation. Similar
arguments hold for constant translations x′ = x + x0 , where x0 is a constant four-vector.
Proof: in order to prove that the action is Lorentz invariant if L is a Lorentz scalar
field, we first give the official definition of a Lorentz scalar field. Consider to this end the
Lorentz transformation xµ → x′µ = Λµν xν , with Λ a continuous Lorentz transformation
tensor (describing rotations and boosts). Then φ(x) ∈ R is called a Lorentz scalar field
6
if it transforms as φ(x) → φ′ (x) = φ(Λ−1x) under the Lorentz transformation, i.e. the
transformed field evaluated at the transformed spacetime point gives the same value as
the original field in the spacetime point prior to the Lorentz transformation. The Jacobian
of this transformation is 1, since det Λ = 1 for a continuous Lorentz transformation.
Therefore, for a Lorentz scalar Lagrangian density L
scalar
L(x) −−−
−→ L′ (x) = L(Λ−1 x) ≡ L(y) ⇒
Z
Z
Z
Z
x = Λy , Jacobian= 1
4
4
4
′
′
S = d x L(x) → S = d x L (x) = d x L(y) ============== d4 y L(y) = S .
Note, though, that the endpoints t1 and t2 of the temporal integration interval will change
under boosts.
1.2.1
Noether’s theorem for continuous symmetries
3 As a next ingredient for setting up quantum field theory we will try to identify conserved currents and “charges” that are present in the theory. These
conserved charges are instrumental in quantizing the theory and finding its particle interpretation.
Consider a field φ(x) that satisfies the Euler–Lagrange equation of L(φ, ∂µ φ) and apply
the infinitesimal continuous transformation
φ(x) → φ′ (x) = φ(x) + α∆φ(x) ,
with α independent of x and infinitesimal .
We speak of a symmetry under this transformation if L(x) changes by a four-divergence:
L(x) → L(x) + α∂µ Gµ (x), since that implies that the equation of motion is left invariant
(cf. the remark on page 5). In that case
∂L
∂L
∆(∂µ φ)
∂φ
∂(∂µ φ)
∂L
∂L ∂L
= α ∂µ
∆φ .
∆φ + α
− ∂µ
∂(∂µ φ)
∂φ
∂(∂µ φ)
α∂µ Gµ = α∆L = α
∆φ +
The second term is zero due to the Euler–Lagrange equation, so we are left with
∂µ
∂L
∆φ − Gµ ≡ ∂µ j µ = 0 ,
∂(∂µ φ)
i.e. j µ is a conserved current. This is trivially extended to cases with more fields and
automatically leads to
Noether’s theorem : for each continuous symmetry there is a conserved current.
7
This theorem has two important consequences:
R
• The “charge” Q(t) = d~x j 0 (x) is conserved globally if ~j(x) vanishes for |~x | → ∞.
Proof: if ~j(x) vanishes for |~x| → ∞ we have
Z
Z
Z
∂j 0 ∂µ j µ = 0
dQ(t)
Gauss
~
~
====== − d~x ∇ · j ====== − d~s · ~j = 0 .
=
d~x
dt
∂t
• More importantly this charge conservation also holds locally!
Proof: following the previous case
Z
Z
Z
d
d
Gauss
0
~
~
d~x j (x) = − d~x ∇ · j ===== −
QV (t) ≡
d~s · ~j .
dt
dt
V
V
S(V )
In other words: any charge leaving the closed volume V must be accounted for by
an explicit flow of the current ~j through the surface S(V ) of V .
Translation symmetry: from imposing the relativity principle we know that L(x) should
be a Lorentz scalar, so under an infinitesimal translation
xν → x′ν = xν − αν
where αν is a constant infinitesimal four-vector
we have
L(x) → L(x + α) ≈ L(x) + αν ∂ν L(x) = L(x) + αν ∂µ g µν L(x) .
The last term is a total four-divergence, so relativistic field theories have translation sym
ν
metry with Gµ (x) = g µν L(x) for all four independent translations labeled by ν. Now
suppose that L depends on an arbitrary collection of fields fa (x) that transform as
ν
fa (x) → fa (x + α) ≈ fa (x) + αν ∂ νfa (x) ≡ fa (x) + αν ∆fa (x) ,
which is valid for all components of viable quantum fields. This results in four conserved
currents:
∂L ∂ νfa − g µν L
(ν = 0, · · · , 3) ,
T µν ≡
∂(∂µ fa )
and hence four conserved charges:
Z
Z
Z
Z
h
i
∂L ˙
00
d~x T =
d~x
fa − L =
d~x πa f˙a − L =
d~x H = H ,
∂ f˙a
Z
Z
Z
∂L j
0j
∇ fa − 0 = − d~x πa ∇jfa ≡ P j .
d~x T = − d~x
∂ f˙a
Summation over a is again implied. The quantity T µν is called the stress-energy tensor or
energy-momentum tensor, H is the physical energy carried by the fields fa , and P j is the
j th component of the physical momentum carried by the fields fa . We will see later that
what we just did does not just hold for scalar fields, but also for any component of a vector,
spinor, ... field.
8
3a The field energy H will play a crucial role in the quantization of free field
theories, since it will feature in the quantum mechanical requirement of having
an energy spectrum that is bounded from below. On top of that it determines
the quantum mechanical time evolution. The field momentum will help us in
determining the particle interpretation of free quantum field theories.
A bit of cosmology (based on “Cosmological Physics” by John A. Peacock).
In general relativity it is important to have an energy-momentum tensor Θµν that is symmetric under the interchange of µ and ν . In the modified Einstein equation including
cosmological constant:
1
(G = Newton’s constant) ,
Rµν − gµν R + Λgµν = − 8π GΘµν
2
this symmetrized energy-momentum tensor describes matter and energy in the universe,
whereas the Ricci tensor Rµν , scalar curvature R = g ρσ Rρσ and cosmological constant Λ
describe the “structure” of spacetime for an empty space (i.e. for the vacuum). In general
it is not guaranteed that T µν is symmetric under the interchange of µ and ν . However,
this can be arranged by adding an appropriate extra term ∂ρ K ρµν with K ρµν = − K µρν
such that Θµν = T µν + ∂ρ K ρµν = Θνµ and ∂µ Θµν = ∂µ T µν + ∂µ ∂ρ K ρµν = ∂µ ∂ρ K ρµν = 0.
For an explicit example, see Ex. 2.1 in the textbook by Peskin & Schroeder.
By bringing the cosmological-constant term to the right-hand side of the modified Einstein
equation, it can be viewed as representing the energy density and pressure (i.e. momentum
current density) of empty space itself. Such a term therefore constitutes the vacuum
contribution to the curvature. Since the vacuum should have no preferred frame, the
cosmological-constant term should be the same for all observers in special relativity, which
is guaranteed by its proportionality to the metric tensor. After all, the metric tensor is
invariant under Lorentz transformations:
g µν → Λµρ Λνσ g ρσ = Λµρ Λνρ = Λµρ (Λ−1 )ρν = g µν .
So, the presence of such a cosmological-constant term does not conflict with any firstprinciple requirements!
For a positive cosmological constant (Λ > 0) the energy density of the vacuum
is positive and the associated pressure is negative, resulting in an accelerated
expansion of empty space as seems to be supported by experiment (see later).
Such a vacuum energy is usually referred to as dark energy.
The reason why the pressure is negative follows from the simple fact that energy is released if the volume of space expands, whereas a positive “pressure on space” would require work to be exerted during the expansion. So, for a positive cosmological constant
the vacuum represents an unlimited energy reservoir , which is tapped when the universe
inflates.
9
Symmetry under rotations and boosts (continuous Lorentz transformations):
under an infinitesimal continuous Lorentz transformation
xρ → x′ρ = Λρσ xσ ≈ xρ + ω ρσ xσ ,
where ω ρσ = − ω σρ ∈ R is an infinitesimal tensor with six independent components. The
Lagrangian is a Lorentz scalar, so
L(x) → L(Λ−1x) ≈ L(x − ωx) ≈ L(x) − ω ρσ xσ ∂ρ L(x)
ω ρρ = 0
===== L(x) − ω ρσ ∂ρ xσ L(x) = L(x) − ωρσ ∂µ g µρ xσ L(x)
1
ωρσ = − ωσρ
======== L(x) − ωρσ ∂µ [g µρ xσ − g µσ xρ ]L(x) .
2
Since L(x) changes by a total four-divergence, relativistic field theories have a symmetry
ρσ
under continuous Lorentz transformations with Gµ (x)
= − [g µρ xσ − g µσ xρ ]L(x) for all
six independent components of ωρσ .
Now consider a Lagrangian for a scalar field φ(x). Such a field transforms as
φ(x) → φ′ (x) = φ(Λ−1x) ≈ φ(x) −
ρσ
1
1
ωρσ [xσ ∂ ρ − xρ ∂ σ ]φ(x) ≡ φ(x) + ωρσ ∆φ(x)
.
2
2
This results in six conserved currents, one for each independent component of ωρσ :
J µρσ (x) =
∂L ρ σ
[x ∂ − xσ ∂ ρ ]φ(x) + [g µρ xσ − g µσ xρ ]L(x) = T µσ (x)xρ − T µρ (x)xσ ,
∂(∂µ φ)
and hence six conserved “charges”:
R • Rotations (ρ, σ = i, j ): J k ≡ 12 ǫijk d~x T 0j (x)xi − T 0i (x)xj , with summation over
the spatial indices i and j implied. This is the k th component of the physical angular
momentum carried by the field φ(x).
R R
• Boosts (ρ, σ = 0, i): K i ≡ d~x T 0i (x)x0 − T 00 (x)xi = x0 P i − d~x xi T 00 (x).
d 0 i R
Conservation of these three “charges” implies that
x P − d~x xi T 00 (x) =
dt
R
d R
i 00
i
00
d~x x T (x) = 0. Since d~x T (x) = H, this equation can be interpreted
P −
dt
as saying that the “centre-of-energy” of the field travels at constant velocity, in
analogy with the movement of the centre-of-mass of a free classical system.
3a The angular momentum of a field depends on the type of field and will thus
be useful after quantization. It will help us to determine the intrinsic spin of
the particles described by the free quantum field theory that corresponds to a
given wave equation. As a result of the relativity principle, each type of wave
equation will give rise to a specific particle spin.
10
Abelian internal symmetry (“global U (1) gauge symmetry”): an internal symmetry involves a transformation of the fields that acts in the same way at every spacetime
point, whereas abelian implies multiplying all fields by a constant phase factor. Consider
a complex scalar field φ(x) that satisfies the Euler–Lagrange equations of the Lagrangian
L(φ, φ∗, ∂µ φ, ∂µ φ∗ ) = (∂µ φ)(∂ µ φ∗ ) − m2 φφ∗ .
This Lagrangian is invariant under the continuous transformation φ → eiα φ , φ∗ → e−iα φ∗ ,
where α ∈ R is a constant. This implies that under the infinitesimal version of this
transformation, i.e.
φ → φ + α(iφ) ≡ φ + α∆φ
and
φ∗ → φ∗ + α(−iφ∗ ) ≡ φ∗ + α∆φ∗ ,
we get ∆L = 0 ⇒ Gµ = 0. As a result the current
j µ = iφ∂ µ φ∗ − iφ∗ ∂ µ φ = i (∂ µ φ∗ )φ − φ∗ ∂ µ φ
is conserved. For an extended example of a gauge symmetry see Ex. 3.
3b We will see later that the conserved charge arising from currents of this type
have the interpretation of electric charge or particle number. The associated
U(1) gauge symmetry will feature prominently in a symmetry-based description
of electromagnetic interactions.
Symmetries versus unobservable quantities: the above-given symmetries are in fact
all related to quantities that are fundamentally unobservable.
3 The abelian internal symmetry is linked to the unobservability of the absolute
phase of a QM wave function. Translation and rotation symmetry are the
result of the unobservability of the absolute position and direction in spacetime.
Symmetry under boosts is related to the unobservability of the absolute velocity
of a chosen reference frame.
1.3
The free Klein-Gordon theory (real case, § 2.3 in the book)
We start our tour of the relativistic quantum-field-theory world with the simplest example:
the quantum field theory for real scalar fields that satisfy the free Klein-Gordon (KG)
equation. The classical Lagrangian for a real scalar field φ(x) that satisfies the free KG
equation is given by
1
1
∂L
Euler-Lagrange
L = (∂µ φ)(∂ µ φ) − m2 φ2 =========⇒ ( + m2 )φ(x) = 0 , π =
= φ̇ .
2
2
∂ φ̇
The corresponding time-independent Hamiltonian reads (cf. page 6)
Z
h1
1 ~ 2 1 2 2i
2
π + (∇φ) + m φ .
H = d~x
2
2
2
11
4a Question: how should we quantize such a classical field theory?
1) Canonical quantization: in principle we could approach this in the same way as in the
case of the quantization of Newtonian mechanics: the dynamical coordinates and associated
conjugate momenta become operators that satisfy canonical commutation relations. In the
Schrödinger picture this reads
• Discrete quantum mechanics: q̂j , p̂k = iδjk 1̂ ,
q̂j , q̂k = p̂j , p̂k = 0 .
• Continuous quantum field theories: φ̂j (~x ), π̂k (~y ) = iδjk δ(~x − ~y )1̂ ,
φ̂j (~x ), φ̂k (~y ) = π̂j (~x ), π̂k (~y ) = 0 .
Subsequently, the fully covariant (time-dependent) versions of these commutation relations
can be obtained by switching to the Heisenberg picture. This type of quantization procedure is called canonical quantization.
2) Quantizing an infinite number of linear harmonic oscillators: in a general quantum field theory the spectrum of Ĥ is hard to find, since it involves an infinite number of
degrees of freedom that in general do not evolve independently. However, in the case of free
theories each degree of freedom does evolve independently. The reason behind this is that
the corresponding equations of motion are linear wave equations, with all quantum fields
as well as their individual components satisfying the KG equation. This latter requirement
is needed in order to implement particle-wave duality in the right way by giving rise to
the correct relation between energy and momentum for the free particles described by the
theory. Consider now such a field component f (~x, t) ∈ R with ( + m2 )f (~x, t) = 0. In
order to decouple the degrees of freedom we use the momentum representation (Fourier
decomposition)
Z
d~p i~p·~x
e g(~p, t) ,
f (~x, t) ≡
(2π)3
so that the KG equation changes into
∂2
2
2
+
(~
p
+
m
)
g(~p, t) = 0
∂t2
for each Fourier-mode p~ . This means that g(~p, t) solves the equation of motion of a harp
monic oscillator vibrating at a frequency ωp~ ≡ p~ 2 + m2 . The most general solution to
the KG equation will therefore be a linear superposition of simple harmonic oscillators, each
with a different amplitude and frequency. So, in order to quantize f (~x, t), one simply has to
quantize the infinite number of oscillators in terms of raising (creation) and lowering (annihilation) operators. The associated harmonic energy quanta are interpreted as particles.
12
Next it will be proven that both procedures are actually equivalent.
Comparing both procedures: let’s first recall how the quantization of a linear harmonic
oscillator goes. Consider to this end the corresponding Hamilton operator
1
1
1
p̂2
+ mω 2 x̂2 ≡ P̂ 2 + ω 2Q̂2
with
Q̂, P̂ = i 1̂ ,
2m
2
2
2
√
√
using P̂ = p̂/ m and Q̂ = x̂ m. Next we introduce a lowering operator â and raising
operator â† according to
Ĥ =
â + â†
Q̂ ≡ √
2ω
â − â†
P̂ ≡ − iω √
.
2ω
From this the fundamental bosonic commutation relation â, â† = 1̂ follows and
,
1
ω (â† â + ââ† ) = ω (â† â + 21 1̂) ≡ ω (n̂ + 12 1̂) ,
2
where n̂ = â† â can be interpreted as a counting operator. Using Ĥ, â† = ω â†, the energy
eigenvalues En and eigenfunctions |ni of this Hamilton operator can be obtained:
Ĥ =
En = (n + 21 )ω
(â† )n
|ni ≡ √ |0i
n!
,
(n = 0, 1, · · ·) .
Based on this we use the following ansatz for the quantized KG field and its conjugate
momentum in terms of a continuous set of oscillator modes labeled by p~ :
φ̂(~x ) =
Z
†
d~p âp~ + â−~p i~p·~x
p
e
=
(2π)3
2ωp~
π̂(~x ) = − i
Z
Z
d~p
1 † −i~
p·~
x
i~
p·~
x
p
= φ̂† (~x ) ,
e
â
e
+
â
p
~
p
~
(2π)3 2ωp~
âp~ − â†−~p i~p·~x
d~p
e
= −i
ωp~ p
(2π)3
2ωp~
Z
d~p
(2π)3
r
ωp~ † −i~
p·~
x
i~
p·~
x
= π̂ † (~x ) ,
âp~ e − âp~ e
2
with âp~ , â†p~ ′ = (2π)3 δ(~p − ~p ′ ) 1̂ and all other commutators 0.
Let’s now see whether we have succeeded in properly quantizing and decoupling the free
real KG theory. From the fundamental bosonic commutation relations for creation and
annihilation operators it follows that
Z
r
i d~p d~p ′ ωp~ ′ i(~p·~x+~p ′·~y ) e
âp~ + â†−~p , âp~ ′ − â†−~p ′
φ̂(~x ), π̂(~y ) = −
6
2 (2π)
ωp~
Z
i
=
1̂ d~p ei~p·(~x−~y ) = iδ(~x − ~y ) 1̂ ,
(2π)3
in agreement with canonical quantization.
13
Energy spectrum and zero-point energy: the Hamilton operator of the free real KG
theory now reads
Z
1 2 1 ~ 2 1 2 2
π̂ + (∇φ̂) + m φ̂
Ĥ = d~x
2
2
2
=
Z
d~x
Z
′
d~p d~p ′ ei~x·(~p +~p ) h
− ωp~ ωp~ ′ (âp~ − â†−~p )(âp~ ′ − â†−~p ′ )
√
6
(2π) 4 ωp~ ωp~ ′
Z
+ (m2 − ~p · p~ ′ )(âp~ + â†−~p )(âp~ ′ + â†−~p ′ )
d~p 1
ωp~ (âp~ â†p~ + â†−~p â−~p )
(2π)3 2
Z
Z
d~p 1
d~p
p
~→−p
~ in 2nd term
†
â
+
ω
â
ωp~ (2π)3 δ(~0 )1̂ ,
=============
p
~
p
~
p
~
(2π)3
(2π)3 2
~x integral
=======
i
which is indeed nicely decoupled and properly time-independent. The last term in the final
expression is called the zero-point energy. It is a consequence of the uncertainty principle
and represents the ground-state energy in the absence of any oscillator quanta.
4b Question: have we obtained an energy spectrum that is bounded from below?
From the decoupled form of the Hamilton operator of the free real KG theory we can read
off that
• the energy spectrum is indeed bounded from below by the zero-point energy;
• only positive-energy quanta feature in the Hamilton operator;
• the zero-point energy is infinite:
R
i~
x·~
p
~
– We have (2π) δ(0 ) = d~x e 3
p
~ = ~0
= lim
RL
L→∞ −L
i~
x·~
p
d~x e
p
~ = ~0
= lim
RL
L→∞ −L
d~x = V.
This is an infinity originating from the fact that space is infinite. Such a longdistance infinity is often referred to as an infra-red (IR) divergence , since it is
related to p~ = ~0 .
Z
d~p 1
– The zero-point energy density
ωp~ is still infinite, originating from the
(2π)3 2
|~p | → ∞ limit of the integrand. This type of infinity is called ultra-violet (UV)
divergence , being related to short distances/high frequencies. It is the consequence of our unrealistic assumption that the theory is valid up to arbitrarily
high energies. As we will see later, the p~ -integral should be cut off at a value
where the theory breaks down or a more fancy technique should be used to
quantify the UV infinity if we do not want to introduce a new energy scale.
14
4d The zero-point energy is inessential for the particle interpretation, but it
is measurable in bounded sytems through the Casimir effect (as is explained in
the bachelor course “Kwantummechanica 3”) and it has explicit cosmological
implications in view of the fact that it contributes to the cosmological constant.
About 68% of the energy density in the universe bears the characteristics of a cosmological
constant with energy scale 10−3 eV, which is surprisingly small. With the Planck mass
Mpl = O(1028 eV) being the natural scale of gravity, where ordinary quantum field theory
most likely breaks down, we would expect the energy scale belonging to the cosmological
constant to be O(Mpl ) if it has a gravitational origin. One of the big questions in presentday high-energy physics therefore reads “Why is the cosmological constant so small?”.
The art of covering up: normal ordering.
In most textbooks all issues related to properties of the vacuum of the theory are simply
circumvented by removing vacuum energies, charges, etc. .1 This is done by applying
normal ordering, i.e. bringing all creation operators to the front:
Z
d~p
†
†
†
†
†
†
ωp~ â†p~ âp~ .
â â → N(â â) = â â , ââ → N(ââ ) = â â ⇒ N(Ĥ) =
(2π)3
After quantization the momentum carried by the KG field (cf. page 8) becomes
Z
Z
d~p i
ˆ
~
~
P = − d~x π̂(~x )∇φ̂(~x ) =
(−i~p )(âp~ − â†−~p )(â−~p + â†p~ )
3
(2π) 2
Z
Z
1
d~p
d~p
†
†
†
†
= −
~p (â−~p â−~p − âp~ âp~ + â−~p âp~ − âp~ â−~p ) =
p~ â†p~ âp~ ,
3
3
2 (2π)
(2π)
where in the last step we have taken ~p → −~p in the first term and we have used that
p~ âp~ â−~p , ~p â†−~p â†p~ and p~ (2π)3 δ(~0 ) are all odd under p~ → −~p whereas the integration
is even. This time there is no zero-point contribution and as such there is no need for
normal ordering, which is consistent with the fact that quantum fluctuations should have
no preferred direction.
Particle interpretation of the free real KG theory:
4d In the next step we determine the particle interpretation of the theory,
ˆ
mostly by simply reading it off from N(Ĥ) and P~ .
• Vacuum (ground state): |0i such that h0|0i = 1 and âp~ |0i = 0 for all p~ . Then
ˆ
N(Ĥ)|0i = 0 and P~ |0i = ~0, i.e. the vacuum “has” energy E = 0 and momentum
P~ = ~0.
1
The tacit assumption here is that some underlying (high-scale) physics takes care of this
15
• Excited states: obtained as (constant) ∗ â†p~ â†~q · · · |0i. Then E = ωp~ + ω~q + · · · and
ˆ †
~ â = p~ â† .
P~ = p~ + ~q + · · · , which follows from Ĥ, â†p~ = ωp~ â†p~ and P,
p
~
p
~
√
– For higher excitations â†p~ is replaced by (â†p~ )n / n! .
– The excitations are interpreted as particles.
– In view of the bosonic commutation relations for the associated creation and
annihilation operators these particles are bosons.
– In fact the particles are spin-0 bosons. This follows from Jˆk , â~†0 = 0 for
k = 1, 2, 3. Bearing in mind that a zero-momentum particle does not give rise
to an orbital angular momentum, this indeed implies that the particles in the
real KG theory also carry no intrinsic angular momentum.
Proof: the quantized version of the angular momentum derived on page 10 yields
Z
k †
†
i
j
ijk
ˆ
d~x π̂(x)∇ φ̂(x)x , â~0
J , â~0 = ǫ
Z
Z
1 ijk
j
= ǫ
d~x x
2
Z
Z
1 ijk
j
d~x x
= ǫ
2
d~p d~p ′ i
p
(2π)6
r
d~p d~p ′ i
p
(2π)3
r
ωp~ ′ i~x·(~p +~p ′ ) e
(âp~ ′ − â†−~p ′ )(âp~ + â†−~p ), â~†0
ωp~
ωp~ ′ i~x·(~p +~p ′ ) †
†
′
e
δ(~p )[âp~ + â−~p ] + δ(~p )[âp~ ′ − â−~p ′ ] .
ωp~
The second term in the last expression vanishes trivially. The first term vanishes
as well since i 6= j and consequently the xi integral will be proportional to δ(pi ).
– An example of such a particle is the π 0 pion.
Normalization of states and completeness relation: note that we did not specify
yet what normalization factor to use in the definition of the 1-particle states. Unlike what
is done in non-relativistic quantum mechanics, where the normalization factor is usually
taken to be 1, we will use a relativistically motivated normalization of the 1-particle states:
p
√
√
|~p i ≡
2ωp~ â†p~ |0i ⇒ h~p |~q i = 2 ωp~ ω~q h0|âp~ â†~q |0i = 2 ωp~ ω~q h0| âp~ , â†~q |0i
= 2ωp~ (2π)3 δ(~p − ~q ) .
The latter expression is invariant under continuous Lorentz transformations.
Proof: in order to prove this statement we first derive the important integration identity
Z
Z 4
d~p 1
dp
=
δ(p2 − m2 )Θ(p0 ) ( Lorentz invariant integration measure ) . (1)
3
3
(2π) 2ωp~
(2π)
We get this identity by using that
X δ(x − xj )
δ h(x) =
|h′ (xj )|
j
for h(xj ) = 0 and h′ (xj ) 6= 0 ,
16
which leads to
δ(p0 − ωp~ )
δ(p0 + ωp~ )
δ(p2 − m2 ) = δ p20 − [~p 2 + m2 ] = δ(p20 − ωp~2 ) =
+
.
2ωp~
2ωp~
Since p0 cannot change sign for p2 > 0, the right-hand-side of equation (1) only contains
Lorentz invariant objects. As a result, the expression on the left-hand-side is Lorentz
invariant as well and the same goes for h~p |~q i, since
Z
Z
d~p h~p |~q i
= d~p δ(~p − ~q ) = 1.
(2π)3 2ωp~
The 1-particle completeness relation is then given by
Z
d~p 1
|~
p
ih~
p
|
=
1̂
3
1-particle subspace
(2π) 2ωp~
since
∀
~q
Z
d~p 1
|~p ih~p |~q i =
(2π)3 2ωp~
Z
d~p 2ωp~ (2π)3 δ(~p − ~q )
|~p i = |~q i .
(2π)3
2ωp~
Finally we may ask the question what state is actually created by φ̂(~x ) = φ̂† (~x ). Letting
this operator act on the vacuum one obtains
Z
Z
d~p ei~p·~x
d~p e−i~p·~x
p~ → − p~
†
p
|~p i .
φ̂(~x )|0i =
(âp~ + â−~p )|0i ======
(2π)3 2ωp~
(2π)3 2ωp~
From this it can be concluded that a particle is created at position ~x, since
Z
d~p e−i~p·~x
h~q |~p i = e−i~q·~x
h~q |φ̂(~x )|0i =
(2π)3 2ωp~
is indeed identical to h~q |~x i in non-relativistic quantum mechanics.
1.4
Switching on the time dependence (§ 2.4 in the book)
4c Next we add the time dependence by switching to the Heisenberg picture,
which makes all operators time dependent according to Ô → Ô(t) ≡ eiĤt Ôe−iĤt
as expected from the fact that Ĥ is the generator of time translations. This
implies that the canonical (equal-time) commutation relations have the same
form as in the Schrödinger picture: φ̂(~x, t), π̂(~y , t) = iδ(~x − ~y )1̂ , with all
other commutators being 0.
Short derivation of φ̂(x) = φ̂(~
x, t): we have Ĥ, âp~ = − ωp~ âp~ ⇒ Ĥâp~ = âp~ (Ĥ−ωp~ )
and Ĥ n âp~ = âp~ (Ĥ −ωp~ )n . That means that eiĤt âp~ e−iĤt = âp~ ei(Ĥ−ωp~ )t e−iĤt = âp~ e−iωp~ t
and eiĤt â†p~ e−iĤt = â†p~ eiωp~ t . Applied to φ̂(x) this yields:
Z
1
∂
d~p
† ip·x −ip·x
p
φ̂(x) ,
and π̂(x) =
(âp~ e
+ âp~ e )
φ̂(x) =
3
(2π)
∂t
p0 = ωp~
2ωp~
17
where the first term corresponds to the positive frequency modes and the second term
to the negative frequency modes. This reflects particle-wave duality, with each frequency
mode corresponding to the creation/annihilation of fundamental quanta of the theory.
Analogously:
ˆ ˆ
ˆ
~ âp~ = − p~ âp~ ⇒ e−iP~ ·~x âp~ eiP~ ·~x = âp~ ei~p·~x .
P,
Combining both identities yields
~ˆ
~ˆ
φ̂(x) = ei(Ĥt − P ·~x ) φ̂(0)e−i(Ĥt − P ·~x ) = eiP̂ ·x φ̂(0)e−iP̂ ·x ,
as we would expect from the fact that P̂ µ is the generator of spacetime translations. Next
we invoke the following relativistic requirement.
4c Causality: a measurement performed at one spacetime point y can only
affect a measurement at another spacetime point x whose separation from the
first point is timelike or lightlike, i.e. (x − y)2 ≥ 0.
This latter requirement means that in such cases a particle can physically travel the corresponding spatial distance within the corresponding time period, since (x − y)2 ≥ 0
corresponds to a spacetime separation inside or on the lightcone |~x − ~y | = |x0 − y 0 |. In
the coordinate representation any observable involving scalar particles can be written in
terms of KG fields. So, if φ̂(x), φ̂(y) = 0 for (x − y)2 < 0, then the measurements do
not influence each other for spacelike separations (i.e. outside the lightcone) and causality
is preserved.
For the real KG field we find
Z
d~p d~q
1
† ip·x
† iq·y −ip·x
−iq·y
âp~ e
+ âp~ e , â~q e
+ â~q e
φ̂(x), φ̂(y) =
√
(2π)6 2 ωp~ ω~q
p0 = ωp~ , q0 = ωq~
Z
d~p 1
−ip·(x−y)
ip·(x−y)
e
−
e
1̂ ≡ D(x − y) 1̂ − D(y − x) 1̂ .
=
(2π)3 2ωp~
p0 = ωp~
The function D(x) has the following properties:
1. In the previous expression each of the terms on the left-hand-side of the second line
is Lorentz invariant according to equation (1). As a result, the function D(x) is
Lorentz invariant as well and hence D(x) = D(Λx) ≡ D(x′ ).
2. D(x) = D(−x) if x0 = 0. This follows directly by taking ~p → − p~ in the integration.
Bearing in mind that for (x − y)2 < 0 there exists a Lorentz transformation Λ such that
x′0 − y0′ = 0, we can derive from these two properties that
property 2
0 ======= D(x′ − y ′) − D(y ′ − x′ ) = D Λ(x − y) − D Λ(y − x)
property 1
======= D(x − y) − D(y − x)
18
if (x − y)2 < 0 .
This automatically implies that causality is preserved in the real KG theory because propagation from y to x, given by h0|φ̂(x)φ̂(y)|0i = D(x − y), is indistinguishable from propagation from x to y , given by h0|φ̂(y)φ̂(x)|0i = D(y −x), if (x−y)2 < 0. This sounds weird,
but in the spacelike regime we cannot think of propagation as particle movement. There
is no Lorentz invariant way to order events, since if we have in one frame that x0 − y0 > 0
a Lorentz transformation can yield another frame where x0 − y0 < 0.
4 In fact, quantizing using canonical quantization conditions was already sufficient for properly implementing causality. In spite of its non-covariant form,
there is no preferred treatment of time by quantizing in the canonical way!
Proof: the proof of this statement exploits the fact that φ̂(x), φ̂(y) is Lorentz invariant,
as well as the fact that for (x − y)2 < 0 there exists a Lorentz transformation Λ such that
x′0 − y0′ = 0. Then we can readily obtain the causality requirement
Lor. inv. ′
′
φ̂(x), φ̂(y) ======= φ̂(~x ′ , t′ ), φ̂(~y ′ , t′ ) = eiĤt φ̂(~x ′ ), φ̂(~y ′ ) e−iĤt = 0
for (x − y)2 < 0 as a direct consequence of canonical quantization.
1.5
Quantization of the free complex Klein-Gordon theory
The Lagrangian for a complex scalar field φ(x) satisfying the free KG equation is given by
L = (∂µ φ)(∂ µ φ∗ ) − m2 φφ∗ ,
which contains twice as many degrees of freedom as the Lagrangian of the real KG theory.
√
This can be seen explicitly by writing φ = (φ1 + iφ2 )/ 2 with φ1,2 ∈ R (see Ex. 4). Then
the Lagrangian becomes
L =
1
1
1
1
(∂µ φ1 )(∂ µ φ1 ) − m2 φ21 + (∂µ φ2 )(∂ µ φ2 ) − m2 φ22 .
2
2
2
2
Now we can either treat φ1,2 or φ, φ∗ as independent degrees of freedom. The quantization
goes exactly as before, with √12 (â1, p~ + iâ2, p~ ) ≡ âp~ and √12 (â†1, p~ + iâ†2, p~ ) ≡ b̂†p~ 6= â†p~ . Hence:
φ̂(x) =
Z
d~p
1
† ip·x −ip·x
p
,
(âp~ e
+ b̂p~ e )
(2π)3 2ωp~
p0 = ωp~
where the first term corresponds to particles and the second to so-called antiparticles. The
associated commutators are given by:
âp~ , â†~q
= b̂p~ , b̂†~q = (2π)3 δ(~p − ~q ) 1̂,
19
with all other commutators being 0 .
From these commutation relations we can derive that causality is conserved in the complex
Klein-Gordon theory as well:
φ̂(x), φ̂(y) = φ̂† (x), φ̂† (y) = 0 ,
see before
φ̂(x), φ̂† (y) = D(x − y) 1̂ − D(y − x) 1̂ ======= 0 if (x − y)2 < 0 .
Note that D(x − y) originates from particle propagation, whereas D(y − x) originates from
antiparticle propagation. This brings us to the following important conclusion:
4c the correct causal structure of the complex Klein-Gordon theory hinges on
the combined treatment of particles and antiparticles, since particle propagation
from y to x, h0|φ̂(x)φ̂† (y)|0i = D(x − y), is indistinguishable from antiparticle
propagation from x to y, h0|φ̂† (y)φ̂(x)|0i = D(y − x), if (x − y)2 < 0.
Particle interpretation: as before we can derive the particle interpretation by looking
at the energy, momentum and “charge” operators (see Ex. 4). After quantization these
operators read:
Z
d~p
ωp~ (â†p~ âp~ + b̂†p~ b̂p~ ) + zero-point energy ,
Ĥ =
(2π)3
Z
d~p
ˆ
P~ =
~p (â†p~ âp~ + b̂†p~ b̂p~ ) ,
(2π)3
Z
d~p
Q̂ ∝
(â†p~ âp~ − b̂†p~ b̂p~ ) = N̂particles − N̂antiparticles .
3
(2π)
The zero-point term for the charge operator has to vanish to guarantee Lorentz-invariant
vacuum properties (see the master thesis by Susanne Lepoeter), so normal ordering is a
physical requirement in that case! Since all expressions only contain number operators, we
have
ˆ
Ĥ, P~ = Ĥ, Q̂ = Ĥ, N̂particles = Ĥ, N̂antiparticles = 0 .
4d In free KG theories (in fact in all free theories) energy, momentum, number
of particles and number of antiparticles are all conserved. In interacting theories
the number of particles and the number of antiparticles are no longer separately
conserved, but their difference quite often is.
Now we can read off the particle interpretation of the complex KG theory: it resembles
the one for the real KG theory, with the difference being that for every particle state there
should now also be an antiparticle state with opposite “charge” quantum numbers and the
same 4-momentum quantum numbers. An example of such a scalar particle–antiparticle
combination is given by the π ± pions. The case φ̂ = φ̂† is special in the sense that particle
and antiparticle states coincide, so all “charges” should be 0.
20
Lorentz transformations and φ̂(x): as before φ̂(x) = eiP̂ ·x φ̂(0)e−iP̂ ·x, but what about
p
Lorentz transformations? We know that |~p i = 2ωp~ â†p~ |0i and that a similar expression holds for antiparticle states, so we can use this to define the unitary operator that
implements (active) Lorentz transformations in the Hilbert space of quantum states:
q
p
−
→
Û (Λ)|0i ≡ |0i p
†
→ â → |0i =
|Λpi ≡ Û (Λ)|~p i ⇒ 2ω−
2ωp~ Û(Λ) â†p~ |0i ======== 2ωp~ Û(Λ) â†p~ Û −1 (Λ)|0i
Λp −
Λp
s
→
ω−
Λp †
†
−1
⇒ define: Û (Λ) âp~ Û (Λ) =
â−→ ,
ωp~ Λp
with a similar expression for b̂†p~ . As a result:
Z
d~p 1 q
†
ip·x
−1
→ (â−
→ e−ip·x + b̂−
2ω−
)
Û (Λ) φ̂(x) Û (Λ) =
→e
Λp
Λp
3
Λp
(2π) 2ωp~
Z
d~p ′ 1 p
† ip′ ·Λx
−ip′ ·Λx
′e
′ (âp
) = φ̂(Λx) ,
+
b̂
2ω
~
p
~
p
~′e
(2π)3 2ωp~ ′
R
where the second line is obtained by using that d~p /(2ωp~ ) and e±ip·x are all Lorentz
invariant. This implies that the transformed field creates/destroys antiparticles/particles
at the spacetime point Λx.
p′ = Λp
====
1.6
Inversion of the Klein-Gordon equation (§ 2.4 in the book)
4e For certain physical applications it is important to know the inverse of
the KG equation, for instance for deriving scattering amplitudes or for solving
systems that involve a KG field being coupled to a classical source.
Since a solution to ( + m2 )φ0 (x) = 0 exists, the inversion of the differential operator
( + m2 ) does not exist formally, so it has to be defined. Once we have defined this inverse
( + m2 )−1 properly an appropriate solution to the equation ( + m2 )φ = j is given by
φ = ( + m2 )−1 j + φ0 , given that φ = φ0 in the absence of the source j.
Green’s function: let’s try to find the so-called Green’s function G(x − y), which is the
inverse KG operator ( + m2 )−1 written in coordinate space. By convention this Green’s
function is required to satisfy (x + m2 )G(x − y) ≡ −iδ (4) (x − y) = −iδ(x0 − y 0 )δ(~x − ~y ),
where the right-hand-side represents (up to the conventional factor −i) the unit operator
in coordinate space. In momentum space this becomes
Z 4
Z 4
d p
d p −ip·(x−y)
−ip·(x−y)
4
G(x − y) ≡
G̃(p)
e
and
δ
(x
−
y)
=
e
,
(2π)4
(2π)4
so that
(−p2 + m2 ) G̃(p) = −i
⇒
21
G̃(p) =
p2
i
.
− m2
The problem with defining the inverse of the KG operator is apparent now: it resides in
the fact that p2 − m2 = p20 − (~p 2 + m2 ) = p20 − ωp~2 = 0 for the physical (anti)particles of the
KG theory. In these so-called on-mass-shell (or short: on-shell) situations with p2 = m2
the Fourier coefficient of the Green’s function blows up, thereby leading to an ill-defined
Fourier integral. That brings us to the key question that we have to address if we want to
define a proper Green’s function:
how should we go around the poles of (p2 − m2 )−1 = (p0 − ωp~)−1 (p0 + ωp~ )−1
while performing the Fourier integral?
There are several options for this, reflecting the fact that the Green’s function cannot be
defined uniquely. We mention here two useful possible definitions.
1) The retarded Green’s function: for taking into account influences that lie in the
past it is useful to shift the poles into the lower-half of the complex plane by an infinitesimal
amount −iǫ (see figure 2), where the infinitesimal constant ǫ ∈ R+ should be taken to 0
at the end of the calculation.
Im p0
x0 < y 0
infinite semi circle
*
*
− ωp~ − iǫ
Re p0
+ ωp~ − iǫ
infinite semi circle
x0 > y 0
Figure 2: Complex poles and closed integration contours for the retarded Green’s function.
Using the complex integration contours as indicated in figure 2, the Fourier integration
yields
(
)
Z
−ip·(x−y) −ip·(x−y) d~
p
ie
ie
DR (x − y) = Θ(x0 − y 0 )(−2πi)
+
(2π)4
2ωp~ p0 = ωp~
−2ωp~ p0 = −ωp~
p~ → − ~
p in 2nd term
0
0
================ Θ(x − y )
Z
22
d~p 1
−ip·(x−y)
ip·(x−y) e
−
e
,
p0 = ωp~
(2π)3 2ωp~
which means that
DR (x − y) = Θ(x0 − y 0) D(x − y) − D(y − x)
= Θ(x0 − y 0 )h0| φ̂(x), φ̂† (y) |0i .
Application: consider a real KG field coupled to an external classical source j(x) that is
switched on during a finite time interval. Then
( + m2 ) φ̂(x) = j(x) ∈ R ,
which would correspond to an extra term + j(x)φ(x) in the Lagrangian (resembling a
forced oscillator). Before j(x) is turned on we have
Z
d~p
1
† ip·x −ip·x
p
,
(â
e
+
â
e
)
φ̂(x) = φ̂0 (x) =
p
~
p
~
(2π)3 2ωp~
p0 = ωp~
with φ0 (x) a solution to the free KG equation ( + m2 )φ0 (x) = 0. After j(x) is turned
on we have
Z
φ̂(x) = φ̂0 (x) + i d4 y DR (x − y)j(y)
Z
Z
4
= φ̂0 (x) + i d y
d~p j(y)
0
0
−ip·(x−y)
ip·(x−y) Θ(x
−
y
)
e
−
e
.
(2π)3 2ωp~
p0 = ωp~
If x0 is smaller than the switch-on time of j then Θ(x0 − y 0)j(y) = 0 and only φ̂0 (x)
remains, in agreement with the initial condition we started out with. If x0 is larger than
R
the switch-off time of j, then Θ(x0 − y 0 )j(y) = j(y). Using that d4 y eip·y j(y) ≡ j̃(p)
R
∗
j∈R
and d4 y e−ip·y j(y) ==== j̃ (p) we find in that case that
φ̂(x) =
≡
and
Z
Z
d~p
1
p
(2π)3
2ωp~
(
∗
j̃(p) −ip·x
j̃ (p) ip·x
†
âp~ + i p
e
+ âp~ − i p
e
2ωp~
2ωp~
d~p
1
† ip·x −ip·x
p
(
α̂
e
+
α̂
e
)
p
~
p
~
p0 = ωp~
(2π)3
2ωp~
N(Ĥ) =
Z
)
p0 = ωp~
d~p
ωp~ α̂p†~ α̂p~ ,
(2π)3
with N denoting normal ordering. The operator α̂p~ is a quasi-particle annihilation operator, satisfying
j̃(p)
|0i ≡ λp~ |0i .
α̂p~ |0i = i p
2ωp~
So, the free-particle vacuum state |0i is now a quasi-particle coherent state. Its energy has
23
changed by an amount
∆E0 = h0|N(Ĥ)|0i =
corresponding to h0|
Z
d~p 1
|j̃(p)|2 ,
3
(2π) 2
R d~p
R d~p
R d~p |j̃(p)|2 /2
†
2
quasi-particles.
α̂
α̂
|0i
=
|λ
|
=
p
~
p
~
(2π)3 p~
(2π)3
(2π)3
ωp~
The particle interpretation has changed as a result of the influence of
ternal source! This example shows that particles and quasi-particles are
quantities and that the retarded Green’s functions are handy tools for
with external influences that are switched on during a finite amount of
the exderived
dealing
time.
2) Feynman propagator: an alternative way of shifting the poles is given in figure 3. As
will be worked out in Ex. 5, this pole configuration is equivalent with replacing (p2 − m2 )−1
by (p2 − m2 + iǫ)−1 , where again the infinitesimal constant ǫ ∈ R+ should be taken to 0
at the end of the calculation.
Im p0
x0 < y 0
infinite semi circle
− ωp~ + iǫ
*
*
Re p0
+ ωp~ − iǫ
infinite semi circle
x0 > y 0
Figure 3: Complex poles and closed integration contours for the Feynman propagator.
Using the complex integration contours as indicated in figure 3, the Fourier integration
yields

