Relativistic Quantum Field Theory Part I

Introduction to Theoretical Elementary Particle
Physics:
Relativistic Quantum Field Theory
Part I
Stefan Weinzierl
September 7, 2016
1
Contents
1 Overview
1.1 Literature . . . . . . . . . . . . . . .
1.2 Units . . . . . . . . . . . . . . . . . .
1.3 The fundamental forces . . . . . . . .
1.4 The elementary particles . . . . . . .
1.4.1 Spin 1/2 particles . . . . . . .
1.4.2 Spin 1 particles . . . . . . . .
1.4.3 Spin 0 particles . . . . . . . .
1.5 Experiments . . . . . . . . . . . . . .
1.6 Observed, but not elementary particles
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5
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2 Review of quantum mechanics
2.1 The harmonic oscillator in classical mechanics .
2.2 The harmonic oscillator in quantum mechanics
2.2.1 Operator formalism . . . . . . . . . . .
2.2.2 Path integrals . . . . . . . . . . . . . .
2.2.3 Summary . . . . . . . . . . . . . . . .
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11
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11
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15
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3 Review of special relativity
18
3.1 Four-vectors and the metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 The Lorentz group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 The Poincaré group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4 Review of classical field theory
4.1 The action principle . . . . . .
4.2 Examples of classical fields . .
4.2.1 The Klein-Gordon field
4.2.2 The Dirac field . . . .
4.2.3 The Maxwell field . .
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5 Quantum field theory: The canonical formalism
5.1 The Klein-Gordon field as harmonic oscillators . . . . . . .
5.2 The Schrödinger picture . . . . . . . . . . . . . . . . . . .
5.3 The Fock space . . . . . . . . . . . . . . . . . . . . . . . .
5.4 The Heisenberg picture . . . . . . . . . . . . . . . . . . . .
5.4.1 Causality . . . . . . . . . . . . . . . . . . . . . . .
5.4.2 The Klein-Gordon propagator . . . . . . . . . . . .
5.5 Wick’s theorem . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Interacting fields . . . . . . . . . . . . . . . . . . . . . . .
5.7 Feynman diagrams . . . . . . . . . . . . . . . . . . . . . .
5.8 Summary: Feynman rules for φ4 -theory in momentum space
2
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23
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29
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51
56
6 Cross sections and decay rates
6.1 The S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Properties of the S-matrix . . . . . . . . . . . . . . . . . . . . . . .
6.3 Relation between invariant matrix elements and Feynman diagrams .
6.4 Final formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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58
60
65
65
67
7 Fermions
7.1 The Dirac equation . . . . . . . . .
7.2 Massless spinors . . . . . . . . . .
7.3 Spinorproducts . . . . . . . . . . .
7.4 Massive spinors . . . . . . . . . . .
7.5 Quantisation of fermions . . . . . .
7.6 Feynman rules for fermions . . . . .
7.7 Rules for traces over Dirac matrices
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69
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70
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77
78
8 Quantum field theory via path integrals
8.1 Trouble with the canonical quantisation of gauge bosons . . . .
8.2 Path integrals . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.3 Transition amplitudes as path integrals . . . . . . . . . . . . . .
8.4 Correlation functions . . . . . . . . . . . . . . . . . . . . . . .
8.5 Fermions in the path integral formalism . . . . . . . . . . . . .
8.5.1 Grassmann numbers . . . . . . . . . . . . . . . . . . .
8.5.2 Path integrals with fermions . . . . . . . . . . . . . . .
8.6 The reduction formula of Lehmann, Symanzik and Zimmermann
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79
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95
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109
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9 Gauge theories
9.1 Lie groups und Lie algebras . . . . . . . .
9.2 Special unitary Lie groups . . . . . . . . .
9.3 Yang-Mills theory . . . . . . . . . . . . . .
9.4 Quantisation of gauge theories . . . . . . .
9.5 The Lagrange density for the fermion sector
9.6 Feynman rules for QED and QCD . . . . .
9.6.1 Propagators . . . . . . . . . . . . .
9.6.2 Vertices . . . . . . . . . . . . . . .
9.6.3 List of Feynman rules . . . . . . .
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10 Spontaneously broken gauge theories
112
10.1 The Higgs boson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
10.2 Yukawa couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
10.3 Feynman rules in the electroweak sector . . . . . . . . . . . . . . . . . . . . . . 117
11 Flavour mixing
118
3
12 Summary: The Standard Model
121
4
1 Overview
1.1 Literature
There is no shortage of text books on quantum field theory. I will list a few of them here:
- M. Peskin und D. Schroeder, An Introduction to Quantum Field Theory, Perseus Books, 1995.
- M. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press,
2014.
- M. Srednicki, Quantum Field Theory, Cambridge University Press, 2007.
- D. Bailin und A. Love, Introduction to Gauge Field Theory, A. Hilger, 1986.
- T. Muta, Foundations of Quantum Chromodynamics, World Scientific, 1987.
- M. Böhm, A. Denner and H. Joos, Gauge Theories of the Strong and Electroweak Interactions,
Teubner, 2001.
- C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, 1980.
- J.D. Bjorken and S.D. Drell, Relativistic Quantum Mechanics, McGraw-Hill, 1964.
- J.D. Bjorken and S.D. Drell, Relativistic Quantum Fields, McGraw-Hill, 1965.
1.2 Units
It is common practice in quantum field theory and elementary particle physics to use natural
units. Thus, by convention we set
~ = c = 1.
For example, the equation
E 2 − c2~p2 = m2 c4
simplifies to
E 2 −~p2 = m2 .
We will use this convention in these lectures. To facilitate the transition to this convention, we
will still write in the very beginning of this course the quantities ~ and c explicitly, but soon set
them equal to one as we proceed.
Energy is measured in eV:
1 eV = 1.6021764 · 10−19J.
5
The following prefixes are used:
1 keV
1 MeV
1 GeV
1 TeV
=
=
=
=
103 eV,
106 eV,
109 eV,
1012 eV.
Momenta are measured in eV/c = eV, masses are given in eV/c2 = eV.
From the uncertainty relation
∆x · ∆p ≥
~
2
it follows that lengths are given in ~c · eV−1 = eV−1 .
Cross sections are given in barn:
1 barn = 10−28 m2 .
Commonly used prefixes in particle physics are:
1 nbarn = 10−9 barn,
1 pbarn = 10−12 barn,
1 fbarn = 10−15 barn.
Conversion constant:
(~c)2 = 0.389379292 · 109 GeV2 pbarn.
This is the most important conversion constant. Typical experiments are scattering experiments.
The momenta of the incoming particles are usually given in GeV. If all calculations are performed in units of GeV, then the cross section has units GeV−2 . The conversion constant above
converts it to pbarn.
1.3 The fundamental forces
We know four fundamental forces: the strong force, the weak force, the electro-magnetic force
and the gravitational force. Particle physics deals with the strong, the weak and the electromagnetic force. The gravitational force is negligible against the other three forces at present
energy scales.
The standard model is based on a local gauge theory with gauge group
SU (3) × SU (2) ×U (1)
SU (3) corresponds to the strong interactions, SU (2) to the weak isospin, U (1) to the hypercharge. The symmetry of the subgroup SU (2) × U (1) is spontaneously broken down to the
familiar U (1)el−magn symmetry of electro-magnetic interactions.
6
1.4 The elementary particles
The spin of the particles:
- Fermions have half-integer spin. In the standard model all fermions have spin 1/2. In extensions of the standard model higher spins may occur, e.g. the gravitino with spin 3/2.
- Bosons have integer spin. In the standard model all bosons have either spin 1 (gauge bosons)
or spin 0 (Higgs boson). In extensions of the standard model there might be particles of
higher spin, e.g. a graviton of spin 2.
1.4.1 Spin 1/2 particles
Quarks: Quarks feel the strong, the weak and the electro-magnetic forces. There are six quarks:
up, Qu =
2
3
charm, Qc =
2
3
top, Qt =
2
3
mu < 10 MeV
down, Qd = − 13
mc = 1.15 − 1.35 GeV
strange, Qs = − 31
mt = 174 ± 5 GeV
bottom, Qb = − 13
md < 10 MeV
ms = 80 − 130 MeV
= 4.1 − 4.4 GeV
mMS
b
1S
mb = 4.6 − 4.9 GeV
The different quark types (up, down, strange, charm, bottom, top) are called “flavours”.
Leptons: Leptons do not feel the strong interaction. There are six leptons:
ν e , Qνe = 0
ν µ , Qνµ = 0
ν τ , Qντ = 0
mνe < 3 eV
e, Qe = −1
mνµ < 0.19 MeV
µ, Qµ = −1
mντ < 18.2 MeV
τ, Qτ = −1
me = 511 keV
mµ = 105.7 MeV mτ = 1.78 GeV
Neutrinos are electrically neutral and interact only through the weak force.
The family structure of the standard
lies:


u
 d 


 νe  ,
e
model: The fermions can be grouped into three fami 
t
c
 s   b
 

 νµ  ,  ντ
τ
µ

The families differ only by the masses of their members.
7


.

1.4.2 Spin 1 particles
Within the standard model the mediators of the interactions are spin 1 particles.
The strong interaction: SU(3): The gauge group SU (3) describes the strong interaction. The
number of generators for a group SU (N) is N 2 − 1, therefore there are 8 generators for SU (3),
and hence 8 gauge bosons for the strong interactions. The gauge bosons of the strong interaction
are called gluons. The fields are denoted by
Aaµ ,
where a runs from 1 to 8.
The weak isospin: SU(2): The weak interaction is described by the gauge group SU (2). There
are three generators
Wµ1 ,Wµ2 ,Wµ3 ,
each with two polarisation states. After electro-weak symmetry breaking we use the fields
Wµ+ ,Wµ− , Zµ ,
with three polarisation states. We have the following relations:
1
Wµ± = √ Wµ1 ∓ iWµ2 ,
2
Zµ = − sin θW Bµ + cos θwWµ3 .
The third spin degree of freedom comes from the Higgs mechanism.
The hypercharge: U(1): The last piece of gauge-symmetries within the standard model is given
by an abelian U (1) gauge symmetry, the hypercharge. The field is denoted by
Bµ
After electro-weak symmetry breaking the photon field Aµ is given as a linear combination of Bµ
and Wµ3 :
Aµ = cos θW Bµ + sin θwWµ3 .
Note that
Aµ
Zµ
=
cos θW sin θW
− sin θW cos θW
and that Aµ remains a massless field with two polarisation states.
8
Bµ
Wµ3
Quantum numbers of the fermions in the electro-weak sector: The left-handed components
(uL , dL ) and (νL , eL ) transform as the fundamental representation under the SU (2) group. The
right-handed components uR , dR , νR and eR transform as a singlet under the SU (2) group.
In detail one has, where I3 denotes the third component of the weak isospin, Y the hypercharge
and Q the electric charge:
I3
Y
Q
uL
1
2
1
3
2
3
dL
− 12
1
3
νL
1
2
eL
− 12
I3
Y
Q
uR
0
4
3
2
3
− 31
dR
0 − 23
− 13
−1
0
νR
0
0
0
−1
−1
eR
0
−2
−1
The electric charge is given by the Gell-Mann-Nishijima formula:
Q = I3 +
Y
2
Remark: The table contains a right-handed neutrino, which does not interact with any other
particle.
1.4.3 Spin 0 particles
The Higgs boson: Within the standard model there is a complex scalar field, transforming as the
fundamental representation of SU (2). This field is conventionally parametrised as follows:
!
φ+ (x)
.
φ(x) =
√1 (v + H(x) + iχ(x))
2
φ+ (x) is a complex field (two real components). The three components φ+ (x) and χ(x) are
absorbed as the longitudinal modes of Wµ± and Zµ . H(x) is the Higgs field.
1.5 Experiments
The first experiments were fixed-target experiments (deep inelastic scattering of electrons on
proton targets).
Accelarators:
- LEP, e+ e− , 210 GeV, L = 1032 cm−2 s−1 ;
9
- TEVATRON, p p̄, 1.96 TeV, L = 5 · 1030 cm−2 s−1 ;
- HERA, e− p, e− : 30 GeV, p : 960 GeV, L = 75 · 1030 cm−2 s−1 ;
- LHC, pp, 14 TeV, L = 1034 cm−2 s−1 ;
- Linear Collider (planned), e+ e− , 500 GeV, L = 5 · 1034 cm−2 s−1 ;
Quarks and gluons are not directly observed in these experiments. Instead one observes hadronic
jets. A jet is a bunch of particles moving in the same direction. Particles in a jet are not necessarily elementary.
1.6 Observed, but not elementary particles
Due to confinement, quarks and gluons cannot be observed as free particles. In experiments
we observe particles which are colour-singlets like mesons and baryons. Within the quark
model, mesons are qq̄-states and baryons qqq-states. Mesons and baryons are called collectively hadrons.
Examples are:
Mesons: Pions, kaons, η’s, D-mesons, J/ψ, ...
Baryons: protons, neutrons, Σ, Ξ, ...
10
2 Review of quantum mechanics
2.1 The harmonic oscillator in classical mechanics
Let us start with classical mechanics and recall the Lagrange and Hamilton description of the
non-relativistic harmonic oscillator in classical mechanics. The Lagrange function for the harmonic oscillator reads
L =
1 2 1
mẋ − mω2 x2 .
2
2
From the Euler-Lagrange equation
d δL δL
−
= 0
dt δẋ δx
follows the equation of motion
ẍ + ω2 x2 = 0,
which has the solution
x(t) = Aeiωt + Be−iωt .
Remark: The conjugate momentum is given by
p =
δL
= mẋ,
δẋ
and the Hamilton function reads
1
1
p2 1 2 2
H = pẋ − L = mẋ2 + mω2 x2 =
+ mω x .
2
2
2m 2
2.2 The harmonic oscillator in quantum mechanics
In quantum mechanics the harmonic oscillator is described by a wave function ψ(x,t). This wave
function can be expanded into an orthonormal basis. We denote by |x;t0iH a wave function in the
Heisenberg picture, which is at the time t = t0 an eigenvector of the Heisenberg position operator
x̂H (t) with eigenvalue x:
x̂H (t0) |x;t0 iH = x |x;t0iH .
In general, the position operator does not commute with the Hamilton operator, therefore for
t 6= t0 the state will in general not be an eigenstate of x̂H (t). Furthermore we remark that a state
in the Heisenberg picture is time-independent. The label t0 refers to the time, where the state
|x;t0 i is an eigenvector of the time-dependent position operator x̂H (t). The relation between the
states in the Heisenberg picture and in the Schrödinger picture is
|x;t0 iH = eiĤt |x,t;t0iS .
11
At time t = t0 the Schrödinger state |x,t0 ;t0i is an eigenstate of the Schrödinger position operator
x̂S :
x̂S |x,t0 ;t0iS = x |x,t0 ;t0iS .
We are interested in the transition amplitude
H hx f ;t f |xi ;ti iH ,
which gives us the probability that the system which was in the eigenstate |xi ;ti iH at time ti will
be found in the state |x f ;t f iH at time t f . We discuss the transition amplitude for pedagocical purposes: On the one hand, the transition amplitude can be worked out in non-relativistic quantum
mechanics. On the other hand, the transition amplitude is already close to objects (scattering
amplitudes), which we will study in quantum field theory. We will discuss two methods for
the calculation of the transition amplitude: one method based on operators, the other on path
integrals. Both methods have a generalisation to quantum field theory.
2.2.1 Operator formalism
Let us first compute the transition amplitude with the operator formalism. This is the standard
method treated in most textbooks of quantum mechanics. Here, we present only the main formulae, but do not give any derivations.
The time evolution of the wave function in the Schrödinger picture is given by the Schrödinger
equation
i~
∂
ψ(x,t) = Ĥψ(x,t),
∂t
where the Hamilton operator is given by
Ĥ =
p̂2 1
+ mω2 x̂2 .
2m 2
We make the ansatz
ψ(x,t) = Û(t,ti)|x,ti;ti iS
The evolution operator satisfies the equation
∂
i~ Û(t,ti) = ĤÛ(t,ti).
∂t
If the Hamilton operator is time-independent (as it is for the harmonic oscillator), the solution
for Û(t,ti) is given by
i
Û (t,ti) = exp − (t − ti ) · Ĥ .
~
12
Remark: If the Hamilton operator Ĥ depends on the time t, a formal solution for Û(t,ti) is given
by


Zt
i
Û(t,ti) = T exp −
dt ′Ĥ(t ′ ) .
~
ti
Here, T denotes the time-ordering operator, which orders operators from right to left in nondecreasing time. Expanding the exponential one obtains
i
Û(t,ti) = 1 −
~
Zt
ti
2 Zt
Zt1
i
dt1 Ĥ(t1) +
dt1 Ĥ(t1) dt2Ĥ(t2 )
~
ti
ti
3 Zt
Zt1
Zt2
i
−
dt1 Ĥ(t1) dt2Ĥ(t2 ) dt3 Ĥ(t3) + ....
~
ti
ti
ti
Note that the factor 1/n! disappears.
To determine Û(t,ti)|x,ti;ti iS we expand |x,ti ;ti iS into eigenstates of the Hamilton operator.
These eigenstates will be labelled |ni and we have
Ĥ|ni = En |ni.
Therefore
Û(t,ti)|x,ti;ti iS
i
i
= exp − (t − ti ) · Ĥ |x,ti ;ti iS = ∑ e− ~ (t−ti )En |nihn|x,ti;ti iS .
~
n
To find these eigenstates we define two operators
ωmx̂ − i p̂
â† = √
.
2ωm~
ωmx̂ + i p̂
,
â = √
2ωm~
If we introduce the characteristic length
x0 =
r
~
,
ωm
we can equally write them as
1
â = √
2
x
d
+ x0
,
x0
dx
1
â = √
2
†
x
d
− x0
.
x0
dx
â is called lowering operator or annihilation operator, â† is called raising operator or creation
operator. From
h d
[x̂, p̂] = x̂,
= i~
i dx
13
it follows that
i
h
â, â† = 1.
The Hamilton operator can be rewritten as
1
1 †
†
†
.
~ω â â + ââ = ~ω â â +
Ĥ =
2
2
We call
N̂ = â† â
the number operator and the problem of finding the energy eigenstates is reduced to the problem
of finding the eigenstates of the number operator. We have
nhn|ni = hn|N̂|ni = hn|â† âni = hân|âni ≥ 0.
Therefore n ≥ 0 and the lowest energy state corresponds to n = 0. Since the norm of â|0i vanishes, we have
â|0i = 0,
d
x
+ 2 |0i = 0.
dx x0
A solution is given by
√
|0i =
πx0
One easily shows that
− 12
1
exp −
2
x
x0
2 !
.
• â† |ni is an eigenstate with eigenvalue n + 1.
• â|ni is an eigenstate with eigenvalue n − 1.
Therefore one finds
√ − 1
1 † n
1
|ni = √
|0i = 2n n! πx0 2 exp −
â
2
n!
x
x0
2 !
Hn
where Hn (t) are the Hermite polynomials.
The corresponding energies are given by
1
.
= ~ω n +
2
En
Finally, we get
H hx f ;t f |xi ;ti iH
=
i
∑ S hx f ,t f ;t f |nie− ~ (t f −ti)En hn|x,ti;tiiS.
n
14
x
,
x0
2.2.2 Path integrals
An alternative approach to determine the transition amplitude H hx f ;t f |xi ;ti iH divides the time
interval (t f − ti ) into n + 1 small sub-intervals with time steps at
ti ,t1 ,t2, ...,tn,t f .
At each intermediate time step we insert a complete set of states
Z∞
−∞
dx |x;t j iH H hx;t j | = 1.
Therefore
H hx f ;t f |xi ;ti iH
=
Z∞
−∞
dxn ...
Z∞
−∞
dx1 H hx f ;t f |xn ;tn iH H hxn ;tn |xn−1 ;tn−1 iH ... H hx1 ;t1 |xi ;tiiH .
Let us study H hx j+1 ;t j+1|x j ;t j iH . If the time interval (t j+1 − t j ) is small, we have
H hx j+1 ;t j+1 |x j ;t j iH
=
− i (t j+1 −t j )Ĥ
|x j ,t j ;t j iS
S hx j+1 ,t j+1 ;t j+1 |e ~
i
≈ hx j+1 |1 − (t j+1 − t j )Ĥ|x j i,
~
i
= hx j+1 |e− ~ (t j+1 −t j )Ĥ |x j i
where we denoted the eigenfunctions of x̂S simply by |xi. We have
1
hx j+1 |x j i = δ(x j+1 − x j ) =
~
Z∞
−∞
dpj
i
exp
p j x j+1 − x j .
2π
~
Here we used the integral representation of the Dirac delta distribution:
δ(x − y) =
Z∞
−∞
d p ip(x−y)
e
.
2π
We then obtain
i
hx j+1 |1 − (t j+1 − t j )Ĥ|x j i
~
+
*
~2 d 2
i
1
2 2
= hx j+1 |x j i − (t j+1 − t j ) x j+1 −
+
mω
x
jxj
2m dx2j 2
~
!!
Z∞
p2j 1
dpj
i
i
1
exp
p j x j+1 − x j
+ mω2 x2j
1 − (t j+1 − t j )
=
~
2π
~
~
2m 2
−∞
15
1
=
~
Z∞
1
≈
~
−∞
Z∞
1
≈
~
−∞
Z∞
−∞
dpj
i
i
exp
p j x j+1 − x j
1 − (t j+1 − t j )H(x j , p j )
2π
~
~
i
dpj
i
exp
p j x j+1 − x j − (t j+1 − t j )H(x j , p j )
2π
~
~
dpj
i
exp
(t j+1 − t j ) p j ẋ j − H(x j , p j ) .
2π
~
Note that H(x j , p j ) denotes the Hamilton function, not the Hamilton operator. (The Hamilton
operator is denoted by Ĥ.) Let us set
∆T = i(t j+1 − t j ).
With the help of
Z∞
−∞
w2
1
dy − 1 αy2 +wy
e 2
e 2α
= √
2π
2πα
we may perform the integration over p j :
Z∞
−∞
dpj
exp
2π
∆T
~
p j ẋ j −
p2j
2m
!!
=
r
∆T 1 2
~m
exp
mẋ .
2π∆T
~ 2 j
Therefore
r
m
∆T 1 2 1
2 2
exp
mẋ − mω x j
2π~∆T
~ 2 j 2
r
m
∆T
=
exp
L(x j , ẋ j ) .
2π~∆T
~
i
hx j+1 |1 − (t j+1 − t j )Ĥ|x j i ≈
~
Finally we get
H hx f ;t f |xi ;ti iH
=
Z∞
dxn ...
−∞
=
Z∞
−∞
dx1 H hx f ;t f |xn ;tniH H hxn ;tn |xn−1 ;tn−1iH ... H hx1 ;t1 |xi ;ti iH
m n+1
2
2π~∆T
Z∞
dxn ...
−∞
Z∞
−∞
i
(t j+1 − t j )L(x j , ẋ j ) ,
dx1 ∏ exp
~
j=0
n
with t0 = ti . We rewrite this as
H hx f ,t f |xi ,tiiH
∼
Z

i
D x(t) exp 
~
Zt f
ti

dt L(x(t), ẋ(t)) =
16
Z
i
D x(t) exp S .
~
Note the appearance of the Lagrange function L(x(t), ẋ(t)) and the action
S =
Zt f
dt L(x(t), ẋ(t)).
ti
2.2.3 Summary
The quantum mechanical harmonic oscillator shows already several concepts, which will reappear later in quantum field theory. These are:
• Annihilation and creation operators.
• Transition amplitudes can be expressed as path integrals.
• The appearance of the Lagrange function and the action in the path integral.
17
3 Review of special relativity
3.1 Four-vectors and the metric
Four-vectors: The space-time coordinates (ct, x, y, z) are regarded as the components of a vector
in a four-dimensional space.
x0 = ct, x1 = x, x2 = y, x3 = z.
xµ = (x0 , x1 , x2 , x3 ),
= (x0 ,~x).
Greek indices µ, ν, ..., which take the values 0, 1, 2, 3, are used to denote the components of a
four-vector. Latin indices i, j, ... are used to denote the (spatial) components of a three-vector.
They take the values 1, 2, 3.
The distance between two points in four-dimensional space-time is
sab = (x0a − x0b )2 − (x1a − x1b )2 − (x2a − x2b )2 − (x3a − x3b )2 .
s2ab > 0
time-like distance;
there exists a frame, in which events a and b occur in the same place.
s2ab < 0
space-like distance;
there exists a frame, in which events a and b occur at the same time.
s2ab = 0
ligth-like distance;
light cone
Two events can only be related by causality, if the distance between them is ≥ 0. This follows
directly from the finiteness of the speed of light.
We define the metric tensor gµν by
gµν
The distance is then given by