Z
d~p ie−ip·(x−y) 


− 2πi
if x0 > y 0

4

(2π)
2ω
p
~

p0 = ωp~
DF (x − y) =
,
Z

−ip·(x−y) 
d~p ie


if x0 < y 0

 + 2πi (2π)4 −2ω p
~
p0 = −ωp~
24
which means that
DF (x − y) = Θ(x0 − y 0 )D(x − y) + Θ(y 0 − x0 )D(y − x)
= Θ(x0 − y 0 )h0|φ̂(x)φ̂† (y)|0i + Θ(y 0 − x0 )h0|φ̂†(y)φ̂(x)|0i
≡ h0|T (φ̂(x)φ̂† (y))|0i .
This is the definition of time ordering: the operator at the latest time is put in front. The
Feynman propagator DF (x − y) is the time-ordered propagation amplitude.
4e The Feynman propagator will feature prominently in the derivation of scattering amplitudes in perturbation theory!
25
2
Interacting scalar fields and Feynman diagrams
The next eight lectures cover large parts of Chapters 4 and 7 as well as a few aspects of
Chapter 10 of Peskin & Schroeder.
5 The task that we set ourselves is to investigate the consequences of adding
interactions that couple different Fourier modes and, as such, the associated
particles. This will be quite a bit more complicated than the free theories that
we have encountered in the previous chapter, where the relevant quantities were
diagonal (i.e. decoupled) in the momentum representation and particle numbers were conserved explicitly. Even worse, up to now nobody has been able to
solve general interacting field theories. Therefore we will focus on weakly coupled field theories, which can be investigated by means of perturbation theory.
Causality dictates us to add local terms only, i.e. L̂int (x) and not L̂int (x, y). In order
to investigate what is meant by “weak interactions”, the following interacting real scalar
theory is considered:
L =
1
1
(∂µ φ)(∂ µ φ) − m2 φ2 + Lint
2
2
with
Lint = −
X λn
φn
n!
n≥3
(φ ∈ R) ,
where λn ∈ R is called a coupling constant. Note that Lint = − Hint , since it contains
no derivatives. The corresponding Euler-Lagrange equation is not a simple linear (wave)
equation anymore:
∂µ (∂ µ φ) + m2 φ +
X
n≥3
λn
φn−1 = 0
(n−1)!
⇒
( + m2 )φ = −
X
n≥3
λn
φn−1 .
(n−1)!
Since πφ = ∂0 φ is unaffected by the interaction, the quantum mechanical basis
φ̂(~x ), π̂φ (~y ) = iδ(~x − ~y ) 1̂
and all other commutators being 0
is the same as in the free KG case. Hence, φ̂(~x ) and π̂φ (~x ) can be given the same Fourierdecomposed form as before (cf. page 13). However, since the non-linear φ̂n−1 term contains
for example (â† )n−1 , the number of particles is not conserved anymore as a result of the
interaction. Consequently, also the particle interpretation, which can be obtained from the
Hamilton operator, will be different.
2.1
When are interaction terms small? (§ 4.1 in the book)
5a To answer this question we have to perform a dimensional analysis: the
R
action S = d4 x L is dimensionless, so L must have dimension (mass)4 , or
short “dimension 4”. The shorthand notation for this is [L] = 4.
26
Kinetic term: the kinetic term has the form (∂µ φ)(∂ µ φ). Since [∂µ ] = 1, that means
that [φ] = 1, which is consistent with the dimension of the mass term ∝ m2 φ2 .
Interaction terms: since [φn ] = n, the coupling constants have a dimension [λn ] = 4−n.
So, λn is not dimensionless, except when n = 4. Three cases can be distinguished:
1. Coupling constants with positive mass dimension. Take λ3 as an example. Using
the dimension of the field, we can see that [λ3 ] = +1. In a process at energy scale E
the coupling constant λ3 will enter in the dimensionless combination λ3 /E. The φ3
interaction can therefore be considered weak at high energies (E ≫ λ3 ) and strong at
small energies (E ≪ λ3 ). For the latter reason such interactions are called relevant .
2. Dimensionless coupling constants. For our real scalar theory, the only dimensionless
coupling constant is λ4 since [λ4 ] = 0. The φ4 interaction can be considered weak if
the coupling constant is small (λ4 ≪ 1). Such interactions are called marginal, since
they are equally important at all energy scales.
3. Coupling constants with negative mass dimension. For the coupling constants with
n ≥ 5 we have [λn ≥ 5 ] = 4 − n < 0. In a process at energy scale E the coupling
constants λn ≥ 5 will enter in the dimensionless combinations λn E n−4. The φn ≥ 5 interactions can therefore be considered weak at low energies and strong at high energies.
Because of this suppressed influence on low-energy physics, such interactions are
called irrelevant . Such interactions have their origin in underlying physics that takes
place at higher energy scales.
5a Complication: it is impossible to avoid high energies in quantum field
theory, because of the occurrence of integrals over all momenta at higher orders
in perturbation theory. We have in fact already encountered an example of this
in § 1.3 while discussing the zero-point energy and its infinities.
2.2
Renormalizable versus non-renormalizable theories
Renormalizable theories: a renormalizable theory has the marked property that it is
not sensitive to our lack of knowledge about high-scale physics. It therefore
• keeps its predictive power at all energy scales in spite of the occurrence of highenergy effects in the quantum corrections;
• can be used to make precise theoretical predictions that can be confronted with
experiment;
• does not involve coupling constants with negative mass dimension.
27
Guided by our quest for the ultimate “theory of everything”, the prevalent view in highenergy physics used to be that any sensible theory that describes nature should be renormalizable. However, this requirement is based on the unrealistic assumption that any theory
that attempts to describe aspects of nature has to be valid up to arbitrarily large energies. It
is much more likely that at some energy scale new physics will kick in, causing the original
theory to be incomplete.
Non-renormalizable theories
5b In situations where our present theoretical knowledge proves insufficient or
where we prefer to describe the physics up to a minimum length scale, another
class of theories is particularly useful. These mostly non-renormalizable theories
are obtained by parametrizing our ignorance (scenario 1 discussed below) or by
“integrating out” known/anticipated physics at small length scales (scenario 2
discussed below).
Non-renormalizable theories, scenario 1: unknown new physics.
Suppose that we are starting to observe experimental deviations from our favourite model of
the world, caused by some unknown high-scale physics. If we only have access to this highscale physics through low-energy data (see the Fermi-model example below), we sometimes
have to content ourselves with an incomplete model that describes the physics as seen
through blurry glasses. In that case we only know the physics up to a certain energy scale µ
(i.e. down to a length scale 1/µ) with higher energy scales (i.e. smaller length scales) being
integrated out. This will in general result in a non-renormalizable effective theory that
describes nature up to the energy scale µ and a Lagrangian that will parametrize our lack
of knowledge about the physics that takes place at higher energy scales. Such effective
theories
• have limited predictive power, since the physics at high energy scales E ≫ µ is not
described properly;
• can contain interactions with coupling constants of negative mass dimension, which
would formally lead to UV infinities at higher orders in perturbation theory as a
result of integrals over all momenta (if we would assume the theory to be correct at
all energy scales, . . . which would be incorrect);
• can nevertheless be used to make reliable predictions at O(µ) energies provided that
the unknown high-scale physics resides at an energy scale ΛNP ≫ µ.
• may reveal at which energy scale the unknown high-scale physics must emerge.
28
Non-renormalizable theories, scenario 2: known/anticipated new physics.
The moment we (think to) know the underlying physics model that is responsible for the
observed low-energy phenomena, we can explicitly integrate out the high-energy degrees
of freedom from the model. This results in the same type of effective Lagrangian, but this
time the underlying physics model has left its fingerprints on the coupling constants. For
instance, if the energy/mass scale of the underlying physics resides at ΛNP , then this scale
will act as a natural scaling factor in the couplings. This procedure of explicitly linking
the coupling constants of the effective theory to the parameters of the underlying physics
model is called matching .
µ ✻
fields φ , Φ
mass m, M
high-energy theory
L(φ) + L′ (φ , Φ)
µ=M
matching
µ≪M
low-energy effective theory
L(φ) + Lint (φ)
field φ
mass m ≪ M
Figure 4: Schematic display of a low-energy effective theory containing a light field φ with
mass m, originating from a high-energy theory that also includes a heavy field Φ with
mass M.
Example: the Fermi-model of weak interactions. This probably sounds rather abstract, so let’s have a closer look at the above-given statements by considering an explicit
example. The so-called Fermi-model of weak interactions has in fact started out along the
lines just described. In this example the role of ΛNP is played by the mass MW of the W
boson. As will be explained in courses covering the Standard Model, decay processes like
µ− → νµ e− ν̄e (muon decay) proceed through the exchange of a W boson with a mass of
about 80 GeV between the particles. The associated decay amplitude contains a factor
1/(p2 − MW2 ), originating from the propagator of the W -boson (cf. page 24), and two
factors of g , corresponding to the coupling constant of the weak interactions. However, at
the typical energy scale of the decay process, i.e. E = O(mµ = 0.1 GeV), the momentum
carried by the W boson is much smaller than its mass MW . In that case, the propagator
factor is perceived as having a constant value:
g2
p2 − MW2
2
p2 ≪ M W
−−−−−−−−−
−→
29
−
g2
+ O(p2/MW4 ) .
MW2
In terms of a diagrammatic representation of the physics that goes on in the decay process
(see later) this corresponds to
e−
p
νµ
g
2
W−
g
e−
νµ
ν̄e
ν̄e
2
p ≪ MW
−−−−−−−−−−→
GF
µ−
µ−
On the basis of such “low-energy” decay processes the existence of (effective) 4-particle
interactions was postulated (Fermi, 1932), with the corresponding dimensionful effective
coupling constant (Fermi-coupling) being small in view of the absorbed 1/MW2 suppression factor. This explains the name “weak interactions”, which simply refers to the fact
that these interactions were perceived as weak at low energies. At p2 = O(MW2 ) the weakinteraction physics underlying the W -boson exchange will reveal itself and the weak interactions will no longer be weak.
5b This is of course all hindsight, since in 1932 the correct model for the weak
interactions did not exist yet. In fact, the above argument can be reversed. The
low-energy Fermi-coupling was measured to be of O(10−5 GeV −2 ) ≈ O(Λ−2
),
NP
which correctly signals that the physics underlying the weak interactions must
reveal itself at an energy scale of O(100 GeV).
Planck scale: applying the same reasoning to the even smaller gravitational constant,
i.e. G = O(10−38 GeV−2 ), we would predict that gravity becomes strong at an energy scale
of O(1019 GeV), which is commonly referred to as the Planck scale ΛP .
Generic properties of effective field theories: the philosophy behind effective field
theories is mostly a pragmatic one. If you want to describe certain physical phenomena
quantitatively, it is an overkill to use a physics model that also gives details about experimentally inaccessible phenomena (like strong gravitational effects). In that case it is more
practical to make use of a simpler, effective description that captures the most important
physics of the system without giving unnecessary detail. Additional (small) effects resulting from the more fundamental theory can be taken into account by adding them as small
perturbations (like relativistic corrections in non-relativistic quantum mechanics).
Consider for instance a fundamental theory with dimensionless coupling constants that
describes the world at O(ΛNP ) energies. Assume, for argument’s sake, that this theory
contains a real scalar field φ that describes light particles with mass m ≪ ΛNP and another real scalar field Φ that describes much heavier particles with mass M = O(ΛNP ).
30
The laws of physics at E ≪ ΛNP are best formulated in terms of the light scalar field with
interactions that are produced by the fundamental high-energy theory. After all, the heavy
particles cannot be produced directly at these energies and therefore it is more practical
to remove them from the description (i.e. integrate them out). This results in an effective
n−4
Lagrangian as given before with effective couplings λn = gn /ΛNP
, where gn is a dimensionless coupling constant governed by the high-energy theory. So, the impact of the φn ≥ 5
terms on physics at E ≪ ΛNP is suppressed by factors (E/ΛNP )n−4 .
• The interactions that are most likely to affect low-energy experiments are the renormalizable φ3 and φ4 terms. That is why at sufficiently low energies effective theories
only contain renormalizable interactions.
• The other interactions are suppressed at low energies and can therefore either be
ignored or incorporated as small perturbations. This aspect makes it possible to
include formally non-renormalizable interactions in the theory without spoiling its
predictive power at low energies. At high energies this is not true anymore, but there
the full glory of the underlying high-energy theory should be taken into account.
• Since the impact of the φn terms is extremely small for larger n, it is in general very
tough to figure out the entire high-energy theory from low-energy data alone!
Remark: the physics at different length/energy scales can be related through the so-called
renormalization group (see later). In particular in condensed-matter physics this renormalization group is a powerful analyzing tool, since different condensed-matter phenomena
are quite often governed by different characteristic length scales. As we will see later, also
in high-energy physics the renormalization group will prove very handy. The main difference between the field-theoretical treatments of both branches of physics resides in the
absence of a smallest length scale in high-energy physics, whereas the atomic scale provides
a natural cutoff in condensed-matter physics.
2.3
Perturbation theory (§ 4.2 in the book)
5c Our ultimate aim is to calculate scattering cross sections and decay rates,
from which information can be obtained on the fundamental particles that exist
in nature and their mutual interactions. The following two models will be used
in the remainder of this chapter:
1. φ4 -theory: L = 21 (∂µ φ)(∂ µ φ) − 21 m2 φ2 − 4!λ φ4 with φ ∈ R. This model contains the
type of quartic interaction with dimensionless coupling constant that also features in
the Higgs model.
31
2. Scalar Yukawa theory: L = (∂µ ψ ∗ )(∂ µ ψ) + 21 (∂µ φ)(∂ µ φ) − M 2 ψ ∗ ψ − 12 m2 φ2 − gψ ∗ ψφ
with φ ∈ R and ψ ∈ C. This is a toy model that resembles the Yukawa theory for
the interaction between fermions and scalars, which will be discussed at a later stage.
Apart from spin aspects these two theories differ in the dimension of the coupling
constant, being +1 for the scalar Yukawa theory and 0 for the true Yukawa theory.
Non-relativistic quantum mechanics: in non-relativistic quantum mechanics scattering reactions are characterized by
• asymptotic free (non-interacting) situations at t → ∓ ∞, involving free particles in
beam, target and detector (due to negligible wave-function overlap);
• a collision stage around t = 0 when the colliding particles interact/vanish and new
particles may be produced.
Quantum field theory: we would like to use the same reasoning in quantum field theory,
assuming the initial and final states of the reaction to be free-particle states. In that case
the initial and final states of the reaction would be eigenstates of the Hamilton operator
of the free Klein-Gordon theory, which are therefore also eigenstates of the particle and
antiparticle number operator. In the end we will have to correct for two aspects that are
not taken into account properly in this way (see later):
• bound states may form;
• more importantly, a particle well-separated from the other particles in the reaction is
nevertheless not alone in quantum field theory, being surrounded by a cloud of virtual
particles. It is not possible to switch off interactions in quantum field theory, so we
have to correct for this later.
The Heisenberg picture: let’s ignore these issues for the moment and try to develop a
calculational toolbox based on the asymptotic free situations at t → ± ∞. As mentioned
on page 26, we start out with the same quantum mechanical basis as in the free theory,
so the Schrödinger picture field φ̂(~x ) can be given the same Fourier-decomposed form as
before. The fact that we are dealing with an interacting theory manifests itself through the
time-independent Hamilton operator, which is used in the Heisenberg picture and which
is needed for determining the particle interpretation:
Z
Z
Ĥ = Ĥ0 + Ĥint = Ĥ0 + d~x Ĥint (~x ) = Ĥ0 − d~x L̂int (~x ) .
The interaction Hamiltonian Hint is assumed to be weak compared to the Hamiltonian
H0 of the free theory. In the last step we have used that there are no derivatives in the
interaction, so Hint = − Lint . This leads to Heisenberg fields
φ̂(x) ≡ φ̂(t, ~x ) = eiĤt φ̂(~x )e−iĤt ,
32
where e± iĤt introduces extra creation/annihilation operators as a result of the presence
of Ĥint and therefore changes the particle content and interpretation. The ground state
of the interacting theory will be denoted by |Ωi, which in general does not coincide with
the vacuum state of the free theory (see the example on page 23). For this state we have
Ĥ|Ωi = E0 |Ωi, with E0 the lowest energy level.
The interaction picture: the asymptotic free situation can be described by the freeparticle Hamilton operator Ĥ0 , so the corresponding time-dependent fields are given by
φ̂I (x) = eiĤ0 t φ̂(~x )e−iĤ0 t
and are called interaction-picture fields. This is actually the situation we have encountered
in the previous chapter, i.e. φ̂I (x) = φ̂ free (x). The creation and annihilation operators
have the same meaning as in the free theory, so the ground state is in this case the stable
vacuum |0i of the free theory, with N(Ĥ0 )|0i = 0 after normal ordering.
Switching between pictures: there is an operator that allows you to switch between
interaction picture and Heisenberg picture:
φ̂(x) = eiĤt φ̂(~x )e−iĤt = eiĤt e−iĤ0 t φ̂I (x)eiĤ0 t e−iĤt ≡ Û −1 (t, 0) φ̂I (x) Û (t, 0) .
The operator Û(t, 0) satisfies the differential equation
i
∂
P. & S.
Û (t, 0) = eiĤ0 t (Ĥ − Ĥ0 )e−iĤt = eiĤ0 t Ĥint e−iĤ0 t eiĤ0 t e−iĤt ≡≡≡ ĤI (t) Û (t, 0) ,
∂t
with boundary condition Û(0, 0) = 1̂ and with ĤI (t) only referring to the interaction
term (according to the definition in the textbook of Peskin & Schroeder).
6 This constitutes a natural starting point for a perturbative expansion:
Z t
Û(t ≥ 0, 0) = 1̂ + (−i) dt1 ĤI (t1 )Û(t1 , 0)
0
= 1̂ + (−i)
Z
t
dt1 ĤI (t1 ) + (−i)
0
2
Z
t
dt1
0
Z
t1
0
dt2 ĤI (t1 )ĤI (t2 ) + · · · ,
where the product ĤI (t1 )ĤI (t2 ) in the last term is ordered in time. In Ex. 6 it will be
derived that
Z
Z t
∞
Rt ′
X
(−i)n t
− i 0 dt ĤI (t′ )
dt1 · · · dtn T ĤI (t1 ) · · · ĤI (tn ) ,
≡
Û (t, 0) = T e
n!
0
0
n=0
which can be truncated at the required perturbative order. Such an object is called a
time-ordered exponential. For now we will define time ordering according to
T Ô1 (t1 )Ô2 (t2 )
=

 Ô1 (t1 )Ô2 (t2 )

Ô2 (t2 )Ô1 (t1 )
33
t1 > t2
.
t2 > t1
Later on we will have to extend the definition of time ordering to fermionic operator fields.
Since ĤI (t′ ) consists of interaction-picture fields only, we have succeeded in rewriting φ̂(x)
in terms of free fields through φ̂(x) = Û −1 (t, 0) φ̂I (x) Û (t, 0).
The definition of Û can be extended to arbitrary reference points:
Û(t, t1 ) ≡ eiĤ0 t e−iĤ(t−t1 ) e−iĤ0 t1 = Û(t, 0)Û −1 (t1 , 0) .
∂
Û (t, t1 ) = ĤI (t)Û (t, t1 ), but with
∂t
boundary condition Û(t1 , t1 ) = 1̂. The same procedure as before yields:
Rt ′
−i
dt ĤI (t′ )
(t ≥ t1 ) .
Û(t, t1 ) = T e t1
This operator still satisfies the differential equation i
This operator has the following properties that follow trivially from the above-given definition of Û(t, t1 ):
Û(t1 , t2 )Û (t2 , t3 ) = Û (t1 , t3 )
and
Û (t1 , t3 )Û −1 (t2 , t3 ) = Û (t1 , t2 ) .
Note that we have not used that Ĥ0 and Ĥ are hermitian, by sticking to Û −1
instead of writing Û †. So, Û (t, t1 ) can be generalized to non-hermitian ĤI (t) or
complex-valued time trajectories, as is used in some of the textbooks on quantum
field theory.
2.4
Wick’s theorem (§ 4.3 in the book)
The scattering amplitude for going from a free-particle initial state |ii to a free-particle
final state |f i now takes the form
lim hf |Û(t+ , t− )|ii ≡ hf |Ŝ|ii ≡ hf |(1̂ + i T̂ )|ii .
t± → ± ∞
In this expression the matrix hf |Ŝ|ii is called the S-matrix (scattering matrix), the unit
operator occurring on the right-hand-side corresponds to the case where no scattering takes
place, and T̂ is the transition operator that describes actual scattering.
6a Question: what should be done to calculate such an S-matrix element at
lowest order in perturbation theory?
The clumsy way of calculating S-matrix elements: let’s consider the scalar Yukawa
R
theory, where Ĥint = g d~x ψ̂ † (~x )ψ̂(~x )φ̂(~x ). Remember that ψ is a complex KleinGordon field, i.e. ψ̂ † 6= ψ̂ , whereas φ is a real Klein-Gordon field, i.e. φ̂† = φ̂. Then we have:
Z
R∞ ′
− i −∞ dt ĤI (t′ )
= 1̂ − ig d4 x ψ̂I† (x)ψ̂I (x)φ̂I (x) + O(g 2 ) .
lim Û(t+ , t− ) = T e
t± → ± ∞
34
Consider the following decay process within the scalar Yukawa theory:
φ(~p ) → ψ(~q1 ) + ψ̄(~q2 ) ,
where φ(~p ) denotes a φ-particle with mass m and momentum p~ , whereas ψ(~q1 ) and
ψ̄(~q2 ) denote a ψ-particle and a ψ-antiparticle with mass M and momenta ~q1 and ~q2
respectively. The ingredients for the calculation are:
|ii =
hf | =
p
2ωp~ â†p~ |0i ,
p
2ω~q1 2ω~q2 h0| ĉ~q2 b̂~q1 ,
d~k
1
† ik·x −ik·x
p
φ̂I (x) =
,
e
â
e
+
â
~
√
~
k
k
(2π)3 2ω~k
k0 = ω~k = ~k 2 +m2
Z ~
1
dk1
† ik1 ·x −ik1 ·x
p
ψ̂I (x) =
,
b̂~ e
+ ĉ~k e
√
1
(2π)3 2ω~k1 k1
k10 = ω~k = ~k12 +M 2
1
Z ~
1
dk2
† ik2 ·x −ik2 ·x
p
.
ψ̂I† (x) =
e
ĉ
e
+
b̂
~
√
~k2
(2π)3 2ω~k2 k2
k2 = ω~ = ~k 2 +M 2
Z
0
Using that hf |ii = 0 we get
hf |Ŝ|ii =
p
8ωp~ ω~q1 ω~q2 h0|ĉ~q2 b̂~q1 −ig
Z
k2
2
d4 x ψ̂I† (x)ψ̂I (x)φ̂I (x) â†p~ |0i .
Since the â-, b̂- and ĉ-operators mutually commute, the â~†k term in φ̂I can be commuted
to the left and will annihilate the vacuum. Similarly b̂~k1 in ψ̂I and ĉ~k2 in ψ̂I† can be
commuted to the right and will annihilate the vacuum there, bearing in mind that the
vacuum expectation value of an operator that involves an odd number of ĉ-operators
vanishes trivially. In other words, only the â~k term in φ̂I , the ĉ~†k term in ψ̂I and the b̂~†k
1
term in ψ̂I† will contribute:
Z
4
hf |Ŝ|ii = −ig d x
We know that
and similarly that
ZZZ
†
d~k d~k1 d~k2 ωp~ ω~q1 ω~q2 21 i(k1 +k2 −k)·x
† †
b̂
ĉ
â
e
h0|ĉ
b̂
âp~ |0i .
~
q
~
q
~
2
1
~
~
k
k2 k1
(2π)9
ω~k ω~k1 ω~k2
â~k â†p~ |0i = â~k , â†p~ |0i = (2π)3 δ(~k − p~ )|0i ,
h0| b̂~q1 b̂~†k = h0|(2π)3δ(~k2 − ~q1 )
2
and
h0| ĉ~q2 ĉ~†k = h0|(2π)3δ(~k1 − ~q2 ) .
1
This leads to the following result for the lowest-order decay amplitude:
Z
hf |Ŝ|ii = − ig d4 x ei(q2 +q1 −p)·x h0|0i = − ig (2π)4 δ (4) (q1 + q2 − p) ,
35
2
with g the strength of the interaction that is responsible for the decay. The δ-function
ensures that energy and momentum are conserved in the decay. In the reference frame
of the decaying particle we have: p = (m, ~0) ⇒ ~q1 + ~q2 = ~0 , ω~q1 + ω~q2 = m with
p
ω~qj = ~qj2 + M 2 ≥ M for j = 1, 2. So, the decay is only possible if m ≥ 2M.
The smart way of calculating S-matrix elements:
6b the trick will be to bring all creation operators to the left and all annihilation
operators to the right, with the vacuum state doing the rest. In other words,
in order to calculate S-matrix elements we need a way to rewrite time-ordered
fields in normal-ordered form . . . as will be provided by Wick’s theorem!
Step 1: consider a real Klein-Gordon field
Z
Z
d~p e−ip·x
d~p eip·x †
p
p
φ̂I (x) =
â
+
â ≡ φ̂+
(x) + φ̂−
(x) ,
p
~
I
I
(2π)3 2ωp~
(2π)3 2ωp~ p~
where the first term corresponds to the positive-frequency part and the second term to the
negative-frequency part. The φ̂+
and φ̂−
fields have the following useful property:
I
I
φ̂+
(x)|0i = 0
I
and
h0| φ̂−
(x) = 0 .
I
Since φ̂+
only contains annihilation operators, the fields φ̂+
(x) and φ̂+
(y) commute.
I
I
I
Similarly, φ̂−
only contains creation operators, so the fields φ̂−
(x) and φ̂−
(y) commute
I
I
I
as well. As a result
x0 > y 0 : T φ̂I (x)φ̂I (y) = φ̂+
(x) + φ̂−
(x) φ̂+
(y) + φ̂−
(y)
I
I
I
I
x0 < y 0 :
(y) = N φ̂I (x)φ̂I (y) + D(x − y) 1̂ ,
(x), φ̂−
= N φ̂I (x)φ̂I (y) + φ̂+
I
I
T φ̂I (x)φ̂I (y)
φ̂+
(y) + φ̂−
(y)
I
I
=
φ̂+
(x) + φ̂−
(x)
I
I
−
= N φ̂I (y)φ̂I (x) + φ̂+
(y),
φ̂
(x)
=
N
φ̂
(x)
φ̂
(y)
+ D(y − x) 1̂ .
I
I
I
I
Now we define a so-called contraction:
 +
−

 φ̂I (x), φ̂I (y) = D(x − y) 1̂
φ̂I (x)φ̂I (y) = φ̂I (x)φ̂I (y) ≡

 +
φ̂I (y), φ̂−
(x) = D(y − x) 1̂
I
if x0 > y 0
0
if x < y
0
= DF (x − y) 1̂ ,
with DF (x − y) the Feynman propagator of the free Klein-Gordon theory. With this
definition, the time-ordered expression can be rewritten as
T φ̂I (x)φ̂I (y)
= N φ̂I (x)φ̂I (y) + φ̂I (x)φ̂I (y) .
As a consequence of normal ordering we get, as expected, that
h0|T φ̂I (x)φ̂I (y) |0i = 0 + DF (x − y) .
36
Step 2, Wick’s theorem: let’s for the moment skip the annoying subscript I and use
the shorthand notation φ̂j ≡ φ̂I (xj ) for j = 1, · · · , n. Wick’s theorem then states:
T (φ̂1 · · · φ̂n ) = N(φ̂1 · · · φ̂n + all possible contractions) .
For example:
T (φ̂1 φ̂2 φ̂3 φ̂4 ) = N φ̂1 φ̂2 φ̂3 φ̂4 + φ̂1 φ̂2 φ̂3 φ̂4 + φ̂1 φ̂2 φ̂3 φ̂4 + φ̂1 φ̂2 φ̂3 φ̂4 + φ̂1 φ̂2 φ̂3 φ̂4 + φ̂1 φ̂2 φ̂3 φ̂4
+ φ̂1 φ̂2 φ̂3 φ̂4 + φ̂1 φ̂2 φ̂3 φ̂4 + φ̂1 φ̂2 φ̂3 φ̂4 + φ̂1 φ̂2 φ̂3 φ̂4
,
with N(φ̂1 φ̂2 φ̂3 φ̂4 ) ≡ DF (x1 − x3 )N(φ̂2 φ̂4 ).
The decomposition stated in Wick’s theorem has the following important consequence:
leftover (uncontracted) normal-ordered terms vanish upon taking the vacuum
expectation value!
For example:
h0|T (φ̂1φ̂2 φ̂3 φ̂4 )|0i = DF (x1 − x2 )DF (x3 − x4 ) + DF (x1 − x3 )DF (x2 − x4 )
+ DF (x1 − x4 )DF (x2 − x3 ) .
6b Feynman propagators thus play a central role in the resulting expressions.
Proof of Wick’s theorem: assume that the theorem is correct for all n ≤ m−1, knowing
that it is okay for n = 1, 2. For convenience we take x01 ≥ x02 ≥ · · · ≥ x0m , bearing in mind
that the order of the scalar fields is irrelevant for time ordering and normal ordering. Then
T (φ̂1 · · · φ̂m ) = φ̂1 φ̂2 · · · φ̂m = φ̂1 T (φ̂2 · · · φ̂m )
by assumption
=========== φ̂1 N(φ̂2 · · · φ̂m + all possible contractions of φ̂2 · · · φ̂m )
−
+
−
= (φ̂+
1 + φ̂1 )N(· · · ) = φ̂1 N(· · · ) + N(φ̂1 · · · ) ,
where in the last step we have used that φ̂−
1 contains creation operators only and therefore
already is in the right position. In contrast, φ̂+
1 contains annihilation operators only and
should be placed after all other fields. To get it in normal-ordered form, we need to
commute it past all other fields:
−
+
φ̂+
1 N(· · · ) = N(· · · ) φ̂1 + corrections for all uncontracted φ̂j>1 .
37
For instance:
+
−
+
−
φ̂+
1 N(φ̂2 · · · φ̂m ) = N [φ̂1 , φ̂2 ]φ̂3 · · · φ̂m + φ̂2 [φ̂1 , φ̂3 ]φ̂4 · · · φ̂m + · · ·
−
+ φ̂2 · · · φ̂m−1 [φ̂+
1 , φ̂m ]
x01 ≥ x0j>1
+ N(φ̂2 · · · φ̂m ) φ̂+
1
======= N(φ̂+
1 φ̂2 · · · φ̂m + φ̂1 φ̂2 φ̂3 · · · φ̂m + φ̂1 φ̂2 φ̂3 φ̂4 · · · φ̂m + · · · ) ,
+
where we have used that N(φ̂2 · · · φ̂m ) φ̂+
1 = N(φ̂1 φ̂2 · · · φ̂m ). Consequently
φ̂1 N(φ̂2 · · · φ̂m ) = N(φ̂1 φ̂2 · · · φ̂m + all single contractions of φ̂1 with another φ̂j ) .
The other (contracted) terms can be worked out in an analogous way, completing the
inductive proof of Wick’s theorem.
2.5
Diagrammatic notation: Feynman diagrams (§ 4.4 in the book)
In order to study the implications of Wick’s theorem we will focus here on the interacting
scalar φ4 -theory, with the scalar Yukawa theory being worked out in the exercises.
6c For calculating amplitudes it will prove handy to introduce a diagrammatic
notation, called Feynman diagrams, for time-ordered vacuum expectation values of interaction-picture fields.
Propagator : we start with a diagrammatic notation for contractions
x1
h0|T φ̂I (x1 )φ̂I (x2 ) |0i = φ̂I (x1 )φ̂I (x2 ) = DF (x1 − x2 ) ≡
x2
,
where the solid line represents the contraction (propagator) and the dots at the end of
the line represent the so-called external points in position space. From this it follows, for
example, that
x1
h0|T φ̂I (x1 ) · · · φ̂I (x4 ) |0i
x2
x1
x2
+
=
x3
x4
x1
x2
+
x3
x4
.
x4
x3
Later on we will need more complicated vacuum expectation values of the form
Rt
− i t + dt ĤI (t)
−
|0i ,
lim h0|T φ̂I (x1 ) · · · φ̂I (xn ) e
t± → ±∞
so let’s further develop the diagrammatic notation. We again start with the case n = 2:
Z
h
i
R
Taylor
4
− i d4 x ĤI (x)
h0|T φ̂I (x1 )φ̂I (x2 ) e
|0i ===== h0|T φ̂I (x1 )φ̂I (x2 ) 1̂ − i d x ĤI (x) + · · · |0i.
We can now calculate this quantity up to the required perturbative order.
38
Lowest order:
x1
h0|T φ̂I (x1 )φ̂I (x2 ) |0i = DF (x1 − x2 ) =
x2
.
First order in λ:
h Z
λ 4 i
4
h0|T φ̂I (x1 )φ̂I (x2 ) − i d x φ̂I (x) |0i
4!
Wick
===== 3
− iλ Z
4!
=
− iλ Z
d4 x h0|φ̂I (x1 )φ̂I (x2 )φ̂I (x)φ̂I (x)φ̂I (x)φ̂I (x)|0i
4!
Z
Z
iλ
iλ
4
2
d4 x DF (x1 − x)DF (x2 − x)DF (x − x)
−
DF (x1 − x2 ) d x DF (x − x) −
8
2
+ 12
=
d4 x h0|φ̂I (x1 )φ̂I (x2 )φ̂I (x)φ̂I (x)φ̂I (x)φ̂I (x)|0i
x1
x2
x1
+
x
x2
.
x
Vertex : the spacetime point x that is integrated over is called an internal point or vertex.
R
To such a vertex we assign the analytic expression −iλ d4 x, which is the amplitude for
emission and/or absorption of particles at the spacetime point x, summed over all points
where this can occur. Also notice that we encounter for the first time pieces of diagram
that involve closed loops.
An example of a higher-order term involving three powers in λ:
1 − iλ 3
P
h0|φ̂I (x1 )φ̂I (x2 )
3! 4!
=
=
iλ3
8
Z
Z
4
d x φ̂I φ̂I φ̂I φ̂I
Z
4
d y φ̂I φ̂I φ̂I φ̂I
Z
d4 z φ̂I φ̂I φ̂I φ̂I |0i
Z
Z
4
d x d y d4 z DF (x1 −x)DF (x−x)DF (x−y)DF (x2 −y)DF2 (y−z)DF (z−z)
4
z
x1
x
y
x2
.
R
R
Here d4 x φ̂I φ̂I φ̂I φ̂I is a shorthand notation for d4 x φ̂I (x)φ̂I (x)φ̂I (x)φ̂I (x). The facR
3
− i d4 x ĤI (x)
follows
directly
from
the
expansion
of
e
, whereas the factor P
tor 3!1 −iλ
4!
represents the number of times the contractions can be permuted without changing the
contribution. This permutation factor is a product of the following terms:
39
• 3! from permuting x, y and z ;
• 4 × 3 from the x contractions;
• 4 × 3 from the y contractions;
• 4 × 3 from the z contractions;
From the O(λ) and O(λ3 ) examples we see that the factor 1/n! is cancelled by the n!
permutation factor from interchanging vertices, and that the factors 1/4! are largely compensated by the number of ways the contractions can be placed into φ̂I φ̂I φ̂I φ̂I .
Symmetry factor : we end up with a leftover factor 1/S , with S the symmetry factor that
represents the number of ways in which diagram components can be interchanged without
changing the actual diagram.
Examples:
x1
x2
x1
S=2
S = 23 = 8
x2
S = 3! = 6
x1
x2
.
S = 2 × 3! = 12
R 4
6c The expression h0|T φ̂I (x1 )φ̂I (x2 ) e− i d x ĤI (x) |0i can now be represented
by the sum of all possible Feynman diagrams with two external points, where a
Feynman diagram is a collection (drawing) of propagators, vertices and external
points. The rules for associating analytic expressions with specific pieces of
diagrams are called the Feynman rules of the theory.
Feynman rules for the scalar φ4 -theory in position space:
1. For each propagator
2. For each vertex
x1
x
3. For each external point
x2
insert DF (x1 − x2 ).
R
insert (−iλ) d4 x.
x
insert 1.
4. Divide by the symmetry factor.
6c Given a specific diagram, the complete analytic expression is obtained by
multiplying the above-given analytic expressions for the specific pieces of the
diagram.
Switching to momentum space: usually it is more convenient to work in momentum
space, rather than position space. First we consider the Feynman propagator:
Z 4
x1 p x2
i
dp
−ip·(x1 −x2 )
e
=
,
DF (x1 − x2 ) =
(2π)4 p2 − m2 + iǫ
40
where the sign (direction) of p is arbitrary since DF (x1 − x2 ) = DF (x2 − x1 ) for a scalar
field. In other words, we can assign a four-momentum p and complex factor i/(p2 −m2 +iǫ)
to each propagator, indicating the direction of the momentum flow by an arrow. This arrow
has no deeper meaning than that in φ4 -theory, but in the scalar Yukawa theory it will be
needed to distinguish particles from antiparticles. Using this momentum-flow convention
a vertex corresponds to the following Fourier integral:
p4 p1
→
p3 p2
Z
d4 z e−i(p1 +p2 +p3 −p4 )·z = (2π)4 δ (4) (p1 + p2 + p3 − p4 ) .
On the left-hand-side of this equation the integral follows from the Feynman rule for the
vertex and the exponential factor is caused by the momentum-space expressions for the
Feynman propagators.
6b In momentum space we hence obtain four-dimensional δ-functions that represent energy-momentum conservation at each vertex. These δ-functions can
be used to perform some of the integrals that originate from the Feynman
propagators.
Feynman rules for the scalar φ4 -theory in momentum space:
p
1. For each propagator
2. For each vertex
insert i/(p2 − m2 + iǫ).
insert −iλ.
3. For each external point
p
insert e−ip·x .
x
4. Impose momentum conservation at each vertex.
5. Integrate over each undetermined momentum pj :
6. Divide by the symmetry factor.
R
d4 pj
(2π)4
.
Vacuum bubbles: the pieces of diagram that are disconnected from the external points
are called vacuum bubbles. For example:
Z
d4 x φ̂I (x)φ̂I (x)φ̂I (x)φ̂I (x)
Z
p1
d4 y φ̂I (y)φ̂I (y)φ̂I (y)φ̂I (y)
=
p4 .
p3
p2
The corresponding diagram will give rise to two energy-momentum δ-functions
(2π)4 δ (4) (p1 + p2 )(2π)4 δ (4) (p1 + p2 ) .
41
Upon inserting the first δ-function, the last δ-function will yield δ (4) (0). This represents
the infinite spacetime volume factor that originates from the fact that this vacuum bubble
can occur at any spacetime point! We have in fact already encountered an example of such
an IR divergence in § 1.3 while discussing the infinities of the zero-point energy. Let’s now
label the possible vacuum bubbles by
Vj
∈
(
,
V1
,
,
,
V3
···
)
,
V4
V2
then the following identity holds:
R
P
− i d4 x ĤI (x)
h0|T φ̂I (x1 )φ̂I (x2 ) e
|0i = e j Vj
x1
+
x2
+
x1
x2
x
x1
x2
x
y
+
···
!
. (2)
The part between parantheses on the right-hand-side is the sum of all connected diagrams,
i.e. continuous drawings that connect external points, whereas the exponential factor in
front is the vacuum-bubble contribution. This vacuum-bubble contribution involves no
external points and is therefore given by
R 4
P
Vj
− i d x ĤI (x)
j
e
= h0|T e
|0i .
Note: in the φ4 -theory each vertex has an even number of lines coming together. So,
x1 and x2 must be connected to each other. The reason for this is that internal lines of
a diagram connect two vertices and therefore count as two lines that are attached to a
vertex. As such, a connected piece of diagram involves an even number of external lines
and points.
Proof of identity (2): consider a diagram with nj vacuum bubbles of type Vj and one
connected piece without vacuum bubbles, like
x1
x2
connected piece
.
n1 = 1
n3 = 2
From the Feynman rules it follows that
analytic expression diagram = (analytic expression connected piece) ×
42
Y 1
(Vj )nj ,
nj !
j
where the symmetry factor comes from interchanging the nj copies of Vj . Hence we find
R 4
h0|T φ̂I (x1 )φ̂I (x2 ) e− i d x ĤI (x) |0i = sum of all diagrams
X
=
X
all possible
all {nj }
connected pieces
(analytic expression connected piece) ×
= (sum of all connected diagrams) ×
Y 1
(Vj )nj
nj !
j
X Y 1
(Vj )nj
nj !
j
all {nj }
R 4
= (sum of all connected diagrams) × h0|T e− i d x ĤI (x) |0i .
P
The only thing left to prove is that the last factor is indeed equal to e j Vj :
X 1
X 1
Y
Y X 1
P
(Vj )nj =
(V1 )n1
(V2 )n2 · · ·
e j Vj =
eVj =
nj !
n1 !
n2 !
n
n
j
j
n
1
j
=
2
X Y 1
(Vj )nj .
nj !
j
all {nj }
We can generalize the above-given separation between connected diagrams and vacuum
bubbles to
R 4
h0|T φ̂I (x1 ) · · · φ̂I (xn ) e− i d x ĤI (x) |0i
R 4
= h0|T e− i d x ĤI (x) |0i × (sum of all connected diagrams with n external points) .
x1 x2
For 4, 6, · · · external points this generalized sum will contain diagrams like
that do not have all external points connected to each other.
x3 x4
R
P
4
Remark: we will see later that the sum j Vj = log h0|T e− i d x ĤI (x) |0i of all vacuum
bubbles is actually related to the difference in the ground-state zero-point energies of the
interacting theory and the free theory.
2.6
Scattering amplitudes (§ 4.6 in the book)
7 At this point you might wonder what such time-ordered vacuum expectation values of interaction-picture fields have to do with amplitudes for decay
processes or scattering reactions.
In order to calculate scattering cross sections and decay rates we will have to work out
plane-wave amplitudes of the form out h~p1 p~2 · · · |~kA~kB iin . Here |~kA~kB iin is the so-called
43
“in-state”. In the case of scattering this is a 2-particle momentum state that is constructed
in the far past, also referred to as “the initial state”. Similarly out h~p1 p~2 · · · | is the so-called
“out-state”, which represents the final state particles in the far future, i.e. the particles
that will end up in the detectors of the experiment.
7 Since the detectors are in general not able to resolve positions at the level of
the de Broglie wavelengths of the particles, it is correct to work with plane-wave
states rather than wave packets in order to describe the collision.
The states out h~p1 ~p2 · · · | and |~kA~kB iin are plane-wave states in the Heisenberg picture.
Normally states are time-independent in the Heisenberg picture. However, the in and out
states that we use here are defined as eigenstates of momentum operators that do depend
on time. As such, the in-state contains the time stamp t = t− → − ∞ and the out-state
t = t+ → + ∞. By evolving these states to the eigenstates at t = 0, one unique set of
Heisenberg-picture plane-wave states is obtained:
p1 p~2
out h~
· · · |~kA~kB iin ≡ h~p1 p~2 · · · |Ŝ |~kA~kB i ≡ h~p1 p~2 · · · | 1̂ + i T̂ |~kA~kB i .
Because of the infinite time interval and the way we normalize the states h~p1 p~2 · · · | and
|~kA~kB i, these matrix elements are Lorentz invariant. As was mentioned earlier, the matrix
element h~p1 p~2 · · · |Ŝ |~kA~kB i is called the S-matrix element and is naturally split into two
parts: a part containing 1̂, which corresponds to the case where no scattering takes place,
and a part containing the transition operator T̂ , which describes actual scattering. So, the
latter part contains all the interesting physics.
The matrix element: the next step is to pull out the anticipated energy-momentum
conservation factor according to
h~p1 ~p2 · · · |i T̂ |~kA~kB i ≡ (2π)4 δ (4) kA + kB − [p1 + p2 + · · · ] iM(kA , kB → p1 , p2 , · · · )
≡ (2π)4 δ (4)
P
i
ki −
P
f
pf
iM({ki } → {pf }) ,
where M is called the invariant matrix element (or short: matrix element).2 All fourmomenta occurring in this expression are on-shell, i.e. p2 = m2 with m the physical mass
of the particle. Therefore it suffices to know the three-momenta of the particles and the
reaction state they belong to (i.e. initial or final state) in order to obtain the complete
four-momenta. By means of this split-up the interaction details (“dynamics”) are separated from the momentum details (“kinematics”).
2
Warning : in some textbooks the factor of i is absorbed into the definition of M
44
Rewriting things in free-particle language (without proof, for now): as will be
shown later, the plane-wave states in the interacting theory can be expressed in terms of
free-particle plane-wave states 0 h~p1 p~2 · · · | and |~kA~kB i0 , resulting in
R t+
dt
Ĥ
−
i
(t)
I
|~kA~kB i0 fully connected × factor
h~p1 p~2 · · · |i T̂ |~kA~kB i = lim
p1 ~p2 · · · |T e t−
0 h~
t± → ± ∞
=
p1 p~2
0 h~
and amputated
· · · |T e− i
R
d4 x ĤI (x)
|~kA~kB i0
fully connected
and amputated
× factor ,
where the (not yet specified) factor comes in at loop level. In this way everything has been
translated into free-particle language, but some of the ingredients still need to be specified.
7 The actual proof of the above statement will be postponed until § 2.9, since
we will need to know a bit more about the properties of loop corrections for that
purpose. This proof will be based on the type of time-ordered vacuum expectation
values of interaction-picture fields that we have encountered previously.
In order to get a feeling for the essential ingredients of that proof we will consider an explicit
example. Let’s have a look at the meaning of “fully connected” and “amputated” by considering the S-matrix element belonging to the 2 → 2 process φ(kA )φ(kB ) → φ(p1 )φ(p2 )
in the scalar φ4 -theory.
The O(λ0 ) term:
~ ~
p1 ~p2 |kA kB i0
0 h~
p
= 4 ωp~1 ωp~2 ω~kA ω~kB h0|âp~1 âp~2 â~†k â~†k |0i
A
B
h
i
= 4 ω~kAω~kB (2π)6 δ(~p1 − ~kA )δ(~p2 − ~kB ) + A ↔ B
1
2
diagrammatically
=============
1
2
+
A
B
.
B
A
This O(λ0 ) term is part of the 1̂ term in Ŝ = 1̂ + i T̂ , so it does not contribute to the
matrix element M.
Arrow of time, Peskin & Schroeder style : the external lines without external points indicate the incoming particles, which are placed at the bottom of the diagram in the notation
of Peskin & Schroeder, and outgoing particles, which are placed at the top of the diagram.
In many textbooks these diagrams will be turned by 90◦ with incoming particles on the
left and outgoing ones on the right, i.e. in that case the time-axis points from left to right
rather than from bottom to top.
45
The O(λ) term:
Z
Z
− iλ λ 4 ~ ~
Wick
4
p1 p~2 | d4 x N φ̂4I (x) |~kA~kB i0
p1 p~2 |T − i d x φ̂I (x) |kA kB i0 =====
0 h~
0 h~
4!
4!
Z
− iλ 4
+
p1 ~p2 | d x N 6 φ̂I (x)φ̂I (x)φ̂I (x)φ̂I (x) + 3 φ̂I (x)φ̂I (x)φ̂I (x)φ̂I (x) |~kA~kB i0 .
0 h~
4!
This time terms that are not fully contracted do not vanish automatically:
Z
p
d~p
1
+
~
p
âp~ e−ip·x 2ω~k â~†k |0i = e−ik·x |0i .
φ̂I (x)|k i0 =
3
(2π)
2ωp~
It is now useful to extend the contraction definition with
φ̂I (x)|~k i0 ≡ φ̂+
(x)|~k i0 = e−ik·x |0i
I
~
and
0 h k |φ̂I (x)
≡ 0 h~k |φ̂−
(x) = h0|eik·x .
I
This means that we need additional Feynman rules for contractions of field operators with
external states:
k
=
x
e−ik·x
k
and
eik·x ,
=
x
where e−ik·x is the amplitude for finding a particle with four-momentum k at the vertex
position x. Diagrammatically the O(λ) terms then consist of the following contributions:
• A term with all φ̂I ’s contracted with each other:
iλ
−
8
Z
1
2
d x 0 h~p1 ~p2 |φ̂I (x)φ̂I (x)φ̂I (x)φ̂I (x)|~kA~kB i0 = x
4
1
2
B
A
+ x
A B
This is a part of the 1̂ term in Ŝ = 1̂ + i T̂ , so it does not contribute to the matrix
element M.
• Terms where some φ̂I ’s are contracted with each other and some with the external
states:
Z
iλ
d4 x 0 h~p1 ~p2 |φ̂I (x)φ̂I (x)φ̂I (x)φ̂I (x)|~kA~kB i0 + three similar terms
−
2
1
=
2
+
x
A
1
B
2
1
+
x
B
A
2
+
x
B
1
A
†
2
.
x
A
B
These terms contribute only if there are as many â as â operators left, so one field
should be contracted with an incoming particle state and one with an outgoing partiR
cle state. Again this is part of the 1̂ term in Ŝ = 1̂ + i T̂ , since the integration d4 x
yields a momentum-conserving δ-function at each vertex. Again no contribution to
the matrix element M is obtained.
46
• A term where all φ̂I ’s are contracted with the external states:
− iλ
Z
1
d4 x 0 h~p1 ~p2 |φ̂I (x)φ̂I (x)φ̂I (x)φ̂I (x)|~kA~kB i0
= − iλ
2
=
x
A
Z
B
d4 x e−i(kA +kB −p1 −p2 )·x = − iλ(2π)4 δ (4) (kA + kB − p1 − p2 ) .
This term gives rise to a − λ contribution to the matrix element M!
Fully connected diagrams: the discussion above reflects the following general principle.
7a Only fully connected diagrams, in which all lines are connected to each
other, contribute to the T -matrix and hence to the matrix element M.
At lowest non-vanishing order we find M(kA , kB → p1 , p2 ) = − λ in the scalar φ4 -theory,
= − iλ.
which can be obtained directly from the momentum-space interaction vertex
Including higher-order terms, while keeping the external lines connected:
1
h~p1 ~p2 |i T̂ |~kA~kB i
1
2
=
2
A
B
1
2
A
+
B
1
2
A
B
1
2
+
B
B
+ ···
A
···
1
2
+
+
2
A
B
+
A
1
+
+
1
2
+
A
B
1
2
+
A
B
+ ···
A
B
The three sets of diagrams that occur on the separate lines of this expression are now
discussed individually.
Set 1: these diagrams contribute to M. Beyond leading order diagrams occur that involve
the creation and annihilation of additional “virtual” particles. Such higher-order contributions are called loop corrections.
Set 2: these diagrams involve disconnected vacuum bubbles, which will again exponentiate
to an overall phase factor that is irrelevant for physical observables! These graphs take
into account the energy shift between the ground state of the free theory and the ground
47
state of the interacting theory with respect to which scattering takes place. So, indeed
only fully connected diagrams matter!
Set 3: such diagrams give rise to contributions of the form
p1
p2
l
x
k′
kA
=
y
kB
=
1
2
Z
d4 k ′
i
4
′2
(2π) k − m2 + iǫ
Z
d4 l
i
×
4
2
(2π) l − m2 + iǫ
× (− iλ)(2π)4 δ (4) (kA + k ′ − p1 − p2 )(− iλ)(2π)4 δ (4) (kB − k ′ )
1
i
(− iλ)2 (2π)4 δ (4) (kA + kB − p1 − p2 ) 2
2
kB − m2 + iǫ
Z
d4 l
i
.
4
2
(2π) l − m2 + iǫ
This contribution contains two propagators, DF (x−y) and DF (y −y), and two δ-functions
from the integrals over x and y . It blows up, since external particles are on-shell,
2
i.e. kB
= m2 . In fact, the diagrams
+
+
+
+
···
represent the evolution of |~p i0 in the free theory into |~p i in the interacting theory, which
causes the complex poles of the propagator to shift away from the free-particle positions at
p2 = m2 . As we will see later, this evolution will give rise to overall proportionality factors
in the T -matrix. All this reflects the fact that a particle is never truly free in quantum
field theory, being surrounded by a cloud of virtual particles. In quantum field theory it is
simply not possible to switch off interactions.
The amputation procedure: in order to deal with contributions of the latter type, the
following procedure is used.
7b Starting at the tip of each external leg, find the last point at which the
diagram can be cut by removing a single propagator in such a way that this
separates the leg from the rest of the diagram. The amputation procedure tells
us to cut the diagram at those points.
Contributions to the T -matrix are then obtained as
P
(2π)4 δ (4) kA +kB − f pf iM kA , kB → {pf } = sum of all fully connected amputated
Feynman diagrams in position space, multiplied by appropriate proportionality factors
at loop level.
48
The missing details concerning the amputation procedure will be discussed after we have
seen some properties of loop corrections.
Position-space Feynman rules for matrix elements in the scalar φ4 -theory:
1. For each propagator
2. For each vertex
x1
x
3. For each external line
x2
insert DF (x1 − x2 ).
R
insert (−iλ) d4 x.
p
insert e−ip·x .
x
4. Divide by the symmetry factor.
Formulated in momentum space: in order to deal with plane-wave states it is more
natural to switch from position space to momentum space. As explained before, in momentum space each interaction vertex gives rise to an energy-momentum δ-function. As
we have seen in the example discussed above, one of these δ-functions is the overall energymomentum δ-function of the T -matrix. Therefore, in momentum space one directly obtains
the matrix element as
iM kA , kB → {pf } = sum of all fully connected amputated Feynman diagrams
7c
in momentum space, multiplied by appropriate proportionality factors at loop level.
Momentum-space Feynman rules for matrix elements in the scalar φ4 -theory:
1. For each propagator
2. For each vertex
3. For each external line
p
insert
p2
i
.
− m2 + iǫ
insert −iλ.
p
insert 1.
4. Impose momentum conservation at each vertex.
5. Integrate over each undetermined loop momentum lj :
Z
d 4 lj
.
(2π)4
6. Divide by the symmetry factor.
Momentum-space Feynman rules for the scalar Yukawa theory: for completeness
we also list here the Feynman rules for the scalar Yukawa theory as derived in the exercises.
1. For each φ-propagator
For each ψ-propagator
q
p
i
.
− m2 + iǫ
i
.
insert 2
p − M 2 + iǫ
insert
49
q2
2. For each vertex
insert −ig .
q
3. For each external φ-line
insert 1.
k
For each incoming ψ-line
insert 1, originating from ψ̂ .
insert 1, originating from ψ̂ † .
For each incoming ψ̄-line
For each outgoing ψ-line
p
k
insert 1, originating from ψ̂ † .
insert 1, originating from ψ̂ .
For each outgoing ψ̄-line
p
4. Impose energy-momentum conservation at each vertex.
5. Integrate over each undetermined loop momentum lj :
Z
d 4 lj
.
(2π)4
The following observations can be made. First of all, in contrast to the scalar φ4 -theory no
symmetry factors are needed in the scalar Yukawa theory, since all fields in the interaction
are different. Secondly, whereas the arrows on the dashed φ-lines have no special meaning,
this is not true for the arrows on the solid lines, which correspond to the ψ̂ and ψ̂ † fields.
This arrow is needed for distinguishing particles (ψ) from antiparticles (ψ̄).
7d Drawing convention: draw arrows on the ψ-lines and the ψ̄-lines.
These arrows represent the direction of particle-number flow: particles flow along
the arrow, antiparticles flow against it. In this convention ψ̂ corresponds to an
arrow flowing into a vertex, whereas ψ̂ † corresponds to an arrow flowing out
of a vertex. Since every interaction vertex features both ψ̂ and ψ̂ † , the arrows
link up to form a continuous flow.
2.7
Non-relativistic limit: forces between particles
7e We are now in the position to address our first major question: how do
forces come about in quantum field theory?
To answer this question we compare the lowest-order relativistic matrix element for the
reaction φ(kA )φ(kB ) → φ(p1 )φ(p2 ), i.e.
iM =
p1
p2
kA
kB
50
= − iλ ,
to the non-relativistic amplitude for elastic potential scattering in Born approximation.
Since the matrix element is Lorentz invariant, we are free to choose the center-of-mass
(CM) frame. In this frame ~kA = − ~kB ≡ ~k and p~1 = − ~p2 ≡ p~ with |~k | = |~p | for elastic
scattering. The non-relativistic limit amounts to |~k | , |~p | ≪ m, from which it follows that
ω~k = ωp~ ≈ m + O(~k 2/m). For scattering from states with momenta ± ~k into states with
momenta ± p~ the comparison then reads:
Z
Z
M
k
,
k
→
p
,
p
/2
A
B
1
2
~
~
p |V (~rˆ )|~k iNR = d~r V (~r ) ei(k−~p )·~r ≡ d~r V (~r ) ei∆·~r ≈ −
,
NR h~
2
(2m)
where the factor 1/2 multiplying the matrix element originates from having identical particles in the reaction. Furthermore, it has been used that the relativistic and non-relativistic
momentum states are related according to
|~p i0 =
p
2ωp~ |~p iNR ≈
√
2m |~p iNR ,
resulting in a relative factor (2m)2 . By inverse Fourier transformation one obtains
Z
V (~r ) ≈
~ −M ~
λ
d∆
−i∆·~
r M=−λ
e
=
=
=
=
=
=
δ(~r )
(2π)3 8m2
8m2
for the interaction potential.
7e The scalar φ4-theory involves a so-called contact interaction ∝ δ(~r ),
which refers to the fact that the particles interact in one spacetime point at lowest order.
We can repeat this for ψ(kA )ψ(kB ) → ψ(p1 )ψ(p2 ) scattering in the scalar Yukawa theory.
In that case all external on-shell particles have mass M and the lowest-order matrix element reads (see Ex. 9):
p1
p2
kA
= − ig 2
≈
p1
+
iM =
NR
p2
ig 2
kB
≡ iM1 + iM2
kA
kB
1
1
+
(kA − p1 )2 − m2 + iǫ
(kA − p2 )2 − m2 + iǫ
1
(~k − ~p )2 + m2
+
1
(~k + ~p )2 + m2
,
p
p
using CM momenta and kA0 − p01 = ~k 2 + M 2 − p~ 2 + M 2 ≈ (~k 2 − p~ 2 )/(2M). The +iǫ
terms have been dropped as a result of the fact that the energy components are suppressed.
51
Note that there are two contributions this time, originating from interchanging the final~ it now follows that
state particles (i.e. ~p → −~p ). Using spherical coordinates with ~ez k ∆
1
V (~r ) = −
4M 2
Z
~
~
d∆
~ r ∆ ≡ |∆|
−i∆·~
M
e
=
=
=
=
= − (g/2M)2
1
3
(2π)
(g/2M)2
= −
4π 2 ir
d cos θ
(2π)2
−1
Z∞
∆2 e−i∆r cos θ
d∆
∆2 + m2
0
Z∞
Z∞
ei∆r − e−i∆r
∆ei∆r
(g/2M)2
d∆ ∆
d∆
=
−
∆2 + m2
4π 2 ir
(∆ + im)(∆ − im)
0
(g/2M)2
= −
4π 2 ir
Z1
Z
C
−∞
d∆
∆ei∆r
(g/2M)2 −mr
= −
e
,
(∆ + im)(∆ − im)
4πr
where the integration contour C is given in figure 5.
Im ∆
C
infinite semi circle
* + im
Re ∆
* − im
Figure 5: Closed integration contour for the determination of the Yukawa potential.
7e The scalar Yukawa theory involves an attractive Yukawa interaction between the ψ-particles, which dies off exponentially at 1/m distances. This
length scale (range) is in fact the Compton wavelength of the exchanged virtual
φ-particles, which mediate the interaction.
These virtual particles are short-lived off-shell particles, i.e. p2 6= m2 . In fact, they are too
short-lived for their energy to be measured accurately. Hence the name virtual particles.
Over 1/m distances the energy can fluctuate by O(m), which is sufficient to create the
φ-particles. Over larger distances the energy can fluctuate less, resulting in the exponential
decrease of the force. If the virtual particles are massless (like the photon) then the Yukawa
interaction has an infinite range and changes into the familiar Coulomb potential ∝ 1/r ,
which is not decreasing exponentially.
52
The true Yukawa theory for the interaction between fermions and scalars was used to
describe the interactions between nucleons. In that case the mediating particle is a pion.
It has a mass of about 140 MeV and therefore an associated characteristic length scale of
roughly 1.4 fm, which agrees nicely with the effective range of the nuclear forces.
Forces in quantum field theory: the forces between particles are caused (mediated)
by the exchange of virtual particles! Interactions caused by spin-0 force carriers (such
as the Yukawa interactions) are universally attractive, just like interactions due to the
exchange of spin-2 particles (such as gravity). The exchange of spin-1 particles can result
in both attractive and repulsive interactions, as we know from electromagnetism.
The relevant details of this statement are worked out in Ex. 10 and 11.
2.8
Translation into probabilities (§ 4.5 in the book)
8 At this point we know how to calculate amplitudes for decay processes and
scattering reactions by means of Feynman diagrams and Feynman rules. In the
next step we derive the probabilistic interpretation belonging to these amplitudes.
2.8.1
Decay widths
Consider an initial state consisting of a single particle in the momentum state |~kA i, decaying into a final state consisting of n particles in the momentum state |~p1 · · · p~n i. The
probability density for this decay to occur is given by
|h~p1 · · · ~pn |Ŝ|~kA i|2
,
h~kA |~kA ih~p1 · · · p~n |~p1 · · · p~n i
with
p. 14
h~kA |~kA i = 2E~kA(2π)3 δ(~0 ) ==== 2E~kAV
and
h~p1 · · · p~n |~p1 · · · p~n i =
n
Y
(2Ep~j V ) .
j =1
This is also valid for identical particles in the final state. Finding a set of particles with
the required momenta effectively identifies the particles. Since the initial and final states
are different in a decay process, the S-matrix element is in fact equivalent with the corresponding T -matrix element. In the rest frame of the decaying particle ~kA = ~0 and
E~kA = mA , hence
n
i2
P
|M(kA → {pj })|2 h
|h~p1 · · · p~n |i T̂ |~kA i|2
=
(2π)4 δ (4) kA −
pj
n
Q
2mA V
j =1
h~kA |~kA ih~p1 · · · p~n |~p1 · · · p~n i
j =1
(2π)4 δ(4) (0) = V T
===========
1
(2Ep~j V )
n
P
|M(kA → {pj })|2
VT
.
(2π)4 δ (4) kA −
pj Q
n
2mA V
j =1
(2Ep~j V )
j =1
53
Rt
The linear time factor T = t−+ dt in this expression was to be expected from Fermi’s
Golden Rule! This factor can be divided out in order to obtain the corresponding constant
decay rate.
Next we integrate over all possible momenta of the n final-state particles. This time
it does matter whether there are identical particles in the final state. In order to avoid
double counting we have to restrict the integration to inequivalent configurations or divide
by 1/nk ! factors for any group of nk identical final-state particles. Generically we will
indicate this combinatorial final-state identical-particle factor by Cf . The final expression
for the integrated constant decay rate then becomes
density of states
n
}|
{ (2π)4 δ (4) k − P p |M(k → {p })|2
zY
Z
A
j
A
j
n
d~pj
j =1
V
= Cf
Q
n
3
(2π)
j =1
2mA
2Ep~j V
R
Γn
j =1
1
=
Cf
2mA
z
Z
Lorentz invariant
}|
{
dΠn |M(kA → {pj })|2 ,
in terms of the relativistically invariant n-body phase-space element
Y
n
n
P
1
d~pj
4 (4)
pj ,
(2π)
δ
k
−
dΠn ≡
A
3
(2π) 2Ep~j
j =1
j =1
(3)
which is sometimes denoted by dPSn in other textbooks. This decay rate is called the
partial decay width for the decay mode into the considered n-particle final state.
After summation over all possible final states one obtains the so-called total decay width
1
Γ =
2mA
X
final states
Cf
Z
dΠf |M(kA → {pf })|2 ,
with dΠf corresponding to a given final state.
8a This total decay width is related to the half-life of the decaying particle
through the relation τ = 1/Γ. If the decaying particle is not at rest, the decay width is reduced by a factor mA /E~kA . This leads to an increased half-life
√
τ E~kA /mA = τ / 1 − ~v 2 ≡ γτ , where ~v is the velocity of the decaying particle.
2.8.2
Cross sections for scattering reactions
Consider a beam of B particles hitting a target at rest consisting of A particles. The case
of two colliding particle beams can be obtained from this by an appropriate Lorentz boost.
Let’s start by assuming constant densities ρA and ρB in target and beam.
54
The number of scattering events will be proportional to (ρA ℓA )(ρB ℓB )O , with O the crosssectional overlap area common to both the beam
and the target. The ratio
ℓA
beam
1 # scattering events
# scattering events
≡
(OℓA ρA )(OℓB ρB )/O
NA
NB /O
vB
≡ σ
O
ρB
ρA
defines the cross section σ as the effective area of
ℓB
a chunk taken out of the beam by each particle
in the target. The quantities NA and NB are
the numbers of A and B particles that are reletarget
vant for scattering, i.e. the particles that at some
point in time belong to the overlap between beam and target. All of this can be equally
well formulated in terms of time-related quantities like the scattering rate and the incoming
particle flux: simply replace the number of scattering events by the number of scattering
events per second and ℓB ρB by the flux vB ρB of beam particles. Hence,
σ =
1 scattering rate
NA beam flux
Approximate plane-wave states: in reality ρA and ρB are not constant, since the
colliding particles are described quantum mechanically by wave packets and both beam
and target have a density profile. However,
the studied range of the interaction between the colliding particles is usually
much smaller than the width of the individual wave packets perpendicular to
the beam, which in turn is much maller than the actual diameter of the beam.
Therefore, in good approximation ρA and ρB can be considered as locally constant on
quantum mechanical (i.e. interaction) length scales3 , whereas the density profiles inside
the beam and target can be incorporated properly by averaging over the overlap region:
Z
ℓA ℓB d2x⊥ ρA (x⊥ )ρB (x⊥ ) ≡ NA NB /O .
Here NA and NB are the effective numbers of A and B particles that are relevant for
scattering and x⊥ is the spatial coordinate perpendicular to the beam. From this it
follows that
# scattering events = σ NA NB /O ,
where σ can be calculated for effectively constant values of ρA and ρB corresponding to
approximately plane-wave initial states. By the way, we don’t have to restrict ourselves to
3
These (slowly changing) densities can even be locally zero!
55
the total number of scattering events. In a similar way we can study the cross section for
scattering into the region d~p1 · · · d~pn around the n-particle final-state momentum point
p~1 , · · · , ~pn . This is actually what detectors usually do: they detect particles with energy and
momentum in certain finite bins, which are given by the detector resolution. These bins
cannot resolve the momentum spread of any of the wave packets, just like the detector cells
can in general not resolve the particle positions at the level of the de Broglie wavelengths.
For all practical purposes detectors observe classical point-like particles with well-defined
momenta (in direction and magnitude). So, in the final state it makes sense to use plane
waves as well.
8b Calculating cross sections therefore amounts to calculating transition probabilities in momentum space. These transition probabilities are universal in the