1 0
0
0
 0 −1 0
0 

= 
 0 0 −1 0  .
0 0
0 −1

3
sab =
3
∑ ∑ gµν
µ=0 ν=0
18
µ
xµa − xb (xνa − xνb ) .
Summation convention of Einstein: The symbol of the sum is dropped and it is understood, that
there is an implicit summation over any pair of indices, which occurs twice. Within a pait, one
index has to be an upper index, the other one a lower index. Therefore:
sab = gµν (xa − xb )µ (xa − xb )ν .
We call a four-vector xµ with an upper index a contravariant four-vector, and we call a four-vector
xµ with a lower index a covariant four-vector. The relation between the two is given by
xµ = gµν xν .
Therefore we can write the distance equally as
sab = (xa − xb )µ (xa − xb )µ = (xa − xb )µ (xa − xb )µ .
Remark: The geometry defined by the quadratic form gµν = diag(1, −1, −1, −1) is non-Euclidean.
The special case of a four-dimensional space with metric diag(1, −1, −1, −1) is often called
Minkowski space.
3.2 The Lorentz group
Axioms for a group: Let G be a non-empty set with a binary operation. G is called a group, if it
satisfies the axioms
• Associativity: a · (b · c) = (a · b) · c.
• Existence of a neutral element: e · a = a.
• Existence of an inverse: a−1 · a = e.
Definition of the Lorentz group: Matrix group, which leaves the metric tensor gµν = diag(1, −1, −1, −1)
invariant:
ΛT gΛ = g,
The same equation with indices:
µ
Λ σ gµν Λντ = gστ .
This group is denoted O(1, 3). It is easy to show that
(det Λ)2 = 1,
and therefore
det Λ = ±1.
19
If we have in addition det Λ = 1 the corresponding group is called the “proper” Lorentz group
and denoted SO(1, 3).
One further distinguishes the cases whether the time direction is conserved or reversed. If
Λ00 ≥ 1,
the time direction is conserved and the corresponding group is called the orthochronous Lorentz
group. If on the other hand
Λ00 ≤ −1,
then the time direction is reversed. Remark:
0
Λ ≥ 1
0
µ
follows from Λ σ gµν Λντ = gστ for σ = τ = 0:
Λ00
2
3 j 2
−∑ Λ0
= 1.
j=1
To summarise: The Lorentz group consists of four components, depending on which values the
quantities
det Λ and Λ00
take. The “proper orthochronous Lorentz group” is defined by
µ
Λ σ gµν Λντ = gστ , det Λ = 1, Λ00 ≥ 1,
and contains the identity. Elements of the group correspond to rotations in four-dimensional
Minkowski space. Each rotation can be decomposed into rotations in the planes xy, yz, zx, tx, ty
and tz. A spatial rotation in the xy-plane is given by


1
0
0
0
 0 cos φ − sin φ 0 
µ

Λ ν = 
 0 sin φ cos φ 0  ,
0
0
0
1
and similar for the yz- and zx-planes. A boost in the tx-plane is given by


cosh φ sinh φ 0 0
 sinh φ cosh φ 0 0 
µ
,
Λ ν = 
 0
0
1 0 
0
0
0 1
with
v
c
1
cosh φ = q
= γ,
2
1 − vc2
sinh φ = q
= βγ,
2
1 − vc2
20
where we used the standard abreviations
1
.
γ= q
v2
1 − c2
v
β= ,
c
Elements of the other three components can be obtained from an element of the proper orthochronous Lorentz group and a discrete transformation of time reversal


−1 0 0 0
 0 1 0 0 
µ

Λ ν = 
 0 0 1 0 
0 0 0 1
and / or spatial inversion
µ
Λ
ν

1 0
0
0
 0 −1 0
0 

= 
 0 0 −1 0  .
0 0
0 −1

Tensors: T µ1 µ2 ...µr is called a tensor if it transforms under Lorentz transformations as
T
′ µ µ ...µ
r
1 2
µ1
µ2
µr
ν1 ν2 ...νr
.
ν1 Λ ν2 ...Λ νr T
= Λ
The number r is called the rank of the tensor.
Pseudo-tensors: Pseudo-tensors transform under elements of the proper orthochronous Lorentz
group as tensors. Under the discrete transformations of time reversal and spatial inversion there
is however an additional minus sign. Pseudo-tensors of rank zero are called pseudo-scalars,
pseudo-tensors of rank one are called axial vectors.
Examples:
Rank 1: Positon vector xµ , momentum vector pµ .
Rank 2: Metric tensor gµν .
Rank 4: Total anti-symmetric tensor (Levi-Civita tensor) εµνρσ . The total anti-symmetric tensor
is defined by
ε0123
εµνρσ
εµνρσ
εµνρσ
=
=
=
=
1,
1 if (µ, ν, ρ, σ) is an even permutation of (0, 1, 2, 3),
−1 if (µ, ν, ρ, σ) is an odd permutation of (0, 1, 2, 3),
0 otherwise.
The total anti-symmetric tensor is a pseudo-tensor, its components are unchanged under time
reversal and spatial inversion.
21
3.3 The Poincaré group
The Poincaré group consists of elements of the Lorentz group and translations. The group elements act on four vectors according to the following transformation law :
′
µ
x µ = Λ ν xν + aµ .
Λ describes rotations in four dimensional space-time (e.g. ordinary rotations on the spacial
components plus boosts) whereas a describes translations.
The group multiplication law is given by
{a1 , Λ1 }{a2 , Λ2 } = {a1 + Λ1 a2 , Λ1 Λ2 }.
The generators of the Poincaré group can be realised as differential operators :
Pµ = i∂µ ,
Mµν = i xµ ∂ν − xν ∂µ .
The algebra of the Poincaré group is given by
Mµν , Mρσ = −i gµρ Mνσ − gνρ Mµσ + gµσ Mρν − gνσ Mρµ ,
Mµν , Pσ = i gνσ Pµ − gµσ Pν ,
Pµ , Pν = 0.
The Poincaré algebra is a Lie algebra, but it is not semi-simple, since it has an Abelian non-trivial
ideal (Pµ ).
Casimir operators are M 2 and W 2 where
M 2 = Pµ Pµ ,
1
W µ = εµνρσ Pν Mρσ .
2
W µ is called the Lubanski-Pauli vector.
22
4 Review of classical field theory
4.1 The action principle
From classical mechanics we are familiar with Hamilton’s principle of the least action for systems with a finite number of degrees of freedom. This principle generalises to systems with
infinite many degrees of freedom:
• Let us start with a countable number of degrees of freedom. We will label the coordinates
by qi , where i ∈ Z. As an example we consider a model of an elastic rod, described by
points of mass m, separated at rest by a distance a and connected by massless springs with
spring constant k. The kinetic and potential energies are
1
mq̇2i ,
∑
2 i
1
V =
k (qi+1 − qi )2 .
2∑
i
T =
The Lagrange function is then given by
1
1 L = T −V = ∑ mq̇2i − k (qi+1 − qi )2 = ∑ a
2 i
2 i
!
qi+1 − qi 2
m 2
.
q̇ − ka
a i
a
In the last expression we already arranged the terms in such a way that it will be easy to
take the continuum limit. The action is a usual
Zt2
S [~q(t)] =
t1
˙
dt L ~q(t),~q(t)
,
where ~q(t) is an infinite-dimensional vector. The equations of motion follow as usual from
δS = 0
and lead to the Euler-Lagrange equations
d δL δL
−
= 0,
dt δq̇i δqi
for i ∈ Z.
• Let us now consider a non-countable number of degrees of freedom. Instead of the integer
i we will label them by x. Furthermore it will be convenient to change the notation from qx
to q(x) and from qx (t) to q(t, x). In the example above we take the limit a → 0 and we set
m
= µ,
a→0 a
lim
23
lim ka = Y.
a→0
Furthermore the expression (qi+1 − qi )/a becomes in the continuum limit
∂q(t, x)
q(t, x + a) − q(t, x)
=
.
a→0
a
∂x
lim
We obtain for the Lagrange function
" #
Z∞
1
∂q(t, x) 2
∂q(t, x) 2
L =
−Y
.
dx µ
2
∂t
∂x
−∞
The expression
" #
1
∂q(t, x) 2
∂q(t, x) 2
L =
µ
−Y
2
∂t
∂x
is called the Lagrange density or Lagrangian. The action is given by
S =
Zt2
t1
dt
Z∞
dx L .
−∞
We may view q(t, x) as a classical field in a (1 + 1)-dimensional space-time. Note that
the Lagrange density is a function of the time derivative ∂q/∂t and the spatial derivative
∂q/∂x.
We now consider the generalisation to four-dimensional space-time. We assume that the Lagrange density is a function of the field ψ(x) and its first derivative ∂µ ψ(x):
L ψ(x), ∂µ ψ(x) .
The motivation is as follows: Classical mechanics suggests to include a dependence on ψ(x) and
the time derivative ∂0 ψ(x). Lorentz invariance instructs us not to single out a specific direction
in space-time. We therefore allow a dependence on all first derivatives ∂µ ψ(x). In principle we
could include also second or higher derivatives, however for our purpose it will be sufficient to
restrict us to first derivatives. The action is given by
S =
Zt2
dt
Z
d3x L =
t1
From the principle of least action
δS = 0,
24
1
c
Z
d 4x L .
we can derive the Euler-Lagrange equations as in classical mechanics, if we assume that the
variation of the field vanishes on two hyper-surfaces t = t1 and t = t2 :
Zt2
Z
#
∂
L
∂
L
δ ∂µ ψ(x)
δψ +
dt d 3 x
δS =
∂ψ
∂ ∂µ ψ(x)
t1
"
#
Zt2 Z
∂
L
∂
L
∂µ (δψ(x))
dt d 3 x
=
δψ +
∂ψ
∂ ∂µ ψ(x)
t1
#
"
Zt2 Z
∂L
∂L
δψ.
− ∂µ
=
dt d 3 x
∂ψ
∂ ∂µ ψ(x)
"
t1
Therefore
∂L
∂L
= 0.
− ∂µ
∂ψ
∂ ∂µ ψ(x)
This equation is also called the field equation.
4.2 Examples of classical fields
In the following we will review the most important cases of fields: These are the Klein-Gordon
field, describing particles of spin 0, the Dirac field, describing particles of spin 1/2 and the
Maxwell field, describing particles of spin 1. We review the properties, when these fields are
treated as classical fields, i.e. fields which satisfy a specific partial differential equation, called
field equation. In the case of spin 0, the field equation is the Klein-Gordon equation, in the case
of spin 1/2 the field equation is the Dirac equation and in the case of spin 1 the field equations
are the (inhomogeneous) Maxwell equations.
4.2.1 The Klein-Gordon field
The Lagrange density for a real scalar field:
L (φ, ∂µ φ) =
1
1
∂µ φ (∂µ φ) − m2 φ2 .
2
2
The corresponding Euler-Lagrange equation yields the Klein-Gordon equation:
✷ + m2 φ = 0.
The Lagrange density for a complex scalar field reads
L (φ, φ∗ , ∂µ φ, ∂µ φ∗ ) =
25
∂µ φ∗ (∂µ φ) − m2 φ∗ φ.
In the complex case we treat φ and φ∗ as two independent fields. Variation with respect to φ∗
yields
✷ + m2 φ = 0,
variation with respect to φ gives
✷ + m2 φ∗ = 0.
4.2.2 The Dirac field
Although Dirac spinors are associated with the quantum mechanical description of spin, we may
ignore this fact for the moment and simply take a Dirac spinor as a four-component vector with
complex entries. A Dirac field associates to every space-time point x a four-component spinor
ψα (x), where α takes the values α ∈ {1, 2, 3, 4}. Note that α is not a Lorentz index, it is a Dirac
index. (We use greek letters from the beginning of the greek alphabet for spinor indices and
greek letters from the middle of the greek alphabet for Lorentz indices.)
Let us define the Dirac matrices. These are (4×4)-matrices, which satisfy the anti-commutation
rules
{γµ , γν } = 2gµν 1.
(The anti-commutator of two matrices is {A, B} = AB + BA.) In addition, there is a fifth matrix
γ5 , defined by
γ5 = iγ0 γ1 γ2 γ3 =
i
εµνρσ γµ γν γρ γσ ,
24
which satisfies
{γµ , γ5 } = 0.
There are various representations for the Dirac matrices, a convenient choice is the Weyl representation. In order to define the Weyl representation we first recall the Pauli matrices. These are
2 × 2-matrices, given by
0 1
0 −i
1 0
σx =
, σy =
, σz =
.
1 0
i 0
0 −1
We write ~σ = (σx , σy , σz ). Next, we define the 4-dimensional σµ -matrices (and σ̄µ -matrices):
µ
σ̄µȦB = (1,~σ) .
σAḂ = (1, −~σ) ,
Here 1 denotes the 2 × 2-identity matrix. There are four 2 × 2-matrices σ0AḂ , σ1AḂ , σ2AḂ and σ3AḂ .
The indices A and Ḃ take values A, Ḃ ∈ {1, 2}. Analog statements hold for σ̄µȦB . We are now in
a position to give the Weyl representation for the Dirac matrices:
0 σµ
1 0
µ
0 1 2 3
γ =
.
,
γ5 = iγ γ γ γ =
0 −1
σ̄µ 0
26
Each entry is a 2 × 2-matrix.
The Lagrange density for the Dirac field depends on four-component spinors ψα (x) and the
Dirac adjoint ψ̄α (x) = ψ† (x)γ0 α :
L (ψ, ψ̄, ∂µ ψ) = iψ̄γµ ∂µ ψ − mψ̄ψ.
The Euler-Lagrange equations yield the Dirac equations
iγµ ∂µ − m ψ = 0,
←
ψ̄ iγµ ∂ µ +m = 0.
The arrow on top of the derivative indicates that the derivative acts to the left.
4.2.3 The Maxwell field
As our final example we consider the electromagnetic field, described by the gauge potential
Aµ (x). The Lagrange density is given by
1
4
L (Aµ, ∂µ Aν) = − Fµν F µν ,
where
Fµν = ∂µ Aν − ∂ν Aµ .
Half of Maxwell´s equations follow from the Euler-Lagrange equations
This yields
δL
δL
= 0.
− ∂ν
δAµ
δ ∂ν Aµ
or equivalently
∂µ F µν = 0,
✷Aµ − ∂µ ∂ν Aν = 0.
Remark: The first set of Maxwell’s equations
∂λ Fµν + ∂µ Fνλ + ∂ν Fλµ = 0
is fulfilled identically with Fµν = ∂µ Aν − ∂ν Aµ .
Gauge invariance: If Aµ = (φ, ~A), then the electric and magnetic fields are given by
~B = ~∇ × ~A,
~E = −~∇φ − ∂ ~A.
∂t
27
Therefore, the potential Aµ determines the electric and magnetic fields. We may ask the reverse
question: Given ~E and ~B, does this define uniquely the potential Aµ ? This is not the case. We
may add to Aµ the divergence of an arbitrary function:
Aµ (x) → Aµ (x) + ∂µ Λ(x).
Since
∂µ ∂ν Λ − ∂ν ∂µ Λ = 0,
this leaves the field stregth
F µν