sense that they are independent of details of the experiment, such as the properties of the beams, the targets or the preparation of the initial-state particles.
The differential cross section: consider an initial state consisting of one target particle and one beam particle in the momentum state |~kA , ~kB i scattering into a final state
consisting of n particles in a momentum state with momenta inside the bin d~p1 · · · d~pn
around p~1 , · · · , ~pn . In analogy with the calculation in § 2.8.1, the corresponding differential
transition probability per unit time and per unit flux is given by
dσ =
1
1
|M(kA , kB → {pj })|2 dΠn ,
F 4E~kAE~kBV
which is usually referred to as the differential cross section. As explained in § 2.8.1 this
result for dσ is also valid for identical particles in the final state. In this expression F
stands for the flux associated with the incoming beam particle:
|~vA − ~vB | ~v = p~/E | ~kA /E~kA − ~kB /E~kB |
1
|~vrel | =
,
======
F =
V
V
V
where we have chosen ~ez along the beam axis. Furthermore, we have used that the fourmomentum of a massive particle reads pµ0 = (m, ~0 ) in its rest frame, which becomes
E0 = m , p~0 = ~0
pµ = γ (E0 + ~v · p~0 ), γ (~p0 + E0~v ) ========= (mγ, mγ~v ) upon boosting with velocity v
along the ~ep -direction. We therefore find
dσ =
|M(kA , kB → {pj })|2 dΠn
4|E~ ~kA − E~ ~kB |
kB
kA
for the differential cross section.
8b The so-called flux factor 14 |E~kB ~kA − E~kA~kB |−1 is invariant under boosts
along the beam direction and the same goes for the differential cross section dσ,
as expected for a cross-sectional area perpendicular to the beam.
56
2.8.3
CM kinematics and Mandelstam variables for 2 → 2 reactions
Consider a 2 → 2 reaction with matrix element M kA , kB → p1 , p2 . In the CM frame
with the z-direction along the beam axis the corresponding kinematics reads:
1
~p1
~kA
A
~kB
CM
CM
θ
B
2
z-axis
φ
~p2
after
before
µ
kAµ = (EA , 0, 0, k) , kB
= (EB , 0, 0, −k)
pµ1 = (E1 , ~p ) , pµ2 = (E2 , −~p ) .
Hence, the two final-state particles are produced back-to-back in the CM frame. Written
in compact notation the CM momenta and energies are given by
k =
⇒
q
2
EA,B
− m2A,B
,
p = |~p | =
EA,B
2
ECM
+ m2A,B − m2B,A
=
2ECM
E1,2
2
ECM
+ m21,2 − m22,1
=
2ECM
q
2
E1,2
− m21,2
,
,
and
EA + EB = E1 + E2 ≡ ECM
k =
p
2 − m2 − m2 )2 − 4m2 m2
(ECM
A
B
A B
,
2ECM
p =
p
2 − m2 − m2 )2 − 4m2 m2
(ECM
1
2
1 2
.
2ECM
The matrix element is Lorentz invariant, so it can be expressed in terms of invariant
combinations of the particle momenta. Since only three out of four particle momenta are
independent, this leaves six kinematical variables: the squared masses of the four particles,
three so-called Mandelstam variables that combine two of the particle momenta and one
condition. We start with the Mandelstam variable
2
s ≡ (kA + kB )2 = (p1 + p2 )2 = ECM
.
In order to guarantee that both k, p ≥ 0 and EA,B ≥ mA,B this variable has to satisfy the
inequalities s ≥ (mA + mB )2 and s ≥ (m1 + m2 )2 . The expressions for the CM energies
and momenta then become
s + m21,2 − m22,1
s + m2A,B − m2B,A
√
√
, E1,2 =
,
EA,B =
2 s
2 s
p
p
(s − m2A − m2B )2 − 4m2A m2B
(s − m21 − m22 )2 − 4m21 m22
√
√
and
p =
.
k =
2 s
2 s
57
The other two Mandelstam variables are
t ≡ (kA − p1 )2 = (kB − p2 )2
u ≡ (kA − p2 )2 = (kB − p1 )2 ,
and
which contain the angular dependence of the reaction through
2~kA · ~p1 = 2k~ez · ~p = 2kp cos θ
2~kA · ~p2 = − 2k~ez · p~ = − 2kp cos θ .
and
These three Mandelstam variables satisfy the energy-momentum conservation condition
s + t + u = m2A + m2B + m21 + m22 .
A few conventions: in general 2 → 2 reactions the most similar initial- and final-state
particles are combined into the t-variable. For instance, in the reaction e+ e− → µ+ µ−
one should combine the momenta of the electron (e− ) and muon (µ− ), or equivalently the
momenta of the positron (e+ ) and antimuon (µ+ ). A reaction channel is referred to as
s-channel (or t-channel, or u-channel) if the Mandelstam variable s (or t, or u) features in
the propagator at lowest order.
1
s-channel:
u-channel:
t-channel:
A
B
2
A
A
The 2-body phase-space element: in the CM frame, where ~kA = − ~kB ≡ k~ez , the
beam flux reads
kECM
k(EA + EB )
≡
.
FCM =
EA EB V
EA EB V
The differential cross section for a 2 → 2 reaction in the CM frame therefore becomes
dσCM =
|M(kA , kB → p1 , p2 )|2 dΠ2
.
4kECM
2
8b Note that the differential cross section falls of as 1/ECM
at high energies.
This is a destructive interference effect caused by probing the relevant interaction length scale with particles that have a much smaller de Broglie wavelength.
In analogy with equation (3) the Lorentz invariant phase-space element for two final-state
particles becomes
Z
dΠ2 =
CM
===
Z
Z
CM: δ(ECM −E1 −E2 ) δ(~
p1 +~
p2 )
d~p1 1
(2π)3 2E1
Z
z
}|
{
d~p2 1
4 (4)
(2π) δ (kA + kB − p1 − p2 )
(2π)3 2E2
dp
p2
16π 2 E1 E2
Z
dΩ δ(ECM − E1 − E2 ) .
58
p
p
Replacing the integration variable p by E1 + E2 = p2 + m21 + p2 + m22 this becomes
Z
Z
Z
p2
d(E1 + E2 )
dΩ δ(ECM − E1 − E2 )
dΠ2 =
16π 2 E1 E2 p/E1 + p/E2
Z
Z 2π Z 1
p
p
=
dΩ =
dφ
d cos θ ,
16π 2 ECM
16π 2 ECM 0
−1
where θ is the polar scattering angle with respect to the beam axis and φ the azimuthal
scattering angle around the beam axis (as displayed in the figure at the start of this
paragraph). From this the following angular differential cross section can be obtained:
dσ 2
p
|M
k
,
k
→
p
,
p
=
| .
(4)
A
B
1
2
2
dΩ CM
64π 2 kECM
In view of rotational symmetry about the z-axis there will be no φ-dependence and the
φ-integral will straightforwardly yield a factor 2π . Once we also integrate over θ to obtain
the total cross section σ, one has to restrict this integration to inequivalent configurations
or multiply by the appropriate final-state identical-particle factor Cf .
To give a simple example, we again consider the process
φ(kA ) + φ(kB ) → φ(p1 ) + φ(p2 )
in the scalar φ4 -theory. As we have seen on page 47, the lowest-order matrix element for
this process is given by −λ. Hence,
Z
dσ dσ λ2
|M|2 p
λ2
λ2
1
=
dΩ
=
=
=
and
σ
=
,
2 k
2
dΩ CM
64π 2 ECM
64π 2ECM
64π 2 s
2
dΩ CM
32πs
where the factor 1/2 occurring in the last expression is the identical-particle factor for two
identical final-state particles. Further examples of cross sections for 2 → 2 reactions can
for instance be found in Ex. 11.
2.9 Dealing with states in the interacting theory (§7.1 in the book)
9 In order to close the gaps that were left behind during previous steps, we now
have to address some of the non-perturbative properties of the interacting theory.
We study these issues by considering n-point correlation functions (Green’s functions) in
the scalar φ4 -theory:
hΩ|T φ̂(x1 ) · · · φ̂(xn ) |Ωi
(n)
.
G (x1 , · · · , xn ) ≡
hΩ|Ωi
Here φ̂(x1 ), · · · , φ̂(xn ) are Heisenberg fields in the interacting theory and |Ωi is the ground
state of the interacting theory, which satisfies
Ĥ |Ωi = E0 |Ωi
and
59
hΩ|Ωi = 1 .
These Green’s functions play an important role in the derivation of scattering amplitudes
and are interesting objects in their own right, for instance for studying density perturbations. Without loss of generality we can take x01 = t1 ≥ x02 = t2 ≥ · · · ≥ x0n = tn ,
so that
hΩ|T φ̂(x1 ) · · · φ̂(xn ) |Ωi = hΩ|φ̂(x1 ) · · · φ̂(xn )|Ωi
Û (tn−1 ,tn )
Û(t1 ,t2 )
}|
{
}|
{
z
z
p. 33
−1
−1
−1
==== hΩ|Û (t1 , 0) φ̂I (x1 ) Û (t1 , 0) Û (t2 , 0) · · · Û (tn−1 , 0) Û (tn , 0) φ̂I (xn ) Û (tn , 0)|Ωi .
Projecting on the free-particle vacuum: for an arbitrary state |ψi it will prove handy
to consider
Ĥ0 |0i ≡ 0
completeness relation for Ĥ X
hψ|Û(0, t− )|0i ====== hψ|eiĤt− |0i ====================
hψ|eiĤt− |nihn|0i
n
= eiE0 t− hψ|ΩihΩ|0i +
X
n 6= Ω
eiEn t− hψ|nihn|0i
and subsequently take the limit t− → − ∞. The “summation” over the excited states
{|ni =
6 |Ωi} is just a shorthand notation, in fact it will involve an integration over energy
(see later). Provided that there is a finite energy gap between the ground state |Ωi and
the excited states |n 6= Ωi, as is for instance the case for massive excitations, we can employ
the Riemann–Lebesgue lemma. This lemma states that
Z v1
dv f (v) eiβv = 0
lim
β →± ∞
v0
for any integrable function f and any compact or non-compact interval [v0 , v1 ]. Using this
lemma one finds for an arbitrary state |ψi the identity
lim
t− → − ∞
e−iE0 t− hψ|Û(0, t− )|0i
= hψ|Ωi +
hΩ|0i
lim
t− → − ∞
X
ei(En −E0 )t−
n 6= Ω
hψ|nihn|0i
= hψ|Ωi .
hΩ|0i
Similarly we can derive the identity
lim
t+ → + ∞
eiE0 t+ h0|Û(t+ , 0)|ψi
= hΩ|ψi .
h0|Ωi
This procedure closely resembles Fermi’s Golden Rule for time-dependent perturbation
theory. By supplying |0i with the right frequency factor and waiting long enough, only
the |Ωi component of |0i survives as a result of destructive phase interference.
Note: on pages 86 and 87 of the textbook by Peskin & Schroeder the same
identities are obtained by tilting the time axis according to t → t(1 − iǫ) with
ǫ ∈ R infinitesimal. This procedure is closely related to the iǫ prescription for
obtaining the Feynman propagator in § 1.6.
60
Inserting these identities in the numerator and denominator of the Green’s functions yields
G(n) (x1 , · · · , xn )
=
=
lim
t± → ± ∞
lim
t± → ± ∞
Û (tn ,t− )
Û(t+ ,t1 )
}|
{
}|
{
z
z
h0| Û(t+ , 0) Û −1 (t1 , 0) φ̂I (x1 )Û(t1 , t2 ) · · · Û (tn−1 , tn )φ̂I (xn ) Û (tn , 0) Û(0, t− ) |0i
h0|Û(t+ , 0) Û(0, t− )|0i
h0|T φ̂I (x1 ) · · · φ̂I (xn ) Û (t+ , t− ) |0i
h0|T φ̂I (x1 ) · · · φ̂I (xn ) Ŝ |0i
=
,
h0|Û(t+ , t− )|0i
h0|Ŝ|0i
9a linking Green’s functions and the time-ordered vacuum expectation values
with vacuum bubbles removed that we studied in § 2.5.
Interpretation of the vacuum bubbles: we have seen that
P
e j Vj
z
}|
{
iE0 (t+ −t− )
lim e
h0|Û(t+ , t− )|0i = h0|ΩihΩ|ΩihΩ|0i = |hΩ|0i|2 .
t± → ± ∞
P
j Vj
From this it follows that e
∝ e−iE0 (t+ −t− ) = e−iE0 T . The sum of all vacuum bubbles is
therefore related to the difference in the ground-state zero-point energies of the interacting
theory and the free theory, the latter of which was defined to be 0 in the discussion above.
Bearing in mind that Vj contains an infinite spacetime factor (2π)4 δ (4) (0) = V T , the
energy density of the ground state of the interacting theory reads
P X Im(Vj )
Im
E0
j Vj
= −
= −
.
V
VT
(2π)4 δ (4) (0)
j
The long-distance infinity from the infinite extent of spacetime has been removed in this
way, leaving behind the UV infinity that reflects our ignorance about the physics governing
the ultra-high-energy regime.
2.9.1
Källén–Lehmann spectral representation
9b In the free theory h0|T φ̂I (x)φ̂†I (y) |0i could be interpreted as the amplitude for a particle to propagate from y to x. The question now is: how should
the corresponding 2-point Green’s function hΩ|T φ̂(x)φ̂† (y) |Ωi be interpreted
in the interacting theory? This question is related to the particle interpretation
of the interacting theory.
Complete set of interacting states: we start out by having a generic look at the excited
states of the interacting theory, with the corresponding energies being defined relative to
ˆ
the ground-state energy E0 . This analysis will be based on the fact that Ĥ, P~ = 0,
ˆ
which implies that there is a simultaneous set of eigenfunctions of Ĥ−E0 1̂ and P~ . These
states can consist of an arbitrary number of particles or they can even be bound states.
61
1) Zero-momentum states: let {|λ~0 i} be the set of excited eigenstates of Ĥ with vanˆ
ishing total three-momentum, i.e. P~ |λ~0 i = ~0. These simultaneous eigenvalues of Ĥ −E0 1̂
ˆ
and P~ can be combined into the four-vector pµ0 = (mλ , ~0 ), where mλ > 0 is the “mass”
associated with the particular zero-momentum state.
ˆ
2) Finite-momentum states: the generator of spacetime translations P̂ µ ≡ (Ĥ−E0 1̂, P~ )
transforms as a contravariant four-vector under boosts: Û −1 (Λ) P̂ µ Û (Λ) = Λµν P̂ ν . This
implies that all boosts of the states |λ~0 i have all possible total three-momenta p~ and are
p
p~ 2 + m2λ . The other way round,
also eigenstates of Ĥ − E0 1̂ with energy Ep~ (λ) ≡
any eigenstate with explicit three-momentum can be boosted to a zero-momentum eigenstate provided that mλ > 0. The sets of eigenvalues pµ = (E −E0 , p~ ) are thus organized
into hyperboloids, as shown in the figure below. The lowest-lying isolated hyperboloid
corresponds to the “1-particle” states of the interacting theory, whereas the other ones
correspond to possible bound states. Above a certain threshold value of mλ a continuum
of “multiparticle” states starts (see later).
E −E0
“multiparticle”
continuum
moving “particle”
bound states
mph
“particle” at rest
|~
p|
Proof of the boost statement : consider the Lorentz transformation Λ that transforms
pµ0 = (mλ , ~0 ) into pµ = Λµν pν0 = (Ep~ (λ), p~ ). Then |λp~i ≡ Û (Λ)|λ~0i indeed satisfies
P̂ µ |λp~ i = Û (Λ) Û −1 (Λ)P̂ µ Û (Λ)|λ~0i = Λµν Û(Λ)P̂ ν |λ~0 i = Λµν pν0 Û (Λ)|λ~0i = pµ |λp~i .
By reversing the argument, the reversed statement can be proven as well, bearing in mind
that E −E0 > 0 for the excited states so that the combined four-momentum eigenvalues
pµ = (E −E0 , p~ ) have to satisfy p2 ≥ 0.
Completeness relation: in the interacting theory we can therefore use the following
completeness relation associated with this complete set of states:
X Z d~p |λp~ ihλp~ |
1̂ = |ΩihΩ| +
,
(2π)3 2Ep~ (λ)
λ
where the first term corresponds to the ground state and the second one to all excited states.
62
The 2-point Green’s function: next we take x0 > y 0 and insert the above-given completeness relation into the 2-point Green’s function. This results in the following split-up:
hΩ|T φ̂(x)φ̂† (y) |Ωi = hΩ|φ̂(x)|ΩihΩ|φ̂† (y)|Ωi
X Z d~p
1
+
hΩ|φ̂(x)|λp~ihλp~ |φ̂† (y)|Ωi .
3
(2π) 2Ep~ (λ)
λ
In the absence of preferred directions in the universe, the ground state |Ωi should be
invariant under spacetime translations and Lorentz transformations, i.e. eiP̂ ·x |Ωi = |Ωi
and Û (Λ)|Ωi = |Ωi. Therefore
p.18
hΩ|φ̂(x)|λp~ i ==== hΩ|eiP̂ ·x φ̂(0)e−iP̂ ·x |λp~ i = e−ip·x hΩ|φ̂(0)|λp~i
0
p =Ep~ (λ)
= e−ip·x hΩ|Û −1 (Λ)Û(Λ)φ̂(0)Û −1 (Λ)Û(Λ)|λp~i
p0 =Ep~ (λ)
p.21
==== e−ip·x hΩ|φ̂(Λ0)|λΛp
i
~ p0 =Ep~ (λ)
and similarly
~ = ~0
take Λ such that Λp
================ e−ip·x hΩ|φ̂(0)|λ~0 i
p0 =Ep~ (λ)
hΩ|φ̂(x)|Ωi = hΩ|φ̂(0)|Ωi ≡ v .
The ground-state expectation value v, which in the literature is sloppily called the “vacuum expectation value” or short vev of the field φ̂, usually is taken to be 0. If this is not
the case then one should reformulate the theory in terms of the field φ̂′ (x) = φ̂(x) − v ,
which has a vanishing vev. The rest goes in the same way as described below. Leaving out
the vev we now obtain
Z
X
d~p e−ip·(x−y) 2
†
|hΩ|φ̂(0)|λ~0 i|
hΩ|φ̂(x)φ̂ (y)|Ωi =
(2π)3 2Ep~ (λ) 0
p =Ep~ (λ)
λ
x0 > y 0 , p. 24
========
X
λ
|hΩ|φ̂(0)|λ~0 i|
2
Z
d4 p ie−ip·(x−y)
.
(2π)4 p2 − m2λ + iǫ
The integral on the last line we recognize as the Feynman propagator belonging to a
“φ-particle” with mass mλ , i.e. DF (x − y; m2λ).
9b The particle interpretation has in fact changed in the interacting theory
from free particles to dressed particles (quasi-particles), so the “particles” we
are dealing with here are not the particles that we know from the free theory!
Källén–Lehmann spectral representation: a similar procedure can be applied in the
case that x0 < y 0. Combining both cases one arrives at the so-called Källén–Lehmann
spectral representation of the 2-point Green’s function:
Z ∞
ds
†
ρ(s)DF (x − y; s) ,
hΩ|T φ̂(x)φ̂ (y) |Ωi =
0 2π
63
where the function ρ(s) in the squared invariant mass s is a positive spectral density
function given by
X
ρ(s) =
2π δ(s − m2λ ) |hΩ|φ̂(0)|λ~0 i|2 .
λ
The states in the interacting theory that describe a single dressed particle correspond to
an isolated δ-function in the spectral density function:
2
ρ1-part. (s) = 2π δ(s − m2ph ) hΩ|φ̂(0)|λ~0 i1-part. ≡ 2πZ δ(s − m2ph ) .
9b The field-strength/wave-function renormalization Z is the probability for
φ̂† (0) to create a state that describes a single dressed particle from the ground
state, whereas mph is the observable physical mass of the dressed particle, being the energy eigenvalue in its rest frame. This physical (dressed) mass is in
general not equal to the (bare) mass parameter m occurring in the Lagrangian,
which is not observable directly!
In momentum space: the Källén–Lehmann spectral representation trivially reads
Z ∞
Z
i
ds
4
ip·x
†
d x e hΩ|T φ̂(x)φ̂ (0) |Ωi =
ρ(s) 2
p − s + iǫ
0 2π
Z ∞
iZ
i
ds
= 2
+
ρ(s) 2
2
p − mph + iǫ
p − s + iǫ
∼sth 2π
in momentum space, where sth denotes the threshold for the creation of the continuum
of “multiparticle” states. The fact that the last integral does not start exactly at sth is
caused by the possible existence of multiparticle bound states. Graphically the analytic
(pole/cut) structure in the complex p2 -plane can be depicted as follows:
Im p2
“1-particle”
pole
bound-state
poles
sth
m2ph
branch cut
Re p2
(continuum of poles)
Figure 6: Poles and cuts of the 2-point Green’s function.
Interacting theory vs free theory:
• In the interacting theory |hΩ|φ̂(0)|λ~0i|2 = |hΩ|φ̂(0)|λp~i|2 represents the probability
for the field φ̂† (0) to create a given dressed state from the ground state, with the
factor Z being the associated probability for creating a “1-dressed-particle” state.
64
The factor Z differs from unity since in the interacting theory φ̂† (0) can also create
“multiparticle” intermediate states with a continuous mass spectrum, unlike in the
free theory.
• In the free theory ρ(s) = 2π δ(s − m2 ) and Z = 1, since
p
h~p |φ̂I (0)|0i = h0| 2Ep~ âp~
†
Z
†
d~q b̂~q + â~q
p
|0i = h0|0i = 1 .
(2π)3
2E~q
For x0 > 0 the quantity
Z
d4 x eip·x h0|T φ̂(x)φ̂† (0) |0i =
p2
i
− m2 + iǫ
is interpreted as the amplitude for a particle to propagate from 0 to x.
2.9.2
2-point Green’s functions in momentum space (§ 6.3 and 7.1 in the book)
9c Question: does all this also follow from an explicit diagrammatic calculation within perturbation theory?
In order to address this question we consider the 2-point Green’s function for ψ-particles
in the scalar Yukawa theory (with tadpole diagrams excluded, as will be explained later):
Z
d4 x eip·x hΩ|T ψ̂(x)ψ̂ † (0) |Ωi =
p
+
p
p − ℓ1
p
ℓ1
+
···
i
i
i
2
−
iΣ
(p
)
+
+ ··· ,
2
p2 − M 2 + iǫ
p2 − M 2 + iǫ
p2 − M 2 + iǫ
=
where
2
− iΣ2 (p ) = (−ig)
2
Z
d 4 ℓ1
i
i
2
4
2
2
(2π) ℓ1 − M + iǫ (p − ℓ1) − m2 + iǫ
is the so-called ψ-particle self-energy at O(g 2 ). Since the corresponding diagram involves
one loop and therefore one energy-momentum integration, we usually refer to this selfenergy as the 1-loop self-energy.
9c There are two main approaches to calculate such an integral:
1. perform the ℓ10 -integration in the complex plane, involving four complex
poles, and work out the resulting ~ℓ1 -integration;
2. apply the following two calculational tricks.
65
Trick 1: use Feynman parameters. Writing the denominators in the integral as
D1 ≡ ℓ21 − M 2 + iǫ
D2 ≡ (p − ℓ1)2 − m2 + iǫ ,
and
we can combine the two denominators into
α2 = 1
1
1
1
1 1
1
=
=
−
D1 D 2
D1 − D2 D 2 D1
D1 − D2 α2 D2 + (1 − α2 )D1 α2 = 0
=
Z
1
0
1
=
dα2
[α2 D2 + (1 − α2 )D1 ]2
Z
1
dα1
0
Z
0
1
dα2 δ(α1 + α2 − 1)
1
.
(α1 D1 + α2 D2 )2
The parameters α1,2 are called Feynman parameters. Inserting the specific expressions for
the denominators we then obtain
Z 1
Z 4
−2
d ℓ1 2
2
2
− iΣ2 (p ) = g
dα2
ℓ1 − 2α2 p · ℓ1 + α2 p2 − α2 m2 − (1 − α2 )M 2 + iǫ
4
(2π)
0
Z 1
Z
d4 ℓ
1
2
,
≡ g
dα2
4
2
(2π) (ℓ − ∆ + iǫ)2
0
with
ℓ ≡ ℓ 1 − α2 p
and
∆ ≡ α2 m2 + (1 − α2 )M 2 − α2 (1 − α2 )p2 .
We have gained the following in this first step:
• The original integrand had four poles in the complex ℓ10 -plane, whereas now we have
only two poles in the complex ℓ0 -plane.
• The integrand has become spherically invariant, implying that integrals with an odd
numerator in ℓ should vanish, i.e.
Z
Z
4
2
d ℓ f (ℓ )ℓµ =
d4 ℓ f (ℓ2 )ℓµ ℓν ℓρ = · · · = 0 .
In contrast, integrals with an even numerator in ℓ can be simplified. For instance
Z
Z
0 if µ 6= ν gµν
4
2
d ℓ f (ℓ )ℓµ ℓν =======
d4 ℓ f (ℓ2 )ℓ2 ,
4
using that
Z
Z
Z
Z
4
4
4
2
2
2
2
2
2
d ℓ f (ℓ )(ℓ1 ) =
d ℓ f (ℓ )(ℓ2 ) =
d ℓ f (ℓ )(ℓ3 ) = − d4 ℓ f (ℓ2 )(ℓ0 )2 .
These properties will in particular prove important for non-scalar particles.
• The trick works equally well for an arbitrary number of propagators occurring in the
loop:
Z 1
Z 1
(n − 1)! δ(α1 + · · · + αn − 1)
1
=
.
dα1 · · · dαn
D1 · · · D n
(α1 D1 + · · · + αn Dn )n
0
0
66
Trick 2: perform Wick rotation. In order to perform the ℓ0 -part of the integral
R 4 2
d ℓ (ℓ − ∆ + iǫ)−j /(2π)4 the integration contour C indicated in figure 7 is used. Since
the poles are situated outside the integration contour in the complex ℓ0 -plane, the integral
along the real ℓ0 -axis is transformed into an integral along the imaginary axis.
Im ℓ0
C
infinite quarter-circle
−
p
~
ℓ 2 +∆−iǫ
*
+
p
Re ℓ0
*
2
~
ℓ +∆−iǫ
infinite quarter-circle
Figure 7: Closed integration contour used for performing Wick rotation.
In this way a Minkowskian integral can be transformed into a Euclidean one:
−i∞
Z−∞
Z∞
Z∞
Z
ℓ0 ≡ i ℓ0E
0
0
0
dℓ → − dℓ ====== −i dℓE = i dℓ0E
−∞
∞
i∞
and
−∞
Z
~ ~
ℓ ≡ ℓE
d~ℓ ====
=
Z
d~ℓE .
This results in
Z
d4 ℓ
i
1
=
(2π)4 (ℓ2 − ∆ + iǫ)j
(2π)4
Z∞ Z
dℓ0E d~ℓE −∞
i
(−1)j
=
16π 4
Z
d4 ℓE
(ℓ2E + ∆ − iǫ)j
i
=
(−1)j
16π 4
Z
2π
i
(−1)j
=
16π 2
Z
∞
dθ1
0
0
Z
1
j
− (ℓ0 )2 − ~ℓ 2 − ∆ + iǫ
π
dθ2 sin(θ2 )
0
E
Z
0
π
1
dθ3 sin (θ3 )
2
ℓ2E
,
dℓE 2
(ℓE + ∆ − iǫ)j
2
67
E
2
Z
0
∞
dℓ2E
ℓ2E
(ℓ2E + ∆ − iǫ)j
where the norm ℓ2E = (ℓ0E )2 + (ℓ1E )2 + (ℓ2E )2 + (ℓ3E )2 is positive definite in Euclidean space.
In the penultimate step it was used that in an n-dimensional Euclidean space the transition
to spherical coordinates is given by
Z
Z ∞
Z 2π Z π
Z π
n−1
d~r f (r) =
dr r f (r)
dθ1 dθ2 sin(θ2 ) · · · dθn−1 sinn−2 (θn−1 )
0
0
2π n/2
=
Γ(n/2)
Z
0
0
∞
dr r n−1 f (r) ,
0
where the gamma function Γ(z) satisfies
Γ(1/2) =
√
π
,
Γ(1) = 1
and
Γ(z + 1) = z Γ(z) .
The result after applying both tricks:
ig 2
− iΣ2 (p ) =
16π 2
Z
ig 2
=
16π 2
Z
2
1
dα2
0
Z
ℓ2E
(ℓ2E + ∆ − iǫ)2
∞
0
dℓ2E
1
0
dα2 − 1 − log ∆(α2 ) − iǫ + UV infinity ,
where the infinity originates from the large-momentum regime ℓ2E → ∞. The logarithm
log(z) ≡ log |z|eiφ
= log |z| + iφ
gives rise to a branch cut for z ∈ IR− , since log(−x ± iǫ) = log(xe±iπ ) = log(x) ± iπ for
x > 0. This corresponds to situations where ∆(α2 ) = α22 p2 + α2 (m2 − M 2 − p2 ) + M 2 < 0
on the interval α2 ∈ [0, 1]. Since ∆(α2 = 1) = m2 and ∆(α2 = 0) = M 2 , this happens
when ∆(α2 ) = 0 has both roots
p
p2 + M 2 − m2 ± (p2 + M 2 − m2 )2 − 4p2 M 2
α2 =
2p2
q
p2 + M 2 − m2 ±
p2 − (M +m)2 p2 − (M −m)2
=
2p2
on the interval α2 ∈ [0, 1], which results in the requirement that p2 > (M + m)2 .
9c There is a minimal value p2min = (M + m)2 of p2 for which the branch
cut of the 2-point Green’s function in the scalar Yukawa theory starts, being
the threshold for the creation of a two-particle state with masses M and m.
This is precisely what we would expect based on the Källén–Lehmann spectral
representation.
68
Dyson series: to all orders in perturbation theory the 2-point Green’s function (a.k.a. the
full propagator or dressed propagator) is given by the Dyson series
Z
d4 x eip·x hΩ|T ψ̂(x)ψ̂ † (0) |Ωi
p
=
p
≡
+
p
p
p
1PI
p
+
p
1PI
p
1PI
+ ··· ,
where
1PI
≡ − iΣ(p2 ) =
+
+
+
+ ···
is the collection of all 1-particle irreducible self-energy diagrams. Diagrams are called
1-particle irreducible if they cannot be split in two by removing a single line.
The single-particle pole and physical mass: the Dyson series is in fact a geometric
series, which can be summed according to
Z
p
p
d4 x eip·x hΩ|T ψ̂(x)ψ̂ † (0) |Ωi =
=
=
i
i
i
2
−
iΣ(p
)
+
+ ···
2
2
2
2
2
p − M + iǫ
p − M + iǫ
p − M 2 + iǫ
p2
−
M2
i
.
− Σ(p2 ) + iǫ
The full propagator has a simple pole located at the physical mass Mph , which is shifted
away from M by the self-energy:
h
i
2
2
2
2
2 = 0 ⇒ Mph
− M 2 − Σ(Mph
) = 0.
p − M − Σ(p ) 2
2
p = Mph
Close to this pole the denominator of the full propagator can be expanded according to
2
2
2 2
p2 − M 2 − Σ(p2 ) ≈ (p2 − Mph
) 1 − Σ′ (Mph
) + O [p2 − Mph
]
for
2
p2 ≈ Mph
,
where Σ′ (p2 ) stands for the derivative of the self-energy with respect to p2 .
9c Just like in the Källén–Lehmann spectral representation, the full propa2
gator has a single-particle pole of the form iZ/(p2 − Mph
+ iǫ) with residue
′
2
Z = 1/ 1 − Σ (Mph ) . This observed close connection to the non-perturbative
analytic structure of the 2-point Green’s function serves as justification for our
procedure, which involved summing the geometric series outside its formal radius of convergence.
69
2.9.3
Deriving n-particle matrix elements from n-point Green’s functions
For real scalar fields φ̂(x) we have seen that
Z
4
ip·x
d xe
hΩ|T φ̂(x)φ̂(0) |Ωi
p2 → m2ph
^
p2
iZ
,
− m2ph + iǫ
by which is meant that the quantities on either side have the same single-particle poles and
residues at the physical mass squared m2ph . The wave-function renormalization factor Z
can be obtained straightforwardly from the 2-point Green’s function in momentum space
by multiplying by (p2 − m2ph )/i and taking the limit p2 → m2ph .
9d We now wish to use this single-particle pole structure to obtain the asymptotic “in” and “out” states of the theory and in particular their matrix elements.
Consider to this end
Z
d4 x eip·x hΩ|T φ̂(x)φ̂(z1 )φ̂(z2 ) · · · |Ωi
with
Z
Z∞
ZT+
ZT−
dx0 = dx0 + dx0 + dx0 ,
T+
T−
−∞
where T− < min zj0 and T+ > max zj0 .
What can we say about the pole structure of this integrated Green’s function?
• The integration region x0 ∈ [T− , T+ ]: since the temporal integration interval is
bounded and the integrand has no p0 -poles, the result of the integral is an analytic
function in p0 without any poles.
• The other two integration regions: the integrand still has no poles, but the integration
intervals are unbounded. Therefore singularities in p0 may develop upon integration!
The integration interval x0 ∈ [T+ , ∞): we again insert the completeness relation for Ĥ
and assume that the field φ̂(x) has a vanishing vev. If hΩ|φ̂(x)|Ωi = v 6= 0, then φ̂(x)
should be rewritten as φ̂(x) ≡ v + φ̂′ (x) and the particle interpretation should be obtained
from φ̂′ (x) rather than φ̂(x). The integral then takes the form
Z∞ Z
XZ
p ·~
x
ip0x0 −i~
0
e
dx d~x e
λ
T+
d~q
1
hΩ|
φ̂(x)|λ
ihλ
|T
φ̂(z
)
φ̂(z
)
·
·
·
|Ωi
q
~
q
~
1
2
(2π)3 2E~q (λ)
Z∞ Z
XZ
−i~
p ·~
x
0
====
dx d~x e
p. 63
T+
λ
p
Z(λ)
d~q
0 0
hλ
|T
φ̂(z
)
φ̂(z
)
·
·
·
|Ωi ei~q ·~x eix [p −Eq~ (λ)] ,
q
~
1
2
3
(2π) 2E~q (λ)
70
p. 63
where hΩ|φ̂(x)|λ~qi ==== e−iq·x hΩ|φ̂(0)|λ~0i
q 0 = E~q (λ)
p
Z(λ) e−iq·x ≡
q 0 = Eq~ (λ)
. The phase of
hΩ|φ̂(0)|λ~0i does not matter in this context, since it can be absorbed in the definition of
|λ~0 i. Now the Riemann–Lebesgue lemma can be invoked, which states that the larger x0
becomes the sharper this integral is peaked around p0 = E~q (λ). This fact can be quantified
0
explicitly by adding a damping factor e−ǫ x (with infinitesimal ǫ > 0) to the integral, in
order to ensure that it is well-defined. This procedure is equivalent with the iǫ prescription
for obtaining the Feynman propagator in § 1.6 and the tilted time axis prescription in the
textbook by Peskin & Schroeder. After performing the trivial ~x integration we get
X
λ
=
p
Z(λ)
hλp~ |T φ̂(z1 )φ̂(z2 ) · · · |Ωi
2Ep~ (λ)
X
λ
Z∞
0 0
dx0 eix [p −Ep~ (λ)+iǫ]
T+
p
0
Z(λ)
ieiT+ [p −Ep~ (λ)+iǫ]
hλp~ |T φ̂(z1 )φ̂(z2 ) · · · |Ωi 0
,
2Ep~ (λ)
p − Ep~ (λ) + iǫ
which corresponds to isolated 1-particle poles, isolated bound-state poles or multiparticle
branch-cut poles. Subsequently we note that
0
i
i
= 2
2
2
2
p − mλ + iǫ
p0 − Ep~ (λ) + iǫ
and
1
ieiT+ [p −Ep~ (λ)+iǫ]
2Ep~ (λ) p0 − Ep~ (λ) + iǫ
have the same residues at the pole p0 = Ep~ (λ) − iǫ.
Z
The 1-particle state in the q
far future corresponds to an isolated pole at the
on-shell energy p0 = Ep~ = p~ 2 + m2ph :
d4 x eip·x hΩ|T φ̂(x)φ̂(z1 )φ̂(z2 ) · · · |Ωi
p0 → Ep~
^
√
i Z
p2 − m2ph + iǫ
p |T
out h~
φ̂(z1 )φ̂(z2 ) · · · |Ωi ,
using the notation |~p iout ≡ |λp~i1-part. for a 1-particle eigenstate with momentum p~ that is
created at asymptotically large times.
The integration interval x0 ∈ (−∞, T− ]: in this case the steps are similar to the ones for
the previous integration interval. The following changes should be made though: the
0
0
damping factor e−ǫ x should be replaced by e+ǫ x , φ̂(x) is now situated at the end of the
operator chain, e−iq·x should be replaced by e+iq·x and the pole energy p0 = Ep~ (λ) − iǫ
now changes to p0 = − Ep~ (λ) + iǫ.
Z
The 1-particle state in the
q far past corresponds to an isolated pole at the on-shell
0
energy p = − Ep~ = − p~ 2 + m2ph :
4
ip·x
d xe
hΩ|T φ̂(x)φ̂(z1 )φ̂(z2 ) · · · |Ωi
√
i Z
hΩ|T
φ̂(z
)
φ̂(z
)
·
·
·
|−~p iin .
1
2
2
^ p2 − mph + iǫ
p0 → − Ep~
71
LSZ reduction formula: the procedure described above can actually be worked out for
situations with as many 1-particle poles as there are field operators in the Green’s function.
This leads to the so-called LSZ (H. Lehmann, K. Symanzik, W. Zimmermann) reduction
formula:
Y
Y
n′ Z
n Z
4
4
−ikj ′ ·yj ′
ipj ·xj
hΩ|T φ̂(x1 ) · · · φ̂(xn ) φ̂(y1 ) · · · φ̂(yn′ ) |Ωi
d yj ′ e
d xj e
j′= 1
j =1
~
p0j → Ep~j
^
kj0′
→ E~k
j′
Y
n
j =1
~
hp
~1 ··· p
~n |Ŝ |k1 ··· kn′ i
√
√
Y
z
n′
}|
{
i Z
i Z
~
~
′
i
h~
p
·
·
·
p
~
|
k
·
·
·
k
out 1
n 1
n in ,
2
2
p2j − m2ph + iǫ
k
′ − mph + iǫ
j
′
j =1
(5)
where the use of e−ikj′ ·yj′ ensures that the particles in the “in” state have positive energy.
The S-matrix element involving n′ particles in the “in” state and n particles in the
“out” state can be obtained from the corresponding (n + n′ )-point Green’s function by
extracting the leading singularities in the energies kj0′ and p0j , which coincide with the
situations where the external particles become on-shell.
9d The pole structure of the Green’s functions emerging at asymptotic times contains all
relevant information about the scattering amplitudes of the theory! To select the required
information one should project on the right singularities by using appropriate plane waves.
Wave packets instead of plane waves:
• In the asymptotic treatment of multiparticle states it is better to use normalized wave
packets. In that case x is constrained to lie within a small band about the trajectory
of a particle with momentum p~ , with the spatial extent of the band being determined
by the wave packet. In this way the particles do not interfere and can effectively be
considered free at asymptotic times, unlike plane-wave states. Therefore we formally
should have made the replacement
Z
Z
Z
d~k
0 0
~
4
ip·x
d4 x eip x ϕ(~k )e−ik·~x ,
d xe
→
3
(2π)
with ϕ(~k ) a function that is peaked around p~ , and we should have taken the limit
of a sharply peaked wave packet ϕ(~k ) → (2π)3 δ(~k − p~ ) at the end of the calculation.
• A 1-particle wave packet spreads out differently than a multiparticle wave packet, so
the overlap between them goes to zero as the elapsed time goes to infinity. Although
φ̂(x) creates some multiparticle states, we can “select” the 1-particle state that we
want by using an appropriate wave packet. By waiting long enough we can make
the multiparticle contribution to the matrix element as small as we like (cf. Fermi’s
Golden Rule for time-dependent perturbation theory).
72
• An n-particle asymptotic state is created/annihilated by n field operators that are
constrained to lie in distant wave packets and therefore are effectively localized.
Under these conditions an n-particle excitation in the continuum can be represented
by n distinct (independent) 1-particle excitations of the ground state.
Translated in terms of Feynman diagrams: in order to investigate the implications
of the LSZ reduction formula we consider the 4-point Green’s function
Z
Z
Z
Z
4
4
4
−ikA ·y1
ip2 ·x2
ip1 ·x1
d4 y2 e−ikB ·y2 hΩ|T φ̂(x1 )φ̂(x2 ) φ̂(y1 )φ̂(y2 ) |Ωi
d y1 e
d x2 e
d x1 e
in the scalar φ4 -theory. From this we want to derive the T -matrix element for the scattering
process φ(kA )φ(kB ) → φ(p1 )φ(p2 ). To this end we need to consider the contributions from
fully connected diagrams, as was explained in § 2.6. These diagrams can be represented
generically by
p1
p2
p2
p1
amp
kA kB
kA
kB
The blob in the centre of the diagram represents the sum of all amputated 4-point diagrams:
1
2
B
1
2
2
+
=
amp
A
1
A
B
1
2
A
A
B
1
2
+
+
B
+
B
A
··· .
The shaded circles indicate that the corresponding full propagators
p
p
=
should be used, where
− iΣ(p2 ) =
1PI
=
i
p2 − m2 − Σ(p2 )
+
+
+ ···
represents the 1-particle irreducible scalar self-energy diagrams in φ4 -theory. Near the
physical particle pole p2 = m2ph the full propagator can be expanded according to
p2 − m2ph
+ O [p2−m2ph ]2 .
p2−m2−Σ(p2 ) ≈ (p2−m2ph ) 1−Σ′ (m2ph ) + O [p2−m2ph ]2 ≡
Z
73
As a result, the sum of all fully connected diagrams contains a product of four poles:
p21
iZ
iZ
iZ
iZ
,
2
2
2
2
2
2
− mph p2 − mph kA − mph kB − m2ph
multiplying the amputated 4-point diagrams. According to the LSZ reduction formula (5)
the T -matrix element for the scattering process φ(kA )φ(kB ) → φ(p1 )φ(p2 ) thus reads
1
√
h~p1 p~2 |i T̂ |~kA~kB i = ( Z )4
2
,
amp
A
B
with all external momenta being on-shell.
Any 4-point diagram that is not fully connected, like the
one displayed in the figure on the right, does not contain
a product of four poles. Such diagrams are therefore projected out in the transition from the Green’s function to
the T -matrix.
kA
kB
kA
kB
9d This completes the derivation of the connection between scattering matrix elements and fully connected amputated Feynman diagrams that was given
on page 49 of these lecture notes. In fact we have also obtained the missing
ingredient in the Feynman rules for the scalar φ4 -theory on page 49.
Multiply the sum of all possible fully connected amputated Feynman diagrams in posi√
′
tion/momentum space by a factor ( Z )n+n for n+n′ external particles.
2.9.4
The optical theorem (§ 7.3 in the book)
From the unitarity of the S-operator it follows that
Ŝ = 1̂+i T̂
Ŝ †Ŝ = 1̂ ====⇒ (1̂− i T̂ † )(1̂+ i T̂ ) = 1̂+ i(T̂ − T̂ † )+ T̂ † T̂ = 1̂ ⇒ − i(T̂ − T̂ † ) = T̂ † T̂ .
In order to investigate the implications of this equation we consider the scattering process
φ(kA )φ(kB ) → φ(p1 )φ(p2 ) in the scalar φ4 -theory:
− i h~p1 p~2 |T̂ |~kA~kB i + i h~p1 ~p2 |T̂ † |~kA~kB i = h~p1 p~2 |T̂ † T̂ |~kA~kB i
n Z
X 1 Y
d~qj
1
=
h~p1 ~p2 |T̂ † |{~qj }ih{~qj }|T̂ |~kA~kB i ,
3 2E
n!
(2π)
j
n
j =1
74
where in the last step a complete set of intermediate plane-wave states has been inserted.
In terms of matrix elements this becomes:
− iM(kA , kB → p1 , p2 ) + iM∗ (p1 , p2 → kA , kB )
X 1 Z
dΠn M∗ p1 , p2 → {qj } M kA , kB → {qj } ,
=
n!
n
containing the n-body phase-space element that was defined in equation (3). Using the
abbreviations a ≡ kA , kB , b ≡ p1 , p2 and f ≡ {qj } this results in the generalized optical
theorem
X Z
∗
Cf dΠf M∗ (b → f )M(a → f ) ,
− iM(a → b) + iM (b → a) =
f
where Cf stands for the combinatorial identical-particle factor belonging to the state f
(i.e. the factors 1/n! in this φ4 example). This generalized optical theorem is equally
valid for initial/final states consisting of one particle or more than two particles. In more
complicated theories the summation on the right-hand-side of the optical theorem runs
over all possible sets of “final-state” particles that can be created by the initial state a.
Specialized to forward scattering, i.e. p1 = kA and p2 = kB (⇒ a = b), this yields the
optical theorem in its standard form:
X Z
Cf dΠf |M(a → f )|2 = inverse flux factor ∗ σtot (a → anything) ,
2ImM(a → a) =
f
where the inverse flux factor reads 4ECM |~k | in the CM frame of the reaction.
9e The optical theorem expresses the total cross section for scattering in terms
of the attenuation (reduction) of the forward-going wave as the beams pass
through each other. This is caused by the destructive interference between the
scattered wave and the beam.
Diagrammatic example for φ4 -theory at first non-trivial order: in Ex. 12 it is
worked out that
q