0
Ex
Ey
Ez
 −E x
0 −Bz
By 

= 
y
z
 −E
B
0 −Bx 
−E z −By
Bx
0

and hence the electric and magnetic fields unchanged. Therefore we may impose additional
conditions on the gauge potential. A common choice is the covariant Lorenz gauge
∂µ Aµ = 0.
A variational problem in the presence of constraints is solved with Lagrange multipliers. This
leads us to the Lagrange density
1
4
L (Aµ, ∂µ Aν ) = − Fµν F µν −
and the equation of motion
2
1
∂µ Aµ
2ξ
1 µ
✷A − 1 −
∂ ∂ν Aν = 0,
ξ
1
∂µ ∂ν Aν = 0.
✷gµν − 1 −
ξ
µ
28
5 Quantum field theory: The canonical formalism
There are several approaches to quantum field theory. In this course we will focus on two of
them: The canonical formalism and the path integral formalism. We start with the canonical
formalism.
Within the canonical approach a system is described by a state vector and operators acting
on the state vectors. An important operator will be the Hamilton operator (or Hamiltonian).
The Hamilton operator can be obtained by analogy with classical field theory. In classical field
theory we have a Hamilton density, consisting of the fields and derivatives of the fields. Promoting the fields to operators gives us the Hamilton operator of quantum field theory.
Within the Schrödinger picture, the time evolution of a state is governed by the Hamilton
operator. We will also consider the Heisenberg picture, where the time-dependence is carried
by the operators, while the states are time-independent. Finally, a third picture will be useful for
practical calculations: the interaction picture.
Remark: Classical physics is concerned with classical point-like particles and classical fields.
In quantum mechanics we describe particles by a wave function. This is often called “first quantisation”. If fields are present, they are treated classically. We would like to treat (matter) particles and (force) fields, which also have a particle nature, on equal footing. This is achieved
in quantum field theory, where fields are described by operators. This is often called “second
quantisation”. However we should add a warning here: The expression “second quantisation”
might give wrong allusions. Also in quantum field theory we quantise only once. In fact we will
soon see that we may treat a quantum field as a collection of infinite many harmonic oscillators.
Each harmonic oscillator is quantised as in quantum mechanics.
5.1 The Klein-Gordon field as harmonic oscillators
We start with the simplest example, a real scalar field as a classical field with Lagrange density
1
1
∂µ φ (∂µ φ) − m2 φ2 .
2
2
L (φ, ∂µ φ) =
We define the canonical conjugated momentum field by
π(x) =
∂L
.
∂φ̇(x)
For the Klein-Gordon field we find
π(x) = φ̇(x).
The Hamilton function is given by
H =
Z
3
d x π(x)φ̇(x) − L =
Z
29
1 2 1 ~ 2 1 2 2
d x π +
∇φ + m φ .
2
2
2
3
We write
H =
Z
d3x H ,
H = π(x)φ̇(x) − L =
1 2 1 ~ 2 1 2 2
∇φ + m φ .
π +
2
2
2
H is called the Hamiltonian. The energy-momentum tensor is given by
T µν =
∂L ν
1
1
∂ φ − gµν L = (∂µ φ)(∂ν φ) − gµν ∂ρ φ (∂ρ φ) + gµν m2 φ2
∂(∂µ φ)
2
2
If the Lagrange density does not depend explicitly on x, i.e. the x-dependence is only through
φ(x), then Noether’s theorem implies
∂µ T µν = 0.
The four “conserved charges” are
H =
and
Pi =
Z
Z
d 3 x T 00 =
d 3 x T 0i = −
Z
d 3x H
Z
d 3 x π∂i φ.
The quantity
−π~∇φ
is called the momentum density and
~P = −
Z
d 3 x π~∇φ
the total momentum of the classical field.
Please note the difference between the canonical conjugated momentum field π(x) and the momentum density [−π(x)~∇φ(x)]. These are not equal. This can already be seen by the fact that the
former is a scalar quantity, wheras the latter is vector-valued.
Let us write the classical Klein-Gordon field as a Fourier integral with respect to ~x:
φ(t,~x) =
Z
d 3 p i~p·~x
e φ(t,~p).
(2π)3
Then the Klein-Gordon equation becomes
2 ∂
2
2
+ |~p| + m
φ(t,~p) = 0.
∂t 2
30
This is the same as the equation of a harmonic oscillator with frequency
q
|~p|2 + m2 .
ω~p =
For a harmonic oscillator we know how to make the transition from the classical picture to the
quantum world. The Hamilton operator for a quantum-mechanical harmonic osciallator with this
frequency is given by
Ĥ =
1 2 1 2 2
π̂ + ω φ̂ .
2 ~p 2 ~p ~p
The corresponding creation and annihilation operators are
r
r
ω~p
ω~p
i
i
†
π̂~p ,
â~p =
π̂~p .
φ̂~p + p
φ̂~p − p
â~p =
2
2
2ω~p
2ω~p
We may solve these equations for φ̂~p and π̂~p and obtain
1 †
p
â~p + â~p ,
φ̂~p =
2ω~p
π̂~p = −i
r
ω~p †
â~p − â~p .
2
Note that the operators φ̂~p and π̂~p correspond to a single momentum mode ~p.
5.2 The Schrödinger picture
We are now in a position to present a quantum field as an operator. The basic idea is to take a
linear superposition of momentum modes, where each individual momentum mode behaves like
a quantum mechanical harmonic oscillator.
We start with the Schrödinger picture, where the operators do not depend on time. In the
Schrödinger picture we denote operators by Ô(~x), or ÔS (~x), if we want to emphasize that an
operator refers to the Schrödinger picture. In the Schrödinger picture, operators may depend on
the spatial coordinates ~x, but not on the time coordinate t.
In the Schrödinger picture the time dependence is carried by the states. States are denoted by
|X i. If we want to emphasize that the states carry the time-dependence we will write |X ,ti. If we
further want to emphasize that a state refers to the Schrödinger picture we will put a subscript S
and write |X ,tiS.
We start from
q̂i , p̂ j = iδi j ,
q̂i , q̂ j = p̂i , p̂ j = 0,
which for a continous system becomes
φ̂(~x), π̂(~y) = iδ3 (~x −~y),
φ̂(~x), φ̂(~y) = [π̂(~x), π̂(~y)] = 0.
31
These are called the canonical commutation relations. We write a Fourier representation for
the fields:
Z
1 i~p·~x
d3 p
† −i~p·~x
p
â~p e + â~p e
,
φ̂(~x) =
(2π)3 2ω~p
r Z
ω~p
d3 p
† −i~p·~x
i~p·~x
â
e
−
â
e
.
(−i)
π̂(~x) =
~
p
~p
(2π)3
2
Note that we are considering a real field. Therefore we would like to have that φ̂(~x) is self-adjoint:
†
φ̂(~x)
= φ̂(~x).
In addition we would like to have
†
= â.
â†
These two conditions explain the sign in the exponent e−i~p·~x in the term proportional to â~†p in the
Fourier representation of φ̂(~x). The inverse formulae read
!
r
Z
ω
i
~
p
â~p =
d 3x
φ̂(~x) + p
π̂(~x) e−i~p·~x ,
2
2ω~p
!
r
Z
ω
i
~
p
φ̂(~x) − p
â~†p =
d 3x
π̂(~x) ei~p·~x .
2
2ω~p
The commutation relation becomes
i
h
†
â~p , â~q = (2π)3δ3 (~p −~q).
In detail:
i
h
†
â~p , â~q =
#
!
!
r
Z
ω~p
ω~q
i
i
−i~p·~x
3
i~q·~y
φ̂(~x) + p
φ̂(~y) − p
π̂(~x) e
, d y
π̂(~y) e
d x
2
2
2ω~p
2ω~q
!
s
s
Z
ω
ω
i
i
~q
~p
d 3 x d 3 ye−i~p·~x ei~q·~y −
π̂(~x), φ̂(~y)
φ̂(~x), π̂(~y) +
2 ω~q
2 ω~p
s
s !
Z
ω~p 1 ω~q
3
3 −i~p·~x i~q·~y 1
+
δ3 (~x −~y)
d x d ye
e
2 ω~q 2 ω~p
s
s !Z
ω~p
ω~q
+
d 3 xe−i(~p−~q)·~x
ω~q
ω~p
s !
s
ω~q
ω~p
+
(2π)3δ3 (~p −~q) = (2π)3 δ3 (~p −~q).
ω~q
ω~p
"Z
=
Z
=
Z
=
1
2
=
1
2
3
r
32
The remaining commutation relations for the creation and annihilation operators are
i
h
â~p , â~q = â~†p , â~†q = 0.
Let us summarise: The equivalence of the canonical commutation relations
φ̂(~x), π̂(~y) = iδ3 (~x −~y),
φ̂(~x), φ̂(~y) = [π̂(~x), π̂(~y)] = 0
in momentum space are the relations
i
h
†
â~p , â~q = (2π)3 δ3 (~p −~q),
i
h
† †
â~p , â~q = â~p , â~q = 0.
The Hamiltonian becomes
Z
Z
i
1h
d3 p
1 2 1 ~ 2 1 2 2
†
†
3
∇φ̂ + m φ̂ =
ω~p â~p â~p + â~p , â~p
Ĥ =
d x π̂ +
2
2
2
(2π)3
2
The second term is proportional to δ3 (~0) and gives an infinite constant. Such a term can be
expected: A single harmonic oscillator has the ground state energy 12 ω, summing over an infinite
number of harmonic oscillators yields an infinite ground state energy. As experiments can only
measure energy differences from the ground state, we will ignore this term. Expressing the
Hamiltonian in terms of annihilation and creation operators we will encounter at intermediate
stages expressions proportional to â~p â~p and â~†p â~†p . These expressions are in addition proportional
to ω~2p − |~p|2 − m2 = 0 and vanish therefore.
The commutation relations of the Hamiltonian with the annihilation and creation operators
are
i
h
Ĥ, â~q = −ω~q â~q .
Ĥ, â~q† = ω~q â~†q ,
Let us look at the total momentum operator
~Pˆ = −
Z
d x π̂(~x)~∇φ̂(~x) =
3
Z
i
d3 p
1h
†
†
~p â~p â~p + â~p , â~p .
(2π)3
2
Again, the second term will give a contribution proportional to δ3 (~0), which we will ignore. In
order to show the equality for the last equal sign in the equation above, we will encounter the
following integrals:
Z
d3 p
3
(2π)
~p a~p a−~p and
Z
d3 p
3
(2π)
~p a~†p a†−~p .
The integrands of these integrals are anti-symmetric under ~p → −~p. Therefore these integrals
vanish. With the same reasoning one can argue that ~pδ3 (~0) is anti-symmetric under ~p → −~p and
therefore the infinite constant has to vanish after integration over d 3 p.
33
Let us look at the commutation relations of ~Pˆ with â~†q and â~q . We have
Z
Z
h
i
h
i
d3 p
d3 p †
ˆ â† =
†
† †
†
†
~P,
=
~
p
â
â
â
−
â
â
â
~
p
â
â
,
â
~p ~q
~q
~p ~p ~q
~q ~p ~p
(2π)3
(2π)3 ~p
Z
d3 p
=
~pâ† (2π)3 δ3 (~p −~q) = ~qâ~q† .
(2π)3 ~p
A similar relation (with an additional minus sign) holds for â~q . Thus we have
i
h
i
h
ˆ â† = ~qâ† ,
ˆ â
~P,
~P,
= −~qâ~q .
~q
~q
~q
We may combine the Hamilton operator Ĥ and the three-momentum operator ~Pˆ into a fourmomentum operator P̂µ :
Z
h
i
3p
d
1
ˆ
†
†
µ
µ
P̂ = Ĥ, ~P =
p â~p â~p + â~p , â~p ,
(2π)3
2
p
with pµ = (ω~p ,~p) and ω~p = |~p|2 + m2 . From now on we will write
q
E~p = ω~p = + |~p|2 + m2
Note that the energy is always positive. We have
i
h
µ µ †
P̂ , â~q = −qµ â~q .
P̂ , â~q = qµ â~†q ,
Let us now discuss the states. The ground state is defined as the state, which is annihilated by
all annihilation operators. We denote the ground state by |0i. Thus
â~p |0i = 0 for all â~p.
If we drop the infinite constant above, the ground state has energy E = 0 and momentum ~p = 0.
All other states can be obtained by acting on |0i with creation operators. As a first example let
us consider the state â~†p |0i. We have
i
h
P̂µ â~†q |0i = â~†q P̂µ + P̂µ , â~†q |0i = qµ â~q† |0i .
Thus, â~†p |0i is an eigenstate of P̂µ with momentum ~p and energy E~p =
â~†p |0i is called a one-particle state.
Next, consider the state
â~†p1 â~†p2 |0i.
34
p
|~p|2 + m2 . The state
This state is again an eigenstate of P̂µ with momentum ~p1 +~p2 and energy E~p1 + E~p2 . The state
â~†p1 â~†p2 |0i is called a two-particle state. Further note that
â~†p1 â~†p2 |0i = â~†p2 â~†p1 |0i,
since â~†p1 and â~†p2 commute.
Following this pattern we can construct n-particle states as
â~†p1 â~†p2 ...â~†pn |0i
and therefore the full spectrum of the Hamilton operator.
Summary: The spectrum of the Hamilton operator
Z
1 2 1 ~ 2 1 2 2
3
Ĥ = = d x π̂ +
∇φ̂ + m φ̂
2
2
2
is obtained from the ground state |0i by successively applying creation operators â~†p . All states
are reached this way.
Normalisation: For the ground state we choose the normalisation
h0|0i = 1.
For one-particle state we choose the normalisation
h~p|~qi = 2E~p(2π)3 δ3 (~p −~q) .
Therefore
|~pi =
p
2E~pâ~†p |0i .
Remark: This normalisation is Lorentz invariant. Consider the boost
E ′ = γE − βγp3 ,
p′3 = −βγE + γp3 .
From
δ ( f (x) − f (x0 )) =
1
| f ′ (x0 )|
δ (x − x0 )
it follows that
d p′3
p3 dE
3 ′
′
3 ′
′
= δ (~p −~q )γ 1 − β
δ (~p −~q) = δ (~p −~q ) ·
= δ (~p −~q )γ 1 − β
d p3
d p3
E
′
E
= δ3 (~p′ −~q′ ) .
E
3
3
′
′
35
Remark:
Z
d3 p 1
=
(2π)3 2E~p
Z
d4 p
(2π)δ(p2 − m2 )θ(p0 )
(2π)4
is a Lorentz-invariant 3-momentum integral. Therefore, if f (p) is Lorentz-invariant, so is
Z
d3 p 1
f (p).
(2π)3 2E~p
The normalisation of the n-particle state is
p
|~p1 ,~p2 , ...,~pni =
2E~p1 · 2E~p2 · ... · 2E~pn â~†p1 â~†p2 ...â~†pn |0i .
Since a creation operator commutes with any other creation operator, it does not matter in which
order we write the creation operators. In order to have a manifest symmetric expressions we can
equally well write
p
1
â~†pσ â~†pσ ...â~†pσ |0i ,
2E~p1 · 2E~p2 · ... · 2E~pn
|~p1 ,~p2 , ...,~pni =
∑
n
n! σ∈Sn 1 2
where the sum is over n! permutations of the set {1, 2, ..., n}.
5.3 The Fock space
We have already mentioned, that all states of the system can be reached by succesively applying
creation operators â~†p to the ground state |0i. We can formalise this situation. From quantum
mechanics we are familiar with the concept of a Hilbert space. We recall that a Hilbert space is
a vectorspace, which has an inner product and which is complete. As base field we will always
taks the complex numbers C. Let us denote by V1 the Hilbert space of the one-particle states
V1 =
|~pi | ~p ∈ R3 .
For convenience we denote
V0 = C.
We further set
Vn = Sym (V1 )⊗n ,
i.e. Vn is the symmetrised n-fold tensor product of V1 . Symmetrisation means for example that
|~p1 i ⊗ |~p2 i and |~p2 i ⊗ |~p1 i are identified in V2 and we simply write
|~p1 ,~p2 i = |~p1 i ⊗sym |~p2 i = |~p2 i ⊗sym |~p1 i ∈ V2 .
The Fock space of the system is the direct sum of all Vn , where n ∈ N0 .
V =
∞
M
Vn
n=0
= V0 ⊕V1 ⊕V2 ⊕ ...
The Fock space is a Hilbert space. V0 contains the zero-particle states, V1 the one-particles state,
V2 the two-particle states, etc.. Thus the states of our system form a Fock space.
36
5.4 The Heisenberg picture
We now turn to the Heisenberg picture. In this section we will denote for clarity operators in the
Heisenberg picture by ÔH (x) and operators in the Schrödinger picture by ÔS (~x). In later section
we will drop the subscripts H and S and denote the operators in the Heisenberg picture simply
by Ô (x) and the operators in the Schrödinger picture by Ô (~x). Note that the operators in the
Heisenberg picture depend on the four-vector x, while the operators in the Schrödinger picture
depend only on the spatial vector ~x, therefore it is clear what operator is meant.
In the Heisenberg picture, the operators φ̂H (x) and π̂H (x) are now time-dependent:
φ̂H (x) = eiĤt φ̂S (~x)e−iĤt ,
π̂H (x) = eiĤt π̂S (~x)e−iĤt .
If the Hamilton operator is time-independent we have ĤH = ĤS = Ĥ. Furthermore we have


T exp −i
Zt
0
dt ′ Ĥ  = e−iĤt .
The states in the Heisenberg picture are given by
|X iH = eiĤt |X ,tiS .
At t = 0 the states in the Heisenberg picture and the Schrödinger picture agree: |X iH = |X , 0iS.
Sometimes it is useful to generalise this slightly , such that the states in the Heisenberg picture
and the Schrödinger picture agree at t0 . The transformation formulae in this case read
ÔH (x) = eiĤ(t−t0 ) ÔS (~x)e−iĤ(t−t0 ) ,
|X iH = eiĤ(t−t0 ) |X ,tiS .
Note that the Heisenberg operator ÔH depends in this case on t and t0 (to be concrete: ÔH
depends on the difference t − t0 ) and that the Heisenberg state |X iH is always independent of t:
i
∂
|X iH = 0.
∂t
From the Heisenberg equation of motion
i
we find
∂
ÔH = ÔH , Ĥ ,
∂t
∂
φ̂H (x) = iπ̂H (x),
∂t
∂
2
2
~
i π̂H (x) = i ∇ − m φ̂H (x).
∂t
i
37
Combining these two results yields
∂2
2
2
~
φ̂H (x) = ∇ − m φ̂H (x),
∂t 2
or
✷ + m2 φ̂H (x) = 0.
Thus, the operator φ̂H (x) fulfills the Klein-Gordon equation.
For a better understanding we express φ̂H (x) and π̂H (x) in terms of creation and annihilation
operators. From
i
h
Ĥ, â~†p = E~p â~†p ,
Ĥ, â~p = −E~p â~p,
we have
and therefore
Ĥ â~p = â~p Ĥ − E~p ,
Ĥ â~†p = â~†p Ĥ + E~p ,
Ĥ n â~†p = â~†p Ĥ + E~p
Therefore
n
Ĥ n â~p = â~p Ĥ − E~p
,
eiĤt â~†p e−iĤt = â~†peiE~pt ,
From
.
eiĤt â~p e−iĤt = â~p e−iE~pt .
Z
d3 p
1 i~p·~x
† −i~p·~x
p
â
e
+
â
e
,
~p
~p
(2π)3 2E~p
r Z
E~p
d3 p
† −i~p·~x
i~p·~x
π̂S (~x) =
â
e
−
â
e
(−i)
~
p
~p
(2π)3
2
φ̂S (~x) =
we then obtain
n
φ̂H (x) =
Z
π̂H (x) =
∂
φ̂H (x).
∂t
d3 p
1 −ip·x
† ip·x p
,
â e
+ â~p e
0
(2π)3 2E~p ~p
p =E~p
Remark: â~p and â~†p denote always the time-independent Schrödinger-picture ladder operators.
The time-dependent ladder operators in the Heisenberg picture are
â~†p,H (t) = eiĤt â~†p e−iĤt = â~†peiE~pt ,
â~p,H (t) = eiĤt â~p e−iĤt = â~p e−iE~pt .
The subscript H indicates Heisenberg operators.
Remark 2: The above equation makes the duality between particle and wave interpretations of
the quantum field explicit: On the one hand, φ̂H (x) is written as a Hilbert space operator, which
creates and destroys the particles that are the quanta of the field excitations. On the other hand,
φ̂H (x) is written as a linear combination of plane-wave solutions of the Klein-Gordon equation.
38
5.4.1 Causality
From this section onwards we drop the subscript H for the operators in the Heisenberg picture.
Let us consider in the Heisenberg picture the amplitude for a particle to propagate from y to
x:
D(x − y) = 0 φ̂(x)φ̂(y) 0 .
We express φ̂(x) and φ̂(y) in terms of creation and annihilation operators. Since â~q annihilates
the ground state |0i and â~†p when acting to the left annihilates the bra-vector h0|, only the term
E
D 0 â~p â~†q 0 = (2π)3δ3 (~p −~q)
survives and we obtain
Z
d 3 p 1 −ip·(x−y)
φ̂
e
0 (x)φ̂(y) 0 =
(2π)3 2E~p
Remark: This expression is Lorentz-invariant.
Let us first consider the case where the difference x − y is purely in the time-like direction:
x0 − y0 = t, ~x −~y = 0:
D(x − y)
4π
(2π)3
=
t→∞
∼
Z∞
0
e−imt .
|~p|2
−it
e
d|~p| p
2 |~p|2 + m2
√
|~p|2 +m2
1
= 2
4π
Z∞
m
dE
p
E 2 − m2 e−iEt
Let us now consider the case where x − y is purely spatial: x0 − y0 = 0, ~x −~y =~r:
D(x − y) =
Z
d 3 p 1 i~p·~r
2π
e =
3
(2π) 2E~p
(2π)3
1
=
(2π)2
Z∞
0
Z∞
d|~p|
Z1
d cos θ
−1
0
|~p|2 ei|~p|r − e−i|~p|r
−i
d|~p|
=
2E~p
i|~p|r
2(2π)2r
|~p|2 i|~p|r cos θ
e
2E~p
Z∞
−∞
|~p|ei|~p|r
d|~p| p
|~p|2 + m2
This integral has branch cuts on the imaginary axis starting at ±im. We can deform the contour to
go around the upper branch cut, the quarter circles at infinity will give a vanishing contribution.
With the substitution ρ = −i|~p| we obtain
1
4π2 r
Z∞
m
ρe−ρr
p
dρ
ρ 2 − m2
39
r→∞
∼
e−mr .
We find that the propagation amplitude for space-like distances is exponentially vanishing, but
non-zero. Is this a problem with causality ? No, to discuss causality we should not ask whether
particles can propagate over space-like distances, but whether a measurement performed at one
point can affect a measurement at another point whose separation from the first is space-like. The
simplest thing to measure is the field φ̂(x), so let’s have a look at the commutator [φ̂(x), φ̂(y)], if
this commutator vanishes for space-like distances, one measurement cannot affect another one
separated at a space-like distance.
φ̂(x), φ̂(y)
Z
i
d 3q
1
1 h −ip·x
d3 p
† ip·x
† iq·y
−iq·y
p
p
=
â~p e
+ â~p e
, â~q e
+ â~q e
(2π)3 2E~p (2π)3 2E~q
Z
d 3 p 1 −ip·(x−y)
ip·(x−y)
e
−e
=
(2π)3 2E~p
= D(x − y) − D(y − x).
Z
When (x − y)2 < 0, we can perform a Lorentz-transformation on the second term (since each
term is separately Lorentz-invariant), taking
(x − y) → −(x − y).
The two terms are therefore equal and cancel in the sum. Therefore causality is preserved.
Remark: If (x − y)2 > 0 there is no continuous Lorentz-transformation, which takes (x − y) →
−(x − y).
5.4.2 The Klein-Gordon propagator
Let us study the commutator [φ̂(x), φ̂(y)] a little bit further. Since it is a c-number, we have
φ̂(x), φ̂(y) = 0 φ̂(x), φ̂(y) 0 .
Let us assume that x0 > y0 . Then we have
Z
d 3 p 1 −ip·(x−y)
ip·(x−y)
e
−e
0 φ̂(x), φ̂(y) 0 =
(2π)3 2E~p
(
)
Z
d3 p
1 −ip·(x−y) 1 −ip·(x−y) =
e
+
e
(2π)3 2E~p
−2E~p
p0 =E
p0 =−E
=
Z
Z
d3 p
(2π)3
~p
d p0
2πi
−1
p 2 − m2
~p
e−ip·(x−y) .
In the second term of the second line we changed the integration variables as ~p → −~p. In going
from the second to the third line we applied Cauchy’s theorem in the reverse direction: We first
recognise the sum of the two terms in the second line as a sum of residues. Then we re-write the
sum of residues as a contour integral. The p0 integral is to be performed along the contour
40
Im(p0 )
−E~p
E~p
Re(p0 )
The condition x0 > y0 ensures that the half-circle at infinity in the lower complex plane gives a
vanishing contribution. To keep track of the contour we also write
Z
Z
d3 p
d p0
i
0 φ̂(x), φ̂(y) 0 =
e−ip·(x−y) ,
3
2
0
2
2
(2π)
2π (p + iε) − |~p| − m
where ε is an infinitessimal quantity with ε > 0. We define
Z d 4 p
i
e−ip·(x−y) .
DR (x − y) = θ(x0 − y0 ) 0 φ̂(x), φ̂(y) 0 =
(2π)4 (p0 + iε)2 − |~p|2 − m2
DR (x − y) is called the retarded Green’s function. Similar the advanced Green’s function is
defined by
D h
i E Z d 4 p
i
ˆ
ˆ
e−ip·(x−y) ,
DA (x − y) = −θ(y − x ) 0 φ(x), φ(y) 0 =
2
4
(2π) (p0 − iε) − |~p|2 − m2
0
0
where in the last expression the contour integral is now evaluated with the contour
Im(p0 )
−E~p
E~p
Re(p0 )
There are four possible contours to evaluate the p0 -integration. A third one is of particular
importance:
Im(p0 )
−E~p
E~p
41
Re(p0 )
This corresponds to the integral
DF (x − y) =
Z
i
d4 p
e−ip·(x−y) ,
4
2
(2π) p − m2 + iε
Again, the +iε-prescription ensures that the poles are avoided as shown in the figure above.
If x0 > y0 we can close the contour below and obtain D(x − y). If x0 < y0 we close the contour above and obtain D(y − x). Therefore
D(x − y) for x0 > y0
DF (x − y) =
D(y − x) for x0 < y0
= θ(x0 − y0 ) 0 φ̂(x)φ̂(y) 0 + θ(y0 − x0 ) 0 φ̂(y)φ̂(x) 0
= 0 T φ̂(x)φ̂(y) 0
In the last expression the time-ordering symbol T occurs, which orders operators from right to
left in non-decreasing time. Thus
φ̂(x)φ̂(y) for x0 > y0 ,
T φ̂(x)φ̂(y) =
φ̂(y)φ̂(x) for x0 < y0 .
The quantity DF (x − y) is called the Feynman propagator. Let’s have a look at
Z d4 p
i
e−ip·(x−y)
✷ + m DF (x − y) = ✷ + m
4
2
(2π) p − m2
Z
d4 p
i
=
(−p2 + m2 )e−ip·(x−y)
4
2
2
(2π) p − m
= −iδ4 (x − y).
2
2
Therefore, if we look at the Fourier transform D̃F (p)
DF (x − y) =
Z
d 4 p −ip·(x−y)
e
D̃F (p),
(2π)4
we obtain an algebraic equation for D̃F (p):
(p2 − m2 )D̃F (p) = i.
Summary: The Feynman propagator in momentum space is obtained from an algebraic equation.
The integration contour in the p0 -plane is given by the iε-prescription. For the Klein-Gordon
propagator we have
D̃F (p) =
i
p2 − m2 + iε
42
.
5.5 Wick’s theorem
We already had the definition of the time-ordered product: This product orders operators such
that the time does not decrease from right to left.
φ̂(x)φ̂(y) for x0 > y0 ,
T φ̂(x)φ̂(y) =
φ̂(y)φ̂(x) for y0 > x0 .
In addition we introduce the normal product, which orders operators such that all annihilation
operators are on the right of all creation operators:
: â~†p â~q : = â~†p â~q ,
: â~q â~†p : = â~†p â~q .
We notice that the vacuum expectation value of a normal ordered product is zero (unless the
product is empty):
0 : φ̂(x1 )φ̂(x2 )....φ̂(xn ) : 0 = 0.
We further introduce the so-called “contraction”, which is just the vacuum expectation value of
the time-ordered product of two operators (or equivalently the Feynman propagator DF (x − y)).
ˆ
= 0 T φ̂(x)φ̂(y) 0 = DF (x − y) .
φ̂(x)φ(y)
Wick’s theorem states that
T φ̂(x1 )φ̂(x2 )...φ̂(xn ) = : φ̂(x1 )φ̂(x2 )...φ̂(xn ) : + all possible contractions.
Example:
T φ̂(x1 )φ̂(x2 )φ̂(x3 )φ̂(x4 ) =: φ̂(x1 )φ̂(x2 )φ̂(x3 )φ̂(x4 ) :
+ : φ̂(x1 )φ̂(x2 )φ̂(x3 )φ̂(x4 ) : + : φ̂(x1 )φ̂(x2 )φ̂(x3 )φ̂(x4 ) : + : φ̂(x1 )φ̂(x2 )φ̂(x3 )φ̂(x4 ) :
+ : φ̂(x1 )φ̂(x2 )φ̂(x3 )φ̂(x4 ) : + : φ̂(x1 )φ̂(x2 )φ̂(x3 )φ̂(x4 ) : + : φ̂(x1 )φ̂(x2 )φ̂(x3 )φ̂(x4 ) :
+ : φ̂(x1 )φ̂(x2 )φ̂(x3 )φ̂(x4 ) : + : φ̂(x1 )φ̂(x2 )φ̂(x3 )φ̂(x4 ) : + : φ̂(x1 )φ̂(x2 )φ̂(x3 )φ̂(x4 ) :
Proof: We decompose any operator into positive and negative frequency parts:
φ̂(x) = φ̂+ (x) + φ̂− (x),
Z
Z
d3 p
1
1
d3 p
−ip·x
−
+
p
p
â
e
,
φ̂
(x)
=
â† eip·x .
φ̂ (x) =
~p
3
3
(2π)
(2π)
2E~p
2E~p ~p
43
φ̂+ contains only annihilation operators, φ̂− contains only creation operators. We proof Wick’s
theorem by induction. We start at n = 2 and assume x0 > y0 :
T φ̂(x)φ̂(y) = φ̂(x)φ̂(y)
= φ̂+ (x)φ̂+ (y) + φ̂+ (x)φ̂− (y) + φ̂− (x)φ̂+ (y) + φ̂− (x)φ̂−(y)
= φ̂+ (x)φ̂+ (y) + φ̂− (y)φ̂+ (x) + φ̂− (x)φ̂+ (y) + φ̂− (x)φ̂−(y) + φ̂+ (x), φ̂−(y)
= : φ̂(x)φ̂(y) : + φ̂+ (x), φ̂− (y)
= : φ̂(x)φ̂(y) : + 0 φ̂+ (x), φ̂− (y) 0
= : φ̂(x)φ̂(y) : +DF (x − y)
= : φ̂(x)φ̂(y) : + φ̂(x)φ̂(y) .
The proof for the case n = 2 and x0 < y0 is similar. Not let’s assume that it is valid for n − 1
fields. Again we assume x01 > x02 > ... > x0n .
T φ̂(x1 )φ̂(x2 )...φ̂(xn ) = φ(x1 )φ(x2 )...φ(xn)
= φ̂(x1 ) T φ̂(x2 )...φ̂(xn )
= φ̂(x1 ) : φ̂(x2 )...φ̂(xn ) + all contractions not involving φ̂(x1 ) :
= φ̂+ (x1 ) + φ̂− (x1 ) : φ̂(x2 )...φ̂(xn ) + all contractions not involving φ̂(x1 ) :
We want to move φ̂+ (x1 ) and φ̂− (x1 ) inside the normal product. For φ̂− (x1 ) this is easy: φ̂− (x1 )
contains only creation operators, therefore
φ̂− (x1 ) : φ̂(x2 )...φ̂(xn ) : = : φ̂− (x1 )φ̂(x2 )...φ̂(xn ) :,
and we can just move it in to the left of all other operators. On the other hand we have
: φ̂+ (x1 )φ̂(x2 )...φ̂(xn ) : = : φ̂(x2 )...φ̂(xn )φ̂+ (x1 ) :,
therefore
φ̂+ (x1 ) : φ̂(x2 )...φ̂(xn ) : = : φ̂(x2 )φ̂(x3 )...φ̂(xn ) : φ̂+ (x1 ) + φ̂+ (x1 ), : φ̂(x2 )φ̂(x3 )...φ̂(xn ) :
= : φ̂+ (x1 )φ̂(x2 )φ̂(x3 )...φ̂(xn ) : + : φ̂+ (x1 ), φ̂(x2 ) φ̂(x3 )...φ̂(xn ) :
+ : φ̂(x2 ) φ̂+ (x1 ), φ̂(x3 ) ...φ̂(xn ) : + : φ̂(x2 )φ̂(x3 )... φ̂+ (x1 ), φ̂(xn ) :
= : φ̂+ (x1 )φ̂(x2 )φ̂(x3 )...φ̂(xn ) : + :φ̂+ (x1 )φ̂(x2 )φ̂(x3 )...φ̂(xn ) : + :φ̂+ (x1 )φ̂(x2 )φ̂(x3 )...φ̂(xn ) :
+ ... + :φ̂+ (x1 )φ̂(x2 )φ̂(x3 )...φ̂(xn ) :,
which proves Wick’s theorem. Let us now apply Wick’s theorem to the vacuum expectation
value of
0 T φ̂(x1 )φ̂(x2 )φ̂(x3 )φ̂(x4 ) 0 .
44
By construction, the vacuum expectation value of a non-empty normal product vanishes, therefore
0 T φ̂(x1 )φ̂(x2 )φ̂(x3 )φ̂(x4 ) 0 =
+ * +
* =
0 : φ̂(x1 )φ̂(x2 )φ̂(x3 )φ̂(x4 ) : 0 + 0 : φ̂(x1 )φ̂(x2 )φ̂(x3 )φ̂(x4 ) : 0
+
* + 0 : φ̂(x1 )φ̂(x2 )φ̂(x3 )φ̂(x4 ) : 0
= 0 T φ̂(x1 )φ̂(x2 ) 0 0 T φ̂(x3 )φ̂(x4 ) 0 + 0 T φ̂(x1 )φ̂(x3 ) 0 0 T φ̂(x2 )φ̂(x4 ) 0
+ 0 T φ̂(x1 )φ̂(x4 ) 0 0 T φ̂(x2 )φ̂(x3 ) 0
= DF (x1 − x2 )DF (x3 − x4 ) + DF (x1 − x3 )DF (x2 − x4 ) + DF (x1 − x4 )DF (x2 − x3 ).
Graphically,
1
0 T φ̂(x1 )φ̂(x2 )φ̂(x3 )φ̂(x4 ) 0 =
2
1
2
+
4
1
+
3
4
3
2
.
4
3
5.6 Interacting fields
Up to now we considered “free” fields, e.g. fields without any interactions. For the free KleinGordon field we had the Lagrange density
L0 =
1
1
∂µ φ (∂µ φ) − m2 φ2 .
2
2
In this theory, no interactions and no scattering occurs. Let us start to look at more interesting
theories with interactions:
1
λ
1
∂µ φ (∂µ φ) − m2 φ2 − φ4 ,
L =
2
2
4!
where λ is a dimensionless coupling constant. This theory is often called “phi-fourth” theory and
one of the simplest theories with interactions. Obviously,
λ
4!
L = L0 + Lint , Lint = − φ4 .
The “classical” equation of motion for the φ4 theory is
λ
✷ + m2 φ = − φ 3 ,
3!
which cannot be solved by Fourier analysis as the free Klein-Gordon equation. For λ = 0 we
recover the Klein-Gordon equation. If λ is small we may treat the interacting theory by perturbation theory.
45
Let us now look at the quantum theory. In this chapter we use for clarity in the essential places
the subscripts S, H and I, to denote operators in the Schrödinger picture, the Heisenberg picture
and the interaction picture, respectively. As Lint does not involve any derivatives, the definition
of
π(x) =
∂L
∂φ̇
is unaffected by Lint . With the same reasoning, we still impose in the quantum theory equal-time
commutation relations
φ̂S (~x), π̂S (~y) = iδ3 (~x −~y) ,
φ̂S (~x), φ̂S (~y) = [π̂S (~x), π̂S (~y)] = 0.
We can write the full Hamiltonian as
Ĥ = Ĥ0 + Ĥint ,
Z
2 1 2 2
1 2 1
3
Ĥ0 =
d x π̂ + ∇φ̂ + m φ̂ ,
2
2
2
Z
λ
Ĥint =
d 3 x φ4 .
4!
Recall that as long as Ĥ does not depend explicitly on the time, we have Ĥ = ĤS = ĤH . We start
with the study of the two-point correlation function, or the two-point Green’s function
Ω T φ̂H (x)φ̂H (y) Ω .
|Ωi denotes the ground state of the interacting theory, which is in generally different from the
ground state |0i of the free theory. The interaction Hamiltonian enters in two places: In the
definition of |Ωi and in the definition of the Heisenberg field
φ̂H (x) = eiĤ(t−t0 ) φ̂S (~x)e−iĤ(t−t0 ) .
It is easiest to begin with the Heisenberg field φ̂H (x). At any fixed time t0 we can of course
expand the Schrödinger field as before in terms of creation and annihilation operators:
φ̂S (t0 ,~x) =
Z
d3 p
1 i~p·~x
† −i~p·~x
p
â
e
+
â
e
.
~p
~p
(2π)3 2E~p
For t 6= t0 we have in the Heisenberg picture
φ̂H (t,~x) = eiĤ(t−t0 ) φ̂S (t0,~x)e−iĤ(t−t0 ) .
For λ = 0, Ĥ reduces to Ĥ0 :
φ̂H (t,~x)λ=0 = eiĤ0 (t−t0 ) φ̂S (t0,~x)e−iĤ0 (t−t0 ) .
46
When λ is small, this expression will still give the most important part of the time dependence of
φ̂H (x), and thus it is convenient to give this quantity a name: the field in the interaction picture,
φ̂I (x).
φ̂I (x) = eiĤ0 (t−t0 ) φ̂S (t0,~x)e−iĤ0 (t−t0 ) .
The states in the interaction picture are defined by
|X ,tiI = eiĤ0 (t−t0 ) |X ,tiS .
At t = t0, the states in the interaction picture and in the Schrödinger picture agree: |X ,t0iI =
|X ,t0iS . This is in complete analogy with quantum mechanics, where one has apart from the
Schrödinger and Heisenberg picture also the interaction picture. As in the free theory we find
φ̂I (x) =
Z
1 −ip·x′
d3 p
† ip·x′ p
√
â
e
+
â
e
.
0
~p
~p
p =E~p = |~p|2 +m2 ,x′0 =t−t0 ,~x′ =~x
(2π)3 2E~p
The problem is now to express the full Heisenberg field φ̂H (x) in terms of φ̂I (x). We have
φ̂H (x) = eiĤ(t−t0 ) e−iĤ0 (t−t0 ) φ̂I (x)eiĤ0 (t−t0 ) e−iĤ(t−t0 )
= Û † (t,t0)φ̂I (x)Û(t,t0),
where
Û(t,t0) = eiĤ0 (t−t0 ) e−iĤ(t−t0 ) .
We have
Û(t0 ,t0) = 1
and
∂
i Û (t,t0) = eiĤ0 (t−t0 ) Ĥ − Ĥ0 e−iĤ(t−t0 )
∂t
= eiĤ0 (t−t0 ) Ĥint e−iĤ(t−t0 )
= eiĤ0 (t−t0 ) Ĥint e−iĤ0 (t−t0 ) e|iĤ0 (t−t0 ){z
e−iĤ(t−t0}) .
|
{z
}
Û(t,t0 )
ĤI (t)
ĤI (t) is the interaction Hamiltonian written in the interaction picture:
iĤ0 (t−t0 )
ĤI (t) = e
−iĤ0 (t−t0 )
Ĥint e
=
Z
d 3x
λ 4
φ̂ .
4! I
Therefore
∂
i Û(t,t0) = ĤI (t)Û(t,t0), Û (t0,t0) = 1.
∂t
47
A solution is given by
Û(t,t0) = 1 + (−i)
+ (−i)3
Zt
t0
Zt
2
dt1 ĤI (t1) + (−i)
Zt
dt1
t0
dt1
t0
Zt1
dt2
t0
Zt2
Zt1
dt2 ĤI (t1)ĤI (t2)
t0
dt3 ĤI (t1 )ĤI (t2)ĤI (t3) + ...
t0
This solution can be verified by differentiation. The initial condition Û (t0,t0) = 1 is obviously
satisfied. Note that the various factors of ĤI stand in time order, later on the left. Note that
Zt
t0
dt1
Zt1
t0
1
dt2 ĤI (t1)ĤI (t2) =
2
Zt
dt1
Zt
t0
t0
dt2 T ĤI (t1)ĤI (t2) .
Similar
Zt
t0
dt1
Zt1
dt2 ...
t0
tn−1
Z
Zt
t0
t0
1
dtn ĤI (t1)ĤI (t2 )...ĤI (tn ) =
n!
Zt
dt1
dt2...
t0
Zt
t0
dtn T ĤI (t1)ĤI (t2 )...ĤI (tn ) .
Therefore
Û(t,t0) = 1 + (−i)
Zt
t0
(−i)2
dt1 ĤI (t1) +
2!
3 Zt
Zt
Zt
dt1
t0
Zt
dt2 T ĤI (t1)ĤI (t2)
t0
Zt
(−i)
dt1 dt2 dt3 T ĤI (t1 )ĤI (t2)ĤI (t3) + ...
3!
t
t0
t0