kB
kA
q2 2
1
Z
kB
kB
kA
kA
1


− i
2Im  − i
dΠ2 − i
.
 =
2
kB
kA
kA
kB
kA
kA
kB
kB
The factors − i on the left-hand-side are in fact cancelled by the factor i from Wickrotating the loop integral. Note the absence of the lowest-order matrix element on the
left-hand-side, because it has no imaginary part. This is nicely consistent with the righthand-side, which contributes at O(λ2 ) rather than at O(λ).
75
Sources of imaginary parts: the imaginary parts that feature in the optical theorem
originate from the iǫ parts of the propagators. For instance
p2 − m2
iǫ
1
1
=
∓
=
P
∓ iπ δ(p2 − m2 ) ,
p2 − m2 ± iǫ
(p2 − m2 )2 + ǫ2
(p2 − m2 )2 + ǫ2
p2 − m2
where P stands for the principal value. When going from p2 − iǫ to p2 + iǫ there is a
− 2πiδ(p2 − m2 ) jump (discontinuity) in the propagator.
9e Non-vanishing imaginary parts correspond to those situations where intermediate particles inside the loop(s) become on-shell. The associated lines of
the diagram are in that case referred to as being “cut”. The imaginary parts
are the result of branch-cut discontinuities, marking invariant-mass values for
which certain multiparticle intermediate states become physically possible.
The Cutkosky cutting rules (without proof): the discontinuities of an arbitrary
Feynman diagram can be obtained by means of a general method that is based on the
discontinuities of the individual propagators. It involves the following three-step procedure
(usually referred to as the Cutkosky cutting rules):
• cut the diagram in all possible ways, with all cut propagators becoming on-shell
simultaneously;
• replace 1/(p2 − m2 + iǫ) by − 2πiδ(p2 − m2 ) in each cut propagator and perform
the loop integrals;
• sum the contributions of all (kinematically) possible cuts.
2.10
The concept of renormalization (chapter 10 in the book)
Before we close this chapter on interacting scalar field theories, there is one
final issue to be addressed.
As we have already observed in the previous discussion, there still is the issue of UV diR∞
vergences from the loop integrals 0 dℓ2E ℓ2E /(ℓ2E + ∆ − iǫ)j for j ≤ 2.
10 Question: how should we deal with UV divergences that occur at loop level
in the perturbative expansion of interacting quantum field theories, bearing in
mind that physical observables should be finite?
The occurrence of singularities should not come as a surprise, though. Inside the loops
particles of all energies are taken into account as being described by the same theory, i.e. we
treat them as point-particles at all length scales, which is rather unrealistic.
76
Regularization: before we can continue the discussion we first have to quantify the UV
divergence. This is called regularization. An obvious way is the so-called cutoff method:
Z
0
∞
dℓ2E
to be replaced by
−−−−−−−−−−→
Z
0
Λ2
dℓ2E ,
which implies that all Fourier modes with momentum larger than Λ are removed. This
means that the corresponding fields are not allowed to fluctuate too energetically. In
this way we look at the physics through blurry glasses: we are interested in length scales
L>
∼ 1/Λ, but we do not care about length scales L < 1/Λ. This approach reflects that
quantum field theory is in some sense an effective field theory with Λ marking the threshold of our ignorance beyond which quantum field theory ceases to be valid. As such, the
cutoff Λ plays the role that 1/a played in the 1-dimensional quantum chain in Ex. 1,
although Λ does not correspond to a specific energy/mass scale in the theory and should
in fact be taken much larger than any such scale.
10a We speak of a renormalizable quantum field theory if it keeps its predictive
power in spite of its shortcomings at small length scales.
Technically this means that we should be able to absorb all UV divergences of the theory
into a finite number of parameters of the theory (like couplings and masses).
Example: consider the φ4 -process φ(kA )φ(kB ) → φ(p1 )φ(p2 ) at 1-loop order in the CM
frame of the reaction. To make life easy we will neglect the mass of the particles in this
study, which will not affect the outcome. Indicating the relevant invariant-mass scale of
the process by s, the matrix element reads
Λ2 Λ2 λ2
Λ2 Mφφ→φφ (s) = − λ +
+ log
+ log
+ 3 + O(λ3 )
log
32π 2
−s − iǫ
−t
−u
4 Λ2 λ2
=== − λ +
+ iπ + 3 + O(λ3 ) .
+ log
3 log
2
2
32π
s
sin θ
CM
Details of the calculation are worked out in Ex. 12. As we will see later Z = 1 + O(λ2 ), so
√
there will be no 1-loop contribution from the wave-function renormalization factor ( Z )4
in φ4 -theory.
From this result a few interesting observations follow.
1. The Lagrangian parameter (bare coupling) λ is not an observable quantity! The
quantum corrections are an integral part of the effective coupling, which can be
measured through |Mφφ→φφ (s)|2 . This effective coupling is energy-dependent due
to the creation and annihilation of virtual particles (quantum fluctuations) at 1-loop
order. So, the effective strength of the φ4 -interaction changes with energy!
77
2. Mφφ→φφ (s) depends logarithmically on the cutoff at O(λ2 ). A short but sloppy way
of saying this is that “Mφφ→φφ (s) is logarithmically divergent”.
3. |Mφφ→φφ (s)|2 is observable and should therefore be independent of Λ. After all, Λ
can be chosen arbitrarily and as such an observable cannot depend on it. To achieve
this, the unobservable bare coupling λ should depend on the cutoff Λ:
!
4 Λ2 λ
dλ
dMφφ→φφ (s)
+ iπ + 3
+ log
−1+
3 log
=
0 =
dΛ2
dΛ2
16π 2
s
sin2 θ
+
⇒
3λ2 1
+ O(λ3 )
32π 2 Λ2
Λ2 dλ
3
d(1/λ)
=
−
≈ −
2
2
2
d log(Λ )
λ dΛ
32π 2
⇒
Λ2 1
1
3
≈
−
log 2
λ(Λ2 )
λ(µ2 )
32π 2
µ
⇒
λ(Λ2 ) ≈
λ(µ2 )
1−
3 λ(µ2 )
32π 2
log(Λ2 /µ2 )
.
This is an example of what is called a Renormalization Group Equation (RGE),
which tells us that λ(Λ2 ) grows with Λ2 if λ(µ2 ) > 0.
The miracle of vanishing divergences: renormalization
Suppose we measure the above-given effective 4-point coupling at s = µ2 and let’s call
this physical observable λph :
Λ2 4 λ2
2
3 log 2 + log
Mφφ→φφ (s = µ ) ≡ − λph = − λ +
+ iπ + 3 + O(λ3 ) .
32π 2
µ
sin2 θ
The bare coupling λ can then be expressed in terms of the physical coupling λph and the
divergence log(Λ2 /µ2 ) according to
Λ2 4 λ2ph
3 log 2 + log
− λ = − λph −
+ iπ + 3 + O(λ3ph ) .
32π 2
µ
sin2 θ
If we now want to know the effective 4-point coupling at an arbitrary scale s, then we can
simply write
µ2 3λ2ph
+ O(λ3ph ) ,
log
Mφφ→φφ (s) = − λph +
32π 2
s
where the logarithmic term is completely governed by the above-given RGE for λ. This
reflects that the observable effective 4-point coupling should not depend on the choice
of reference scale µ.
10b The reference scale µ labels an equivalence class of parametrizations of
the φ4 -theory and it should not matter which element of the class we choose for
setting up the theory.
78
When expressed in terms of the physical coupling λph , the effective coupling |Mφφ→φφ (s)|2
is independent of the cutoff Λ, as expected for a correct observable! The cutoff dependence
has been absorbed into a redefinition of the unobservable Lagrangian parameter (bare
coupling) λ in terms of the observable physical parameter (effective coupling) λph . In the
literature this physical observable is usually referred to as the renormalized coupling λR ,
although this terminology is a bit strange bearing in mind that the original coupling was
not normalized to begin with. This is an example of the concept of renormalization.
10b Renormalization: express physically measurable quantities in terms of
physically measurable quantities and not in terms of bare Lagrangian parameters.
• For setting up a perturbative expansion, the bare Lagrangian parameters are in fact
not the right parameters. Instead the physically measurable parameters should be
used (cf. the discussion about m and mph in § 2.9.2).
• The occurrence of infinities in the loop integrals is linked to this. Our initial perturbative expansion consisted of taking Λ → ∞ while keeping λ and m finite. From
the renormalization group viewpoint, however, the set (µ = Λ = ∞, λ < ∞, m < ∞)
does not belong to the equivalence class of the φ4 -theory!
• The convergence of the perturbative series can be further improved by using physical quantities at the “right scale”, thereby avoiding large logarithmic factors like
log(µ2 /s) in the example above. This choice of scale has no consequence for all-order
calculations, but it does if the series is truncated at a certain perturbative order.
To complete the story for the scalar φ4 -theory we consider the UV divergences that are
present in the scalar self-energy. This time the mass parameter is essential and therefore
should not be neglected.
Scalar self-energy at O(λ):
ℓ1
2
O(λ)
− iΣ(p ) ====
p
p
cutoff Λ ≫ m
− iλ
=
2
Z
d4 ℓ1
i
2
4
(2π) ℓ1 − m2 + iǫ
− iλ
32π 2
Z
Λ2
−−−−−−−−−−−→
Wick rotation
0
Λ2 ℓ2E
− iλ
2
2
=
Λ − m log
.
dℓE 2
ℓE + m2 − iǫ
32π 2
m2
2
After Dyson summation the full propagator becomes
iZ
i
≡
+ regular terms .
p2 − m2 − Σ(p2 ) + iǫ
p2 − m2ph
79
Since the 1-loop scalar self-energy does not depend on p2, it is absorbed completely into
the physical mass:
Λ2 λ
O(λ)
2
2
2
2
2
2
mph = m + Σ(mph ) ==== m +
Λ − m log
,
32π 2
m2
whereas the residue of the pole remains 1.
Note the strong Λ2 dependence of the scalar mass, which implies that this mass
is very sensitive to high-scale quantum corrections. This is in fact a general
feature of scalar particles, like the Higgs boson: intrinsically the quantum corrections to the mass of a scalar particle are dominated by the highest mass scale
the scalar particle couples to!
Scalar self-energy at O(λ2 ): the residue of the pole is affected at 2-loop level by the
contribution
ℓ2
p
p
(−iλ)2
=
6
ℓ1
Z
d4 ℓ1
(2π)4
Z
d4 ℓ2
i
i
i
2
2
4
2
2
(2π) ℓ1 − m + iǫ ℓ2 − m + iǫ (ℓ1 + ℓ2 + p)2 − m2 + iǫ
= a + bp2 + cp4 + · · · .
To assess the UV behaviour of this diagram we perform naive power counting, which involves treating all loop momenta as being of the same order of magnitude. For ℓ1,2 → ∞
R
ℓE ≤ Λ
we obtain an integral of the order d8 ℓE /ℓE6 −−−
−→ Λ8−6 = Λ2 .
• a = O(Λ2 ) is obtained by setting p = 0;
• b = O(log Λ) is obtained by taking 12 ∂ 2/∂p20 and then setting p = 0. In naive power
counting this logarithmically divergent term corresponds to integrals of order Λ0 .
1 4
∂ /∂p40
4!
• c = O(1) is obtained by taking
and then setting p = 0.
Adding all self-energy contributions and focussing on the diverging terms
p2
−
Z =
m2
i
− Σ(p2 ) + iǫ
→
1
= O(log Λ) ,
1−B
p2
−
m2
m2ph =
i
iZ
≡ 2
+ regular terms ,
2
− A − Bp
p − m2ph
m2 + A
≡ Zm2 + δm2
1−B
,
δm2 =
A
= O(Λ2 ) .
1−B
This leads to an O(Λ2 ) shift in the mass and an O(log Λ) contribution to the wavefunction renormalization, which can be absorbed in the field φ itself.
80
So, UV divergent loop corrections in φ4 -theory are present in Σ(p2 ) and Mφφ→φφ (s), with
Σ(m2ph ) = m2ph − m2 = (Z − 1)m2 + δm2 ≡ m2 δZ + δm2
,
Σ′ (m2ph ) = 1 − 1/Z
Mφφ→φφ (s = µ2 ) = − λph ≡ − Z 2 λ − δλ .
and
The occurrence of the factor Z 2 in the last expression originates from the multiplicative
√
factor ( Z )4 that should be added according to the Feynman rules.
2.10.1
Physical perturbation theory (a.k.a. renormalized perturbation theory)
10b The lowest-order φ4 -theory should have been written in terms of the ex-
perimentally measurable physical parameters mph and λph , and perturbation
theory should have been defined with respect to this lowest-order theory.
This is done as follows: take the original Lagrangian and write
√
φ = φR Z
m2 Z = m2ph − δm2
,
λZ 2 = λph − δλ
,
and
Z ≡ 1 + δZ
so that
L =
1
λ
1
1
λph 4
1
(∂µ φ)(∂ µφ) − m2 φ2 − φ4 = (∂µ φR )(∂ µφR ) − m2ph φ2R −
φ
2
2
4!
2
2
4! R
+
1
δλ 4
1
δZ (∂µ φR )(∂ µφR ) + δm2 φ2R +
φ .
2
2
4! R
We get back the original Lagrangian in terms of renormalized objects (first line) and we
obtain extra interactions that are called counterterms (second line), since their purpose
is to cancel the divergences in the theory. The Feynman rules for the propagators and
vertices including counterterms are now given by
p
p
×
=
p
p2
i
,
− m2ph + iǫ
= −iλph ,
= i(p2 δZ + δm2 ) ,
×
= iδλ .
Renormalization conditions: as an explicit example, the full propagator now reads
i/ p2 − m2ph − ΣR (p2 ) , with the renormalized self-energy given by
2
− iΣR (p ) =
+
×
+
+
81
+
×
+
×
+ ···
The parameters δZ and δm2 can be fixed by imposing the renormalization conditions
ΣR (m2ph ) = 0
and
Σ′R (m2ph ) = 0
⇒
full propagator =
p2
i
+ regular terms .
− m2ph
The pole structure of the full propagator then resembles that of a free particle, so in that
sense the physical 1-particle states have been re-normalized by this procedure. Adding
one more renormalization condition based on Mφφ → φφ in order to fix δλ , we have three
conditions fixing three counterterm parameters. This will in fact be sufficient to make all
observables of the φ4 -theory finite.
10b The scalar φ4 -theory is called renormalizable: “the infinities of the theory
can be absorbed into a finite number of parameters”.
2.10.2
What has happened?
The above procedure seems odd: we calculated something that turned out to be infinite,
then subtracted infinity from our original mass and coupling in an arbitrary way and ended
up with something finite. Moreover, we have added divergent terms to our Lagrangian and
we have suddenly ended up with a scale-dependent coupling. Why would a procedure
consisting of such ill-defined mathematical tricks be legitimate? To see what has really
happened, let us closely examine the starting point of our calculation.
In general, we start with a Lagrangian containing all possible terms that are compatible
with basic assumptions such as relativity, causality, locality, etc. It still contains a few
parameters such as m and λ in the case of φ4 -theory. It is tempting to call them “mass”
and “coupling”, as they turn out to be just that in the classical (i.e. lowest-order) theory.
However, up to this point they are just free parameters. In order to make the theory
predictive, the parameters need to be fixed by a set of measurements: we should calculate
a set of cross sections at a given order in perturbation theory, measure their values and then
fit the parameters so that they reproduce the experimental data. After this procedure, the
theory is completely determined and becomes predictive.
The bare parameters m and λ are only useful in intermediate calculations and will be
replaced by physical (i.e. measured) quantities in the end anyway. So, we might as well
parametrize the theory in terms of the latter. The renormalizability hypothesis is that this
reparametrization of the theory is enough to turn the perturbation expansion into a welldefined expansion. The divergence problem then has nothing to do with the perturbation
expansion itself: we have just chosen unsuitable parameters to perform it. Also, the
fact that our physical coupling is scale-dependent should not surprise us. The physical
reason for this “running” is the existence of quantum fluctuations, which were not there
in the classical theory. These fluctuations correspond to intermediate particle states: at
82
sufficiently high (i.e. relativistic) energies, new particles can be created and annihilated.
As the available energy increases, more and more energetic particles can be created. This
effectively changes the couplings.
Having traded the bare parameters m and λ for renormalized parameters mph and λph ,
let us take a closer look at the internal consistency of the renormalization procedure. We
have introduced the physical coupling at a reference scale µ, but we could equally well have
chosen an energy scale µ′ with corresponding effective coupling λ′ph . Physical processes
should not depend on our choice of reference scale, hence the couplings should be related
in such a way that for any observable O we have O = O(mph , µ, λph) = O(mph , µ′ , λ′ph ).
In other words, there should exist an equivalence class of parametrizations of the theory
and it should not matter which element of the class we choose. This observation clarifies
where the divergences came from: our initial perturbation expansion consisted of taking
Λ → ∞ while keeping m and λ finite. From the viewpoint of the renormalization group,
however, the set (µ = Λ = ∞ , m < ∞ , λ < ∞) does not belong to any equivalence class
of the φ4 -theory.
2.10.3
Superficial degree of divergence and renormalizability
10c The statement at the end of §2.10.1 was a bit premature. In fact we still
have to prove that amplitudes with more than four external particles do not
introduce a new type of infinity that cannot be absorbed into the 2- and 4-point
terms in the Lagrangian.
A 6-point diagram like
will contain singular building blocks like
and
that should become finite
once we perform the afore-mentioned renormalization procedure. The question that remains is whether the overall 6-point diagram can give rise to a new type of infinity. To assess
this we perform naive power counting, i.e. we treat all loop momenta as being of the same
order of magnitude. The outcome of this power counting is called the superficial degree of
divergence D of the diagram, with D = 0 denoting logarithmic divergence.
Consider an amputated diagram with N external lines, P propagators and V vertices.
• Each external line is connected to one vertex, each propagator is connected to two
vertices and in φ4 -theory four lines meet in each vertex. This results in the condition
4V = N + 2P
⇒
P = 2V − N/2
83
and
N = even number .
• The number of loop momenta is given by the number of propagators − the number
of four-momentum δ-functions + 1, since one of the δ-functions corresponds to the
external momenta and will not fix an internal loop momentum. This results in
L = P − V + 1 = V − N/2 + 1
independent loop momenta. So, loop diagrams require V ≥ N/2.
Power counting: assume for argument’s sake that the loop momenta are d-dimensional.
That means that in the context of naive power counting each loop momentum contributes
Λd and each propagator Λ−2. The superficial degree of divergence of the diagram then reads
D = dL − 2P = d(V − N/2 + 1) − 2(2V − N/2) = d + V (d − 4) + N(1 − d/2) ,
whereas the coupling λ has mass dimension [λ] = 4 − d in d dimensions.
Superficially the diagram diverges like ΛD if D > 0 and like log(Λ) if D = 0,
provided it contains a loop. The diagram does not diverge superficially if D < 0.
Let’s now consider a few values for the dimensionality d of spacetime.
d = 4: D = 4 − N is independent of V and [λ] = 0 ⇒ the theory is renormalizable.
Divergences occur at all orders, but only a finite number of amplitudes diverges
superficially (i.e. amplitudes with N = 2 or 4)! The theory keeps its predictive
power in spite of the infinities that occur if we assume it to be valid at all energies.
d = 3: D = 3 − N/2 − V and [λ] = 1 ⇒ the theory is superrenormalizable. At most a
finite number of diagrams diverges superficially (i.e. the diagrams with N = 2 and
V = 1 or V = 2), as the diagrams get less divergent if the loop order is increased!
d = 5: D = 5 − 3N/2 + V and [λ] = − 1 ⇒ the theory is nonrenormalizable. Now all
amplitudes will diverge superficially at a sufficiently high loop order! An infinite
amount of counterterms would be required to remove all divergences, which means
that all predictive power is lost if we assume the theory to be valid at all energies!
10c If we express the superficial degree of divergence in terms of V and N,
the coefficient in front of V determines whether the theory is superrenormalizable (negative coefficient), renormalizable (zero coefficient) or nonrenormalizable (positive coefficient)!
In conclusion: for d > 4 the scalar φ4 -theory is nonrenormalizable and [λ] < 0, for d = 4
it is renormalizable and [λ] = 0, and for d < 4 it is superrenormalizable and [λ] > 0.
These conclusions agree nicely with the general discussion on page 27 of these lecture notes.
84
3
The Dirac field
During the next three and a half lectures Chapter 3 of Peskin & Schroeder will be covered.
We have seen various aspects of scalar theories, describing spin-0 particles. However, most
particles in nature have spin 6= 0.
11a Question: how should we find Lorentz-invariant equations of motion for
fields that do not transform as scalars?
Consider to this end an n-component multiplet field Φa (x) with a = 1, · · · , n, which has
the following linear transformation characteristic under Lorentz transformations:
Lorentz transf.
Φa (x) −−−−−−−−−−→ Mab (Λ)Φb (Λ−1 x)
with summation over the repeated index implied. A compact way of writing this is
Lorentz transf.
Φ(x) −−−−−−−−−−→ M(Λ)Φ(Λ−1 x) .
In the case of scalar fields the transformation matrix M(Λ) was simply the unit matrix. In
order to find different solutions, we make use of the fact that the Lorentz transformations
form a group: Λµν = g µν is the unit element, Λ−1 = ΛT is the inverse, and for Λ1
and Λ2 being Lorentz transformations also Λ3 = Λ2 Λ1 is a Lorentz transformation. The
transformation matrices M(Λ) should reflect this group structure:
M(g) = I
,
M(Λ−1 ) = M −1 (Λ)
and
M(Λ2 Λ1 ) = M(Λ2 )M(Λ1 ) .
To phrase it differently, the transformation matrices M(Λ) should form an n-dimensional
representation of the Lorentz group!
The continuous Lorentz group (rotations and boosts): transformations that lie
infinitesimally close to the identity transformation define a vector space, called the Lie
algebra of the group. The basis vectors for this vector space are called the generators of
the Lie algebra. The Lorentz group has six generators J µν = −J νµ , three for boosts and
three for rotations. These generators are antisymmetric, as a result of Λ−1 = ΛT , and they
satisfy the following set of fundamental commutation relations:
µν ρσ J ,J
= i g νρ J µσ − g µρ J νσ − g νσ J µρ + g µσ J νρ .
The three generators belonging to the boosts and the three generators belonging to the
rotations are given by
1 jkl kl
ǫ J ⇒ J jk = ǫjkl J l
(j , k , l = 1, · · · , 3) ,
2
with summation over the repeated spatial indices implied. The latter generators, which
span the Lie algebra of the rotation group, satisfy the fundamental commutation relations
j k
J , J = iǫjkl J l .
K j ≡ J 0j
respectively
Jj ≡
85
11a In fact it is proven in Ex. 15 that all finite-dimensional representations
of the Lorentz group correspond to pairs of integers or half integers (j+ , j− ),
where both j+ and j− correspond to a representation of the rotation group.
The sum j+ + j− should be interpreted as the spin of the representation, since
it corresponds to the actual rotations contained in the Lorentz group.
A finite Lorentz transformation is then in general given by exp(−iωµν J µν/2), where the
antisymmetric tensor ωµν ∈ IR represents the Lorentz transformation. For instance:
ω12 = − ω21 = δθ ,
rest = 0
⇒



ω µν = 

1 0
0 0
0 1 −δθ 0
0 δθ
1 0
0 0
0 1





for an infinitesimal rotation about the z-axis (see Ex. 14), and
ω01 = − ω10 = δv
,
rest = 0
⇒



ω µν = 

1 δv 0 0
δv 1 0 0
0 0 1 0
0 0 0 1





for an infinitesimal boost along the x-direction (see Ex. 14).
The task at hand is now to find the matrix representations of the generators of
the Lorentz group.
Examples: in Ex. 14 it is proven that
• (J µν )αβ = i(g µα g νβ − g µβ g να ) are the six generators that describe Lorentz transformations of contravariant four-vectors:
Lorentz transf.
xα −−−−−−−−−−→ Λαβ xβ =
exp(−iωµν J µν/2)
α
xβ ≈
β
g αβ −
i
ωµν (J µν )αβ xβ .
2
This implies that g αβ − 2i ωµν (J µν )αβ = g αβ + ω αβ represents the infinitesimal form of
the Lorentz transformation matrix Λαβ , as is indeed the case.
• J µν = i(xµ ∂ ν − xν ∂ µ ) are the six generators in coordinate space, which describe the
infinitesimal Lorentz transformations of scalar fields
Lorentz transf.
φ(x) −−−−−−−−−
−→ φ(Λ−1x) ≈ φ(x) −
as derived on page 10.
86
1
ωρσ [xσ ∂ ρ − xρ ∂ σ ]φ(x) ,
2
Dirac’s trick: introduce four n × n matrices γ µ that are referred to as
the γ-matrices of Dirac, which satisfy the Dirac algebra (Clifford algebra)
µ ν
γ ,γ
≡ γ µ γ ν + γ ν γ µ = 2g µν In ,
11b
with In the n×n unit matrix. In Ex. 14 it is proven that this implies that the
n×n matrices S µν ≡ 4i γ µ , γ ν form a representation of the generators J µν of
the Lorentz group.4
Four-dimensional solution to the Dirac algebra: since there are no solutions for
n = 2 or 3, the first solution can be found for n = 4. Written in 2 × 2 block form in terms
of the 2 × 2 unit matrix I2 and the Pauli spin matrices
!
!
!
0
1
0
−
i
1
0
σ1 =
, σ2 =
and
σ3 =
,
1 0
i 0
0 −1
the solution reads
0 I2
γ0 =
I2
0
!
γj =
and
0
σj
− σj
0
!
(j = 1, 2, 3)
in the Weyl representation, which is also known as the chiral representation. In fact there
is an infinite number of such four-dimensional representations, since for any invertable 4×4
matrix V also V γ µ V −1 is a solution. In the Weyl representation the generators of the
Lorentz group have a block-diagonal form. The generators for boosts are given by
!
j
σ
0
i
i
i
γ 0, γ j = γ 0γ j = −
(j = 1, 2, 3) ,
S 0j =
4
2
2
0 − σj
whereas the generators S 1 , S 2 and S 3 for rotations follow from
!
!
j k
1 l
σ
,
σ
0
σ
0
i
i
j
=
6
k
2
γj , γk = −
≡ ǫjkl S l
S jk ====
j k = ǫjkl
1 l
4
4
0
σ ,σ
0 2σ
Sl =
⇒
1 l
σ
2
0
0
1 l
σ
2
!
≡
1 l
Σ
2
(j, k = 1, 2, 3)
(l = 1, 2, 3) .
The generators for rotations look like twice replicated two-dimensional representations of
the rotation group. We will come back to this point later on. As a result of the properties
(γ 0 )† = γ 0
,
(γ j )† = − γ j
(j = 1, 2, 3)
⇒
(γ µ )† = γ 0 γ µ γ 0 ,
the generators of the Lorentz group satisfy
i µ † ν †
i
(γ ) , (γ ) = γ 0 S µν γ 0 .
(S µν )† = − (γ ν )† , (γ µ )† =
4
4
4
In fact this is true for any spacetime dimensionality
87
This means that the generators of rotations are hermitian, since (S jk )† = S jk , indicating
that rotations preserve normalization. On the other hand, the generators of boosts are
non-hermitian, since (S 0j )† = − S 0j , indicating that boosts do not preserve normalization
owing to the Lorentz contraction of spatial volumes.
Dirac spinors and adjoint Dirac spinors: a four-component field ψ(x) that Lorentz
transforms according to this four-dimensional representation of the Lorentz group is called
a Dirac spinor:
Lorentz transf.
ψ(x) −−−−−−−−−−→ Λ1/2 ψ(Λ−1 x)
with
Λ1/2 = exp(−iωµν S µν/2) .
The adjoint Dirac spinor ψ̄(x) is defined as
ψ̄(x) ≡ ψ † (x)γ 0
and therefore transforms as
Lorentz transf.
ψ̄(x) −−−−−−−−−
−→ ψ̄(Λ−1 x)Λ−1
1/2 ,
since
(S µν )† = γ 0 S µν γ 0
γ 0 Λ†1/2 γ 0 = γ 0 exp iωµν [S µν ]†/2 γ 0 ===========
= exp(iωµν S µν/2) = Λ−1
1/2 .
Using the important γ-matrix property
µ ρσ γ ,S
=
i
i µ ρ σ
γ ,γ γ
= (γ µ γ ρ γ σ − γ ρ γ σ γ µ ) = i(g µρ γ σ − g µσ γ ρ )
2
2
= i(g ρµ g σν − g ρν g σµ )γ ν = (J ρσ )µν γ ν ,
the following infinitesimal Lorentz-transformation identity holds up to O(ω):
I4 +
i
i
i
ωρσ S ρσ γ µ I4 − ωαβ S αβ ≈ g µν − ωρσ (J ρσ )µν γ ν .
2
2
2
This reflects that for finite transformations
µ
µ ν
Λ−1
1/2 γ Λ1/2 = Λ ν γ ,
which indicates that γ µ transforms like a contravariant four-vector provided it is properly
contracted with Dirac spinors and adjoint Dirac spinors.
11d Consequently, ψ(x), γ µ ∂µ ψ(x), γ µ γ ν ∂µ ∂ν ψ(x), · · · are good building blocks
for constructing a Lorentz-invariant wave equation for Dirac spinors, whereas
ψ̄(x)ψ(x), ψ̄(x)γ µ ∂µ ψ(x), · · · are scalar building blocks for obtaining the corresponding Lagrangian.
88
3.1
Towards the Dirac equation (§ 3.2 and 3.4 in the book)
11d
Dirac-field bilinears (currents): the interesting objects in spinor
space are of the form ψ̄ Γψ , with Γ a 4 × 4 matrix that consists of a sequence
of γ-matrices. These objects are called bilinears or currents. They will be
needed to construct Lagrangians that include interactions with other fields, like
ψ̄(x)γ µ ψ(x)Aµ (x) for interactions with a vector field and ψ̄(x)γ µ γ ν ψ(x)hµν (x)
for interactions with a tensor field. A basis for Γ that satisfies Γ† = γ 0 Γγ 0 is
given by the following combinations of γ-matrices:
I4 , γ µ , σ µν =
where
i µ ν µ 5
γ , γ , γ γ , iγ 5 ,
2
γ 5 ≡ iγ 0 γ 1 γ 2 γ 3 = −
i µνρσ
ǫ
γµ γν γρ γσ
4!
in terms of the totally antisymmetric tensor
ǫµνρσ


+1



=
−1



 0
if (µνρσ) = even permutation of (0123)
if (µνρσ) = odd permutation of (0123) .
else
Properties of γ 5 : the properties of the matrix γ 5 will prove important for the description
of weak interactions. They read:
(γ 5 )† = γ 5
⇒
,
(γ 5 )2 = I4
5 µν γ ,S
= 0
⇒
5 µ
γ ,γ
= 0
and
(µ = 0, · · · , 3)
5
γ , Λ1/2 = 0 .
This means that γ 5 is a “Lorentz scalar” if it is properly contracted with Dirac spinors
and adjoint Dirac spinors. Since γ 5 commutes with the generators of Lorentz transformations in spinor space, eigenvectors of γ 5 corresponding to different eigenvalues transform
independently (i.e. without mixing).
11c According to Schur’s lemma this implies that the Dirac representation of the
Lorentz group is reducible, i.e. we should be able to write it in terms of two independent lower-dimensional chiral representations.
In the Weyl representation of the γ-matrices, the matrix γ 5 has the following form in terms
of 2 × 2 blocks:
!
−
I
0
2
γ5 =
.
0 I2
89
As a result,
PR
1
≡ (I4 + γ 5 ) =
2
0
0
0 I2
!
and
PL
I2 0
1
≡ (I4 − γ 5 ) =
2
are (chiral) projection operators on 2-dimensional vectors ψR and ψL :
!
!
ψL
0
ψ ≡
→ PR ψ =
and
PL ψ =
ψR
ψR
0
ψL
0
!
0
!
,
which are eigenvectors of γ 5 corresponding to the chirality eigenvalues +1 and −1.
In terms of these right-handed Weyl spinors ψR and left-handed Weyl spinors ψL the infinitesimal Lorentz transformations of ψ can be rewritten as (cf. Ex. 15 and the generators
that are given on page 87)
!
!
[I2 − i ~θ · ~σ /2 − β~ · ~σ /2]ψL
ψL
Lorentz transf.
−−−−−−−−−−→
.
[I2 − i ~θ · ~σ /2 + β~ · ~σ /2]ψR
ψR
The real infinitesimal parameters θ~ and β~ coincide with the parameters δ~
α and δ~v that
were used in Ex. 14. We see that the Weyl spinors transform independently, which indeed
implies that the four-dimensional Dirac representation of the Lorentz group is reducible
and can be split into two two-dimensional representations. For later use we mention the
following identity for the Pauli spin matrices:
σ 2~σ ∗ = − ~σ σ 2
⇒
Lorentz transf.
σ 2 ψL∗ −−−−−−−−−−→ σ 2 [I2 + i ~θ · ~σ ∗/2 − β~ · ~σ ∗/2]ψL∗
= [I2 − i ~θ · ~σ /2 + β~ · ~σ /2]σ 2 ψL∗ ,
which indicates that σ 2 ψL∗ transforms like a right-handed Weyl spinor.
Chirality and currents: from the 4 × 4 matrix basis on the previous page all possible
Γ† = γ 0 Γγ 0
hermitian currents can be obtained as ψ̄ Γψ , since (ψ̄ Γψ)† = ψ † Γ† γ 0 ψ ======== ψ̄ Γψ .
These currents and their associated continuous Lorentz transformations read:
scalar current : jS (x) ≡ ψ̄(x)ψ(x)
Lorentz transf.
−−−−−−−−−−→
jS (Λ−1 x) ,
Lorentz transf.
vector current : jVµ (x) ≡ ψ̄(x)γ µ ψ(x)
−−−−−−−−−−→
tensor current : jTµν (x) ≡ ψ̄(x)σ µν ψ(x)
Lorentz transf.
−−−−−−−−−
−→
Λµα jVα (Λ−1 x) ,
Λµα Λνβ jTαβ (Λ−1 x) ,
Lorentz transf.
axial vector current : jAµ (x) ≡ ψ̄(x)γ µ γ 5 ψ(x)
−−−−−−−−−−→
pseudo scalar current : jP (x) ≡ i ψ̄(x)γ 5 ψ(x)
−−−−−−−−−−→
Λµα jAα (Λ−1 x) ,
Lorentz transf.
−1 5
µ
µ ν
5
making use of the fact that Λ−1
1/2 γ Λ1/2 = Λ ν γ and Λ1/2 γ Λ1/2 = γ .
90
jP (Λ−1 x) ,
Using the chiral projection operators PL/R , the Dirac spinors can be decomposed into
chiral components according to
PL/R ψ(x) ≡ ψL/R
⇒
ψ̄L/R ≡ (ψL/R )† γ 0 = ψ † PL/R γ 0 = ψ † γ 0 PR/L = ψ̄PR/L .
This results in the following chiral decompositions of the currents.
• The scalar current mixes left- and righ-handed Weyl spinors, since
ψ̄ψ = ψ̄(PR + PL )ψ = ψ̄(PR2 + PL2 )ψ = ψ̄L ψR + ψ̄R ψL .
This will prove important for the description of massive spin-1/2 particles.
• The vector current treats left- and righ-handed Weyl spinors on equal footing, since
ψ̄γ µ ψ = ψ̄γ µ (PR2 + PL2 )ψ = ψ̄(PL γ µ PR + PR γ µ PL )ψ = ψ̄R γ µ ψR + ψ̄L γ µ ψL .
This will prove important for vector-like theories, describing for instance the electromagnetic and strong interactions.
• Similarly the tensor current mixes left- and righ-handed Weyl spinors:
ψ̄σ µν ψ = ψ̄L σ µν ψR + ψ̄R σ µν ψL .
This is needed for describing Lorentz transformations, as we have seen already.
• The axial vector current treats left- and righ-handed Weyl spinors in opposite ways:
ψ̄γ µ γ 5 ψ = ψ̄γ µ γ 5 (PR2 + PL2 )ψ = ψ̄(PL γ µ γ 5 PR + PR γ µ γ 5 PL )ψ
γ 5 ψR/L = ± ψR/L
= ψ̄R γ µ ψR − ψ̄L γ µ ψL .
= ψ̄R γ µ γ 5 ψR + ψ̄L γ µ γ 5 ψL ===========
This will prove important for chiral theories, like the one that describes weak interactions.
• Similarly the pseudo scalar current decomposes according to
i ψ̄γ 5 ψ = i ψ̄L γ 5 ψR + i ψ̄R γ 5 ψL = i ψ̄L ψR − i ψ̄R ψL .
This will prove important in describing interactions between spin-0 and spin-1/2
particles.
Handy combinations of such currents are given by the left/right-handed vector currents
µ
jL/R
(x) ≡ ψ̄(x)γ µ PL/R ψ(x) = ψ̄L/R (x)γ µ ψL/R (x) ,
which will feature in the Standard Model of electroweak interactions.
91
Dirac equation: let’s now try to construct a Lorentz-invariant wave
equation that has the Klein-Gordon equation built in. The simplest candidate
is the Dirac equation
11e
(iγ µ ∂µ − m)ψ(x) = 0 .
This is a first order differential equation, whereas the Klein-Gordon equation was a second
order equation. This is possible because γ µ behaves like a vector without actually introducing a preferred direction, which is not possible in scalar theories!
Proof: first of all
0 = (iγ ν ∂ν + m)(iγ µ ∂µ − m)ψ(x) = − (γ ν γ µ ∂ν ∂µ + m2 )ψ(x)
= −
1
2
{γ µ , γ ν } = 2 g µν I4
γ µ , γ ν ∂µ ∂ν + m2 ψ(x) ============ − ( + m2 )ψ(x) ,
so the Klein-Gordon equation is indeed built in! Secondly, under continuous Lorentz transformations a Dirac spinor transforms according to ψ(x) → ψ ′ (x) = Λ1/2 ψ(Λ−1 x). If ψ(x)
satisfies the Dirac equation then it follows that
∀ (iγ µ ∂µ − m)ψ(x) = 0 ⇒ (iγ µ ∂µ − m)Λ1/2 ψ(Λ−1 x) = Λ1/2 (iΛµσ γ σ ∂µ − m)ψ(Λ−1 x)
x
= Λ1/2 iΛµσ γ σ (Λ−1 )νµ (∂ν ψ)(Λ−1x) − mψ(Λ−1 x)
⇒
= Λ1/2 iγ ν ∂ν ψ − mψ (Λ−1 x) = 0
(iγ µ ∂µ − m)ψ ′ (x) = 0 .
If the field ψ(x) satisfies the Dirac equation then so does the Lorentz transformed field
ψ ′ (x), as required for having a Lorentz invariant wave equation.
In the Weyl representation the Dirac equation reads
− mI2
0 = (iγ µ ∂µ −m)ψ =
~)
i(I2 ∂0 − ~σ · ▽
~)
i(I2 ∂0 + ~σ · ▽
− mI2
!
ψL
ψR
!
≡
− mI2
iσ̄ µ ∂µ
iσ µ ∂µ
− mI2
!
using the compact notation
σ µ ≡ (I2 , ~σ )
and
σ̄ µ ≡ (I2 , − ~σ )
⇒
γµ =
0
σµ
σ̄ µ
0
!
.
From this we conclude that
11e the two representations associated with ψL and ψR are mixed by the mass
term in the Dirac equation! In the massless case the Dirac equation splits into
two independent wave equations for ψL and ψR , the so-called Weyl equations
iσ̄ µ ∂µ ψL (x) = 0
and
92
iσ µ ∂µ ψR (x) = 0 .
ψL
ψR
!
The Dirac Lagrangian: the Lagrangian that corresponds to the Dirac equation reads
LDirac(x) = ψ̄(x)(iγ µ ∂µ − m)ψ(x) .
Proof: the Euler-Lagrange equations for the ψ̄ and ψ fields are given by
∂L ∂L
∂µ
= − (iγ µ ∂µ − m)ψ = 0 ,
−
∂(∂µ ψ̄)
∂ ψ̄
∂L ←
∂L
∂µ
−
= ∂µ i ψ̄γ µ + m ψ̄ = ψ̄(i ∂ µ γ µ + m) = 0 ,
∂(∂µ ψ)
∂ψ
which are indeed the Dirac equation and the corresponding adjoint equation
←
0 = (iγ µ ∂µ − m)ψ(x) † γ 0 = − i ∂µ ψ † (x) γ µ † γ 0 − mψ † (x)γ 0 = − ψ̄(x)(i ∂ µ γ µ + m) .
Conserved currents: in preparation for the quantization of the free
Dirac theory and the derivation of its particle interpretation, we have a closer
look at the conserved currents for the solutions ψ(x) of the Dirac equation.
11e
1. The vector current jVµ (x) is conserved.
Dirac eqns.
Proof 1: ∂µ jVµ = (∂µ ψ̄)γ µ ψ + ψ̄γ µ ∂µ ψ ======== im ψ̄ψ − im ψ̄ψ = 0 .
Proof 2: in Ex. 17 an alternative proof is given based on global U(1) invariance.
µ
2. The axial vector current jA
(x) is conserved if m = 0.
Dirac eqns.
Proof 1: ∂µ jAµ = (∂µ ψ̄)γ µ γ 5 ψ − ψ̄γ 5 γ µ ∂µ ψ ======== 2im ψ̄γ 5 ψ = 0 if m = 0.
Proof 2: in Ex. 17 an alternative proof is given based on global chiral invariance.
3. The energy-momentum tensor T µν is conserved.
Only the spacetime coordinates of ψ̄(x) and ψ(x) transform under translations,
i.e. the spinors themselves do not transform. Consequently, the energy-momentum
tensor T µν derived on page 8 will be conserved. This gives rise to four conserved
charges, the field energy
Z
Z
i
h
˙ −L
H =
d~x H =
d~x πψ ψ̇ + ψ̄π
Dirac
ψ̄
and field momentum
P~ = −
with
πψ =
Z
i
h
~ ψ̄)π ,
~ ψ + (▽
d~x πψ ▽
ψ̄
∂LDirac
= i ψ̄γ 0 = iψ †
∂(∂0 ψ)
and πψ̄ =
∂LDirac
= 0.
∂(∂0 ψ̄)
From these conjugate momenta we can read off that out of the eight real
degrees of freedom of the Dirac spinor ψ(x) in fact four belong to the
conjugate momentum.
93
4. Under continuous Lorentz transformations a Dirac spinor transforms as
Lorentz transf.
inf. 1
i
ψ(x) −−−−−−−−−−→ Λ1/2 ψ(Λ−1 x) ≈ I4 − ωρσ S ρσ ψ(x) − ωρσ xσ ∂ ρ − xρ ∂ σ ψ(x) ,
2
2
where the first term is typical for Dirac spinors and the second term is the same
as for scalar fields. Bearing in mind that the Dirac Lagrangian is a Lorentz scalar,
we can generalize the derivation on page 10 to arrive at the following six conserved
Noether currents:
∂LDirac ρ σ
J µρσ (x) =
x ∂ − xσ ∂ ρ − iS ρσ ψ(x) + g µρ xσ − g µσ xρ LDirac(x)
∂(∂µ ψ)
= T µσ xρ − T µρ xσ + ψ̄(x)γ µ S ρσ ψ(x) .
11e The last term in these conserved Noether currents is specific for Dirac
theories. After quantization of the Dirac theory this term will help us to
determine the spin of the particles described by the (free) Dirac field theory.
3.2
Solutions of the free Dirac equation (§ 3.3 in the book)
11e Since solutions of the (free) Dirac equation automatically satisfy the Klein-
Gordon equation, we can use the standard plane-wave (Fourier) decomposition
in order to decouple the degrees of freedom as much as possible.
The positive-enery case: according to this decomposition we introduce
p
ψp (x) ≡ u(p)e−ip·x with p2 = m2 and p0 > 0 ⇒ pµ =
p~ 2 + m2 , p~ ) ≡ (Ep~ , p~ ) .
The spinor u(p) then has to satisfy the Dirac equation in momentum space:
(γ µ pµ − m)u(p) ≡ (p/ − m)u(p) = 0 ,
using Feynman slash notation. The claim is now that u(p) can be written as
!
√
p·σξ
,
u(p) =
√
p · σ̄ ξ
with ξ an arbitrary normalized 2-dimensional vector.
Proof: using that
q
p
p
{σj , σk } = 2δjk I2
0
0
(p · σ)(p · σ̄) = (p I2 − ~p · ~σ )(p I2 + p~ · ~σ ) =========== I2 p20 − p~ 2 = mI2 ,
it easily follows that
(p/ − m)u(p) =
− mI2
p · σ̄
p·σ
− mI2
94
!
√
√
p·σξ
p · σ̄ ξ
!
= 0.
The negative-enery case: similarly we introduce
ψp (x) ≡ v(p)e+ip·x
with again pµ = (Ep~ , p~ )
to get two more independent solutions of the Dirac equation. The spinor v(p) has to
satisfy
− (γ µ pµ + m)v(p) ≡ − (p/ + m)v(p) = 0
and is given by
v(p) =
√
p·ση
√
− p · σ̄ η
!
,
with η another arbitrary normalized 2-dimensional vector.
Helicity: for the normalized base vectors ξ 1, ξ 2 and η 1 , η 2 we could for instance choose
the eigenvectors of ~σ · ~p/|~p | ≡ ~σ · ~ep with eigenvalues +1 , −1. This results in
u1 (p) =
u2 (p) =
and
v 1 (p) =
v 2 (p) =
!
p
|p
~|≫ m p
Ep~ − |~p | ξ 1
−−−
−→
2Ep~
p
Ep~ + |~p | ξ 1
Ep~ + |~p | ξ 2
!
Ep~ − |~p | η 1
p
− Ep~ + |~p | η 1
!
p
p
Ep~ − |~p | ξ 2
p
|p
~|≫ m
p
2Ep~
−−−
−→
|p
~|≫ m
−−−
−→ −
p
0
ξ1
ξ2
0
η1
η2
0
,
!
0
2Ep~
!
p
|p
~|≫ m p
Ep~ + |~p | η 2
2Ep~
−−−
−→
p
− Ep~ − |~p | η 2
!
!
!
,
.
Hence, in the ultrarelativistic limit the chiral states coincide with the eigenstates of the
helicity operator
!
~σ · ~ep
0
1
ˆ
~ =
.
ĥ = ~ep · S
2
0
~σ · ~ep
In that case positive helicity (h = + 1/2) corresponds to right-handed chirality (ψR ) and
negative helicity (h = − 1/2) to left-handed chirality (ψL ).
Helicity is frame dependent if m 6= 0, since ~ep can be flipped by a boost along
that direction. Helicity is frame independent if m = 0. The Lorentz invariance
of helicity for m = 0 is manifest in the notation of Weyl spinors, since ψL/R
live in different representations of the Lorentz group.
95
Normalization and orthogonality of the u and v spinors: from the orthogonality
properties ξ r † ξ s = δrs and η r † η s = δrs of the normalized 2-dimensional base vectors ξ 1, ξ 2
and η 1 , η 2 , it follows that
!
√
s
p
·
σ
ξ
√
√
= ξ r † (p · σ + p · σ̄)ξ s = 2Ep~ δrs ,
ur † (p)us (p) = ξ r † p · σ , ξ r † p · σ̄
√
s
p · σ̄ ξ
r†
s
v (p)v (p) = η
ur † (p)v s (p̃) = ξ
r †√
r †√
p · σ, −η
p·σ, ξ
r †√
r †√
p · σ̄
p · σ̄
√
p · σ ηs
√
− p · σ̄ η s
√
p · σ̄ η s
√
− p · σ ηs
!
!
= η r † (p · σ + p · σ̄)η s = 2Ep~ δrs ,
= 0 = v r † (p̃)us (p) ,
with p̃ µ ≡ (p0 , −~p ) ⇒ p̃ · σ̄ = p · σ and p̃ · σ = p · σ̄ . This is obviously not boost-invariant.
Lorentz invariant contractions are obtained through
!
√
s
p
·
σ̄
ξ
√
√
= 2m δrs ,
ū r (p)us (p) = ur † (p)γ 0 us (p) = ξ r † p · σ , ξ r † p · σ̄
√
p · σ ξs
v̄ r (p)v s (p) = v r † (p)γ 0 v s (p) =
ū r (p)v s (p) = ξ
r †√
p·σ, ξ
r †√
η
r †√
p · σ̄
p · σ, −η
r †√
p · σ̄
!
√
− p · σ̄ η s
= − 2m δrs ,
√
p · σ ηs
!
√
− p · σ̄ η s
= 0 = v̄ r (p)us (p) .
√
s
p·σ η
Polarization sums: for dealing with Feynman diagrams that involve Dirac fermions,
polarization sums (helicity sums) are an essential ingredient. These polarization sums read
!
√
2
2
s
X
X
p
·
σ
ξ
√
√
us (p)ūs (p) =
ξ s † p · σ̄ , ξ s † p · σ
√
p · σ̄ ξ s
s=1
s=1

√

= 
 √
compl.
======
2
X
s=1
p·σ
p · σ̄
2
P
s=1
2
P
s=1
√
ξ s ξ s † p · σ̄
√
ξ s ξ s † p · σ̄
mI2 p · σ
p · σ̄ mI2
!

√
ξ sξ s † p · σ 
s=1

2

P
√
s s †√
p · σ̄
ξ ξ
p·σ
√
p·σ
2
P
s=1
= γ µ pµ + mI4 = p/ + mI4 ,
v s (p)v̄ s (p) = p/ − mI4 ,
where in the third step the completeness relation for the 2-dimensional basis ξ 1 , ξ 2 is used.
96
3.3
Quantization of the free Dirac theory (§ 3.5 in the book)
The same philosophy will be applied as in the Klein-Gordon case. We
11f
diagonalize the Hamiltonian Ĥ of the Dirac theory in its quantized form by
expanding the solutions of the Dirac equation in spatial plane-wave modes,
which are written in terms of creation and annihilation operators. The particle interpretation is obtained by letting these creation operators act on the
vacuum state |0i, which is defined to contain no particles (i.e. positive-energy
quanta) and to have the lowest energy. This leads to the requirement that
the spectrum of Ĥ should be bounded from below. On top of that, we again demand that causality should be preserved for having a viable theory.
Derivation of the operator algebra: step 1. According to the discussion on page 93
†
µ
˙
HDirac (x) = πψ (x)ψ̇(x) + ψ̄(x)π
ψ̄ (x) − LDirac (x) = iψ (x)ψ̇(x) − ψ̄(x)(iγ ∂µ − m)ψ(x)
~ ψ(x) + m ψ̄(x)ψ(x) .
= − i ψ̄(x) ~γ · ▽
In analogy with the approach worked out on page 13 we expand a Schrödinger-picture
solution of the Dirac equation in terms of spatial plane-wave modes, bearing in mind that
ψ̂(x) is non-hermitian and has spinorial degrees of freedom:
ψ̂(~x , t = 0) =
Z
ψ̄ˆ(~x , t = 0) =
Z
2
d~p ei~p ·~x X s s
s† s
p
â
u
(p)
+
b̂
v
(p̃)
,
p
~
−~
p
(2π)3
p0 = Ep~
2Ep~ s = 1
2
d~p e− i~p ·~x X s † s
s
s
p
â
ū
(p)
+
b̂
v̄
(p̃)
= ψ̂ † (~x , t = 0)γ 0 .
−~
p
p
~
(2π)3
p0 = Ep~
2Ep~ s = 1
The difference with the scalar case is the occurrence of the u and v spinors that span
spinor space. The Hamilton operator of the free Dirac theory now reads
Ĥ =
Z
×
h
~ ψ̂(x) + m ψ̄ˆ(x)ψ̂(x)
d~x − i ψ̄ˆ(x) ~γ · ▽
2
X
s,s′ = 1
================
=====
Z
t=0
====
Z
d~x
Z
′
d~p d~p ′ ei~x ·(~p −~p )
p
×
(2π)6 2 Ep~ Ep~ ′
′
′
′
′
âsp~ † ūs (p) + b̂s−~p v̄ s (p̃) (~γ · p~ ′ + m) âsp~ ′ us (p′ ) + b̂s−~p† ′ v s (p̃ ′ )
~x integral , Dirac eqn.
norm.
i
Z
2
s′ s′
d~p 1 X
s† s†
s′ † s′
s
s†
â
u
(p)
+
b̂
v
(p̃)
â
u
(p)
−
b̂
v
(p̃)
−~
p
p
~
p
~
−~
p
(2π)3 2 s,s′ = 1
Z
2
2
d~p X
d~p X
s† s† s
s
Ep~ âp~ âp~ − b̂−~p b̂−~p =
Ep~ âsp~ † âsp~ − b̂sp~ b̂sp~ † ,
3
3
(2π) s = 1
(2π) s = 1
97
where on the third line we have used the Dirac equation through
(~γ · p~ + m)us (p) = − (p/ − m)us (p) + Ep~ γ 0 us (p) = Ep~ γ 0 us (p) ,
(~γ · ~p + m)v s (p̃) = (p̃/ + m)v s (p̃) − Ep~ γ 0 v s (p̃) = − Ep~ γ 0 v s (p̃) .
From this expression for the Hamilton operator of the free Dirac theory we can read off
that
• the energy spectrum is not bounded from below if we use commutation relations like
in the case of scalar theories;
• it does certainly not help if b̂sp~ † is replaced by ĉp~s , since in that case the problem
cannot be solved at all;
• 11f we are forced to impose fermionic anticommutation relations on the creation and
annihilation operators b̂sp~ † and b̂sp~ , being the alternative starting point for setting up
a many-particle quantum theory:
s s′ † s s′ ′ b̂p~ , b̂p~ ′
= (2π)3 δ(~p − p~ ′ )δss′ 1̂ and
b̂p~ , b̂p~ ′ = b̂sp~ † , b̂sp~ ′† = 0 .
Upon implementing these anticommutation relations, the Hamilton operator indeed becomes bounded from below by a zero-point energy:
Ĥ =
Z
2
d~p X
Ep~ âsp~ † âsp~ + b̂sp~ † b̂sp~ − (2π)3 δ(~0 )1̂ .
3
(2π) s = 1
• Again only positive-energy quanta feature in the Hamilton operator.
• This time we find an infinite zero-point energy with opposite sign, which can again
be removed by normal ordering:
b̂† b̂ → N(b̂† b̂) = b̂† b̂ ,
b̂ b̂† → N(b̂ b̂† ) = − b̂† b̂ .
Note the extra minus sign that is required for normal ordering of fermionic operators.
This will also have repercussions on the derivation of Wick’s theorem and the ensuing
Feynman rules.
The opposite-sign fermionic zero-point energy could actually cancel the infinities
originating from bosonic zero-point energies. So, there might be some profound
physical concepts hidden in the zero-point sector . . . !?
Let’s ignore the latter issue from now on and proceed with the operator algebra.
Question: do the operators â , â† also obey fermionic anticommutation relations
or bosonic commutation relations?
98
Derivation of the operator algebra: step 2. To address the previous question we
need to study the causal structure of the theory. Irrespective of the commutator algebra
being fermionic or bosonic, one finds
Ĥ, âsp~ = − Ep~ âsp~
and
Ĥ, b̂sp~
= − Ep~ b̂sp~ .
As explained on page 18 of these lecture notes, we can switch to the Heisenberg picture by
using the identities eiĤt âsp~ e−iĤt = âsp~ e−iEp~ t and eiĤt b̂sp~ e−iĤt = b̂sp~ e−iEp~ t :
ψ̂(x) =
Z
ψ̄ˆ(x) =
Z
2
X
d~p
1
s† s
ip·x s s
−ip·x
p
v
(p)e
â
u
(p)e
+
b̂
,
p
~
p
~
(2π)3
p0 = Ep~
2Ep~ s = 1
2
X
d~p
1
s† s
ip·x
s s
−ip·x p
ū
(p)e
+
b̂
v̄
(p)e
â
.
p
~
p0 = Ep~
(2π)3
2Ep~ s = 1 p~
Based on these operator fields we can have a look at causality. The propagation of positiveenergy particles from y to x is given by
h0|ψ̂a (x)ψ̄ˆb (y)|0i =
=
Z
Z
′
2
d~p d~p ′ e−ip·x+ip ·y X s
s′ ′
s s′ †
p
ua (p)ūb (p )h0|âp~ âp~ ′ |0i
6
p0 = Ep~ , p′0 = Ep~ ′
(2π) 2 Ep~ Ep~ ′ s,s′ = 1
Z
2
d~p e−ip·(x−y)
d~p e−ip·(x−y) X s
s
u
(p)ū
(p)
=
(p
/
+
m)
ab a
b
3
p0 = Ep~
(2π)3
2Ep~
(2π)
2E
p0 = Ep~
p
~
s=1
= (i∂
/x + m)ab D(x − y) ,
where a, b are spinor indices and D(x − y) is given on page 18. This expression is valid
irrespective of the statistics for the â-operators:
′
′
h0|âsp~ âsp~ ′† |0i = h0| (2π)3 δ(~p − p~ ′ )δss′ 1̂ ± âsp~ ′† âsp~ |0i = (2π)3 δ(~p − p~ ′ )δss′ ,
where the +/− sign occurring after the first step refers to bosonic/fermionic statistics.
Similarly the propagation of positive-energy antiparticles from x to y is given by
h0|ψ̄ˆb (y)ψ̂a (x)|0i =
=
Z
Z
′
2
d~p d~p ′ eip·x−ip ·y X s′ ′ s
s′ s †
p
v̄
(p
)v
(p)h0|
b̂
b̂
|0i
′
b
a
p
~ p
~
6
(2π) 2 Ep~ Ep~ ′ ′
p0 = Ep~ , p′0 = Ep~ ′
s,s = 1
Z
2
d~p eip·(x−y) X s
d~p eip·(x−y)
s
va (p)v̄b (p)
=
(p/ − m)ab 3
3
(2π)
2Ep~ s = 1
(2π)
2Ep~
p0 = Ep~
p0 = Ep~
= − (i∂
/x + m)ab D(y − x) .
99
For (x−y)2 < 0 we know that D(x−y) = D(y −x), hence we have to conclude
that h0| ψ̂a (x), ψ̄ˆb (y) |0i =
6 0 and h0| ψ̂a (x), ψ̄ˆb (y) |0i = 0 in that case.
Causality: in the coordinate representation any observable that involves Dirac particles
contains an even number of spinor fields, i.e. as many ψ as ψ̄ fields, since such an observ
able should have no open spinor indices. So, if either ψ̂a (x), ψ̄ˆb (y) = ψ̂a (x), ψ̂b (y) = 0
or ψ̂a (x), ψ̄ˆb (y) = ψ̂a (x), ψ̂b (y) = 0 for (x − y)2 < 0, then measurements do not
influence each other for spacelike separations and causality is preserved! As we have seen
above, the first option cannot be achieved but the second option is possible. Based on the
previous discussion,
ψ̂a (x), ψ̄ˆb (y) =
2 X
s s′ † d~p d~p ′
1
s
s′ ′
−ip·x+ip′·y
p
u
(p)ū
(p
)
e
âp~ , âp~ ′
a
b
(2π)6 2 Ep~ Ep~ ′ ′
s,s = 1
Z
′
′ ′ ′
′ ′ + vas (p)v̄bs (p′ ) eip·x−ip ·y b̂sp~ †, b̂sp~ ′ + usa (p)v̄bs (p′ ) e−ip·x−ip ·y âsp~ , b̂sp~ ′
+
′
′
vas (p)ūsb (p′ ) eip·x+ip ·y
and
ψ̂a (x), ψ̂b (y) =
Z
s † s′ † b̂p~ , âp~ ′
p0 = Ep~ , p′0 = Ep~ ′
= 0
2 X
s s′ 1
d~p d~p ′
s
s′ ′
−ip·x−ip′·y
p
u
(p)u
(p
)
e
âp~ , âp~ ′
a
b
(2π)6 2 Ep~ Ep~ ′ ′
s,s = 1
′ ′ ′
′ ′
′ + vas (p)vbs (p′ ) eip·x+ip ·y b̂sp~ †, b̂sp~ ′† + usa (p)vbs (p′ ) e−ip·x+ip ·y âsp~ , b̂sp~ ′†
+
′
′
vas (p)usb (p′ ) eip·x−ip ·y
s † s′ b̂p~ , âp~ ′ p0 = Ep~ , p′0 = Ep~ ′
= 0
is guaranteed for spacelike separations (x − y)2 < 0 if
s s′ † ′ âp~ , âp~ ′ = (2π)3 δ(~p −~p ′ )δss′ 1̂ = b̂sp~ , b̂sp~ ′† ,
with all other anticommutators being 0 .
Note: the anticommutation relation for â and â† follows from the first two terms in the
first expression, bearing in mind the anticommutation relation for b̂ and b̂† as well as the
equality D(x − y) = D(y − x) for (x − y)2 < 0.
In the free Dirac theory both particles and antiparticles have to be fermions.
On top of that, the creation and annihilation operators for particles anticommute with those for antiparticles. This implies that particles and antiparticles
are versions of the same object, differing merely by the quantum number charge
(as we will see below).
11f
100
Canonical equal-time anticommutation relations: from the fundamental fermionic
anticommutation relations for creation and annihilation operators it follows that
Z
i
p. 99
1̂ h i~p ·(~x−~y )
d~p
− i~
p ·(~
x−~
y)
ˆ
ψ̂a (~x , t), ψ̄b (~y , t) ====
e
(p
/
+
m)
+
e
(p
/
−
m)
ab
ab p0 = Ep~
(2π)3 2Ep~
p
~→−p
~
in 2nd term
==============
Z
d~p i~p ·(~x−~y ) 0
e
(γ )ab 1̂ = (γ 0 )ab δ(~x − ~y ) 1̂
(2π)3
X
(γ 0 )2 = I
ψ̂a (~x , t), i ψ̄ˆb (~y , t) (γ 0 )bc ====== iδac δ(~x − ~y ) 1̂
ψ̂a (~x , t), π̂ψc (~y , t) =
⇒
b
and
ψ̂a (~x , t), ψ̂c (~y , t) = π̂ψa (~x , t), π̂ψc (~y , t) = 0 .
The quantization of the free Dirac theory could equally well have been
performed by imposing canonical equal-time anticommutation relations for the
fields and their corresponding conjugate momenta.
11f
Completing the particle interpretation: what else do we know about the particles
and antiparticles in the Dirac theory?
• After quantization the momentum carried by the Dirac field becomes (cf. page 93)
Z
Z
ˆ
~ ) ψ̂(x)
~
~
P = − d~x π̂ψ (x)∇ψ̂(x) =
d~x ψ̂ † (x)(−i ▽
=
Z
Z
2
2
X
X
d~p
d~p
s† s
s†
s
p~
(âp~ âp~ + b̂−~p b̂−~p ) =
p~
(âsp~ † âsp~ + b̂sp~ † b̂sp~ ) ,
3
3
(2π)
(2π)
s=1
s=1
just like in the scalar case.
• After quantization the conserved charge (called particle number) originating from
the global U(1) gauge symmetry becomes
Q̂ =
⇒
Z
d~x
ĵV0 (x)
N(Q̂) =
=
Z
Z
†
d~x ψ̂ (x) ψ̂(x) =
Z
2
d~p X s † s
(âp~ âp~ + b̂s−~p b̂s−~†p )
3
(2π) s = 1
2
d~p X s † s
(âp~ âp~ − b̂sp~ † b̂sp~ ) ,
3
(2π) s = 1
which implies that particles/antiparticles have particle number +/− 1. Multiplied
by the electromagnetic charge q of the particles this yields the total charge operator
for interactions with electromagnetic fields (see later). So, in that case we can read
off that particles and antiparticles have opposite charge.
101
• Based on the discussion on page 94, the total spin operator in the Dirac theory is
given by
!
Z
1
~
σ
0
~ˆ =
ψ̂(x) .
S
d~x ψ̂ † (x) 2 1
0 2 ~σ
Just like in the previous case, the order of the b̂sp~ and b̂sp~ † operators results in
opposite spin quantum numbers for antiparticles if we would set η s = ξ s in the
v and u spinors.
This allows us to read off the particle content of the free Dirac theory. We already know
that for anticommuting creation and annihilation operators there exists a groundstate
(vacuum state) |0i such that h0|0i = 1 and âsp~ |0i = b̂sp~ |0i = 0 for all p~ and s. Then
ˆ
N(Ĥ)|0i = 0, P~ |0i = ~0 and N(Q̂)|0i = 0, i.e. the vacuum “has” energy E = 0,
momentum P~ = ~0 and charge Q = 0. From this groundstate the 1-particle excitations
can be obtained as âsp~ † |0i and b̂sp~ † |0i, corresponding to an energy Ep~ , a momentum p~ ,
spin 1/2 and opposite charge/particle number. In view of the fermionic anticommutation
relations, â† creates fermionic particles and b̂† creates fermionic antiparticles.
11g
We can summarize the particle interpretation of the free Dirac theory as
follows: âsp~ † creates particles with energy Ep~ , momentum ~p , spin 1/2, charge q
and polarization appropriate to ξ s , whereas b̂sp~ † creates antiparticles with energy Ep~ , momentum p~ , spin 1/2, charge −q and polarization opposite to η s .
Hence, if m = 0 then an antiparticle with helicity ±1/2 has chirality ∓ 1.
Inversion of the Dirac equation: the retarded Green’s function is obtained by
SR (x − y) ab ≡ θ(x0 − y 0 )h0| ψ̂a (x), ψ̄ˆb (y) |0i
p. 99
==== θ(x0 − y 0)(i∂
/x + m)ab D(x − y) − D(y − x) = (i∂
/x + m)ab DR (x − y) .
We have used in the last step that ∂0 θ(x0 − y 0) = δ(x0 − y 0) causes D(x − y) − D(y − x)
to vanish according to property 2 on page 19, which implies that it is safe to interchange
the order of θ(x0 − y 0) and (i∂
/x + m)ab .
Proof that this Green’s function indeed inverts the Dirac equation:
(i∂
/x −m)SR (x−y) = (i∂
/x −m)(i∂
/x +m)DR (x−y) = − (+m2 )DR (x−y)I4 = iδ (4) (x−y)I4 .
In Fourier language this inversion reads:
Z 4
Z 4
d p
d p −ip·(x−y)
−ip·(x−y)
(i∂
/x − m)SR (x − y) =
(p/ − m) S̃R (p)e
= i
e
I4
4
(2π)
(2π)4
⇒
S̃R (p) =
i
i(p/ + m)
≡
,
2
2
p −m
p/ − m
102
with the same prescription to go around the complex poles as in the Klein-Gordon case.
Similarly the Feynman prescription yields the Feynman propagator
SF (x − y)
ab
=
Z
=