0

Zt


= T exp −i dt ′ ĤI (t ′ ) .


+
t0
Therefore we have expressed the full field φ̂H in terms of φ̂I ;
φ̂H (x) = Û † (t,t0)φ̂I (x)Û(t,t0),
We can generalise the evolution operator to take as the second argument values other than our
reference time t0 :



Zt


Û(t,t ′) = T exp −i dt ′′ ĤI (t ′′) .


t′
48
Û(t,t ′) satisfies
∂
i Û(t,t ′) = ĤI (t)Û(t,t ′),
∂t
Û(t,t) = 1,
and the identities
Û(t1,t2 )Û(t2,t3) = Û (t1,t3),
Û(t1 ,t3)Û † (t2,t3) = Û (t1,t2).
Û(t,t ′) is unitary:
Û(t,t ′)Û † (t,t ′) = Û(t,t) = 1.
We further have
Û(t,t ′) = Û(t,t0)Û † (t ′,t0)
′
′
= eiĤ0 (t−t0 ) e−iĤ(t−t ) e−iĤ0 (t −t0 ) ,
In particular we have
Û(t0 , −T ) = e−iĤ(t0 −(−T )) e−iĤ0 ((−T )−t0 ) .
Let us now discuss the ground state |Ωi. Imagine starting with |0i, the ground state of Ĥ0 and
evolving through time with Ĥ:
e−iĤT |0i =
full T
∑ e−iEn
n
|nfull ihnfull |0i.
Here we inserted a complete set of states. Enfull and |nfull i are the eigenvalues and eigenstates of
the full Hamiltonian Ĥ. The eigenvalues and eigenstates of the free Hamiltonian will be denoted
by En and |ni. We define the zero of the energy by
Ĥ0 |0i = 0.
We single out the ground state |Ωi = |0full i:
full T
e−iĤT |0i = e−iE0
full T
|ΩihΩ|0i + ∑ e−iEn
n6=0
|nfull ihnfull |0i,
where E0full = hΩ|Ĥ|Ωi. Since Enfull > E0full for n > 0 the additional terms disappear, if we send
T → ∞(1 − iε). Therefore
|Ωi =
lim
T →∞(1−iε)
−iE0full T
e
49
hΩ|0i
−1
e−iĤT |0i.
Since T is now very large, we can shift it by a small constant:
|Ωi =
=
=
lim
T →∞(1−iε)
lim
T →∞(1−iε)
lim
T →∞(1−iε)
−1
−iE0full (T +t0 )
e
hΩ|0i
e−iĤ(T +t0 ) |0i
−1
−iE0full (t0 −(−T ))
e−iĤ(t0 −(−T )) e−iĤ0 ((−T )−t0 ) |0i
e
hΩ|0i
−1
full
e−iE0 (t0 −(−T )) hΩ|0i
Û(t0 , −T )|0i
In the second line we have used Ĥ0 |0i = 0. Similar, we can express hΩ| as
hΩ| =
lim
T →∞(1−iε)
−1
full
h0|Û(T,t0) e−iE0 (T −t0 ) h0|Ωi
.
Therefore we have for x0 > y0 > t0 (we drop the subscript H for φ̂(x) = φ̂H (x)):
Ω T φ̂(x)φ̂(y) Ω =
=
=
lim
T →∞(1−iε)
lim
T →∞(1−iε)
h0|Û(T,t0)Û †(x0 ,t0)φ̂I (x)Û(x0 ,t0)Û †(y0 ,t0)φ̂I (x)Û(y0 ,t0)Û(t0 , −T )|0i
full
e−2iE0 T hΩ|0ih0|Ωi
h0|Û(T, x0 )φ̂I (x)Û(x0 , y0 )φ̂I (x)Û(y0 , −T )|0i
full T
e−2iE0
|h0|Ωi|2
Now
1 = hΩ|Ωi =
h0|Û(T,t0)Û(t0, −T )|0i
−2iE0full T
e
2
|h0|Ωi|
=
h0|Û(T, −T )|0i
full T
e−2iE0
|h0|Ωi|2
.
Therefore we have for x0 > y0
Ω T φ̂(x)φ̂(y) Ω =
h0|Û(T, x0 )φ̂I (x)Û(x0 , y0 )φ̂I (x)Û(y0 , −T )|0i
.
T →∞(1−iε)
h0|Û(T, −T )|0i
lim
We notice that all fields are in time-order. The final formula, valid for any x0 and y0 reads
T φ̂I (x)φ̂I (y)Û(T, −T ) 0
0
Ω T φ̂(x)φ̂(y) Ω =
lim
T →∞(1−iε)
0 Û(T, −T ) 0
RT
0 T φ̂I (x)φ̂I (y) exp −i dt ĤI (t) 0
−T
.
=
lim
T
R
T →∞(1−iε)
0 T exp −i dt ĤI (t) 0
−T
50
There is an obvious generalisation to more than two fields:
Ω T φ̂(x1 )φ̂(x2 )...φ̂(xn ) Ω =
RT
0 T φ̂I (x1 )φ̂I (x2 )...φ̂I (xn ) exp −i dt ĤI (t) 0
−T
.
=
lim
T
R
T →∞(1−iε)
0 T exp −i dt ĤI (t) 0
−T
5.7 Feynman diagrams
We have already seen in the free field theory, that we can represent certain expressions graphically, e.g.
1
2
1
2
1
2
0 T φ̂(x1 )φ̂(x2 )φ̂(x3 )φ̂(x4 ) 0 =
+
+
.
4
3
4
3
4
3
This describes two non-interacting particles. Things get more interesting in the interacting theory
as soons as we have more than one field at the same space-time point. Let’s again look at
RT
0 T φ̂I (x)φ̂I (y) exp −i dt ĤI (t) 0
−T
lim
Ω T φ̂(x)φ̂(y) Ω =
T
R
T →∞(1−iε)
0 T exp −i dt ĤI (t) 0
−T
in the interacting φ4 -theory. Expanding the exponential in the numerator we obtain

 +
* 
T
Z


0 =


0 T φ̂I (x)φ̂I (y) exp −i dt ĤI (t)


−T


 +
* ZT


0 + ...


0 T φ̂I (x)φ̂I (y) 0 + 0 T φ̂I (x)φ̂I (y) −i dt ĤI (t)


−T
Z
Z
λ
= DF (x − y) + 0 T φ̂I (x)φ̂I (y)(−i) dt d 3 z φ̂4I 0 + ...
4!
Z
−iλ
d 4 z 0 T φ̂I (x)φ̂I (y)φ̂I (z)φ̂I (z)φ̂I (z)φ̂I (z) 0 + ...
= DF (x − y) +
4!
For the term proportional to λ we use Wick’ theorem. In total there are 15 possibilities of
contracting the 6 fields with each other. However, only two of them are really different. The
first possibility is to contract φ̂I (x) with φ̂I (y), then we have three possibilities of contracting
the four φ̂I (z) with each other, all of them are equivalent. The second possibility consists in
51
contracting φ̂I (x) with one of the φ̂I (z) (four choices) and φ̂I (y) with one of the remaining φ̂I (z)
(three choices). The two remaining φ̂I (z) are then contracted with each other. Here we have in
total 4 · 3 = 12 possibilities to produce this configuration. Therefore
0 T φ̂I (x)φ̂I (y)φ̂I (z)φ̂I (z)φ̂I (z)φ̂I (z) 0 =
3DF (x − y)DF (z − z)DF (z − z) + 12DF (x − z)DF (y − z)DF (z − z).
Graphically:


3x
y



z 
 + 12 
x
z
y


.
Lines in this diagram are called propagators, internal points, where four lines meet, are called
vertices. Therefore
Z
−iλ
d 4 z 0 T φ̂I (x)φ̂I (y)φ̂I (z)φ̂I (z)φ̂I (z)φ̂I (z) 0 =
4!
Z
1
1
4
(−iλ) d z DF (x − y)DF (z − z)DF (z − z) + DF (x − z)DF (y − z)DF (z − z) .
8
2
The numbers 8 and 2 are called the symmetry factors of the diagrams. The symmetry factor is
given by a factor 4! for each vertex in the diagram. One then divides by the number of ways one
obtains from Wick’s theorem the same diagram (3 and 12 in the above example).
Alternatively, one can compute the symmetry factor as follows: S is the order of the permutation group of the internal lines and vertices leaving the diagram unchanged when the external
lines are fixed.
Examples:
S = 2,
S = 2 · 2 · 2 = 8,
S = 3! = 6,
52
S = 2.
It is common practice to let a Feynman diagram represent all possible contractions leading to this
diagram with the appropriate symmetry factor included. Therefore
= (−iλ)
x
z
y
1
2
Z
d 4 zDF (x − z)DF (y − z)DF (z − z)
We can now formulate the Feynman rules for φ4 -theory in position space:
1. For each propagator,
y = DF (x − y).
x
2. For each vertex,
z
= (−iλ)
Z
d 4 z.
3. For each external point,
= 1.
x
4. Divivde by the symmetry factor S.
In practice one works in momentum space. As an example we consider the following Fourier
transform
Z
Z
Z
1
4 ipx
4 iqy
d xe
d ye (−iλ)
d 4 zDF (x − z)DF (y − z)DF (z − z) =
2
Z 4
Z
Z
Z
Z
Z
d p1 −ip1 (x−z) d 4 p2 −ip2 (y−z) d 4 p3 −ip3 (z−z)
1
4
4
4 i(px+qy)
d x d y d ze
e
e
e
(−iλ)
2
(2π)4
(2π)4
(2π)4
×D̃F (p1 )D̃F (p2 )D̃F (p3 )
Z
Z
Z
1
4
4
= (−iλ)
d p1 d p2 d 4 p3 δ4 (p1 − p)δ4 (p2 − q)δ4 (p1 + p2 − p3 + p3 )
2
×D̃F (p1 )D̃F (p2 )D̃F (p3 )
Z 4
1
d p3
4 4
= (−iλ) (2π) δ (p + q)D̃F (p)D̃F (q)
D̃F (p3 )
2
(2π)4
53
We see that we obtain one delta-function, enforcing energy-momentum conservation of the incoming and outgoing momenta. Further at each vertex, energy and momentum are conserved.
Finally we have to integrate over any unrestricted four-momentum. We use the notation
Gnwith vacuum graphs (x1 , ..., xn) = 0 T φ̂I (x1 )...φ̂I (xn ) 0
for the Green function with n external legs. The sub-script “with vacuum graphs” will be explained at the end of this section. We define the Fourier transform of Gnwith vacuum graphs (x1 , ..., xn)
by
Gnwith vacuum graphs (x1 , ..., xn) =
Z
d 4 p1 d 4 pn −i ∑ p j x j
...
e
(2π)4 δ (p1 + ... + pn ) G̃nwith vacuum graphs (p1 , ..., pn)
4
4
(2π) (2π)
Remark: This is not the standard Fourier transform, as we have factored out the term
(2π)4 δ (p1 + ... + pn ) .
This term expresses overall momentum conservation and is always there, therefore it is convenient to factor it out. With these conventions we can now state the Feynman rules for φ4 -theory
in momentum space to compute G̃nwith vacuum graphs (p1 , ..., pn):
1. For each propagator,
→
p
i
=
p2 − m2 + iε
2. For each vertex,
z
= (−iλ).
←
p
= 1.
3. For each external point,
4. Impose momentum conservation at each vertex.
5. Integrate over each undetermined momentum;
Z
d4 p
(2π)4
54
.
6. Divivde by the symmetry factor S.
Remark: In the original formula in position space we had
ZT
lim
T →∞(1−iε)
−T
dt
Z
d 3z
which we wrote for simplicity as
Z
d 4 z.
Still we have to keep in mind that for the time-component we have to integrate over a slightly
distorted contour. When going to momentum space this is equivalent to integrate over the p0 component along the following contour:
Im(p0 )
−E~p
Re(p0 )
E~p
This is just the Feynman prescription.
Let us go back to our original formula
RT
0 T φ̂I (x)φ̂I (y) exp −i dt ĤI (t) 0
−T
lim
Ω T φ̂H (x)φ̂H (y) Ω =
T
R
T →∞(1−iε)
0 T exp −i dt ĤI (t) 0
−T
In the numerator we had contributions like
(−iλ)
The piece
Z