 (i∂
/x + m)ab D(x − y) = h0|ψ̂a (x) ψ̄ˆb (y)|0i
d4 p i(p/ + m)ab −ip·(x−y)
e
= (i∂
/x + m)ab DF (x − y)
(2π)4 p2 − m2 + iǫ

 (i∂
/x + m)ab D(y − x) = − h0|ψ̄ˆb (y) ψ̂a (x)|0i
if x0 > y 0
if x0 < y 0
≡ h0|T (ψ̂a (x) ψ̄ˆb (y))|0i .
Note the extra minus sign in the definition of time ordering for fermionic fields. Just
like in the Klein-Gordon case the Feynman propagator SF (x − y) ab is the time-ordered
propagation amplitude, which will play a crucial role in the Feynman rules for fermions.
Lorentz transformations and ψ̂(x): just like in the Klein-Gordon case the 1-particle
p
states are normalized according to |~p , si ≡ 2Ep~ âsp~ † |0i, with a similar expression holding
for 1-antiparticle states. Using this definition we can define the unitary operator that
implements (active) Lorentz transformations in the Hilbert space of quantum states:
≡ |0i
z }| {
p
−
→
s†
s † −1
−
→
|Λp, si ≡ Û(Λ)|~p , si ⇒
2EΛp â−→ |0i =
2Ep~ Û (Λ) âp~ Û (Λ) Û (Λ)|0i
Λp
s
→
E−
Λp s †
⇒ define: Û (Λ) âsp~ † Û −1 (Λ) =
â−→ ,
Ep~ Λp
q
provided that we choose the axis of spin quantization to be parallel to the boost/rotation
axis. The transformation property of b̂sp~ † has an analogous form. As a result:
Û (Λ) ψ̂(x) Û
−1
(Λ) =
p′ = Λp
====
⇒
Z
Z
2
X
1 q
d~p
s
−ip·x
→
→ u (p)e
2E−
âs−
+ b̂s−→† v s (p)eip·x
Λp
3
Λp
Λp
(2π) 2Ep~
s=1
1
d~p ′
p
3
(2π)
2Ep~ ′
2
X
s=1
s ′
Λ−1
1/2 u (p )
s ′
Λ−1
1/2 v (p )
z }| {
z }| { ′ s† s
s
s
−1 ′
−ip′ ·Λx
âp~ ′ u (Λ p ) e
+ b̂p~ ′ v (Λ−1 p′ ) eip ·Λx
Û (Λ) ψ̂(x) Û −1 (Λ) = Λ−1
1/2 ψ̂(Λx) ,
R
where the second line is obtained by using that d~p /(2Ep~ ) and e± ip·x are all Lorentz
invariant. This implies that the transformed field creates/destroys antiparticles/particles
at the spacetime point Λx.
103
3.4
Discrete symmetries
11h Apart from the symmetry under continuous Lorentz transformations and
translations, there are two more spacetime symmetries a free Lagrangian should
have in relativistic field theories. These correspond to the discrete Lorentz transformations that complete the Lorentz group:
P
• parity (spatial inversion) P , which reverses the handedness of space: t, ~x == t, −~x .
T
• time reversal T , which interchanges forward and backward light cones: t, ~x == −t, ~x .
In addition it is also useful to consider a non-spacetime discrete operation called charge
conjugation C , which interchanges particles and antiparticles. In particular P and C play
a crucial role in constructing the Standard Model of electroweak interactions. In these
lecture notes we will explicitly consider parity transformations. The details for the other
discrete transformations can be found in the textbook of Peskin & Schroeder.
Parity: this mirror-reflection spacetime transformation is implemented in the Hilbert space
of quantum states by a unitary operator (basis
transformation) P̂ . Its action on the creation
and annihilation operators is such that a state
|~p , si is transformed into a state | −~p , si, provided that the spin is quantized along an arbitrary fixed axis.5 This implies
P̂ âsp~ P̂ † = βa âs−~p
and
P̂ b̂sp~ P̂ † = βb b̂s−~p ,
mirror
p~
−~p
parity
where βa,b are phase factors. Applying P̂ twice
should have no effect on observables in the Dirac theory, which contain even numbers of
2
Dirac fields. Therefore we know that βa,b
= ± 1. In analogy to the case of continuous
Lorentz transformations, the transformation property of the Dirac field under parity then
becomes
†
P̂ ψ̂(x)P̂ =
Z
2
X
1
d~p
(P )
ip·x s
s
−ip·x
∗ s† s
p
v
(p)e
β
â
u
(p)e
+
β
b̂
≡ Λ1/2 ψ̂(x̃) ,
a −~
p
b −~
p
(2π)3
p0 = Ep~
2Ep~ s = 1
with x̃ µ ≡ (x0 , −~x ). Using p̃ µ ≡ (p0 , −~p ) ⇒ p̃ · σ̄ = p · σ and p̃ · σ = p · σ̄ , we can
rewrite the u and v spinors according to
5
If we would instead use spin quantization along the momentum direction, then also the associated
quantum number helicity would be reversed under parity.
104
√
us (p) =
√
p · σ ξs
p · σ̄ ξ
s
!
√
p · σ ηs
√
− p · σ̄ η s
s
v (p) =
√
=
!
√
p̃ · σ̄ ξ s
p̃ · σ ξ
s
!
= γ0
√
p̃ · σ̄ η s
√
− p̃ · σ η s
=
√
!
√
p̃ · σ ξ s
p̃ · σ̄ ξ
= −γ
0
s
!
= γ 0 us (p̃) ,
√
p̃ · σ η s
√
− p̃ · σ̄ η s
!
= − γ 0 v s (p̃) .
Bearing in mind that the integral over p~ and the energy Ep~ are unaffected by the transition
from ~p to p̃~ = − p~ , this leads to
Z
2
X
1
dp̃~
(P )
ip̃·x̃ s 0 s
−ip̃·x̃
∗ s† 0 s
†
p
= Λ1/2 ψ̂(x̃)
γ
v
(p̃)e
β
â
γ
u
(p̃)e
−
β
b̂
P̂ ψ̂(x)P̂ =
a p̃
~
b p̃
~
(2π)3
p̃0 = Ep̃~
2Ep̃~ s = 1
⇒
βb∗ = − βa
(P )
Λ1/2 = βa γ 0 .
and
The transformation property of ψ̄ˆ(x) then follows:
P̂ , γ 0 = 0
P̂ ψ̄ˆ(x)P̂ † = P̂ ψ̂ † (x)γ 0 P̂ † =========
†
(P ) −1
P̂ ψ̂(x)P̂ † γ 0 = βa∗ ψ̄ˆ(x̃)γ 0 = ψ̄ˆ(x̃)Λ1/2 ,
since P̂ acts on the Hilbert space of quantum states and not on spinor space. The effective
transformation properties of the γ-matrices then read
(P ) −1
(P )
β ∗ βa = 1
a
==== γ 0 γ µ γ 0 = γµ ,
Λ1/2 γ µ Λ1/2 ==
which we can write as
(P ) −1
(P )
Λ1/2 γ µ Λ1/2 = (ΛP )µν γ ν
with

1 0
0
0
 0 −1 0
0 

= 
 0 0 −1 0  = spatial inversion
0 0
0 −1

(ΛP )µν
in analogy with the continuous Lorentz transformations. Furthermore
(P ) −1
(P )
(P )
Λ1/2 γ 5 Λ1/2 = γ 0 γ 5 γ 0 = − γ 5 = det(Λ1/2 )γ 5 .
Now we have all ingredients for deriving the transformation properties of the normalordered currents that are the basic building blocks for observables:
scalar current : N ĵS (x)
vector current : N ĵVµ (x)
tensor current : N ĵTµν (x)
P
−
−→ N ĵS (x̃) ,
P
−
−→ N ĵµV (x̃) ,
P
T
−
−→ N ĵµν
(x̃) ,
axial vector current : N ĵAµ (x)
pseudo scalar current : N ĵP (x)
105
P
−
−→ − N ĵµA (x̃) ,
P
−
−→ − N ĵP (x̃) .
These transformation properties actually follow from the fact that left/right-handed fields
are transformed into right/left-handed fields under partity. Note that the phase factor βa
does not occur in any of these transformation properties. Therefore we might just as well
set βa = − βb = 1, resulting in the following textbook statement:
11h
in the Dirac theory particles and antiparticles have opposite intrinsic parity.
P
−→ ∂ µ , the free Dirac Lagrangian is evidently invariant under parity.
Since ∂µ −
Charge conjugation and time reversal: after similar sets of steps it can be derived
how the normal-ordered currents transform under charge conjugation and time reversal.
These topics will not be discussed in these lecture notes. The interested reader is referred
to p. 67–71 in the textbook of Peskin & Schroeder.
Interacting relativistic field theories: the free Dirac Lagrangian is invariant under
all three discrete symmetries. For interacting theories involving Dirac fields, however, the
following holds:
• electromagnetic, strong and gravitational interactions are P - and C-invariant;
• weak interactions violate P - and C-invariance (maximally) in the Standard Model,
but preserve the combined CP -invariance;
• rare processes involving K-mesons violate CP -invariance: within the Standard Model
this leads to the requirement that there should be at least three families of fermions;
• all interacting relativistic field theories should be CP T -invariant in order to have a
theory that preserves causality and that has a Lorentz-scalar hermitian Lagrangian.
106
4
Interacting Dirac fields and Feynman diagrams
The next lecture covers § 4.7 of Peskin & Schroeder.
12 We have already seen in detail how the Feynman rules come about in scalar
theories. Next we move on to theories that involve Dirac fermions. In that case
the interaction Hamiltonian will contain an even number of spinor fields in
order to have a Lorentz invariant action.
4.1
Wick’s theorem for fermions
The first thing we have to do is to generalize Wick’s theorem. We start with the propagator
using explicit spinor indices a and b:
Z 4
d p i(p/ + m)ab −ip·(x−y)
ˆ
e
h0|T ψ̂a (x) ψ̄b (y) |0i = SF (x − y) ab =
(2π)4 p2 − m2 + iǫ
=

ˆ

 + h0|ψ̂a(x) ψ̄b (y)|0i

 − h0|ψ̄ˆ (y) ψ̂ (x)|0i
b
a
if x0 > y 0
if x0 < y 0
= − h0|T ψ̄ˆb (y) ψ̂a (x) |0i,
which involves time-ordered fields. For Wick’s theorem we will have to generalize the
definition of time ordering to cases with more fields. Define time ordering to pick up one
minus sign for each interchange of fermionic operators: e.g. for x03 > x01 > x04 > x02
T ψ̂a1 (x1 )ψ̂a2 (x2 )ψ̂a3 (x3 )ψ̂a4 (x4 ) = (−1)3 ψ̂a3 (x3 )ψ̂a1 (x1 )ψ̂a4 (x4 )ψ̂a2 (x2 ) .
Similarly the definition of normal ordering is generalized for more than two fermionic operators according to
N(âsp~ âr~q â~tl † ) = (−1)2 â~tl † âsp~ âr~q = (−1)3 â~tl † âr~q âsp~ ,
where again each interchange of fermionic operators gives rise to a minus sign.
12a In the proof of Wick’s theorem the order of the creation and annihilation
operators will matter this time.
Based on these generalizations of time ordering and normal ordering, we can extend the
definition of contractions (see Ex. 18):
with
T ψ̂a (x) ψ̄ˆb (y) = N ψ̂a (x) ψ̄ˆb (y) + ψ̂a (x) ψ̄ˆb (y) ,
ψ̂a (x) ψ̄ˆb (y) = − ψ̄ˆb (y) ψ̂a (x) ≡
 ˆ− +

 + ψ̂a (x), ψ̄b (y)


− ψ̄ˆb+ (y), ψ̂a− (x)
107
if x0 > y 0
=
if x0 < y 0
SF (x − y)
ab
1̂
where ψ̂ + and ψ̂ − correspond to the positive and negative frequency parts. Furthermore
ψ̂a (x) ψ̂b (y) = ψ̄â (x) ψ̄ˆb (y) = 0 ,
since these fields anticommute.
Wick’s theorem for fermionic fields: let’s again skip the subscript I that we would
normally use to indicate (free) interaction picture fields. Wick’s theorem then states
T ψ̂a1 (x1 ) · · · ψ̂an (xn )
as before, with for example
= N ψ̂a1 (x1 ) · · · ψ̂an (xn ) + all possible contractions ,
N ψ̂a1 (x1 )ψ̂a2 (x2 )ψ̄â3 (x3 )ψ̄â4 (x4 ) ≡ − ψ̂a1 (x1 ) ψ̄â3 (x3 )N ψ̂a2 (x2 ) ψ̄â4 (x4 )
= − N ψ̂a2 (x2 ) ψ̄â4 (x4 ) SF (x1 − x3 ) a1 a3 .
The proof of this version of Wick’s theorem (see Ex. 18) proceeds in a way similar to the
one that was worked out for scalar fields.
4.2
Feynman rules for the Yukawa theory
In order to assess the consequences of the fermionic version of Wick’s theorem we consider
the Yukawa theory for the interactions between fermions and scalars. The Lagrangian of
the Yukawa theory is given by
LYukawa = ψ̄(iγ µ ∂µ − mψ )ψ +
1
1
(∂µ φ)(∂ µ φ) − m2φ φ2 − g ψ̄ψφ ≡ LDirac + LKG + Lint ,
2
2
with φ a real scalar field and ψ a Dirac field. This gives rise to the following interaction
R
term in the Hamilton operator of the Yukawa theory: Ĥint = g d~x ψ̄ˆ(x)ψ̂(x)φ̂(x).
4.2.1
Implications of Fermi statistics
In order to study the consequences of fermionic minus signs we consider the ψ-fermion
scattering reaction
ψ(kA , sA )ψ(kB , sB ) → ψ(p1 , r1 )ψ(p2 , r2 )
at lowest order in perturbation theory, with the momenta and spin states of the particles
indicated between parentheses. In chapter 2 we have seen that the corresponding T -matrix
element is given by (skipping spin labels)
R 4
h~p1 ~p2 |i T̂ |~kA~kB i = 0 h~p1 p~2 |T e− i d x ĤI (x) |~kA~kB i0 fully connected × factor
and amputated
in terms of free-particle plane-wave states and interaction-picture (free) fields. The lowest108
order contribution to the ψ-scattering reaction then reads
p1 ~p2 |T
0 h~
(−ig)2 Z
2!
d x ψ̄Î (x)ψ̂I (x)φ̂I (x)
4
Z
ˆ
d y ψ̄I (y)ψ̂I (y)φ̂I (y) |~kA~kB i0 .
4
In order to perform the corresponding calculation we have to define
ψ̂I (x)|~k, si0 ≡ ψ̂I+ (x)|~k, si0 =
Z
d~q e−iq·x X r r p
p
â~q u (q) 2E~k â~sk † |0i = e−ik·x us (k)|0i ,
3
(2π)
2E~q r
with similar expressions for other initial and final states.
12b Since ψ̂I contains â and b̂† operators, it can be contracted with a fermion
state on the right (initial state) or an antifermion state on the left (final state).
The opposite holds for ψ̄Î , since it contains b̂ and â† operators.
Minus signs from interchanging fermions: for the lowest-order ψ-fermion-scattering
T -matrix elements we obtain
R
R
(−ig)2
2! 0 h~p1 p~2 | d4 x ψ̄Î (x)ψ̂I (x)φ̂I (x) d4 y ψ̄Î (y)ψ̂I (y)φ̂I (y)|~kA~kB i0
2!
= −
Z
−ig 2
d4 q
(2π)4 δ (4) (kA − p1 − q)(2π)4 δ (4) (kB − p2 + q) ×
(2π)4 q 2 − m2φ + iǫ
× ūs1 (p1 )usA (kA ) ūs2 (p2 )usB (kB )
=
s1
ig 2
ū (p1 )usA (kA ) ūs2 (p2 )usB (kB ) (2π)4 δ (4) (kA + kB − p1 − p2 )
2
2
(kA − p1 ) − mφ + iǫ
≡ (2π)4 δ (4) (kA + kB − p1 − p2 ) iM1
and
R
R
(−ig)2
2! 0 h~p1 p~2 | d4 x ψ̄Î (x)ψ̂I (x)φ̂I (x) d4 y ψ̄Î (y)ψ̂I (y)φ̂I (y)|~kA~kB i0
2!
s2
s1
−ig 2
sA
sB
=
ū
(p
)u
(k
)
ū
(p
)u
(k
)
(2π)4 δ (4) (kA + kB − p1 − p2 )
2
A
1
B
2
2
(kA − p2 ) − mφ + iǫ
≡ (2π)4 δ (4) (kA + kB − p1 − p2 ) iM2 ,
which have opposite signs. Note that we have used here a definition for the two-fermion
initial and final states: |~kA , sA ; ~kB , sB i ∝ â~skA † â~skB † |0i and h~p1 , r1 ; p~2 , r2 | ∝ h0| ârp~11 ârp~22 .
A
109
B
These T -matrix elements correspond to the following Feynman diagrams:
p1
iM1 =
q = kA −p1
kA
p2
p2
and
kB
iM2 =
kA
q = kA −p2
p1
.
kB
In these Feynman diagrams solid lines are used to indicate the fermions and dashed ones
to indicate the scalar particles.
12b Due to Fermi statistics the second contribution has a relative minus sign
with respect to the first one, since it involves the interchange of two fermions.
The overall sign depends on the definitions of the multiparticle states, for instance one
might define h~p1 ~p2 | ∝ h0| âp~2 âp~1 instead of ∝ h0| âp~1 âp~2 .
Minus signs from closed fermion loops: the Feynman rules for fermions result in a
factor −1 for closed fermion loops, as well as a trace of a product of Dirac matrices. We
find for instance that
∝
X
ψ̄âI (x1 )ψ̂aI (x1 ) ψ̄ˆbI (x2 )ψ̂bI (x2 ) ψ̄ˆcI (x3 )ψ̂cI (x3 ) ψ̄ˆdI (x4 )ψ̂dI (x4 )
a,b,c,d
= −
X
ψ̂dI (x4 )ψ̄âI (x1 ) ψ̂aI (x1 )ψ̄ˆbI (x2 ) ψ̂bI (x2 )ψ̄ˆcI (x3 ) ψ̂cI (x3 )ψ̄ˆdI (x4 )
a,b,c,d
= − Tr SF (x4 − x1 )SF (x1 − x2 )SF (x2 − x3 )SF (x3 − x4 ) .
In order to figure out how the matrices in the propagators should be contracted, we have
used explicit (repeated) spinor labels during intermediate steps.
12b This sign difference between fermionic and bosonic loops has important im-
plications for the high-energy behaviour of the fundamental interactions: strong
interactions are asymptotically free, electromagnetic interactions are not.
4.2.2
Drawing convention
12c In analogy with the conventions for the scalar Yukawa theory, we also
draw arrows on the fermion lines in the actual Yukawa theory. These arrows
represent the direction of particle-number flow: particles flow along the arrow,
antiparticles flow against it. In this convention ψ̂ corresponds to an arrow
flowing into a vertex, whereas ψ̄ˆ corresponds to an arrow flowing out of a
vertex. Since every interaction vertex features both ψ̂ and ψ̄ˆ, the arrows link
up to form a continuous flow. But this time there is more to it!
110
Consider for example a Feynman diagram like
p1
p2
= − g2
kA
kB
= − ig
2
= − ig
2
=
X
a,b
p1 ~p2 |
0 h~
R
R
d4 x ψ̄âI (x)ψ̂aI (x)φ̂I (x) d4 y ψ̄ˆbI (y)ψ̂bI (y)φ̂I (y)|~kA~kB i0
Z
Z
Z
X
d4 q (q/ + mψ )ab − iq·(x−y)
4
i(p2 −kB )·x
d x d ye
v̄a (kB )
e
ub (kA ) ei(p1 −kA )·y
4 q 2 − m2 + iǫ
(2π)
ψ
a,b
Z
d4 q v̄(kB )(q/ + mψ )u(kA )
(2π)4 δ (4) (p2 − q − kB ) (2π)4 δ (4) (p1 + q − kA )
2
4
2
(2π)
q − mψ + iǫ
4
−ig 2
v̄(kB )(k/A − p/1 + mψ )u(kA ) (2π)4 δ (4) (kA + kB − p1 − p2 ) .
(kA − p1 )2 − m2ψ + iǫ
In order to figure out how the spinors and matrices should be contracted, we have used
explicit (repeated) spinor labels during intermediate steps.
12c Here we see the importance of introducing the arrow convention: the spinor
indices are in this way always contracted along the fermion line, with the arrow
indicating the reversed order. Phrased differently, you should insert γ-matrices
and spinors while going against the arrow of particle-number flow!
4.2.3
Källén–Lehmann spectral representation for fermions
The non-perturbative analysis of the 2-point Green’s function follows the same steps as in
the scalar case with just a few obvious modifications:
P s
Z
iZ2
u (p)ūs (p)
s
d4 x eip·x hΩ|T ψ̂(x)ψ̄ˆ(0) |Ωi =
+ remainder containing other states
p2 − m2ph + iǫ
=
iZ2 (p/ + mph )
+ ··· .
p2 − m2ph + iǫ
Like before, Z2 represents the probability for the fermionic quantum field to create or
annihilate an exact “1-dressed particle” eigenstate of Ĥ from the ground state, with mph
denoting its observable physical mass:
p
p
hΩ|ψ̂(0)|~k , si = us (k) Z2 , hΩ|ψ̄ˆ(0)|~k , si = v̄ s (k) Z2 ,
p
p
h~p , r|ψ̄ˆ(0)|Ωi = ūr (p) Z2 , h~p , r|ψ̂(0)|Ωi = v r (p) Z2 .
More details will be worked out in the next chapter.
111
4.2.4
Momentum-space Feynman rules for the Yukawa theory
The Feynman rules that we have obtained for the Yukawa theory in the previous sections
can be summarized by the following list:
q
1. For each scalar propagator
For each fermion propagator
2. For each vertex
insert
p
a
b
q2
insert
i
.
− m2φ + iǫ
i(p/ + mψ )ab
.
p2 − m2ψ + iǫ
insert −ig .
3. For each external scalar line
q
insert
√
insert v̄ s (k) Z2 , originating from ψ̄ˆ .
For each incoming antifermion line
For each outgoing fermion line
Z.
√
insert us (k) Z2 , originating from ψ̂ .
k
For each incoming fermion line
√
k
√
insert ū r (p) Z2 , originating from ψ̄ˆ .
p
√
insert v r (p) Z2 , originating from ψ̂ .
For each outgoing antifermion line
p
4. Impose energy-momentum conservation at each vertex.
5. Integrate over each undetermined loop momentum lj :
Z
d 4 lj
.
(2π)4
6. Figure out the relative signs of the diagrams, caused by interchanging fermions.
7. Insert γ-matrices and spinors while going against the arrow of particle-number flow.
8. Each fermion loop receives a minus sign and involves a trace over spinor space.
The following observations can be made. First of all, no symmetry factors are needed in
the Yukawa theory since all fields in the interaction are different. Secondly, as can be seen
from the propagator, the sign (direction) of the momentum matters for fermions. Finally,
each distinct type of particle in the theory will have its own wave-function renormalization
√
√
factor, i.e. Z for the scalar particles and Z2 for the fermions.
112
4.3
How to calculate squared amplitudes
12d The final expressions for amplitudes that involve external fermions typi-
cally feature subexpressions starting with a ū or v̄ spinor, followed by a chain
of contracted matrices in spinor space, and closed by a u or v spinor. How
should we calculate squared amplitudes of that form?
In order to answer this question we select a typical term that features in |M|2 :
∗
ū(p)Γ1 u(k) ū(p)Γ2 u(k) ,
where the first factor originates from M and the second one from M∗ . Here Γ1,2 denote
arbitrary 4×4 matrices in spinor space.
Step 1: expressing things in terms of traces in spinor space.
We can make use of the identity
ū(p)Γ2 u(k)
∗
= ū(k)γ 0 Γ†2 γ 0 u(p)
to rewrite the expression given above in terms of traces in spinor space:
∗
ū(p)Γ1 u(k) ū(p)Γ2 u(k)
= ū(p)Γ1 u(k) ū(k)γ 0 Γ†2 γ 0 u(p)
= Tr
u(p)ū(p) Γ1 u(k)ū(k) γ 0 Γ†2 γ 0 .
Looking at the trace in the last line, the various combinations of Dirac spinors occurring
between square brackets are in fact 4×4 matrices in spinor space.
Step 2: employing polarization sums.
• If we are not able to produce polarized beams or to measure the polarization of the
final-state particles, then we have to average over the initial-state polarizations and
to sum over the final-state polarizations:
0 † 0
Tr u(p)ū(p) Γ1 u(k)ū(k) γ Γ2 γ
P s
0 † 0
1 P r
r
s
Tr
→
u (p)ū (p) Γ1
u (k)ū (k) γ Γ2 γ
2
r
s
1 Tr [p/ + mψ ] Γ1 [k/ + mψ ]γ 0 Γ†2 γ 0 ,
=
2
if we assume that one of the fermions is an initial-state fermion and the other one
a final-state fermion. The final trace can be worked out using the trace technology
developed in Ex. 16.
113
• If we are able to polarize the beams or measure polarization, then we can use a similar
trick provided that we project the Dirac spinors on the correct polarization states:
us (k) → P us (k)
⇒ ūs (k) → ūs (k)γ 0 P † γ 0 ≡ ūs (k) P̄
ur (p) → P ′ ur (p)
⇒ ūr (p) → ūr (p)γ 0 P ′† γ 0 ≡ ūr (p) P̄ ′ .
This allows us to perform the spin sums as before, but this time without plugging in
the spin average factor 1/2. The spin projection matrices P, P̄ and P ′ , P̄ ′ will select
the correct states! This time we get
′
′
0 † 0
Tr P [p/ + mψ ] P̄ Γ1 P [k/ + mψ ] P̄ γ Γ2 γ .
Once we know the spin projection matrices, the resulting traces can be calculated.
• If we are interested in polarized cross sections at high energies, then mψ can be
neglected with respect to the energy of the fermions. On top of that it will in
that case make sense to consider helicity states as our polarization states of choice.
After all, helicity eigenstates and chirality eigenstates coincide and are not mixed by
Lorentz transformations if mψ = 0. In that case the spin projections become
us (k) → PR/L us (k) , ūs (k) → ūs (k)PL/R
for helicity +/− fermions ,
v s (k) → PR/L v s (k) , v̄ s (k) → v̄ s (k)PL/R
for helicity −/+ antifermions ,
where it is used that P̄L/R = PR/L . The indicated chiralities reflect the fact that
particles and antiparticles have an opposite definition for their polarization states.
12d To summarize: calculating cross sections involving fermions simply boils
down to working out a collection of traces, irrespective of the fact whether one
is able to polarize the beams and/or measure final-state polarization.
Trace technology: the most important trace identities have been worked out in Ex. 16.
The relevant part that we need later on is summarized by
Tr(γ µ1 · · · γ µ2n+1 ) = Tr(odd number of γ-matrices) = 0 ,
Tr(γ µ1 · · · γ µ2n+1 γ 5 ) = 0 ,
Tr(I4 ) = 4 ,
Tr(γ µ γ ν ) = 4g µν
Tr(γ 5 ) = Tr(γ µ γ ν γ 5 ) = 0 ,
,
Tr(γ µ γ ν γ ρ γ σ ) = 4(g µν g ρσ − g µρ g νσ + g µσ g νρ ) ,
Tr(γ µ γ ν γ ρ γ σ γ 5 ) = − 4iǫµνρσ ,
with ǫµναβ ǫµνρσ = 2δ ασ δ βρ − 2δ αρ δ βσ .
114
5
Quantum Electrodynamics (QED)
During the last two lectures material will be covered that is not treated in this form in the
textbook of Peskin & Schroeder.
13 In this last chapter we will have a look at electromagnetic interactions of
matter particles. This will be used as motivation for the gauge principle, which
introduces the concept of gauge bosons as fundamental force carriers.
5.1
Electromagnetism
We start with the derivation of Maxwell’s equations in vacuum in covariant form. For an
electromagnetic field in vacuum with charge density ρc (t, ~x ) ≡ ρc (x) ∈ IR and current
density ~jc (t, ~x ) ≡ ~jc (x) ∈ IR3 the Maxwell equations read:
∂ ~
~ × E(x)
~
▽
= − B(x)
,
∂t
~ · B(x)
~
▽
= 0 ,
~ · E(x)
~
▽
= ρc (x) ,
∂ ~
~ × B(x)
~
▽
E(x) + ~jc (x) ,
=
∂t
~ ~x ) ≡ E(x)
~
~ ~x ) ≡ B(x)
~
where E(t,
∈ IR3 and B(t,
∈ IR3 are the electric and magnetic fields.
Next we introduce the electromagnetic 4-vector potential
Aµ (x) ≡
such that
~
φ(x) , A(x)
,
~ φ(x) − ∂ A(x)
~
~
,
E(x)
= −▽
∂t
~ × A(x)
~
~
.
B(x)
= ▽
In this way the two Maxwell equations on the first line are satisfied automatically, since
~ × ▽
~ φ(x) = ~0 . The other two Maxwell equations can be
~ · ▽
~ × A(x)
~
▽
= 0 and ▽
rewritten as
∂ ~ ~ ~ · ▽
~ φ(x) ▽ · A(x) − ▽
∂t
∂2
∂
~ · A(x)
~ 2 φ(x) − ∂ ▽
~
▽
+
−
φ(x)
=
∂t2
∂t
∂t
ρc (x) = −
and
∂ ~
∂2 ~
~ × ▽
~ × A(x)
~
~jc (x) = ▽
▽φ(x)
+ 2 A(x) +
∂t
∂t
∂2
∂
~ · A(x)
~ ▽
~ 2 A(x)
~
~
▽
▽
+
+
−
φ(x)
,
=
∂t2
∂t
115
using the identity
general ~ ~
~ 2A(x)
~ × A(x)
~ × ▽
~
~
~
▽
.
−▽
====== ▽ ▽ · A(x)
Defining the electromagnetic 4-current density
jcµ (x) ≡
ρc (x) , ~jc (x)
Maxwell’s equations can be cast in the form of the covariant electromagnetic wave equation
Aν (x) − ∂ ν ∂µ Aµ (x) = jcν (x) .
Gauge freedom: the vector potential Aµ (x) is not fixed completely by its relation to the
electric and magnetic fields. For an arbitrary scalar function χ(x) that vanishes sufficiently
quickly as |~x | → ∞, the transformed vector potential
Aµ (x) → A′ µ (x) = Aµ (x) + ∂ µ χ(x)
gives rise to the same electric and magnetic fields and therefore describes the same physics.
13a The associated freedom to choose the vector potential is called the gauge freedom.
Since the current density jcν (x) is a physical observable, the field-derivative combination
Aν (x) − ∂ ν ∂µ Aµ (x) should be gauge invariant, i.e. independent of the choice of gauge.
Proof: introduce the electromagnetic field tensor (see Ex. 2)


0
− E 1 (x) − E 2 (x) − E 3 (x)


3
2
 E 1 (x)

0
−
B
(x)
B
(x)


µν
µ ν
ν µ
νµ
F (x) = ∂ A (x) − ∂ A (x) = − F (x) = 
,
1
 E 2 (x) B3 (x)
0
− B (x) 


3
2
1
E (x) − B (x) B (x)
0
then the electromagnetic wave equation can be rewritten as
jcν (x) = ∂µ ∂ µAν (x) − ∂ ν∂µ Aµ (x) = ∂µ ∂ µAν (x) − ∂ νAµ (x)
= ∂µ F µν (x) .
Since the electromagnetic field tensor is gauge invariant, i.e.
F ′ µν (x) = ∂ µA′ ν (x) − ∂ νA′ µ (x) = ∂ µ Aν (x) + ∂ ν χ(x) − ∂ ν Aµ (x) + ∂ µ χ(x) = F µν (x) ,
the same holds for ∂µ F µν (x) = jcν (x).
Local charge conservation: from the electromagnetic wave equation one can derive that
∂ν jcν (x) = ∂ν ∂µ F µν (x) = 0 ,
since F µν (x) = − F νµ (x). Hence,
13a the current density jcν (x) is a conserved current and the electric charge
R
V
R
d~x jc0 (x) = d~x ρc (x) is conserved locally.
V
116
Electromagnetic Lagrangian: the Lagrangian density belonging to the electromagnetic
wave equation is given by
Le.m. (x) = −
1
Fµν (x)F µν (x) − jcµ (x)Aµ (x) .
4
Proof: first we consider
∂Le.m.
∂(∂µ Aν )
=
Ex.2
−
==== −
∂
1
1
∂
Fρσ F ρσ − Fρσ
F ρσ
4 ∂(∂µ Aν )
4
∂(∂µ Aν )
1 µ ν
1
(g ρ g σ − g νρ g µσ )F ρσ − Fρσ (g µρ g νσ − g νρg µσ ) = − F µν .
4
4
As a result, the Euler-Lagrange equation for the field Aν (x) indeed reads
− ∂µ F µν (x) + jcν (x) = 0
5.2
⇒
∂µ F µν (x) = jcν (x) .
QED and the gauge principle
For Dirac fermions (matter particles) with charge q the electromagnetic current density
is given by jcν, Dirac (x) = q ψ̄(x)γ ν ψ(x), since
• this current is indeed conserved (cf. page 93);
• after normal ordering the 0th component can indeed be identified with the total
charge density (cf. page 101):
Z
Z
2
d~p X s † s
0
(âp~ âp~ − b̂sp~ † b̂sp~ ) ,
d~x N ĵc , Dirac (x) = q
3
(2π) s = 1
counting particles with charge q and antiparticles with charge −q .
Minimal substitution and QED: the Lagrangian density of Dirac fermions with charge
q in an electromagnetic field is obtained by applying the
QM
13b minimal substitution prescription pµ → pµ −q Aµ ===⇒ i∂ µ → i∂ µ −q Aµ
to the Lagrangian density LDirac (x) of the free Dirac theory and by subsequently adding
the kinetic pure electromagnetic term LMaxwell (x) = − 41 Fµν (x)F µν (x). This results in the
Lagrangian density for Quantum Electrodynamics (QED):
LQED (x) = ψ̄(x)(i∂
/ − m)ψ(x) −
1
Fµν (x)F µν (x) − q ψ̄(x)γ µ ψ(x)Aµ (x) ,
4
containing the aforementioned interaction term
Lint (x) = − q ψ̄(x)γ µ ψ(x)Aµ (x) = − jcµ, Dirac (x)Aµ (x) .
117
As in the case of the Yukawa interactions, also the local electromagnetic interactions between the matter particles are mediated by force carriers. This was to be expected, bearing
in mind that charged objects are observed to interact while being at non-zero disctance!
Since [ψ] = [ψ̄] = 3/2 and [Aµ ] = 1, the electric charge q ∈ IR is a dimensionless coupling
constant. We will see later that this dimensionless coupling constant indeed implies that
QED is a renormalizable theory.
QED from a symmetry principle: gauge invariance and the gauge principle.
Alternatively we could start from the free Dirac Lagrangian
LDirac(x) = i ψ̄(x)γ µ ∂µ ψ(x) − m ψ̄(x)ψ(x) ,
which is invariant under the global gauge transformation (abelian U(1) transformation)
ψ(x) → ψ ′ (x) = eiα ψ(x) ,
ψ̄(x) → ψ̄ ′ (x) = e− iα ψ̄(x)
(α ∈ IR independent of xµ ) .
According to Noether’s theorem this global gauge symmetry can be associated with a
conserved current and charge. In non-relativistic quantum mechanics this global gauge
invariance of a free-fermion system simply underlines the unobservability of the absolute
phase of a wave function: only relative phases are observable through interference.
13c
The gauge principle: in the context of relativistic gauge theories, which should
be local, it is now postulated that this gauge invariance should also hold locally.
Consider to this end the local gauge transformation
ψ(x) → ψ ′ (x) = eiα(x) ψ(x) ,
ψ̄(x) → ψ̄ ′ (x) = e− iα(x) ψ̄(x)
(α(x) a real scalar field) .
The requirement of local gauge invariance 6 has profound consequences, since the kinetic
term transforms as
i ψ̄(x)γ µ ∂µ ψ(x) → i ψ̄(x)e− iα(x) γ µ ∂µ eiα(x) ψ(x) = i ψ̄(x)γ µ ∂µ ψ(x) − ψ̄(x)γ µ ψ(x) ∂µ α(x)
and therefore is not invariant under local gauge transformations. The last term, which
involves the covariant vector field ∂µ α(x), explicitly spoils the invariance. So, we need
to replace the ordinary derivative ∂µ by a gauge covariant derivative (or short: covariant
derivative) Dµ such that
Dµ ψ(x) → Dµ′ ψ ′ (x) = eiα(x) Dµ ψ(x) ,
causing Dµ ψ(x) and ψ(x) to transform similarly under local gauge transformations! This
can be achieved by
Dµ ≡ ∂µ + igAµ (x) ,
6
with
Aµ (x) → A′µ (x) = Aµ (x) −
1
∂µ α(x) ,
g
See the bachelor thesis of Pim van Oirschot for more details and extra motivation
118
where g is a gauge coupling and Aµ (x) a gauge field. In view of the Lorentz transformation property of ∂µ α(x), this gauge field should be a covariant vector field. Its transformation property resembles a gauge transformation for the electromagnetic vector potential
with χ(x) = − α(x)/g.
Proof:
1
Dµ′ ψ ′ (x) = ∂µ + ig Aµ (x) − ∂µ α(x) eiα(x) ψ(x)
g
= eiα(x) ∂µ ψ(x) + iψ(x)∂µ α(x) + igAµ (x)ψ(x) − iψ(x)∂µ α(x) = eiα(x) Dµ ψ(x) .
This means that the Lagrangian
i ψ̄(x)γ µ Dµ ψ(x) − m ψ̄(x)ψ(x) = i ψ̄(x)γ µ ∂µ ψ(x) − m ψ̄(x)ψ(x) − g ψ̄(x)γ µ ψ(x)Aµ (x)
is locally gauge invariant. It contains the gauge interaction
Lint (x) = − g ψ̄(x)γ µ ψ(x)Aµ (x) ,
which involves a gauge field that is coupled to a conserved current. Finally we can add
the gauge-invariant kinetic term LMaxwell (x) = − 14 Fµν (x)F µν (x) for a free gauge field,
where the field tensor Fµν (x) is defined as
igFµν (x) ≡ Dµ , Dν = ∂µ + igAµ (x) ∂ν + igAν (x) − ∂ν + igAν (x) ∂µ + igAµ (x)
= ig ∂µ Aν (x) − ∂ν Aµ (x) .
In conclusion, for g = |e| we find the same Lagrangian LQED as obtained by minimal
substitution for a particle with charge +|e|. For a general charge q = Q|e| one has to
modify the gauge transformation according to eiα(x) → eiQα(x) and the covariant derivative according to Dµ → ∂µ + iQ|e|Aµ (x) = ∂µ + iq Aµ (x). Such a rescaling leaves the
transformation property of the gauge field unaffected, but changes the interaction strength
from |e| to q .
Massless gauge fields: a massive gauge field would correspond to an extra mass term
+ 21 MA2 Aµ (x)Aµ (x) in the Lagrangian, which is obviously not gauge invariant. A theory
that is manifestly invariant under local gauge transformations requires the gauge bosons
described by Aµ (x) to be massless, i.e. MA = 0. So, in order to give mass to gauge bosons
an additional mechanism is required in the context of gauge theories.
13c Going beyond QED: motivated by the success of describing QED through
the gauge principle, this postulate will later on be extended to other types of
gauge transformations in order to describe other fundamental interactions in
119
nature, i.e. the strong and weak interactions. The associated extended gauge
interactions will describe the fundamental interactions between matter fermions
as being mediated by gauge bosons, just like we have just worked out for the electromagnetic interactions that are mediated by photons. In order to find the right
group structure for the extended gauge transformations, we will be guided by experimental observations of particle interactions and charge conservation laws!
5.2.1
Quantization of the free electromagnetic theory
The gauge freedom of the electromagnetic vector potential complicates the usual
quantization procedure. The reason for this lies in the following observations.
The electromagnetic gauge freedom revisited:
• By choosing an appropriate χ(x) it is possible to cast Aµ (x) in such a form that
~ · A(x)
~
= A0 (x) = 0 is satisfied. In this form we see
the Coulomb condition ▽
immediately that Aµ (x) has in fact only two physical degrees of freedom! These are
the degrees of freedom that should be quantized in the corresponding quantum field
theory . . . however, the Coulomb condition is not Lorentz invariant and therefore
leads to Feynman rules that are rather unpleasant.
• Lorentz invariance is manifest, resulting in simple Feynman rules, if we choose χ(x)
such that the Lorenz condition ∂ · A(x) = 0 is satisfied. In this form we do not
see straightaway that Aµ (x) has two physical degrees of freedom. One would expect three physical degrees of freedom in view of the Lorenz condition ∂ · A(x) = 0,
but there is still more gauge freedom left as a result of the gauge transformation
Aµ (x) → A′µ (x) = Aµ (x) + ∂µ χ′ (x) with χ′ (x) = 0. In other words:
we have to tread carefully while quantizing the free electromagnetic theory!
• What we are doing here is called gauge fixing. The gauge freedom for non-constant
χ(x) reflects the redundancy in our description of electromagnetism: the gaugetransformed fields describe the same physics and are therefore to be identified. This
can be traced back to the electromagnetic wave equation
Aν (x) − ∂ ν ∂µ Aµ (x) = (g νµ − ∂ ν ∂µ )Aµ (x) = jcν (x) ,
where the differential operator (g νµ − ∂ ν ∂µ ) is not invertible in the Green’s function sense as (g νµ − ∂ ν ∂µ )∂ µ χ(x) = 0 for arbitrary χ(x). Given an initial field
configuration Aµ (t0 , ~x ) we cannot unambiguously determine Aµ (t, ~x ), since Aµ (x)
and Aµ (x) + ∂ µ χ(x) are not distinguishable .
120
13a To summarize: Aµ (x) is not a physical object as it contains redundant
information. All fields that are linked by a gauge transformation form an
equivalence class and are therefore to be identified: the physics is uniquely
described by selecting a representative of each equivalence class. Different
configurations of these representatives are called different gauges. By fixing
the gauge the redundancy is removed and an unambiguous electromagnetic
evolution is obtained. We can choose freely here, but some choices will
prove more handy for certain problems than others.
The problem of quantizing the free electromagnetic field: the first thing to note
in this context is that the Lagrangian density LMaxwell (x) = − 41 Fµν (x)F µν (x) for a free
electromagnetic field is not suited for canonical quantization:
π µ (x) =
∂LMaxwell
= − F 0µ (x)
∂(∂0 Aµ )
⇒
π j (x) = E j (x) and π 0 (x) = 0 .
This implies that the 0th component of Aµ (x) is not dynamical and therefore cannot be
quantized. This is no big surprise, since only two out of four degrees of freedom are actually
physical.
13c If we want to quantize all components of Aµ (x), we can exploit the freedom
to choose a gauge by adding a so-called gauge-fixing term to the Lagrangian that
makes A0 (x) dynamical. Subsequently we will have to constrain the physical
states of the theory in accordance with the gauge choice in order to restore
Maxwell’s theory in the classical limit (i.e. in terms of expectation values for
physical states).
To this end we apply a method that resembles the Lagrange-multiplier method in classical mechanics. Since we would like to impose the manifestly invariant Lorenz condition
∂ · A(x) = 0, this amounts to
L(x) = −
2
λ
1
∂ · A(x) .
Fµν (x)F µν (x) −
4
2
For calculational convenience we will choose λ = 1, which is known as ’t Hooft-Feynman
gauge fixing. In that case the Euler-Lagrange equation takes the form of the massless
KG equation Aµ (x) = 0 for all four components of the electromagnetic potential. This
implies that we can quantize as in the massless scalar case:
Â†µ (x) = Âµ (x)
========
=
Âµ (x) =
Z
3
d~p
1 X r r
r† r∗
−ip·x
ip·x p
â ǫ (p)e
+ âp~ ǫµ (p)e
.
(2π)3 2Ep~ r=0 p~ µ
p0 = Ep~ = |~
p|
Note that we overquantize by quantizing all four degrees of freedom in our attempt to
maintain manifest covariance and thereby achieve calculational simplicity (as we will see
121
later on). The associated four polarization vectors ǫ0µ (p), · · ·, ǫ3µ (p) span Minkowski space:
r=0
→
ǫ00 (p) = 1 , ~ǫ 0 (p) = ~0 (“scalar polarization”) ,
r = 1, 2
→
ǫr0 (p) = 0 , ~ǫ r (p) · p~ = 0 (physical transverse polarization states) ,
r=3
→
ǫ30 (p) = 0 , ~ǫ 3 (p) =
p~
p~
=
Ep~
|~p |
(“longitudinal polarization”) ,
′
′
with the normalization condition ǫr (p) · ǫr ∗ (p) = g rr . In order to be consistent with the
Lorentz covariant canonical quantization condition
Âµ (~x , t), π̂ ν (~y , t) = igµν δ(~x − ~y ) 1̂ ,
r
the creation and annihilation operators âr†
~ of the massless electromagnetic energy
p
~ and âp
quanta (photons = antiphotons) satisfy the bosonic quantization conditions
′ ′
ârp~ , ârp~ ′† = − (2π)3 g rr δ(~p − p~ ′ ) 1̂
and
′ ′ ârp~ , ârp~ ′ = ârp~ † , ârp~ ′† = 0 ,
leading to a normal-ordered Hamilton operator of the form
N(Ĥ) = N
R
µ
d~x π̂ (x)∂0 Âµ (x) − L̂(x)
=
Z
3
X
d~p
r† r
0† 0
E
â
â
−
â
â
.
p
~
~
~
p
~ p
p
~ p
(2π)3
r=1
The weird minus signs in the Hamilton operator and in one of the commutation relations
is a consequence of the wrong sign of the kinetic term for the unphysical scalar A0 -field in
the Lagrangian density:
kinetic terms
L(x) −−−−−−−−−−→
1
1 ~˙ 2
A − Ȧ20 .
2
2
The energy spectrum thus seems unbounded from below. However, that is caused by unphysical modes of the theory! So, . . .
Decoupling unphysical states (see Ex. 20): next we need to decouple unphysical (nontransverse) asymptotic Fock states from physical ones, which describe transverse photons
~ · A(x)
~
= A0 (x) = 0. This means that the unphysical states should yield
that satisfy ▽
vanishing contributions to observable quantities. This can be achieved by demanding the
physical asymptotic states |ψi in Fock space to satisfy hψ|∂ · Â(x)|ψi = 0, implying that
the Lorenz condition (and therefore Maxwell’s equations) are valid as expectation values
for physical Fock states. In this way quantum field theory reproduces the classical theory
in the classical limit. In Fock space this requirement boils down to the condition that
â0p~ |ψi = â3p~ |ψi for all values of the momentum p~ . In Ex. 20 it will be proven that this
gives rise to the identity hψ|Âµ (x)|ψi = hψ|Âtrans
(x)|ψi + ∂µ χ(x) , with χ(x) = 0. Since
µ
1,2
ǫ (p) · p = 0, the Lorenz condition hψ|∂ · Â(x)|ψi = 0 automatically follows.
122
13c The aforementioned identity in fact reflects the remaining gauge arbitrari-
ness of the classical electromagnetic field hψ|Âµ (x)|ψi in the Lorenz gauge,
which causes the effective number of degrees of freedom to be two. The latter
can be seen directly if we choose χ(x) to vanish, resulting in a purely transverse
electromagnetic field.
A direct consequence of the condition â0p~ |ψi = â3p~ |ψi is that the physical states |ψi in
Fock space have a positive definite energy expectation value, as required:
Z
2
X
d~p
r
hψ|N(Ĥ)|ψi =
Ep~
hψ|âr†
~ |ψi ≥ 0 .
p
~ âp
(2π)3
r=1
1. External (asymptotic) photons are physical. Therefore only the transverse
degrees of freedom feature, as expected for massles spin-1 particles.
2. Internal (virtual) photons are not required to be physical. All four degrees
of freedom of Aµ (x) are needed, since an internal photon is not subject to
influences from the physical (asymptotic) states. It is precisely this gauge
freedom for internal photons that we are trying to use to our advantage.
Feynman propagator and polarization sum: for performing Feynman-diagram calculations we need two objects. Let’s start as usual with the propagation of photons from
y to x:
Z
′
3
d~p d~p ′ e−ip·x+ip ·y X r
r′ ∗ ′
r r′ †
p
ǫ
(p)ǫ
(p
)h0|â
â
h0|Âµ (x)Âν (y)|0i =
′ |0i
µ
ν
p
~
p
~
(2π)6 2 Ep~ Ep~ ′ r,r′ = 0
p0 = |~
p | , p′0 = |~
p ′|
= −
Z
3
X
d~p e−ip·(x−y) 0
0∗
r
r∗
ǫµ (p)ǫν (p) −
ǫµ (p)ǫν (p) = − gµν D(x − y) ,
(2π)3
2Ep~
p0 = |~
p|
r=1
where in the last step it was used that the polarization vectors ǫ1µ (p), · · ·, ǫ3µ (p) span
3-dimensional space and therefore satisfy a completeness relation. In analogy with the
Klein-Gordon case, the Feynman propagator of the free electromagnetic theory reads:
Z 4
d p −igµν −ip·(x−y)
e
= − gµν DF (x − y; m2 = 0) ,
h0|T Âµ (x)Âν (y) |0i =
(2π)4 p2 + iǫ
This is what we have gained by overquantizing:
14a the propagator for internal (virtual) photons has become extremely simple
and manifestly Lorentz covariant in the ’t Hooft-Feynman gauge!
In order to be able to deal with external (physical) photons one also needs the so-called
polarization sum for transverse photons:
2
X
r=1
ǫrµ (p)ǫrν ∗ (p) = − gµν − ǫ3µ (p)ǫ3ν ∗ (p) + ǫ0µ (p)ǫ0ν ∗ (p) ,
123
which can be rewritten by introducing
nµ ≡ ǫ0µ (p)
into
2
X
r=1
ǫ3µ (p) =
⇒
pµ
pµ
− nµ =
− nµ
Ep~
n·p
pµ pν
pµ nν + nµ pν
+
.
2
(n · p)
n·p
ǫrµ (p)ǫrν ∗ (p) = − gµν −
Such a complicated expression is unavoidable for external photons, but we managed to
keep it out of the Feynman propagator!
5.3
Feynman rules for QED (§ 4.8 in the book)
In order to obtain the full set of momentum-space Feynman rules for QED we simply have
to supplement the Feynman rules for fermions, which were given in the context of the
Yukawa theory, by the following four photonic Feynman rules:
µ
1. For each photon propagator
ν
insert
q
2. For each QED vertex
µ
−igµν
.
q 2 + iǫ
insert −iq γ µ .
µ
3. For each incoming photon line
√
= Âµ (x)|~p , ri0 insert ǫrµ (p) Z3 (r = 1, 2).
p
For each outgoing photon line
µ
p
√
= 0 h~p, r|Âµ(x) insert ǫrµ∗ (p) Z3 (r = 1, 2).
The following remarks are in order. First of all, the polarization vectors featuring in
√
the last two Feynman rules are transverse (physical) ones and Z3 is the wave-function
renormalization factor for photons. Secondly, the sign (direction) of the momentum in the
photon propagator does not matter, like in the scalar case. Finally, the γ-matrix occurring
in the QED vertex will be wedged between Dirac spinors, with the Dirac indices contracted
as usual along the fermion line against the arrow.
(x)|ψi + ∂µ χ(x) with χ(x) = 0,
14a Remark: since hψ|Âµ (x)|ψi = hψ|Âtrans
µ
we can always add to ǫrµ (p) a term ∝ pµ with p2 = 0 without changing the
physics outcome (see § 5.5).
124
5.4
Full fermion propagator (§ 7.1 in the book)
To all orders in perturbation theory the full fermion propagator in QED is given by the
Dyson series
Z
d4 x eip·x hΩ|T ψ̂(x)ψ̄ˆ(0) |Ωi
p
=
+
p
≡
p
p
p
1PI
p
+
p
1PI
p
1PI
+ ··· ,
where
1PI
≡ − iΣ(p/) =
+
+
+
+ ···
is the collection of all 1-particle irreducible fermion self-energy diagrams. This Dyson series
can again be summed up as a geometric series:
Z
d4 x eip·x hΩ|T ψ̂(x)ψ̄ˆ(0) |Ωi
=
=
=
p
p
i(p/ + m)
i(p/ + m)
i(p/ + m) −
iΣ(p
/
)
+
+ ···
p2 − m2 + iǫ
p2 − m2 + iǫ
p2 − m2 + iǫ
i
≡ S(p) ,
p/ − m − Σ(p/)
using that Σ(p/) = ΣS (p2 ) m + ΣV (p2 ) p/ commutes with p/ and the mass parameter m in
the Lagrangian. The full propagator has a simple pole located at the physical mass mph ,
which is shifted away from m by the fermion self-energy:
h
i
p/ − m − Σ(p/) = 0 ⇒ mph − m − Σ(p/ = mph ) = 0 .
p/ = mph
Close to this pole the denominator of the full propagator can be expanded according to
p/ − m − Σ(p/) ≈ (p/ − mph ) 1 − Σ′ (p/ = mph ) + O [p/ − mph ]2
for
p/ ≈ mph ,
where Σ′ (p/) stands for the derivative of the fermion self-energy with respect to p/. Just like
in the Källén–Lehmann spectral representation, the full propagator has a single-particle
pole of the form iZ2 (p/ + mph )/(p2 −m2ph + iǫ) with Z2 = 1/ 1 −Σ′ (p/ = mph ) (see p.111).
The fermion self-energy: in order to find out whether the fermion self-energy is more
difficult to calculate we consider the 1-loop contribution in QED. Indicating the photon
125
mass by λ we then obtain
p − ℓ1
p
p
ℓ1
= − iΣ2 (p/) = (−iq)
= −q
2
2
Z
Z
d4 ℓ1 µ i(ℓ/1 + m) ν
−igµν
γ
γ
(2π)4 ℓ21 − m2 + iǫ (p −ℓ1)2 − λ2 + iǫ λ ↓ 0
d4 ℓ1
4m − 2ℓ/1
2
4
2
(2π) ℓ1 − m + iǫ (p −ℓ1)2 − λ2 + iǫ
p. 66 , ℓ1 = ℓ + α2 p
============== − q
= −q
2
Z
0
2
Z
1
dα2
0
1
dα2 (4m − 2α2 p/)
Z
Z
d4 ℓ 4m − 2ℓ/ − 2α2 p/
(2π)4 (ℓ2 − ∆ + iǫ)2
d4 ℓ
1
,
(2π)4 (ℓ2 − ∆ + iǫ)2
with
∆ = α2 λ2 + (1 − α2 )m2 − α2 (1 − α2 )p2
just like in the scalar case. In the second line of this expression we have used that
γ µ (ℓ/1 + m)γµ = (m − /ℓ 1 )γ µ γµ + 2ℓ/1 = 4m − 2ℓ/1 .
The threshold for the creation of a fermion–photon 2-particle state is here situated at
p2 = (m + λ)2 , which approaches m2 in the limit λ ↓ 0 for massless photons. The rest of
the calculation, including the regularization of the UV divergence, goes like in the scalar
case worked out in § 2.9.2. Note that the fermion mass receives a UV-divergent shift
Σ2 (p/ = mph ) ∝ mph log(Λ2 /m2ph ).
14b Fermion masses are naturally protected against high-scale quantum correc-
tions: if there would be no coupling between left- and right-handed Dirac fields
in the Dirac Lagrangian (i.e. m = 0), then no such coupling can be induced by
the perturbative vector-current QED corrections! Fermion masses are protected
by the invariance under chiral transformations of the massless theory.
5.5
The Ward –Takahashi identity in QED (§ 7.4 in the book)
14b Question: how does the gauge invariance of QED manifest itself in Green’s
functions and scattering amplitudes?
In order to answer this question, we consider a QED diagram to which we want to attach
an additional photon with momentum k . Upon contraction of this photon line with the
corresponding momentum k a special identity can be derived that is related to the U(1)
gauge symmetry.
126
Step 1: how can the photon be attached to an arbitrary diagram involving (anti)fermions
and photons?
• The photon cannot be attached to a photon, since it has charge 0.
• The photon can be attached to a fermion line that connects two external points or
to a fermion loop.
Step 2: consider an arbitrary fermion line with j photons attached to it and all photon
momenta defined to be incoming. Graphically this can be represented by
ℓj
ℓj−1
ℓj−2
νj
νj−1
kj
where ℓi = ℓ0 +
i
P
n=1
ℓ2
ℓ1
ℓ0
ν2
kj−1
k2
ν1
k1
kn . This line can either flow between external points or close into a
loop (which means that l0 = lj ) and the photons can either be on-shell or virtual. There
will be j + 1 places to insert the extra photon with momentum k , for example between
photons i and i +1:
k
µ
ℓi−1
kµ ∗
ℓi +k ℓi
νi
= ···
h
i
i
i
i
···
−iq k/
−iq γ νi
/ℓ i + k/ − m
/ℓ i − m
/ℓ i−1 − m
h = ··· q
i
i
i
i
νi
··· ,
−iq γ
−
/ℓ i − m
/ℓ i + k/ − m
/ℓ i−1 − m
where we have used that k/ = /ℓ i + k/ − m − (ℓ/i − m) in the last step. Insertion between
photons i −1 and i gives in a similar way:
···
h
i
i
i
i
νi
··· .
−iq γ q
−
/ℓ i + k/ − m
/ℓ i−1 − m
/ℓ i−1 + k/ − m
Note that the second term of the ith insertion cancels the first term of the (i−1)th insertion.
Finally we have to sum over all possible insertions along the fermion line. This causes all
127
terms to cancel pairwise except for two unpaired terms at the very end of the chain:
k
j
X
i=0
µ
kµ ∗
=
ℓj +k
ℓi +k
ℓj
q ∗
ℓ0
ℓj−1
kj
−q ∗
ℓi
ℓ1
kj−1
k2
ℓj +k ℓj−1 +k
kj
ℓ0
k1
ℓ1 +k
kj−1
k2
ℓ0 +k
k1
Consequence 1: if the fermion line is part of an on-shell matrix element and connects two
of the external states, then the corresponding amputation procedure removes both terms
on the right-hand-side. This is caused by the fact that one of the endpoints gives rise to a
shifted 1-particle pole, i.e. 1/(ℓ2j − m2 ) instead of 1/[(ℓj + k)2 − m2 ] or 1/[(ℓ0 + k)2 − m2 ]
instead of 1/(ℓ20 − m2 ).
Consequence 2: if the fermion line closes in itself to form a loop (i.e. ℓ0 = ℓj + k), then
the two terms on the right-hand-side give rise to the integrals
−q
j+1
Z
d4 ℓ0
(2π)4
Tr
1
1
1
γ νj
γ νj−1 · · ·
γ ν1
/ℓ 0 − m
/ℓ j−1 − m
/ℓ 1 − m
1
1
1
νj
νj−1
ν1
= 0,
γ
γ
···
γ
− Tr
/ℓ 0 + k/ − m
/ℓ j−1 + k/ − m
/ℓ 1 + k/ − m
if we are allowed to change the integration variable from ℓ0 to ℓ0 + k in the first term!
Consequence 3: the Ward –Takahashi identity for Green’s functions reads
q1
kµ ∗
qn
µ
q1
=
q
qi −k
qn
n
X
− q
i=1
k
p1
pn
q1
p1
pn
qn
n
X
i=1
p1
pn
pi +k
128
where the blobs represent all possible diagrams and photon insertions. In formula language
this can be written compactly as
X
µ
kµ G (k; p1 , · · · , pn ; q1 , · · · , qn ) = q
G(p1 , · · · , pn ; q1 , · · · , qi−1 , qi − k, qi+1 , · · · , qn )
i
− G(p1 , · · · , pi−1 , pi + k, pi+1 , · · · , pn ; q1 , · · · , qn ) .
14b This is the diagrammatic identity that imposes the U(1) gauge symmetry
and associated electric charge conservation on quantum mechanical amplitudes!
Example of a Ward–Takahashi identity:
p +k
p +k
kµ ∗
µ
k
=
p
kµ ∗
µ
amp
k
p
Ward – Takahashi
= S(p + k) −iq kµ Γµ (p + k, p) S(p) =========== q S(p) − q S(p + k)
⇒ −ikµ Γµ (p + k, p) = S −1 (p + k) − S −1 (p) = −i k/ + Σ(p/) − Σ(p/ + k/) .
Here S(p) is the full fermion propagator, Σ(p/) the corresponding 1-particle irreducible
self-energy and −iq Γµ (p + k, p) the sum of all amputated 3-point diagrams contributing
to the QED vertex. Hence, Γµ (p + k, p) is given by γ µ at lowest order in perturbation
theory, which is indeed in agreement with the Ward–Takahashi identity.
5.6
The photon propagator (§ 7.5 in the book)
The Ward –Takahashi identity has important implications for the properties of
the photon propagator.
Transversality: the 1-particle irreducible photon self-energy
iΠµν (k) ≡
µ
1PI
ν
k
k
satisfies the Ward –Takahashi identity (transversality condition)
kµ Πµν (k) = 0 .
In view of Lorentz covariance Πµν (k) can be decomposed into only two possible terms, a
term ∝ g µν and a term ∝ k µ k ν . Therefore the Ward–Takahashi identity translates into
the condition
Πµν (k) = (k 2 g µν − k µ k ν )Π(k 2 ) ,
129
with Π(k 2 ) regular at k 2 = 0 since a pole at k 2 = 0 would imply the existence of a singlemassless-particle intermediate state. As a result, the full photon propagator is of the form
µ
ν
k
=
+
+
1PI
1PI
1PI
k
=
−igσν
−igµν
−igµρ 2 ρσ
i(k g − k ρ k σ )Π(k 2 ) 2
+ 2
+ ···
2
k + iǫ
k + iǫ
k + iǫ
= −
=
+ ···
k2
i
kµ kν
i
(gµν − kµ kν /k 2 ) 1 + Π(k 2 ) + · · · − 2
+ iǫ
k + iǫ k 2
k k −i(gµν − kµ kν /k 2 )
i
µ ν
− 2
.
2
2
k + iǫ
k2
(k + iǫ) 1 − Π(k )
Mass of the photon: consider an arbitrary internal photon line
q ′ = Q′ |e|
q = Q |e|
µ
ν
on-shell
k
on-shell
k
The kµ and kν terms in the full propagator yield a vanishing contribution due to the
Ward –Takahashi identity for on-shell amplitudes. Hence,
µ
ν
k
k
effectively
−−−−−−−−−−→
−igµν
,
(k 2 + iǫ) 1 − Π(k 2 )
−1
which has a pole at k = 0 with residue Z3 ≡ 1 − Π(0) . As a result of the Ward –
Takahashi identity, which in turn is a consequence of the gauge symmetry, mphoton = 0 to
all orders in perturbation theory:
2
14b
the local U(1) gauge symmetry protects the photon from becoming
massive through quantum corrections.
Observable charge: consider the same amplitude as before for
p
• low |k 2 | ⇒ e → e Z3 , which is the finite physically observable charge obtained
from the singular quantities e and Z3 ;
• high |k 2 | ⇒
⇒
−igµν e2
−igµν Z3 e2
−igµν e2
→
=
k2
k 2 1 − Π(k 2 )
k 2 1 − Z3 Π(k 2 ) − Π(0)
e2
Z α
3
,
= α → α(k 2 ) =
4π
1 − Z3 Π(k 2 ) − Π(0)
130
where the factor Z3 in front of Π(k 2 )−Π(0) turns e2 inside the photon self-energy
into the finite combination Z3 e2 .
14d The electromagnetic fine structure constant becomes a running coupling,
i.e. a coupling that changes with invariant mass. In fact it becomes larger
with increasing invariant mass, causing the exchanged (virtual) photon to
propagate more easily through spacetime.
The physical picture behind this is that virtual fermion-antifermion pairs that are
created from the vacuum partially screen the charges of the interacting particles
(vacuum polarization), resulting in a lower effective charge. For larger |k 2 | more of
the polarization cloud is penetrated and hence more of the actual charge can be felt.
All couplings in the Standard Model of electroweak interactions are in fact running couplings. As can be seen in the plot, the behaviour of the hypercharge coupling, indicated
by U(1), resembles the one for QED. However, due to bosonic loop effects the couplings
of the weak interactions, indicated by SU(2),
and strong interactions, indicated by SU(3),
actually become weaker for increasing invariant mass.
UV divergences: at 1-loop order the photon self-energy in QED is given by
iΠµν
2 (k)
≡
µ
k+ℓ1
k
= − 4q
2
p. 66
ℓ1
Z
==== − 4q
ℓ = ℓ1 +α2 k
ν
= (−1) q
k
Z
Tr(γ µ [ℓ/1 + m]γ ν [ℓ/1 + k/ + m])
d4 ℓ1
(2π)4 [(ℓ1 + k)2 − m2 + iǫ][ℓ21 − m2 + iǫ]
d4 ℓ1 ℓµ1 (ℓ1 + k)ν + ℓν1 (ℓ1 + k)µ + g µν [m2 − ℓ21 − ℓ1 · k]
(2π)4
[(ℓ1 + k)2 − m2 + iǫ][ℓ21 − m2 + iǫ]
2
Z
d4 ℓ1
(2π)4
======= − 4q
Z1
dα2
0
2
Z1
dα2
0
i
==== − 4q 2
16π 2
p. 67
2
Z1
0
Z
2ℓµ1 ℓν1 + ℓµ1 k ν + ℓν1 k µ + g µν [m2 − ℓ21 − ℓ1 · k]
(ℓ21 − m2 + 2α2 ℓ1 · k + α2 k 2 + iǫ)2
d4 ℓ 2ℓµ ℓν + g µν (∆ − ℓ2 ) + (k 2 g µν − k µ k ν )2α2 (1− α2 )
(2π)4
(ℓ2 − ∆ + iǫ)2
Z∞
g µν (∆ + ℓ2E /2) + (k 2 g µν − k µ k ν )2α2 (1− α2 )
dα2 dℓ2E ℓ2E
,
(ℓ2E + ∆ − iǫ)2
0
131
where ∆ = m2 − α2 (1 − α2 )k 2 . The resulting integral is clearly divergent.
Transversality lost: if we were to regularize (quantify) the UV divergence in the usual
way by means of a cutoff Λ, then Πµν
2 (k) would contain a leading singularity that is proR 2 2
2 µν
µν Λ
portional to g 0 dℓE = Λ g . This has disastrous consequences, since it violates the
transversality requirement and gives the photon an infinite mass. After all, a Λ2 g µν term
in Πµν (k) gives rise to a Λ2 /k 2 contribution to Π(k 2 ) and therefore shifts the pole of
k 2 1 − Π(k 2 ) away from k 2 = 0.
Question: what has happened here?
In fact the fermion-loop Ward –Takahashi identity on p. 128 has been invalidated, since we
are actually not allowed to shift the integration variable without consequences when using
the cutoff method.
14e We need another regularization scheme that preserves the fundamental U(1)
symmetry, otherwise the results cannot be trusted. Dimensional regularization
(’t Hooft –Veltman, 1972): compute Feynman diagrams as analytic functions
of the dimensionality of spacetime. Use to this end a d-dimensional Minkowski
space consisting of one time dimension and d−1 spatial dimensions.
• For sufficiently small d any loop integral will converge in the UV domain
and the fermion-loop Ward –Takahashi identity is retained for all d.
• The final expressions for observables are then obtained as d → 4 limits.
Examples of integrals calculated with dimensional regularization (DREG):
Z
DREG
d4 ℓE
1
−
−→
2
4
2
(2π) (ℓE + ∆)
Z
∆d/2−2
==========
(4π)d/2 Γ(d/2)
z = ∆/(∆ + ℓ2E )
1
1
2π d/2 1
dd ℓE
p. 67, 68
=
=
=
=
=
=
(2π)d (ℓ2E + ∆)2
(2π)d Γ(d/2) 2
Z1
dz z 1−d/2 (1−z)d/2−1 =
Z∞
(ℓ2 )d/2−1
dℓ2E 2E
(ℓE + ∆)2
0
∆d/2−2
Γ(2 − d/2) .
(4π)d/2
0
Here we have used the integral identity
Z1
dz z b−1 (1−z)c−1 =
Γ(b)Γ(c)
Γ(b + c)
0
in terms of the gamma function Γ(z), which satisfies
Γ(1/2) =
√
π
,
Γ(1) = 1
132
and
Γ(z + 1) = z Γ(z) .
This time Γ(2− d/2) represents the UV singularities, since the gamma function Γ(z) has
poles at z = 0, −1, −2, · · · and therefore Γ(2 − d/2) has poles at d = 4, 6, 8, · · · :
d≈4
Γ(2 − d/2) ≈ −
2
− γE + O(d − 4)
d−4
with
γE = 0.5772 = Euler’s constant .
In a similar way one finds
Z
DREG
d4 ℓE
ℓ2E
∆d/2−1
−
−→
(2π)4 (ℓ2E + ∆)2
(4π)d/2 Γ(d/2)
Z1
dz z
−d/2
(1−z)
0
d/2
d∆
=
2− d
Z
d4 ℓE
1
.
2
4
(2π) (ℓE + ∆)2
Transversality restored: returning to the integrand on page 131, we see that the nontransverse term indeed vanishes:
DREG
2ℓµ ℓν + g µν (∆ − ℓ2 ) −
−→ g µν ∆ − ℓ2 [1 − 2/d]
Wick
−
−→ g µν ∆ + ℓ2E [1 − 2/d]
integrals
−−−
−→ 0 ,
as required by gauge symmetry. So, dimensional regularization is a viable way of dealing
with UV divergences in the context of gauge symmetries. This regularization method was
used successfully by ’t Hooft and Veltman to prove the renormalizability of the Standard
Model of electroweak interactions, for which they were awarded the Nobel Prize in 1999.
133

Download Report

Reader for the course Quantum Field Theory W.J.P. Beenakker

Paperzz.com

Your Paperzz