1

d 4 z DF (x − y)DF (z − z)DF (z − z) =  x
8
z
55
y

z 
.
is called a vacuum diagram. It is not connected to the external points x and y. Examples for
vacuum diagrams are:
,
,
It is clear that the denominator in the formula for Ω T φ̂H (x)φ̂H (y) Ω produces only vacuum
graphs. Let us label the values of the various vacuum graphs by Vi , and the values of the pieces
connected to x and y by D j . If a diagram consists of a piece D j and n1 vacuum graphs of type V1 ,
n2 vacuum graphs V2 , ..., then the value for the total diagram is
Dj ∏
i
1
(Vi )ni
ni !
The factor ni ! is the symmetry factor for ni copies of Vi . Then the numerator of the formula for
the two-point correlation function is
1
∑ ∑ D j ∏ ni! (Vi)ni ,
j {ni }
i
where {ni } means all ordered sets {n1 , n2 , ...} of non-negative integers. We have the following
exponentiation:
1
∑ ∑ D j ∏ ni! (Vi)ni
j {ni }
=
i
1
∑ D j ∏ ∑ ni! (Vi)ni
j
=
i
∑ D j ∏ exp (Vi)
i
j
=
ni
∑Dj
j
!
exp
∑ Vi
i
On the other hand we find for the denominator
 +

* 
T
Z


= exp
0 T exp −i dt ĤI (t) 0


−T
!
∑ Vi
i
!
.
Therefore the exponential cancels between numerator and denominator and we finally obtain
Ω T φ̂H (x)φ̂H (y) Ω = sum of all diagrams without any vacuum graphs.
5.8 Summary: Feynman rules for φ4-theory in momentum space
φ4 -theory is defined by the Lagrangian
L =
1
1
λ
∂µ φ (∂µ φ) − m2 φ2 − φ4 .
2
2
4!
56
We are interested in
Gn (x1 , ..., xn) =
and it’s Fourier transform defined through
Gn (x1 , ..., xn) =
Z
Ω T φ̂H (x1 )...φ̂H (xn ) Ω
d 4 p1 d 4 pn −i ∑ p j x j
...
e
(2π)4 δ (p1 + ... + pn ) G̃n (p1 , ..., pn)
(2π)4 (2π)4
(Previously we had considered Gnwith vacuum graphs , as discussed at the end of the last section, Gn and
Gnwith vacuum graphs differ by vacuum graphs.)
We have the following Feynman rules in momentum space: G̃n (p1 , ..., pn) is given as the sum of
all diagrams without any vacuum graphs with the following rules:
1. For each propagator,
→
p
=
i
p2 − m2 + iε
2. For each vertex,
z
= −iλ.
←
p
= 1.
3. For each external point,
4. Impose momentum conservation at each vertex.
5. Integrate over each undetermined momentum;
Z
d4 p
(2π)4
6. Divivde by the symmetry factor S.
57
.
6 Cross sections and decay rates
We have learned how to compute an abstract quantity, – the n-point correlation function –,
through Feynman diagrams. We now want to relate this quantity to quantities, which can be
measured: Cross sections and decay rates.
Elementary particle experiments are often scattering experiments: One collides two beams of
particles with well-defined momenta, and observes what comes out. The likelihood of any particular final state can be expressed in terms of the cross section. This is a quantity intrinsic to the
colliding particles and therefore allows comparison of two different experiments with different
beam sizes and intensities.
The cross section is defined as follows: Consider a bunch of particles of type A with density
ρA (particles per unit volume). Denote the length of the bunch by lA . Assume further that the
bunch is moving in the positive z-direction. Consider a second bunch of particles of type B with
number density ρB and bunch length lB , moving along the negative z-direction. Denote further by
F the cross-sectional area common to the two bunches. We expect the total number of scattering
events to be proportional to ρA , lA , ρB , lB and F. The cross section, denoted by σ, is just the total
number of events divided by all these quantities:
σ =
Number of scattering events
.
ρA lA ρB lB F
Obviously, this definition is symmetric between A and B, therefore we could have worked in any
other reference frame. In the early days of particle physics, scattering experiments were often
performed as fixed-target experiments. For fixed-target experiments it is common practice to
use the laboratory frame lab frame. In the lab frame one sort of particles is at rest, say A,
while the other sort moves with velocity v towards the target. Modern scattering experiments are
collider experiments. Here the centre-of-mass frame is more convenient.
The cross section has units of area.
In real beams, ρA and ρB are not constant, the particle density is generally larger at the center of
the beam than at the edges and one obtains
Number of scattering events = σ lA lB
Z
d 2 x ρA (x)ρB (x).
If the densities are constant, we simply have
Number of scattering events =
σNA NB
,
F
where NA and NB are the total number of A and B particles. NA and NB are given by
NA = ρA lA F,
NB = ρB lB F.
Cross sections for many different processes may be relevant to a single scattering experiment.
In e+ e− collisions, for example, one can measure the cross section for the production of µ+ µ− ,
58
τ+ τ− , µ+ µ− γ, etc.. If one is interested in the cross section for the production of a specific final
state (for example exactly one µ+ and one µ− ) one speaks about an exclusive cross section, If
on the other hand several final states may contribute to the cross section, one speaks about an
inclusive cross section. An example for a typical inclusive cross section would be one µ+ plus
anything else.
Usually we wish to measure not only what the final-state particles are, but also the momenta
with which they come out. For that we define the differential cross section
d3 p
dσ
3
1 ...d pn
which is the quantity when integrated over any small d 3 p1 ...d 3 pn gives the cross section for
scattering into that region of final-state phase space. Note that the various final-state momenta are
not all independent: Four components will always be constrained by momentum conservation.
In the simplest case of only two final-state particles this leaves only two unconstrained variables,
usually taken to be the angles θ and φ of the momentum of one of the particles. Integrating
dσ
d 3 p1 d 3 p2
over the four constrained momentum components then leaves us with the usual differential cross
section
dσ
.
dΩ
A second measurable quantity is the decay rate of an unstable particle. The decay rate Γ of an
unstable particle A, which is assumed to be at rest, is defined as follows:
Γ =
Number of decays per unit time
.
Number of A particles present
The lifetime τ is the reciprocal of the sum of the decay widths into all possible final states. The
particle’s half-life time is τ · ln 2.
In non-relativistic quantum mechanics, an unstable atomic state shows up in scattering experiments as a resonance. Near the resonance energy E0 , the scattering amplitude is given by the
Breit-Wigner-formula
f (E) ∼
1
E − E0 + iΓ/2
The cross section therefore has a peak of the form
σ ∼
1
(E − E0 )2 + Γ2 /4
59
The width of the resonance peak is equal to the decay rate of the unstable state. The Breit-Wigner
formula also applies to relativistic quantum mechanics. In an amplitude involving an unstable
particle of momentum p and mass m we have
1
1
≈
p2 − m2 + imΓ
2E~p p0 − E~p + i
.
m/E~p Γ/2
Remark: A correct treatment of unstable particles in higher orders in perturbation theory is a
topic of current interest.
6.1 The S-matrix
To calculate the cross section we set up wavepackets representing the initial-state particles,
evolve this initial state for a very long time with the time-evolution operator exp(−iĤt) of the
full interacting field theory and overlap the resulting final state with wavepackets representing
some desired set of final-state particles.
A wavepacket is given by
|φi =
Z
E
d 3k
1
~k) ~k ,
p
φ(
(2π)3 2E~k
where |~ki is a one-particle state of momentum ~k in the full interacting theory and the weight
function φ(~k) is peaked around ~p.
We recall that we chose as normalisation for the one-particle states
h~p|~qi = 2E~p(2π)3 δ3 (~p −~q) .
If we further require that
Z
then we have
d 3 k ~ 2
φ k = 1,
(2π)3
hφ|φi = 1.
In a typical scattering experiment one scatters two initial-state particles and one observes, say,
n final-state particles. This leads us to the definition of in- and out-states. The in-state |φA φB i
consists of the two particles A and B in the initial state, the out-state |φ1 φ2 ...φni of the n particles in the final state. In the far past (t → −∞), the in-state is described by two wave-packets,
corresponding to two well-sepatated single particle states and we may describe the in-state in
the far past as a free state. As time proceeds, the two particles start to interact and scatter. The
situation for the out-state is analogous: In the far future (t → ∞), the out-state is described by
60
n wave-packets, corresponding to n well-sepatated single particle states and we may describe in
the far future the out-state as a free state. States which are described in the far past or far future
as free states are called asymptotic states. We will denote a state, which corresponds at t → −∞
to a free n-particle state as an in-state
|φ1 φ2 ...φn iin = |φ1 φ2 ...φn ;t0 = −∞iH
and a state, which corresponds at t → ∞ to a free n-particle state as an out-state
|φ1 φ2 ...φniout = |φ1 φ2 ...φn ;t0 = ∞iH .
The notation on the right-hand-side indicates that the Heisenberg state |φ1 φ2 ...φn;t0iH coincides
at t0 ∈ {±∞} with a free n-particle state. The probability we wish to compute is
P = |out hφ1 φ2 ...φn |φA φB iin |2
The in-state |φA φB iin has a nice interpretation as two localised wavepackets in the far past, but
due to the evolution with the full interacting Hamiltonian in e−iĤt , the wave packets will in
general not stay intact as time proceeds. In a similar way, the out-state out hφ1 φ2 ...φn | has a nice
interpretation as n localised wavepackets in the far future, however evolving this state backwards
in time with the full interacting Hamiltonian will in general not leave the wavepackets intact.
What is important is that the in- and out-states may have a non-zero overlap, which we wish to
compute.
Let us first consider the in-states: Consider particle A at rest and particle B along the z-axis.
It is important to take the transverse displacement into account. If the wave packet of particle B
is not centered at x = y = 0 in the transverse plane, but at ~b, this yields a factor exp(−i~b ·~kB ). |~b|
is called the impact parameter.
|φA φB iin =
Z
d 3 kA
(2π)3
Z
d 3 kB φA (~kA )φB (~kB ) −i~b·~kB ~ ~ E
q
e
kA kB .
(2π)3 (2E )(2E )
in
~kA
~kB
The wave packet of particle A is centered around pA in momentum space, the one of particle B is
centered around pB . In a similar way we could write for the out-states
!
Z 3
d q f φ f (~q f )
out hφ1 φ2 ...φn | =
∏ (2π)3 p2E~q out h~q1~q2 ...~qn| .
f
f
Again, the wave packet for particle f is centered around p f in momentum space. However, in order to avoid some technical difficulties it is simpler to work for the out-state not with wavepackets
but to use a state with definite momenta
p1~p2 ...~pn |
out h~
and to put in various normalisation factors (essentially a factor 1/(2E~p f ) for every final-state
particle) by hand.
61
Let us recall the connection with the Schrödinger picture:
|φiH = eiĤt |φ,tiS .
If we would like to indicate that a state has specific properties at a particular time t0 (e.g. for the
case at hand, if a state corresponds at t = t0 to a n-particle state with momenta ~p1 , ~p2 , ..., ~pn ) we
will put an extra argument and write
|φ;t0iH = eiĤt |φ,t;t0iS .
We have
~ ~
p1~p2 ...~pn|kA kB iin
out h~
=
=
lim
T →∞
lim
~ ~
D
E
~p1~p2 ...~pn , T ; T e−iĤ(2T ) ~kA~kB , −T ; −T
p1~p2 ...~pn ; T |kA kB ; −T iH
H h~
T →∞ S
S
The state |~p1~p2 ...~pn ,t0;t0 iS corresponds at t = t0 to a n-particle state with momenta ~p1 , ~p2 , ...,
~pn and we will simply write |~p1~p2 ...~pn i. Thus
D
E
−iĤ(2T ) ~ ~
~
~
h~
p
~
p
...~
p
|
k
k
i
=
lim
~
p
~
p
...~
p
e
k
k
out 1 2
n A B in
1 2
n
A B .
T →∞
We define the S-matrix by
p1~p2 ...~pn |~kA~kB iin =
out h~
E
D
~p1~p2 ...~pn Ŝ~kA~kB .
If the particles do not interact at all, Ŝ is the identity operator. Even if the theory contains
interactions, the particles have some probability of missing each other. To isolate the interesting
part of the Ŝ-matrix – that is, the part due to interactions – we define the T̂ -matrix by
!
Ŝ = 1 + i(2π)4δ4 kA + kB − ∑ p f
T̂ .
f
Of special interest are the matrix elements of T̂ :
iA (kA kB → p1 p2 ...pn ) = h~p1~p2 ...~pn |iT̂ |~kA~kB i
The probability we want to compute is now
P =
∏
f
Z
d3 p f 1
(2π)3 2E~p f
!
|out h~p1~p2 ...~pn |φA φB iin |2 .
The bracket in front gives the correct normalisation factor for the out-state. The in-state still
depends on the impact parameter ~b. To obtain the cross section we have to integrate over all
impact parameters:
σ =
Z
d 2 b P(~b).
62
Therefore
dσ
d 3 p1 d 3 p2
...
(2π)3 (2π)3
1
∏ 2E~p
=
f
f
!Z
−i~b·(~kB −~kB′ )
e

d 2b 
∏
i=A,B
Z
d 3 ki
(2π)3

Z
3
′
∗
′
~
~
d ki φi (ki ) 
φ (k )
qi i
q
3
2E~k′
2E~ki (2π)
i
p1~p2 ...~pn |kA kB iin out h~p1~p2 ...~pn|kA′ kB′ i∗in .
out h~
If we are not interested in the trivial case where no scattering case takes place, we can drop the
identity from the S-matrix and write
p1~p2 ...~pn |kA kB iin
out h~
p1~p2 ...~pn |kA′ kB′ iin
out h~
= (2π)4δ4 (kA + kB − ∑ p f )iA (kA kB → p1 p2 ...pn) ,
f
4 4
= (2π) δ
(kA′ + kB′ −
∑ p f )iA
f
kA′ kB′ → p1 p2 ...pn .
′
We can use the second of these delta functions, together with the δ2 (kB⊥ − kB⊥ ) to perform all six
k′ integrals. We have
Z
d 3 kA′
(2π)3
Z
Z
d 3 kB′
2 2 ⊥
⊥′
)(2π)4δ4 (kA′ + kB′ − kA − kB ) =
(2π)
δ
(k
−
k
B
B
(2π)3
dkAz ′
Z
dkBz ′ δ(EA′ + EB′ − EA − EB )δ(kAz ′ + kBz ′ − kAz − kBz )
Z
=
dkAz ′ δ(EA′ + EB′ − EA − EB )kz ′ =kz +kz −kz ′
=
B
A
B
A
1
1
= kz
.
z =
k EAA − EBB |vA − vB |
The difference |vA − vB | is the relative velocity of the beams in the lab frame. The initial wave
packets are localised in momentum space, centered around pA and pB . We evaluate all factors,
which are smooth functions of kA and kB at pA and pB . We obtain
!
1
dσ
|A (pA pB → p1 p2 ...pn )|2
=
∏ 2E~p
d 3 p1 d 3 p2
2EA 2EB |vA − vB |
f
f
3
3 ...
(2π) (2π)
Z
d 3 kA
(2π)3
Z
d 3 kB ~ 2 ~ 2
φA (kA ) φB (kB ) (2π)4δ4 (kA + kB − ∑ p f ).
(2π)3
f
We can simplify this formula further by noting that real detectors cannot resolve small variations
of the incoming and outcoming momenta. Therefore we can approximate δ(kA + kB − ∑ p f ) by
δ(pA + pB − ∑ p f ) and we obtain
!
dσ
1
|A (pA pB → p1 p2 ...pn )|2
=
(2π)4 δ4 (pA + pB − ∑ p f ).
∏ 2E~p
d 3 p1 d 3 p2
2E
2E
|v
−
v
|
A
B A
B
f
f
f
3
3 ...
(2π) (2π)
63
Note that all dependence on the shape of the wave packets has disappeared.
By a similar reasoning we find for the decay rate
!
1
|A (pA → p1 p2 ...pn )|2
dΓ
=
(2π)4δ4 (pA − ∑ p f ).
∏ 2E~p
d 3 p1 d 3 p2
2m
A
f
f
3
3 ...
(2π) (2π)
To obtain the total cross section or the total decay rate we integrate over all final states. If the
final state contains n identical particles, we have to be careful not to count the same final state
more then once. One possibility is to impose a strict ordering on the energies of the final state
particles
E1 > E2 > ... > En ,
a different (and in practice the preferred option) is to integrate without any restriction and to
divide the result by a factor of 1/n!. Therefore
σ=
1
2EA 2EB |vA − vB |
1
n!
Z
d3 p f
∏ (2π)32E~p
f
f
1 1
Γ=
2mA n!
Z
!
(2π)4δ4 (pA + pB − ∑ p f ) |A (pA pB → p1 p2 ...pn )|2 ,
d3 p f
∏ (2π)32E~p
f
f
!
(2π)4 δ4 (pA − ∑ p f ) |A (pA → p1 p2 ...pn )|2 .
We denote the Lorentz-invariant phase-space element by
!
d3 p f
1
(2π)4 δ4 (Q − ∑ p f )
dφ(Q; p1 , p2 , ..., pn) =
∏
3
n! f (2π) 2E~p f
!
d4 p f
1
2
2
=
(2π)δ(p f − m )θ(E~p f ) (2π)4δ4 (Q − ∑ p f ).
∏
4
n! f (2π)
The flux factor 1/(4EA EB |va − vB |) is not Lorentz invariant. For beams along the z-axis the flux
factor is invariant under boosts along the z-axis and we have for beams along the z-axis
1
1
=
,
2EA 2EB |vA − vB |
2K(Q2 )
where
2
2K(Q ) = 2
r
Q2 − (m1 + m2 )2
Q2 − (m1 − m2 )2
= 2Q2 for massless particles,
and Q2 = (pA + pB )2 .
64
6.2 Properties of the S-matrix
In this paragraph we will denote a generic in-state by
E
|iiin = ~k1~k2 ...~km ,
in
and a generic out-state by
| f iout = |~p1~p2 ...~pn iout .
We defined the S-matrix by
out h f |iiin =
and therefore
E
D f Ŝ i = lim f e−iĤ(2T ) i ,
T →∞
Ŝ =
lim e−iĤ(2T ) .
T →∞
Since Ĥ is hermitian, the S-matrix is unitary:
Ŝ−1 = Ŝ† .
Let us further denote
Sfi =
f Ŝ i .
The Hilbert spaces of the in- and out-states are isomorphic Fock spaces. Thus we may express
any in-state |iiin as a linear combination of out-states | f iout and vice versa. The relation is
provided by the S-matrix:
†
| f iout = ∑ S f i |iiin .
|iiin = ∑ S f i | f iout ,
i
f
6.3 Relation between invariant matrix elements and Feynman diagrams
Here we sketch only a heuristic argument for the relation of the S-matrix to Feynman diagrams
and postpone the proof to later. The proof is based on the reduction formula of Lehmann,
Symanzik and Zimmermann. The S-matrix was defined by
E
D
−iĤ(2T ) p1~p2 ...~pn |~pA~pB iin = ~p1~p2 ...~pn Ŝ ~pA~pB = lim ~p1~p2 ...~pn e
~pA~pB
out h~
T →∞
To compute this quantity we would like to replace the states |~pA~pB i and |~p1~p2 ...~pni, which are
states in the full interacting theory, with their counterparts in the free theory with Hamiltonian
Ĥ0 . We will denote the corresponding states in the free theory by |~pA~pB i0 and |~p1~p2 ...~pn i0 . For
the vacuum state we already found such a formula
−1
full
e−iE0 T hΩ|0i
e−iĤT |0i.
|Ωi =
lim
T →∞(1−iε)
65
Now we look for a relation of the form
|~pA~pB i ∼
lim
T →∞(1−iε)
e−iĤT |~pA~pB i0 ,
where we have omitted the prefactor. We change to the interaction picture:
|~pA~pB iin = eiĤ(−T −t0 ) |~pA~pB , −T iS = eiĤ(−T −t0 ) e−iĤ0 (−T −t0 ) |~pA~pB , −T iI
= Û † (−T,t0) |~pA~pB , −T iI ,
|~p1~p2 ...~pniout = eiĤ(T −t0 ) |~p1~p2 ...~pn, T iS = eiĤ(T −t0 ) e−iĤ0 (T −t0 ) |~p1~p2 ...~pn, T iI
= Û † (T,t0) |~p1~p2 ...~pn , T iI ,
and therefore
p1~p2 ...~pn, T |~pA~pB iin =
out h~
lim
T →∞(1−iε)
We have
I
D
E
~p1~p2 ...~pn Û (T,t0) Û † (−T,t0) ~pA~pB , −T
I



ZT


†


Û (T,t0) Û (−T,t0) = Û (T, −T ) = T exp −i dt ĤI (t)
.


−T
The T in front of the curly brackets denotes the time-ordered product. Following the same
arguments we used to derive the relation between |Ωi and |0i, one arrives at


p1~p2 ...~pn |~pA~pB iin
out h~
∼
lim
T →∞(1−iε)
p1~p2 ...~pn|T
0 h~
exp −i
ZT
−T
dt ĤI (t) |~pA~pB i0 .
Here we neglected all proportionality factors. We will do a more careful derivation later on,
including the proportionality factors.
It can be shown that the non-trivial part of the S-matrix can be computed as follows




ZT
lim 0 h~p1~p2 ...|T exp −i dtHI (t) |~pA~pB i0 
~p1~p2 ...~pn iT̂ ~pA~pB =
T →∞(1−iε)
connected ,
−T
amputated
A Feynman diagram is called connected, if it does not have disjoint pieces.
A Feynman diagram is called amputated, if for all external lines, the corresponding propagator
is removed.
66
6.4 Final formula
Let us now summarise and present the final “master”-formula for the computation of cross sections and decay rates. Up to now we only considered scalar particles. There are a few minor
modifications if the involved particles have additional degrees of freedom, like spin or colour
degrees of freedom. We haven’t treated spin and other degrees of freedom in detail yet, but later
we will see that
|A |2
involves a sum over all spins and other degrees of freedom. This is fine for final-state particles,
but for initial-state particles we have to average over these degrees of freedom. If the initial-state
particles have spin, this brings an additional factor of
1
nspin (A)nspin (B)
for scattering reactions and a factor
1
nspin (A)
for the decay rate. The quantity nspin (A) denotes the number of spin degrees of freedom of
particle A. Similar, if the initial-state particles have colour charges (i.e. are quarks or gluons),
this brings an additional factor of
1
ncolour (A)ncolour (B)
for scattering reactions, where ncolour (A) denotes the number of colour degrees of freedom of
particle A.
To calculate the cross section at an collider with no initial-state hadrons (e.g. an electronpositron collider):
σ =
1
1
1
2
2K(Q ) nspin (A) nspin (B)
Z
dφ (pA + pB ; p1 , ..., pn) |A (pA pB → p1 p2 ...)|2 ,
wherep2K(Q2 ) is the flux factor and we have 2K(Q2 ) = 2Q2 for massless incoming particles.
Q = (pA + pB )2 is the centre-of-mass energy. For a decay rate we have
Γ =
1
1
2mA nspin (A)
Z
dφ (pA ; p1 , ..., pn) |A (pA → p1 p2 ...)|2 .
The phase-space measure is given by
1
dφ(Q; p1 , p2 , ..., pn) =
∏ j n j!
d3 p f
∏ (2π)32E~p
f
f
67
!
(2π)4 δ4 (Q − ∑ p f ),
if the final state contains n j identical particles of type j. If the colliding particles are not elementary (like protons or antiprotons), we have to include the probability of finding the elementary
particle A inside the proton or antiproton. If the proton has momentum p p one usually specifies
the probability of finding a parton with momentum fraction x by the parton distirbution function
f (x)
The parton has then the momentum
pA = xp p .
For the cross section we have to integrate over all possible momentum fractions and the formula
for a hadron-hadron collider becomes
σ =
Z
Z
dxA f (xA )
Z
dxB f (xB )
1
1
1
1
2K(ŝ) nspin (A) nspin (B) ncolour (A)ncolour (B)
dφ (pA + pB ; p1 , ..., pn) |A (pA pB → p1 p2 ...)|2 .
ncolour (A) and ncolour (B) are the number of colour degrees of the initial state particles. We further
have
ŝ = (pa + pB )2 = (xA p pA + xB p pB )2 = xA xB (p pA + p pB )2 .
The last equality holds if p2pA = p2pB = 0. At high energies, like the LHC experiments, masses of
the order of 1 GeV do not play an essential role and the proton can be treated approximately as
massless.
68
7 Fermions
Up to now we considered only spin zero particles. In this section we study quantum field theories
with spin 1/2 particles. But before we embark on quantisation of these theories, we first discuss
solutions of the classical theory.
7.1 The Dirac equation
The Lagrange density for a (classical)
Dirac field depends on four-component spinors ψα (x)
†
0
(α = 1, 2, 3, 4) and ψ̄α (x) = ψ (x)γ α :
L (ψ, ψ̄, ∂µ ψ) = iψ̄(x)γµ ∂µ ψ(x) − mψ̄(x)ψ(x)
Here, the (4 × 4)-Dirac matrices satisfy the anti-commutation rules
{γµ , γν } = 2gµν 1, {γµ , γ5 } = 0, γ5 = iγ0 γ1 γ2 γ3 =
i
εµνρσ γµ γν γρ γσ .
24
The Euler-Lagrange equations yield the Dirac equations
iγµ ∂µ − m ψ(x) = 0,
←
µ
ψ̄(x) iγ ∂ µ +m = 0.
For computations it is useful to have an explicit representation of the Dirac matrices. There are
several widely used representations. A particular useful one is the Weyl representation of the
Dirac matrices:
0 σµ
1 0
µ
0 1 2 3
γ =
γ5 = iγ γ γ γ =
σ̄µ 0
0 −1
Here, the 4-dimensional σµ -matrices are defined by
µ
σ̄µȦB = (1,~σ)
σAḂ = (1, −~σ)
and ~σ = (σx , σy , σz ) are the standard Pauli matrices:
0 1
0 −i
1 0
σx =
σy =
σz =
1 0
i 0
0 −1
Let us now look for plane wave solutions of the Dirac equation. We make the Ansatz
u(p)e−ipx , p0 > 0, p2 = m2 , incoming fermion,
ψ(x) =
v(p)e+ipx , p0 > 0, p2 = m2 , outgoing anti-fermion.
u(p) describes incoming particles, v(p) describes outgoing anti-particles. Similar,
ū(p)e+ipx , p0 > 0, p2 = m2 , outgoing fermion,
ψ̄(x) =
v̄(p)e−ipx , p0 > 0, p2 = m2 , incoming anti-fermion,
69
where
ū(p) = u† (p)γ0 ,
v̄(p) = v† (p)γ0 .
ū(p) describes outgoing particles, v̄(p) describes incoming anti-particles. Then
(p/ − m) u(p) = 0,
ū(p) (p/ − m) = 0,
(p/ + m) v(p) = 0,
v̄(p) (p/ + m) = 0,
We will find that there is more then one solution for u(p) (and the other spinors ū(p), v(p), v̄(p)).
We will label the various solutions with λ. The degeneracy is related to the additional spin degree
of freedom and we find for a spin 1/2 particles 2 solutions. We require that the two solutions
satisfy the orthogonality relations
ū(p, λ̄)u(p, λ) = 2mδλ̄λ ,
v̄(p, λ̄)v(p, λ) = −2mδλ̄λ ,
ū(p, λ̄)v(p, λ) = v̄(λ̄)u(λ) = 0,
and the completeness relation
∑ u(p, λ)ū(p, λ)
= p/ + m,
∑ v(p, λ)v̄(p, λ)
= p/ − m.
λ
λ
7.2 Massless spinors
Let us now try to find explicit solutions for the spinors u(p), v(p), ū(p) and v̄(p). The simplest
case is the one of a massless fermion:
m = 0.
In this case the Dirac equation for the u- and the v-spinors are identical and it is sufficient to
consider
p/u(p) = 0,
ū(p)p/ = 0.
In the Weyl representation p/ is given by
p/ =
0
pµ σµ
0
pµ σ̄µ
,
therefore the 4 × 4-matrix equation for u(p) (or ū(p)) decouples into two 2 × 2-matrix equations.
We introduce the following notation: Four-component Dirac spinors are constructed out of two
Weyl spinors as follows:
pA
|p+i
u+ (p)
=
u(p) =
.
=
|p−i
u− (p)
pḂ
70
Bra-spinors are given by
u(p) = (hp−| , hp+|) = pA , pḂ = (ū− (p), ū+(p)) ,
If we define the helicity projection operators
1
(1 + γ5 ) =
=
2
1
=
(1 − γ5 ) =
2
P+
P−
1 0
0 0
0 0
0 1
,
,
then
u± (p) = P± u(p),
ū± (p) = ū(p)P∓.
The two solutions of the Dirac equation
p/u(p, λ) = 0
are then
u(p, +) = u+ (p),
u(p, −) = u− (p).
We now have to solve
pµ σ̄µ |p+i = 0,
hp+| pµ σ̄µ = 0,
pµ σµ |p−i = 0,
hp−| pµ σµ = 0.
It it convenient to express the four-vector pµ = (p0 , p1 , p2 , p3 ) in terms of light-cone coordinates:
p+ = p0 + p3 , p− = p0 − p3 , p⊥ = p1 + ip2 , p⊥∗ = p1 − ip2 .
Note that p⊥∗ does not involve a complex conjugation of p1 or p2 . For null-vectors one has
p⊥∗ p⊥ = p+ p−
Then the equation for the ket-spinors becomes
p+ p⊥∗
p− −p⊥∗
|p+i = 0,
|p−i = 0,
p⊥ p−
−p⊥
p+
Solutions for ket-spinors are
iα+
|p+i = c+ e
−p⊥∗
p+
−iα−
,
|p−i = c− e
71
p+
p⊥
.
Solutions for bra-spinors are
′
hp+| = c+ e−iα+ (−p⊥ , p+ ) ,
′
hp−| = c− eiα− (p+ , p⊥∗ ) .
c+ and c− are real positive numbers. We assumed that |p+i and hp + | have the same normalisation factor c+ , and a similar assumption was made for |p−i and hp − |. We normalise the
massless spinors such that
hp ± |γµ |p±i = 2pµ .
On the l.h.s we obtain
′
p + σ̄µ p+ = c2+ ei(α+ −α+ ) p+ 2pµ ,
′
hp − |γµ |p−i = p − σµ p− = c2− e−i(α− −α− ) p+ 2pµ .
hp + |γµ |p+i =
Since we assumed p0 > 0 we have p+ ≥ 0. It follows that the constants c+ and c− are given by
1
c+ = c− = p
.
|p+ |
For the phases one has
α′+ = α+ mod 2π,
α′− = α− mod 2π.
In the following we will use
α+ = α′+ = α− = α′− = 0.
Therefore
|p+i = pA = p
1
−p⊥∗
p+
Ḃ
1
p+
p⊥
,
|p−i = p = p
|p+ |
1
hp+| = pḂ = p
(−p⊥ , p+ ) .
|p+ |
,
|p+ |
1
hp−| = pA = p
(p+ , p⊥∗ ) ,
|p+ |
Let us now introduce the 2-dimensional antisymmetric tensor:
0 1
εAB =
,
εBA = −εAB
−1 0
Furthermore:
εAB = εȦḂ = εAB = εȦḂ .
We then find the following relations for raising and lowering a spinor index A or Ḃ:
pA = εAB pB ,
pȦ = εȦḂ pḂ ,
pḂ = pȦ εȦḂ ,
pB = pA εAB .
Note that raising an index is done by left-multiplication, whereas lowering is performed by rightmultiplication. Therefore the relations between the different spinors are as follows:
72
hp − | =
εAB
pA
|p+i = pA
hermitian conj.
hermitian conj.
εȦḂ
|p−i = pȦ
hp + | = pȦ
Note that the relations p†A = pȦ and pA† = pȦ are valid for real momenta pµ = (p0 , p1 , p2 , p3 ).
If on the other hand (p0 , p1 , p2 , p3 ) are complex, these relations will in general not hold. In the
latter case pA and pȦ are considered as independent quantities. The reason, why the relations
p†A = pȦ and pA† = pȦ do not hold in the complex case lies in the definition of p⊥∗ : We defined
p⊥∗ as p⊥∗ = p1 − ip2 , and not as p1 − ip∗2 . With the definition p⊥∗ = p1 − ip2 we have that p⊥∗
is a holomorphic function of p1 and p2 . There are applications where holomorphicity is more
important than nice properties under hermitian conjugation.
7.3 Spinorproducts
We can define two spinorproducts
1
(p⊥∗ q+ − p+ q⊥∗ ) ,
|p+ | |q+ |
1
(p⊥ q+ − p+ q⊥ ) .
[qp] = hq + |p−i = qḂ pḂ = p
|p+ | |q+ |
hpqi = hp − |q+i = pA qA = p
We have
hpqi [qp] = 2p · q.
Therefore
|hpqi| = |[qp]| =
The spinorproducts are anti-symmetric:
p
2p · q.
hqpi = − hpqi ,
[pq] = − [qp] .
From the Schouten identity for the 2-dimensional antisymmetric tensor
εAB εCD + εAC εDB + εAD εBC = 0
one derives
hp1 p2 ihp3 p4 i = hp1 p4 ihp3 p2 i + hp1 p3 ihp2 p4 i
[p1 p2 ] [p3 p4 ] = [p1 p4 ] [p3 p2 ] + [p1 p3 ] [p2 p4 ]
73
Fierz identity:
hp1 + |γµ |p2 +ihp3 − |γµ |p4 −i = 2[p1 p4 ]hp3 p2 i
Useful formulas in the bra-ket notation:
hp ± |γµ1 ...γµ2n+1 |q±i = hq ∓ |γµ2n+1 ...γµ1 |p∓i
hp ± |γµ1 ...γµ2n |q∓i = −hq ± |γµ2n ...γµ1 |p∓i
7.4 Massive spinors
As in the massless case, a massive spinor satisfying the Dirac equation has a two-fold degeneracy.
We will label the two different eigenvectors by “+” and “-”. Let p be a massive four-vector with
p2 = m2 , and let q be an arbitrary light-like vector. With the help of q we can construct a light-like
vector p♭ associated to p:
p♭ = p −
We define
p2
q.
2p · q
1
(p/ − m) |q−i,
hp♭ + |q−i
1
v(p, −) = ♭
(p/ − m) |q+i.
hp − |q+i
1
(p/ + m) |q−i,
hp♭ + |q−i
1
u(p, −) = ♭
(p/ + m) |q+i,
hp − |q+i
v(p, +) =
u(p, +) =
For the conjugate spinors we have
1
hq − | (p/ + m) ,
hq − |p♭ +i
1
ū(p, −) =
hq + | (p/ + m) ,
hq + |p♭ −i
1
hq − | (p/ − m) ,
hq − |p♭ +i
1
v̄(p, −) =
hq + | (p/ − m) .
hq + |p♭ −i
ū(p, +) =
v̄(p, +) =
These spinors satisfy the Dirac equations
(p/ − m) u(λ) = 0,
ū(λ) (p/ − m) = 0,
(p/ + m) v(λ) = 0,
v̄(λ) (p/ + m) = 0,
the orthogonality relations
ū(λ̄)u(λ) = 2mδλ̄λ ,
v̄(λ̄)v(λ) = −2mδλ̄λ ,
ū(λ̄)v(λ) = v̄(λ̄)u(λ) = 0,
and the completeness relation
∑ u(λ)ū(λ)
= p/ + m,
∑ v(λ)v̄(λ)
= p/ − m.
λ
λ
74
7.5 Quantisation of fermions
We start from the Lagrangian
L (ψ, ψ̄, ∂µ ψ) = iψ̄(x)γµ ∂µ ψ(x) − mψ̄(x)ψ(x)
The canonical momentum conjugate to ψ is
∂L
= iψ̄γ0 = iψ† .
∂∂0 ψ
Thus the Hamilton function is
Z
Z
h
i Z
h
i
h
i
3
†
3
3
†
0
0
~
~
H =
d x iψ ∂0 ψ − L = d x ψ̄ −i~γ · ∇ + m ψ = d x ψ −iγ ~γ · ∇ + mγ ψ.
Here
~γ = γ1 , γ2 , γ3 ,
If we define
~∇ = (∂1 , ∂2 , ∂3 ) .
~α = γ0~γ,
β = γ0 ,
we obtain
H =
Z
h
i
~
~
d x ψ −iα · ∇ + mβ ψ.
3
†
Let us expand the field ψ(x) in a set of eigenfunctions of
hD = −i~α · ~∇ + mβ
From the solution of the free Dirac equation we already know that
h
i
−i~α · ~∇ + mβ u(p, λ)e−ip·x = 0.
Therefore u(p, λ)ei~p·~x are eigenfunctions of hD with eigenvalues E~p . Similarly, the functions
v(p, λ)e−i~p·~x are eigenfunctions of hD with eigenvalues −E~p . These form a complete set of
eigenfunctions, since for any ~p there are two u’s and two v’s, giving us four eigenvectors of the
4 × 4 matrix hD . We write for the field operators in the Heisenberg picture
Z
d3 p
1
†
−ip·x
ip·x
p
ψ̂H (x) =
â~p,λ u(p, λ)e
+ b̂~p,λ v(p, λ)e
,
(2π)3 2E~p ∑
λ
Z
1
d3 p
†
−ip·x
ip·x
ˆ
p
ψ̄H (x) =
b̂
v̄(p,
λ)e
+
â
ū(p,
λ)e
.
~p,λ
~p,λ
(2π)3 2E~p ∑
λ
75
The creation and annihilation operators obey the anticommutation rules
o
n
o n
â~p,λ , â~†q,λ′ = b̂~p,λ , b̂~†q,λ′ = (2π)3δ3 (~p −~q)δλλ′ .
All other anti-commutators vanish:
o o
n
n
â~p,λ , â~q,λ′ = â~†p,λ , â~q†,λ′ = b̂~p,λ , b̂~q,λ′ = b̂~†p,λ , b̂~†q,λ′ = 0,
o n
o n
o
n
â~p,λ , b̂~q,λ′ = â~p,λ , b̂~q†,λ′ = â~†p,λ , b̂~q,λ′ = â~†p,λ, b̂~†q,λ′ = 0.
The anticommutation relations for ψ̂S and ψ̂†S in the Schrödinger picture (or equal-time anticommutation relations in the Heisenberg picture) are then
o
n
ψ̂S (~x), ψ̂†S (~y) = δ3 (~x −~y),
n
o
†
†
ψ̂S (~x), ψ̂S (~y) = 0.
{ψ̂S (~x), ψ̂S (~y)} =
The vacuum |0i is defined to be the state such
â~p,λ |0i = b̂~p,λ |0i = 0.
The Hamiltonian can be written as
Ĥ =
Z
d3 p
†
†
â
â
+
b̂
b̂
E
~p
~p,λ ~p,λ
~p,λ ~p,λ ,
(2π)3 ∑
λ
where an infinite constant has been dropped. The momentum operator is
~Pˆ =
Z
Z
d3 p
†
†
d 3 x ψ̂†S −i~∇ ψ̂S =
~
p
â
â
+
b̂
b̂
~p,λ ~p,λ
~p,λ ~p,λ
(2π)3 ∑
λ
Thus both â~†p,λ and b̂~†p,λ create particles with energy +E~p and momentum ~p. We will refer to the
particles created by â~†p,λ as fermions and to those created by b̂~†p,λ as antifermions.
The one-particle states
|~p, λi =
p
2E~pâ~†p,λ |0i
are defined so that their inner product
~p, λ|~q, λ′ = 2E~p (2π)3δ3 (~p −~q)δλλ′
is Lorentz invariant.
76
7.6 Feynman rules for fermions
To apply Wick’s theorem we have to generalise the definitions of the time-ordered product and
the normal product towards anticommuting operators. We make the following defintions:
ˆ (y) for x0 > y0
ψ̂(x)ψ̄
ˆ (y) =
T ψ̂(x)ψ̄
ˆ (y)ψ̂(x) for y0 > x0
−ψ̄
For the normal product we define
: â~†p â~q : = â~†pâ~q ,
: â~q â~†p : = −â~†p â~q ,
where â~†p and â~q are creation and annihilation operators for fermions. Therefore a minus sign
occurs everytime we have to exchange two fermionic operators.
The Lagrange density for the Yukawa theory:
LYukawa = LDirac + LKlein−Gordon − gψ̄ψφ,
LDirac = iψ̄γµ ∂µ ψ − mψ̄ψ,
LKlein−Gordon =
1
1
∂µ φ (∂µ φ) − m2 φ2 .
2
2
We obtain the following Feynman rules in momentum space:
1. For each propagator,
→
p
→
p
=
=
i
p2 − m2 + iε
i(p/ + m)
p2 − m2 + iε
2. For each vertex,
= −ig
3. For each external point,
←
p
= 1
←
p
= ū(p, λ)
77
←
p
= v(p, λ)
→
p
= v̄(p, λ)
→
p
= u(p, λ)
4. Impose momentum conservation at each vertex.
5. Integrate over each undetermined momentum;
Z
d4 p
(2π)4
6. Divivde by the symmetry factor S.
7. For each closed fermion loop a factor of (−1).
7.7 Rules for traces over Dirac matrices
In evaluating the amplitude squared we encounter trace over Dirac matrices:
Theorem 1:
Tr γµ γν = 4gµν
Proof:
Tr γµ γν + Tr γν γµ = 2gµν Tr 1 = 8gµν
Theorem 2:
Tr γµ1 γµ2 ...γµ2n = gµ1 µ2 Tr γµ3 ...γµ2n − gµ1 µ3 Tr γµ2 γµ4 ...γµ2n + gµ1 µ4 Tr γµ2 γµ3 γµ5 ...γµ2n − ...
Proof:
{γµ , γν } = 2gµν
+ cyclic property of the trace
Theorem 3:
Tr γµ1 γµ2 ...γµ2n−1 = 0
Proof:
Tr γµ = Tr γ5 γ5 γµ = −Tr γ5 γµ γ5 = −Tr γµ
Theorem 4:
Tr γµ γν γρ γσ γ5 = 4iεµνρσ.
78
8 Quantum field theory via path integrals
Review of the canonical formalism for the quantisation of field theories: We are given a Lagrange
density
L (φ, ∂µφ) = L0 (φ, ∂µ φ) + Lint(φ, ∂µ φ),
which can be split into a “free” part L0 (bilinear in the fields) and a part Lint describing the interactions. Each term in Lint contains at least three fields.
The momentum density conjugate to φ(x) is given by
π(x) =
∂L
.
∂φ̇(x)
The Hamilton function H and the Hamiltonian H are given by
H =
Z
3
d x π(x)φ̇(x) − L =
Z
d 3x H .
In the canonical formalism the field φ(x) and the conjugate momentum π(x) become operators.
In the Schrödinger picture the operators are time-independent. We postulate canonical (anti)commutation relations. For bosons we require
φ̂S (t0,~x), π̂S (t0 ,~y) = iδ3 (~x −~y),
φ̂S (t0 ,~x), φ̂S (t0 ,~y) = [π̂S (t0,~x), π̂S (t0 ,~y)] = 0.
For fermions we have (recall that the conjugate momentum is ∂L /∂(∂0 ψ) = iψ† ):
o
n
ψ̂S (t0,~x), iψ̂†S (t0,~y) = iδ3 (~x −~y),
n
o
{ψ̂S (t0 ,~x), ψ̂S (t0,~y)} =
iψ̂†S (t0 ,~x), iψ̂†S (t0 ,~y) = 0.
To change from the Schrödinger picture to the Heisenberg picture we have the formulae
φ̂H (t,~x) = eiĤ(t−t0 ) φ̂S (t0,~x)e−iĤ(t−t0 ) ,
π̂H (t,~x) = eiĤ(t−t0 ) π̂S (t0 ,~x)e−iĤ(t−t0 ) .
We can also split the Hamiltonian into a “free” part and a piece describing the interactions:
Ĥ = Ĥ0 + Ĥint .
We define the field operator in the interaction picture as
φ̂I (t,~x) = eiĤ0 (t−t0 ) φ̂S (t0 ,~x)e−iĤ0 (t−t0 ) .
79
We have the following relation between the field operators in the Heisenberg picture and the
interaction picture:
φ̂H (x) = Û † (t,t0)φ̂I (x)Û(t,t0),
where



Zt1




Û(t1,t0 ) = T exp −i dt ĤI (t)
.


t0
8.1 Trouble with the canonical quantisation of gauge bosons
The canonical formalism worked fine for the quantisation of spin 0 and spin 1/2 particles. Let
us now consider spin 1 particles. As a first example we consider the photon field without any
interactions with fermions. The Lagrange density is
1
4
L (Aµ, ∂µ Aν) = − Fµν F µν ,
where
Fµν = ∂µ Aν − ∂ν Aµ .
The canonical momentum conjugate to Aµ is given by
Πµ =
∂L
= −F0µ .
∂Ȧµ
The canonical commutation relation would be given by
Âµ (t0,~x), Π̂ν (t0,~y) = igµν δ3 (~x −~y).
If we set µ = ν = 0 we obtain
On the other hand, we have
Â0 (t0,~x), Π̂0(t0,~y)
= iδ3 (~x −~y).
Π0 (t0 ,~y) = −F00 = 0.
Thus the simple-minded application of the canonical quantisation fails. The problem is related
to the invariance of the Lagrange density under the gauge transformation
Aµ (x) → Aµ (x) + ∂µ Λ(x).
One possibility to circumvent the problem is to eliminate the freedom of gauge transformations
by putting constraints on the field Aµ . One adds a gauge-fixing term to the Lagrange density:
1
4
L (Aµ, ∂µ Aν ) = − Fµν F µν −
80
2
1
∂µ Aµ
2ξ
Remark: The Lagrange density is no longer gauge-invariant. The canonical momentum is now
given by
1
Πµ = −F0µ − g0µ (∂ν Aν ) .
ξ
The above ad-hoc recipe works fine for abelian gauge theories like QED, but fails for non-abelian
gauge theories, which are needed for the strong and weak interactions.
For a deeper understanding of the problem and its solution it is simpler to switch to a second
method for the quantisation of field theories: quantisation in the path-integral formalism.
It should be noted that with an elaborate mathematical formalism it is possible to quantise nonabelian gauge fields also in the canonical formalism.
8.2 Path integrals
We start from the well-known Gaussian integral:
−1/2
(2π)
Z∞
−∞
1 2
dy exp − ay
= a−1/2 .
2
This is a one-dimensional integral. We may generalise this formula to a n-dimensional integral
as follows: Let A be a real symmetric positive definite matrix. Then
−n/2
(2π)
Z∞
−∞
1 T
dy1 ...dyn exp − ~y A~y
= (det A)−1/2 .
2
We recall that the exponential of a matrix A is defined by the Taylor series
∞
exp (A) =
1
∑ n! An
n=0
and that the logarithm of matrix A is defined is defined as a matrix such that
exp (ln A) = A.
We have the formula
ln det A = Tr ln A.
This formula is most easily proved by diagonalising the matrix A. Therefore we may re-write
our n-fold Gaussian integral as
(2π)
−n/2
Z∞
−∞
1 T
1
dy1 ...dyn exp − ~y A~y
= exp − Tr ln A .
2
2
81
We now generalise this formula in two steps: In the first step we go from a finite number of
variables y j with j ∈ {1, ..., n} to an infinite countable number of variables φ j with j ∈ N. In
the second step we go from a countable number of variables φ j to an over-countable number of
variables φ(x) with x ∈ R. In this way we arrive at a path integral:
Z
Z
Z
1
1
dx dy φ(x)A(x, y)φ(y)
= [det A (x, y)]− 2
D φ exp −
2
1
= exp − Tr ln A
2
The factors of (2π) on the left-hand side have been absorbed into the path integral measure D φ.
Let us look at a simple example:


− 21
Z
Zt f
2
2
1
d
d
D η(t) exp −
η(t) − 2 + ω2 η(t) = det − 2 + ω2
2
dt
dt
ti
To calculate the determinant we solve the eigenvalue problem
d2
2
− 2 + ω ψn (t) = λn ψn (t).
dt
One finds
λn =
and therefore
d2
2
det − 2 + ω
=
dt
where B is an (infinite) constant.
nπ
t f −ti
2
+ ω2 ,
sinh ω t f − ti
∏ λn = B ω t f − ti ,
n=1
∞
Note: In practice, path integrals are never calculated explicitly !
Up to now we only considered integrals, where the argument of the exponential function was
quadratic in the integration variables. We consider now Gaussian integrals with a linear term.
For a finite number of variables one finds
Z∞
1
1 T −1
1 T
−n/2
T
= exp − Tr ln A exp
~w A ~w .
dy1 ...dyn exp − ~y A~y + ~w ~y
(2π)
2
2
2
−∞
Remark: A−1 exists because we assumed A to be positive definite.
Generalisation to an infinite number of components:
Z
Z
Z
Z
1
dx dy φ(x)A(x, y)φ(y) + dx J(x)φ(x) =
D φ exp −
2
Z
Z
1
1
−1
exp − Tr ln A exp
dx dy J(x)A (x, y)J(y)
2
2
82
Let us now discuss what happens if we differentiate these formulae with respect to wi or J(x).
We start with the case of a finite number of variables. Differentiation with respect to wi at ~w = ~0
gives:
Z∞
1
∂
∂
=
(2π)−n/2 dy1 ...dyn exp − ~yT A~y + ~wT~y ...
∂wi1 ∂win
2
Z∞
−∞
~w=0
1 T
−n/2
= (2π)
dy1 ...dyn yi1 ...yin exp − ~y A~y
2
−∞
1
1 T −1 ∂
∂
exp − Tr ln A exp
~w A ~w ...
=
∂wi1 ∂win
2
2
~w=0
In the second line we applied the differentiation under the integral, in the third line we only
substituted the result for the Gaussian integral. The main point of this formula is the equality of
the second and third line: A Gaussian integral without a term ~wT~y in the exponent, but with an
extra factor yi1 ...yin in front of the exponent is equal to the derivative of a Gaussian integral with
a linear term with respect to the quantities wi j .
Let us now generalise this to path integrals. We first have to generalise the ordinary derivative
to a functional derivative:
δ
Z [J(x) + εδ(x − y)] − Z [J(x)]
Z [J(x)] = lim
ε→0
δJ(y)
ε
With this definition we have the following generalisation:
Z
Z
Z
Z
1
∂
∂
=
D φ exp −
dx dy φ(x)A(x, y)φ(y) + dx J(x)φ(x) ...
∂J(x1 ) ∂J(xn )
2
J=0
Z
Z
Z
1
=
D φ φ(x1 )...φ(xn) exp −
dx dy φ(x)A(x, y)φ(y)
2
Z
Z
∂
∂
1
1
−1
=
dx dy J(x)A (x, y)J(y) .
...
exp − Tr ln A exp
∂J(x1 ) ∂J(xn)
2
2
J=0
8.3 Transition amplitudes as path integrals
We start from the relation between Schrödinger and Heisenberg operators:
φ̂H (t,~x) = eiĤ(t−t0 ) φ̂S (t0 ,~x)e−iĤ(t−t0 )
Here, t0 is a reference time, where φ̂H (t0 ,~x) = φ̂S (t0,~x). Let |φ(t j ,~x),tiS be a Schrödinger state,
which for t = t j is an eigenstate of the Schrödinger field operator φ̂S (t0,~x) with eigenvalue φ(t j ,~x):
φ̂S (t0,~x)|φ(t j ,~x),t j iS = φ(t j ,~x)|φ(t j ,~x),t j iS
83
Denote the corresponding Heisenberg state by
|φ(t j ,~x)iH = eiĤ(t−t0 ) |φ(t j ,~x),tiS.
|φ(t j ,~x)iH is an eigenstate of φ̂H (t j ,~x) with eigenvalue φ(t j ,~x):
φ̂H (t j ,~x)|φ(t j ,~x)iH = φ(t j ,~x)|φ(t j ,~x)iH .
Proof:
φ̂H (t j ,~x)|φ(t j ,~x)iH =
iĤ(t j −t0 )
e
−iĤ(t j −t0 )
φ̂S (t0,~x)e
iĤ(t j −t0 )
e
|φ(t j ,~x),t j iS
= eiĤ(t j −t0 ) φ̂S (t0 ,~x)|φ(t j ,~x),t j iS = eiĤ(t j −t0 ) φ(t j ,~x)|φ(t j ,~x),t j iS
= φ(t j ,~x)|φ(t j ,~x)iH
As a short-hand notation we write
|φk ,t j i = |φk (t j ,~x)iH ,
|φk i = |φk (t j ,~x),t j iS .
We are interested in the transition amplitude
hφ f ,t f |φi ,ti i.
|φi ,ti i is a state with eigenvalue φi (ti,~x) at t = ti and similar for |φ f ,t f i. We divide the time
interval (t f − ti ) into n + 1 small sub-intervals with time steps at
ti ,t1 ,t2, ...,tn,t f .
At each intermediate time step we insert a complete set of states
Z
Therefore
hφ f ,t f |φi ,ti i =
Z
D φ j (~x) |φ j ,t j ihφ j ,t j | = 1.
D φn (~x)...
Z
D φ1 (~x) hφ f ,t f |φn ,tnihφn ,tn|φn−1 ,tn−1 i...hφ1,t1|φi ,tii.
Let us study hφ j+1 ,t j+1|φ j ,t j i. If the time interval (t j+1 − t j ) is small, we have
iĤ(t j −t0 )
−iĤ(t j+1 −t0 )
e
|φ j ,t j i = hφ j+1 |e−iĤ(t j+1 −t j ) |φ j i
hφ j+1 ,t j+1 |φ j ,t j i = hφ j+1 |e
≈ hφ j+1 |1 − i(t j+1 − t j )Ĥ|φ j i
Let us first consider a simple case where Ĥ is replaced by a function f (φ̂), which depends only
on φ̂, but not on π̂. Then
hφ j+1 | f (φ̂)|φ j i = f (φ j )hφ j+1 |φ j i = f (φ j )δ(φ j+1 − φ j )
84
We rewrite this as
hφ j+1 | f (φ̂)|φ j i = f (φ j )
Z
Z
3
D π j (~x) exp i d x π j (φ j+1 − φ j )
Note that factors of 2π are absorbed into the integral measure. Next we consider the case where
the Hamiltonian is replaced by a function g(π̂), which depends only on π̂. We introduce a complete set of momentum eigenstates and obtain
Z
Z
3
hφ j+1 |g(π̂)|φ j i =
D π j (~x) g(π j ) exp i d x π j (φ j+1 − φ j )
Thus if Ĥ contains only terms of the form f (φ̂) and g(π̂), its matrix element can be written as
Z
Z
3
hφ j+1 |Ĥ(φ̂, π̂)|φ j i =
D π j (~x) H(φ j , π j ) exp i d x π j (φ j+1 − φ j )
In general this formula will not hold for arbitrary Ĥ, since the order of a product φ̂π̂ matters on
the left side (where φ̂ and π̂ appear as operators), but not on the right side.
If this formula holds, the Hamiltonian is said to be in Weyl order. Any Hamiltonian can be
put into a Weyl order by commuting φ̂’s and π̂’s.
Assuming now Weyl order, we find
Z
Z
3
hφ j+1 |1 − i(t j+1 − t j )Ĥ|φ j i =
D π j (~x) 1 − i(t j+1 − t j )H(φ j , π j ) exp i d x π j (φ j+1 − φ j )
Z
Z
3
≈
D π j (~x) exp i
d x π j (φ j+1 − φ j ) − i(t j+1 − t j )H(φ j , π j )
Z
Z
3
=
D π j (~x) exp i(t j+1 − t j ) d x π j φ̇ j − H (φ j , π j ) .
Therefore
hφ f ,t f |φi ,ti i =
=
Z
Z
D φn (~x)...
D φ(t,~x)
Z
Z
D φ1 (~x) hφ f ,t f |φn ,tnihφn ,tn|φn−1 ,tn−1i...hφ1,t1|φi ,tii
Z
Z
3
D π(t,~x) exp i dt d x πφ̇ − H (φ, π) .
In most cases H (φ, π) will be quadratic in π. We can then complete the square, perform a Wick
rotation and integrate over D π(t,~x). For example, for
1 2 1 ~ 2
H = π + ∇φ +V (φ) ,
2
2
85
L =
1
∂µ φ (∂µ φ) −V (φ)
2
we have
Z
One obtains
Z
1 2 1 ~ 2
4
D π(t,~x) exp i d xπφ̇ − π − ∇φ −V (φ)
2
2
Z
Z
2 1 2 1 2
1
4
=
D π(t,~x) exp i d x − π − φ̇ + φ̇ − ~∇φ −V (φ)
2
2
2
Z
2
1
1
= N exp i d 4 x φ̇2 − ~∇φ −V (φ)
2
2
Z
µ
4 1
= N exp i d x ∂µ φ (∂ φ) −V (φ) .
2
Z
Z
Z
Z
3
hφ f ,t f |φi ,tii =
D φ(t,~x) D π(t,~x) exp i dt d x πφ̇ − H (φ, π)
Z
Z
4
=
D φ(x) exp i d x L .
8.4 Correlation functions
In the previous section we found
hφ f ,t f |φi ,ti i =
Z
Z
4
D φ(x) exp i d x L .
The time interval goes from ti to t f , in all other respects this formula is manifestly Lorentz invariant. Any other symmetries that the Lagrangian may have are preserved by the functional
integral as long as the path integral measure is also invariant under these symmetries. This will
be important for internal symmetries related to gauge groups.
We now would like to make the next step and define quantum field theory through path integrals. We have to find a functional formula to compute correlation functions like
hΩ|T φ̂(x1 )φ̂(x2 )|Ωi,
and to show its equivalence with the canonical operator formalism. Let us consider


Z
D φ(x)φ(x1 )φ(x2 ) exp i
ZT
dt
−T
where the boundary conditions on the path integral are
φ(−T,~x) = φa (~x),
86
Z
d 3 x L (φ) ,
φ(T,~x) = φb (~x).
We break up the functional integral as follows:
Z
D φ(x) =
Z
D φ1 (~x)
Z
Z
D φ2 (~x)
D φ(x)
φ(x01 ,~x)=φ1 (~x),φ(x02 ,~x)=φ2 (~x)
R
The main functional integral D φ(x) is now constrained at times x01 and x02 in addition to the
endpoints −T and T . With this decomposition the extra factors φ(x1 ) and φ(x2 ) in the original
path integral become φ1 (~x) and φ2 (~x). If x01 < x02 :


Z
ZT
D φ(x)φ(x1 )φ(x2 ) exp i
=
Z
D φ1 (~x)
Z
dt
−T
Z
d 3 x L (φ) =
Z
D φ2 (~x)
φ(x01 ,~x)=φ1 (~x),φ(x02 ,~x)=φ2 (~x)
=
Z
=
Z
D φ1 (~x) φ1 (~x)
Z
D φ1 (~x) φ1 (~x)
Z

D φ(x) φ(x1 )φ(x2 ) exp i
Z
D φ2 (~x) φ2 (~x)
φ(x01 ,~x)=φ1 (~x),φ(x02 ,~x)=φ2 (~x)
ZT
dt
Z
−T

D φ(x) exp i
ZT
dt
Z
−T
D φ2 (~x) φ2 (~x) hφb , T |φ2 , x02 ihφ2 , x02 |φ1 , x01 ihφ1, x01 |φa , −T i.

d 3 x L (φ)

d 3 x L (φ)
Since
φ̂H (t,~x)|φ(~x),ti = φ(~x)|φ(~x),ti,
we can turn the fields φi (~x) into Heisenberg operators φ̂i (~x). Using in addition the completeness
relation
Z
we obtain
Z
D φ1 (~x) φ1 (~x)
=
=
Z
D φ1 (~x)
Z
Z
D φ(~x) |φ,tihφ,t| = 1,
D φ2 (~x) φ2 (~x) hφb , T |φ2 , x02 ihφ2, x02 |φ1 , x01 ihφ1 , x01 |φa , −T i =
D φ2 (~x) hφb , T |φ̂(x2 )|φ2, x02 ihφ2 , x02 |φ̂(x1 )|φ1 , x01 ihφ1 , x01 |φa , −T i
φb , T φ̂(x2 )φ̂(x1 ) φa , −T .
If we had the order x01 > x02 we would have found
φb , T φ̂(x1 )φ̂(x2 ) φa , −T .
In summary we have shown that
Z
φ(−T,~x) = φa (~x),
φ(T,~x) = φb (~x)

D φ(x) φ(x1 )φ(x2 ) exp i
ZT
dt
−T
87
Z

d 3 x L (φ) =
φb , T T φ̂(x1 )φ̂(x2 ) φa , −T .
If we replace the Heisenberg states by Schrödinger states
|φ(~x),ti = eiĤt |φ(t,~x),tiS = eiĤt |φ(~x)i,
we have
D
E
φb e−iĤT T φ̂(x1 )φ̂(x2 ) eiĤ(−T ) φa .
As in the canonical operator formalism we can now send T → ∞(1 − iε) to project out the ground
state
e−iĤT |φa i =
lim e−iE0 T |Ωi.
∑ e−iEnT |nihn|φai → hΩ|φai T →∞(1−iε)
n
The phase and the overlap factor drop out if we divide by the same quantity without the field
insertions and we obtain the final formula
R 4
R
D
φ(x)
φ(x
)φ(x
)
exp
i
d
x
L
(φ)
1
2
R
R
Ω T φ̂(x1 )φ̂(x2 ) Ω =
lim
.
D φ(x) exp [i d 4 x L (φ)]
T →∞(1−iε)
This expresses the two-point correlation function in terms of path integrals. For higher correlation functions one obtains
R 4
R
D
φ(x)
φ(x
)...φ(x
)
exp
i
d
x
L
(φ)
1
n
R
R
.
lim
Ω T φ̂(x1 )...φ̂(xn ) Ω =
D φ(x) exp [i d 4 x L (φ)]
T →∞(1−iε)
Let us now introduce the “generating functional”
Z[J(x)] = N
Z
Z
4
D φ(x) exp i d x L (φ) + J(x)φ(x) .
We have
Ω T φ̂(x1 )...φ̂(xn ) Ω =
(−i)n δn Z[J(x)] .
Z[0] δJ(x1 )...δJ(xn) J=0
The functional Z[J] generates all Green functions:
Z[J] = Z[0] ∑
n
in
n!
Z
d 4 x1 ...d 4xn Ω T φ̂(x1 )...φ̂(xn ) Ω J(x1 )...J(xn).
8.5 Fermions in the path integral formalism
8.5.1 Grassmann numbers
Ordinary number commute:
xi , x j = 0.
88
The Grassmann algebra consists of anti-commuting numbers
θi , θ j = 0.
The differentiation is defined by
d
θ = 1.
dθ
d
1 = 0,
dθ
If we have several Grassmann variables, we first bring the Grassmann variable we want to differentiate immediately after the differentiation operator. Thus we have
∂
θ1 ...θ j ...θm = (−1) j−1 θ1 ...θ̂ j ...θm ,
∂θ j
where the hat indicates that the corresponding variable has to be omitted. Note that
∂ ∂
∂ ∂
F = −
F.
∂θi ∂θ j
∂θ j ∂θi
The Taylor expansion of a function F(θ) depending on a Grassmann variable θ is given by
F(θ) = F0 + F1 θ.
The differential dθ is also a Grassmann variable:
{θ, dθ} = 0.
Integration over a Grassmann variable is defined by
Z
Z
dθ = 0,
dθθ = 1.
Multiple integrals are defined by iteration:
Z
dθ1 dθ2 F(θ1 , θ2 ) =
Z
dθ1
Z
dθ2 F(θ1 , θ2 ) .
Let us now consider
Z
dθ1 dθ2 ...dθnd θ̄1 d θ̄2 ...d θ̄n exp θ̄i Ai j θ j =
Z
n
1
θ̄i Ai j θ j
=
dθ1 dθ2 ...dθnd θ̄1 d θ̄2 ...d θ̄n 1 + θ̄i Ai j θ j + ... +
n!
Z
n
1
=
dθ1 dθ2 ...dθnd θ̄1 d θ̄2 ...d θ̄n
θ̄i Ai j θ j
n!
Z
n(n−1)
1
=
dθ1 dθ2 ...dθnd θ̄1 d θ̄2 ...d θ̄n (−1) 2
∑ Ai1 j1 ...Ain jn θ̄i1 ...θ̄in θ j1 ...θ jn
n!
i1 ,...,in , j1 ,..., jn
89
=
Z
dθ1 dθ2 ...dθnd θ̄1 d θ̄2 ...d θ̄n
=
Z
dθ1 dθ2 ...dθnd θ̄1 d θ̄2 ...d θ̄n (−1)
n(n−1)
1
(−1) 2
∑ εi1...in ε j1... jn Ai1 j1 ...Ain jn θ̄1...θ̄nθ1...θn
n!
i1 ,...,in , j1 ,..., jn
n(n−1)
2
∑
ε j1... jn A1 j1 ...An jn θ̄1 ...θ̄nθ1 ...θn
j1 ,..., jn
= (−1)
n(n−1)
2
∑
ε j1 ... jn A1 j1 ...An jn
j1 ,..., jn
= (−1)
n(n−1)
2
det A.
The limit n → ∞ yields a path integral over a Grassmann field. We thus arrive at the important
formula
det A ∼
Z
D ψ(x) D ψ̄(x) exp
Z
d 4 x d 4 y ψ̄(x)A(x, y)ψ(y).
8.5.2 Path integrals with fermions
For fermions we considered up to now the Lagrange density of free fermions and the Lagrange
density of interacting fermions in the Yukawa model:
LYukawa = LDirac + LKlein−Gordon − gψ̄ψφ,
LDirac = iψ̄γµ ∂µ ψ − mψ̄ψ,
1
1
∂µ φ (∂µ φ) − m2 φ2 .
2
2
With the help of Grassmann numbers we may write down the generating functional
LKlein−Gordon =
Z[J(x), η(x), η̄(x)] =
N
Z
Z
4
D φ(x)D ψ(x)D ψ̄(x) exp i d x L (φ) + J(x)φ(x) + ψ̄(x)η(x) + η̄(x)ψ(x)
Here ψ̄(x), ψ(x), η̄(x) and η(x) are fields of Grassmann nature. As before we obtain the Green
functions by differentiation. For example:
E
2
D 2 Z[J(x), η(x), η̄(x)] (−i)
δ
ˆ β (x2 ) Ω =
Ω T ψ̂α (x1 )ψ̄
Z[0, 0, 0] δη̄α (x1 )δ(−ηβ (x2 )) J=0
The additional minus sign in the differentiation with respect to η(x2 ) comes from anti-commuting
ψ̄(x)η(x) = −η(x)ψ̄(x).
8.6 The reduction formula of Lehmann, Symanzik and Zimmermann
In the canonical operator formalism we stated the formula that the non-trivial part of the S-matrix
can be computed as follows




ZT
lim 0 h~p1~p2 ...|T exp −i dtHI (t) |~pA~pB i0 
~p1~p2 ... iT̂ ~pA~pB =
.
T →∞(1−iε)
−T
90
connected,amputated
We now derive in the path integral formalism the reduction formula of Lehmann, Symanzik
and Zimmermann, which explains why propagators of external legs are amputated. In the path
integrals formalism we already showed that
R 4
R
D
φ(x)
φ(x
)...φ(x
)
exp
i
d
x
L
(φ)
1
n
R
R
,
Ω T φ̂(x1 )...φ̂(xn ) Ω =
D φ(x) exp [i d 4 x L (φ)]
where the boundary conditions on the path integral are
lim φ(−T,~x) = lim φ(T,~x) = 0.
T →∞
T →∞
For the computation of scattering amplitudes we would like to have as boundary condition not
the vacuum but an n particle state. If we assume that interaction are only relevant within a finite
volume, we can take this n particle state as the superposition of n non-interacting one-particle
states. We call such a state an asymptotic state. Asymptotic fields are solutions of the noninteracting theory, e.g free fields. The general solution for the free field theory is given as a
Fourier expansion:
Z
d3k 1 −ikx
†
ikx
√
φasymp (x) =
a(k)e
+ a (k)e
(2π)3 2k0
Note that a(k) and a† (k) are here c-numbers, not operators. With boundary conditions at the
remote past/future:
t → +∞ :
t → −∞ :
Z
d 3k
1
√
a(k)e−ikx ,
3 2k
(2π)
0
Z
3
d k 1 †
√ a (k)eikx .
φasymp (x) =
3 2k
(2π)
0
φasymp (x) =
If we give k0 a small imaginary part k0 → k0 − iε, we can use the general formula for both cases.
If we consider a scalar field theory, the asymptotic field satisfies the Klein-Gordon equation
✷ + m2 φasymp (x) = 0.
Consider now:
Zasymp [J] =
Z
lim φ=φasymp
Z
4
D φ exp i d x L (φ) + J(x)φ(x)
With
L (φ) = L0 (φ) + Lint(φ),
Z
n
Z
∞
1
4
4
i d x Lint (φ) ,
exp i d x Lint (φ) = ∑
n=0 n!
Z
Z
δ
4
4
iφ(x) exp i d y L0 (φ) + J(y)φ(y) =
exp i d y L0 (φ) + J(y)φ(y) ,
δJ(x)
91
one arrives at
Z
δ
4
Zasymp [J] = exp i d xLint −i
δJ(x)
Z
lim φ=φasymp
Z
4
D φ exp i d y L0 (φ) + J(y)φ(y) .
Let us now define the free-field functional Zasymp,0 [J]
Z
Z
4
Zasymp,0 [J] =
D φ̃ exp i d x L0 (φ̃) + J(x)φ̃(x)
lim φ̃=φasymp
and change the integration variables accoridng to
φ̃(x) = φ(x) + φasymp (x).
Then
Z
Zasymp,0 [J] =
lim φ=0
Z
4
D φ exp i d x L0 (φ + φasymp ) + J(x)φ(x) + J(x)φasymp (x) .
Note that now the boundary conditions are
lim φ = 0.
t→±∞
Consider now the example of a scalar field
L0 =
1
1
∂µ φ ∂µ φ − m2 φ2 .
2
2
Then
i
Z
4
d x L0 φ + φasymp
= i
Z
= i
Z
1
d x L0 (φ) − φasymp ✷ + m2 φasymp
2
1
1
2
2
− φ ✷ + m φasymp −
✷ + m φasymp φ
2
2
4
d 4 x L0 (φ) ,
since φasymp satisfies the Klein-Gordon equation:
✷ + m2 φasymp = 0.
In this case we have
Zasymp,0 [J] =
Z
lim φ=0
Z
Z
4
4
D φ exp i d y J(y)φasymp (y) exp i d x L0 (φ) + J(x)φ(x) .
92
We now write
Z
lim φ=0
Z
Z
i
4
4
4
d y d z J(y)∆(y, z)J(z) .
D φ exp i d x L0 (φ) + J(x)φ(x) = exp −
2
For the example discussed above (scalar field theory) we have
Z
∆(x − y) =
d 4 p −ip(x−y)
1
e
.
4
2
(2π)
p − m2 + iε
Since
✷x + m2 ∆(x − y) = −δ(x − y)
and
Z
i
δ
4
4
d y d z J(y)∆(y, z)J(z) =
exp −
δJ(x)
2
Z
Z
i
4
4
4
−i d w ∆(x − w)J(w) exp −
d y d z J(y)∆(y, z)J(z) ,
2
we have
✷x + m
2
Z
Z
i
i
δ
4
4
4
4
d y d z J(y)∆(y, z)J(z) = iJ(x) exp −
d y d z J(y)∆(y, z)J(z) .
exp −
δJ(x)
2
2
Therefore
Z
4
2
δ
δJ(x)
d x φasymp (x) · ✷x + m
Z
δ
4
× exp i d yLint −i
δJ(y)
Z
Z
4
×
D φ exp i d z (L0 (φ) + J(z)φ(z)) .
Zasymp [J] = exp
lim φ=0
Let us now define
Z [J] =
Z
lim φ=0
Z
4
D φ exp i d z L (φ) + J(z)φ(z) .
Define the Green functions as functional derivatives of Z[J]:
Gn (x1 , ..., xn) = 0 T φ̂(x1 )...φ̂(xn ) 0
(−i)n
δn Z [J]
=
Z[0] δJ(x1 )...δJ(xn) J=0
93
Then
in
Zasymp [0] = ∑
n!
Z
d 4 x1 ...d 4xn φasymp (x1 )...φasymp (xn ) ✷x1 + m2 ... ✷xn + m2 Gn (x1 , ..., xn).
Define now the Fourier transform of the Green functions by
n
G (x1 , ..., xn) =
Z
d 4 p1 d 4 pn −i ∑ p j x j
...
e
(2π)4 δ4 (p1 + ... + pn ) G̃n (p1 , ..., pn)
(2π)4 (2π)4
and the truncated (amputated) Green function in momentum space by
G̃ntrunc (p1 , ..., pn)
=
i
2
p 1 − m2
−1
...
i
2
p n − m2
−1
G̃n (p1 , ..., pn).
Then
Zasymp [0] =
1
∑ n!
Z
Z
d4 p j
(2π)4 δ4
(2π)4
∑ pk
G̃ntrunc (p1 , ..., pn)
d 4 x1 ...d 4 xn e−i ∑ p j x j φasymp (x1 )...φasymp (xn ).
Consider
Z
Z
Z
i
d 3k 1 h
−ikx
†
ikx
√
a(k)e
+
a
(k)e
(2π)3 2k0
Z
i
2π h
=
d3k √
a(k)δ4 (p + k) + a† (k)δ4 (p − k)
2k0
d 4 x e−ipx φasymp (x) =
d 4 x e−ipx
For k0 > 0 and p0 > 0 only the second term contributes. For k0 > 0 and p0 < 0 only the first term
contributes. One obtains
1
Zasymp [0] = ∑
n!
Z
d3 p j
1
p
(2π)4 δ4
3
(2π)
|2p0 |
94
∑ pk
G̃ntrunc (p1 , ..., pn) ∏ a(−pi ) ∏ a† (p j ).
9 Gauge theories
We have seen that electrodynamics can be described by a gauge potential Aµ (x).
1
4
L (Aµ, ∂µ Aν) = − Fµν F µν ,
where
Fµν = ∂µ Aν − ∂ν Aµ .
This Lagrange density is invariant under the gauge transformation
Aµ (x) → Aµ (x) − ∂µ Λ(x).
We can write this gauge transformation also as
with
Aµ (x) → U (x) Aµ (x) + i∂µ U †(x),
U (x) = e−iΛ(x) .
The gauge symmetry is given by a U (1) group: This is an abelian group, whose elements can be
parameterised with a coordinate ϕ as follows:
e−iϕ ,
0 ≤ ϕ < 2π.
This is obviously a group:
e−iϕ1 · e−iϕ2 = e−i(ϕ1 +ϕ2 ) ,
−1
e−iϕ
= eiϕ .
It is also a one-dimensional compact manifold (e.g. the circle line).
9.1 Lie groups und Lie algebras
A Lie group is a group G which is also an analytic manifold such that the mapping (a, b) → ab−1
of the product manifold G × G into G is analytic.
A Lie algebra over a commutative ring K is a K-module A together with a mapping x ⊗ y → [x, y]
such that for x, y, z ∈ A:
[x, x] = 0,
[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0.
Elements of a Lie group are written in terms of the generators as
g = exp (−iT a αa ) .
95
The generators T a satisfy a Lie algebra, e.g. the commutators of generators are linear combinations of the generators, i.e.
[T a , T b ] = i f abc T c .
Note that in the mathematical literature the convention for the definition of the generators is usually such that no explicit factors of i appear in the formulae above. The convention used in the
physical literature (which is adopted here) ensures that the generators for the unitary groups are
hermitian matrices.
Examples of Lie groups:
- GL(n, R), GL(n, C): group of non-singular n × n matrices with n2 real parameters (GL(n, R)),
respectively 2n2 real parameters (GL(n, C)).
- SL(n, R), SL(n, C): det A = 1; SL(n, R) has n2 − 1 real parameters; SL(n, C) has 2(n2 − 1) real
parameters.
- O(n) : RRT = 1.
- SO(n): RRT = 1 and det R = 1.
- U (n): UU † = 1; n2 real parameters.
- SU (n): UU † = 1 and detU = 1; n2 − 1 real parameters.
- Sp(n): Invariance group of
n
∑
j=1
x j y j+n − x j+n y j .
A Lie algebra is simple if it is non-Abelian and has no non-trivial ideals.
A Lie algebra is called semi-simple if it has no non-trivial Abelian ideals.
A simple Lie algebra is also semi-simple.
Semi-simple groups are a direct product of simple groups. The compact simple Lie algebras are
An
Bn
Cn
Dn
=
=
=
=
SU (n + 1),
SO(2n + 1),
Sp(n),
SO(2n).
The exceptional groups are
E6 , E7 , E8 , F4 , G2 .
96
9.2 Special unitary Lie groups
We discuss here SU (N). The standard normalisation is
1
Tr T a T b = TR δab = δab
2
As an example we consider first the group SU (2). This group has three generators I 1, I 2 und I 3 ,
which are proportional to the Pauli matrices:
1 0 1
1 0 −i
1 1 0
1
2
3
I =
, I =
, I =
.
2 1 0
2 i 0
2 0 −1
As a further example we consider SU (3). Here we have eight generators λa , a = 1, ..., 8, which
are called Gell-Mann matrices.






0 1 0
0 −i 0
1 0 0
1
1
1
λ1 =  1 0 0  , λ2 =  i 0 0  , λ3 =  0 −1 0  ,
2
2
2
0 0 0
0 0 0
0 0 0






0 0 1
0 0 −i
0 0 0
1
1
1
λ4 =  0 0 0  , λ5 =  0 0 0  , λ6 =  0 0 1  ,
2
2
2
1 0 0
i 0 0
0 1 0




0 0 0
1 0 0
1
1
λ7 =  0 0 −i  , λ8 = √  0 1 0  .
2
2 3
0 i 0
0 0 −2
The Fierz identity reads for SU (N):
Tiaj Tkla
1
1
δil δ jk − δi j δkl .
=
2
N
Proof: T a and the unit matrix form a basis of the N × N hermitian matrices, therefore any hermitian matrix A can be written as
A = c0 1 + ca T a .
The constants c0 and ca are determined using the normalisation condition and the fact that the
T a are traceless:
1
Tr(A) ,
N
= 2Tr (T a A) .
c0 =
ca
Therefore
Alk
2Tiaj Tkla +
1
δi j δkl − δil δ jk
N
97
= 0.
This has to hold for an arbitrary A, therefore the Fierz identity follows. Useful formulae involving
traces:
1
1
a
a
Tr (XY ) − Tr (X ) Tr (Y ) ,
Tr (T X ) Tr (T Y ) =
2
N
1
1
a
a
Tr (X ) Tr(Y ) − Tr (XY ) .
Tr (T X T Y ) =
2
N
From
[T a , T b ] = i f abc T c
one derives by multiplying with T d and taking the trace:
i
h i f abc = 2 Tr T a T b T c − Tr T b T a T c
This yields an expression of the structure constants in terms of the matrices of the fundamental
representation. We can now calculate for the group SU (N) the fundamental and the adjoint
Casimirs:
N2 − 1
δi j ,
2N
= Nδad .
(T a T a )i j = CF δi j =
f abc f dbc = CA δad
9.3 Yang-Mills theory
C.N. Yang and R.L.Mills1 suggested 1954 a generalisation towards non-abelian gauge groups.
The field strength tensor is now given by
a
Fµν
= ∂µ Aaν − ∂ν Aaµ + g f abc Abµ Acν ,
where a is an index running from 1 to N 2 − 1 for a SU (N) gauge group. The Lagrange density
reads:
1 a aµν
F .
L = − Fµν
4
The Lagrange density is invariant under the local transformations
i
a a
a a
T Aµ (x) → U (x) T Aµ (x) + ∂µ U †(x)
g
with
U (x) = exp (−iT a θa (x)) .
The action is given by the integral over the Lagrange density:
S =
1 C.N.
Z
Yang and R.L. Mills, Phys. Rev. 96, (1954), 191
98
d 4x L
9.4 Quantisation of gauge theories
Let us examine closer the quantisation of gauge theories. The Lagrange density is given by
1
4
a aµν
F ,
LQCD = − Fµν
with
a
Fµν
= ∂µ Aaν − ∂ν Aaµ + g f abc Abµ Acν .
This Lagrange density is nicely invariant under local gauge transformations
i
a a
a a
T Aµ (x) → U (x) T Aµ (x) + ∂µ U †(x),
g
but we are not yet happy: If we try to calculate the gluon propagator we have to invert a certain
matrix and it turns out that this matrix is singular. Something is going wrong. Let´s look again
at the generating functional:
Z
Z
4
a
µ
a
Z[J] =
D Aµ (x) exp i d x L + Aµ (x)Ja (x)
The path integral is over all possible gauge field configurations, even the ones which are just
related by a gauge transformation. These configuration describe the same physics and it is sufficient to count them only ones. Technically this is done as follows: Let us denote a gauge
transformation by
U (x) = exp −iT b θb (x) .
The gauge transformation is therefore completely specified by the functions θb (x). We denote
by Aaµ (x, θb ) the gauge field configuration obtained from Aaµ (x) through the gauge transforamtion
U (x):
i
a a
a a
T Aµ (x, θb ) = U (x) T Aµ (x) + ∂µ U †(x),
g
Aaµ (x, θb ) and Aaµ (x) are therefore gauge-equivalent configurations. From all gauge-equivalent
configurations we are going to pick the one, which satisfies for a given Gµ and Ba (x) the equation
Gµ Aaµ (x, θb ) = Ba (x).
Let us assume that this equation gives a unique solution θb for a given Aaµ . (This is not necessarily
always fulfilled, cases where a unique solution may not exist are known as the Gribov ambiguity.)
Let ~α = (α1 , ..., αn) a n-dimensional vector and let the
gi = gi (α1 , .., αn), i = 1, ..., n,
99
be functions of α j . Then
Z
n
∏ dα j
j=1
!
!
n
∂gi
∏ δ (gi(α1, ..., αn)) det ∂α j
i=1
= 1.
Proof: We change the variables from α j to
βi = gi (α1 , .., αn).
Then
Z
n
∏ dα j
j=1
!
!
n
∂gi
∏ δ (gi(α1, ..., αn)) det ∂α j
i=1
=
Z
n
∏ dβ j
j=1
!
n
∏ δ (βi)
i=1
!
= 1.
We generalise this to the continuum. For a gauge theory with a single generator we obtain:
µ
Z
δG Aµ (x, θ(x))
µ
D θ(x) δ G Aµ (x, θ(x)) − B(x) det
= 1.
δθ(y)
For a gauge theory with n generators we find
Z
∏ D θb (x)
b
δn Gµ Aaµ (x, θb (x)) − Ba (x) det MG = 1,
where
δGµ Aaµ (x, θc (x))
.
=
δθb (y)
(MG (x, y))ab
Remark: θb are coordinates of the Lie group:
b
U = exp −iT θb
As the Lie group is also a manifold, we can integrate over the manifold. With the coordinates
above, the invariant meausure is given by
∏ D θb .
b
The integral measure dg is called a left invariant meausure, if
Z
dg f (g0 g) =
Z
dg f (g)
for arbitrary elements g and g0 of the group G. A meausure is called right invariant, if
Z
dg f (gg0) =
100
Z
dg f (g).
In general, right and left invariant meausures are not necessarily equal. However, it is known that
they are equal for compact groups, simple groups and semi-simple groups.
Remark 2: If the gauge fixing condition is chosen such that
Gµ Aaµ (x, θb ) − Ba (x) = 0
is linear in θ, then the functional derivative
δGµ Aaµ (x, θc (x))
δθb (y)
will be independent of θ and we may take the determinant in front of the integral
det MG
Z
∏ D θb (x)
b
δn Gµ Aaµ (x, θb (x)) − Ba (x)
We now consider
Z[0] =
Z
D Aaµ (x)
= 1.
Z
4
a
exp i d x L (Aµ(x))
and insert the gauge-fixing equation
Z[0] =
=
Z
D Aaµ (x)
Z
∏ D θb(x) δn
b
Z
4
a
× exp i d x L (Aµ(x))
=
Z
∏ D θb(x)
b
Z
Gµ Aaµ (x, θb (x)) − Ba (x) det MG (Aaµ (x, θb (x)))
D Aaµ (x) det MG (Aaµ (x, θb (x))) δn Gµ Aaµ (x, θb (x)) − Ba (x)
Z
4
a
× exp i d x L (Aµ(x, θb (x)))
=
=
Z
∏ D θb(x)
b
Z
D Aaµ (x, θb (x)) det MG (Aaµ (x, θb (x))) δn Gµ Aaµ (x, θb (x)) − Ba (x)
Z
4
a
× exp i d x L (Aµ(x, θb (x)))
!Z
Z
∏ D θb (x)
D Aaµ (x) det MG (Aaµ (x)) δn
b
Gµ Aaµ (x) − Ba (x)
Z
4
a
exp i d x L (Aµ (x))
Here we used the gauge invariance of the action, of det MG and of D Aaµ (x). The integral over all
gauge-transformations
Z
∏ D θb(x)
b
101
is the just an irrelevant prefactor, which we neglect in the following. We then obtain
Z
Z
a
a
n
µ a
a
4
a
Z[0] =
D Aµ (x) det MG (Aµ (x)) δ G Aµ (x) − B (x) exp i d x L (Aµ (x))
This functional still depends on Ba (x). As we are not interested in any particular choice of Ba (x),
we average over Ba (x) with weight
Z
i
4
a
d x (B (x)Ba (x))
exp −
2ξ
and obtain
Z
Z
i
4
a
D B (x) Z[0] exp −
d x (B (x)Ba (x)) =
2ξ
a
Z
Z
1 a
4
a
=
D B (x)
δ
exp i d x L (Aµ (x)) − B (x)Ba (x)
2ξ
Z
Z
1
Gµ Aaµ (x) (Gν Aν a (x)) .
=
D Aaµ (x) det MG exp i d 4 x L (Aaµ(x)) −
2ξ
a
Z
D Aaµ (x) det MG
n
Gµ Aaµ (x) − Ba (x)
We now consider as new generating functional
Z
Z
ν
1
a
4
a
µ a
a
µ
Z[J] =
D Aµ (x) det MG exp i d x L (Aµ (x)) −
G Aµ (x) (G Aν a (x)) + Aµ (x)Ja (x) .
2ξ
We observe that Z[J] contains a term we could expect from a naive fixing of the gauge
−
In addition, the determinant
1
Gµ Aaµ (x) (Gν Aν a (x)) .
2ξ
det MG
appears in front of the exponent.
Various gauges are:
• Lorenz gauge or covariant gauge: Gµ = ∂µ .
(MG (x, y))ab = δab ✷ − g f abc ∂µ Acµ δ4 (x − y)
• Coulomb gauge: Gµ = (0, ~∇).
(MG (x, y))ab =
δab ∇2 − g f abc~Ac~∇ δ4 (x − y)
102
• Axial gauge: Gµ = nµ , where nµ is a constant four-vector.
(MG (x, y))ab = δab n · ∂ − g f abc n · Ac δ4 (x − y)
We would like to exponentiate the determinant. In the treatment of fermions within the pathintegral formalism we had the formula
Z
det A ∼
We write this as
det A =
Z
D ψ(x) D ψ̄(x) exp
Z
D ψ(x) D ψ̄(x) exp −i
d 4 x d 4 y ψ̄(x)A(x, y)ψ(y).
Z
d 4 x d 4 y ψ̄(x)A(x, y)ψ(y).
Applying this to det MG :
det MG =
Z
Z
4
a
ab
b
D c (x) D c̄ (x) exp i d x c̄ (x) −MG c (x) .
b
a
Specialising to the covariant gauge Gµ = ∂µ one obtains
Z
Z
b
b
a
4
a
ab
abc µ c
det MG =
D c (x) D c̄ (x) exp i d x c̄ (x) −δ ✷ + g f ∂ Aµ c (x)
Z
Z
b
b
a
4
a
µ ab
=
D c (x) D c̄ (x) exp i d x c̄ (x) −∂ Dµ c (x) ,
where
Dab
= δab ∂µ − g f abc Acµ
µ
is the covariant derivative. In the Lorenz gauge we obtain finally
Z
D Aaµ (x)
Z
Z[J, ξ, ξ̄] =
D cb (x) D c̄a (x)
Z
1 µ a ν a
4
a
a
µ ab
exp i d x L (Aµ(x)) −
∂ Aµ (x) (∂ Aν (x)) + c̄ (x) −∂ Dµ cb (x)
2ξ
i
+Aaµ (x)Jaµ (x) + c̄a (x)ξa (x) + ξ̄a (x)cb (x) .
The quantity
LGF = −
is called the gauge-fixing term, the quantity
1 µ a ν a
∂ Aµ (∂ Aν )
2ξ
b
LFP = −c̄a ∂µ Dab
µ c
the Faddeev-Popov term.
103
9.5 The Lagrange density for the fermion sector
Example: The Lagrange density for a free electron (e.g. no interactions) is given by
LF = ψ̄(iγµ∂µ − me )ψ.
We are now looking for a Lagrange density for the fermionic sector, which remains invariant
under gauge transformations. Recall that under a gauge transformation a fermion field ψi (x)
transforms as
Ui j (x) = exp (−iT a θa (x)) ,
ψi (x) → Ui j (x)ψ j (x),
ψ̄i (x) → ψ̄ j (x)U ji† (x).
θa (x) depends on the space-time coordinates x. For an infinitessimal gauge transformation we
have
ψi (x) → (1 − iT a θa (x)) ψ j (x).
We immediately see that a fermion mass term
−mψ̄(x)ψ(x)
is invariant under gauge transformations. (Note however that in the standard model the fermion
masses are generated through the Yukawa couplings to the Higgs field.) But the term involving
derivatives is not gauge invariant:
iψ̄(x)γµ ∂µ ψ(x) → iψ̄(x)U † (x)γµ ∂µ (U (x)ψ(x)) = iψ̄(x)γµ ∂µ ψ(x) + iψ̄(x)γµ U † (x)∂µU (x) ψ(x) .
{z
}
|
extra
The solution comes in the form of the covariant derivative
where the gauge field transforms as
T a Aaµ (x)
Then the combination
Dµ = ∂µ − igT a Aaµ (x),
i
a a
→ U (x) T Aµ (x) + ∂µ U †(x).
g
iψ̄(x)γµ Dµ ψ(x) = iψ̄(x)γµ ∂µ − igT a Aaµ (x) ψ(x)
i
a a
†
†
µ
→ iψ̄(x)U (x)γ ∂µ − igU (x) T Aµ (x) + ∂µ U (x) U (x)ψ(x)
g
= iψ̄(x)γµ ∂µ ψ(x) + iψ̄(x)γµ U † (x)∂µU (x) ψ(x)
h
i
+gψ̄(x)γµ T a Aaµ (x)ψ(x) + iψ̄(x)γµ ∂µU †(x) U (x) ψ(x)
is invariant.
The Lagrange density for the fermion sector:
Lfermions =
∑
fermions
ψ̄(x) iγµ Dµ − m ψ(x).
104
9.6 Feynman rules for QED and QCD
Expanding the Lagrange density into terms bilinear in the fields and interaction terms. As an
example consider the gluonic part of the Lagrange density:
1
4
a aµν
LQCD = − Fµν
F −
1 µ a 2
(∂ Aµ ) + LFP + Lfermions
2ξ
where
a
Fµν
= ∂µ Aaν − ∂ν Aaµ + g f abc Abµ Acν .
This yields
2 1
1
∂µ Aaν − ∂ν Aaµ − (∂µ Aaµ )2
4
2ξ
1 2 eab a b ecd cµ dν abc
a
bµ cν
−g f
∂µ Aν A A − g f Aµ Aν f A A
4
+LFP + Lfermions .
LQCD = −
Terms bilinear in the fields define the propagators. Terms with three or more fields define interaction vertices.
For QED (photons and electrons) the Lagrange density is given by
LQED = ψ̄(i∂/ − me )ψ −
2 1
1
∂µ Aν − ∂ν Aµ − (∂µ Aµ )2 + eψ̄γµ Aµ ψ.
4
2ξ
In the QED case, the Faddeey-Popov ghosts have no interactions and behave as non-interacting
free particles. We may ignore them in the QED case.
9.6.1 Propagators
Terms bilinear in the fields yield propagators. Consider
Lbilinear (x) =
1
boson,complex
φi (x)Piboson,real
(x)φ j (x) + χ∗i (x)Pi j
(x)χ j (x) + ψ̄i (x)Pifermion
(x)ψ j (x),
j
j
2
where φi denotes a set of real boson fields (one degree of freedom), χi denotes a set of complex
boson fields (two degrees of freedom), and ψi denotes a set of fermion fields. The boson fields
may be scalar or vector fields. P is a matrix operator that may contain derivatives and must have
an inverse. P is taken to be a hermitian operator (real symmetric operator in the case of real
boson fields):
P† = P
105
Define the inverse of P by
∑ Pi j (x)P−1
jl (x − y)
= δil δ4 (x − y),
j
and its Fourier transform by
Pi−1
j (x)
=
Z
d 4 k −ik·x −1
e
P̃i j (k).
(2π)4
Then the propagator is given by
∆F (k)i j =
Z
d 4 xeik·x h0|T (φi (x)φ j (0)|0i = i P̃−1 (k)
The propagator of a scalar particle:
ij
.
From
P(x) = −✷ − m2
we found already
P̃(k) =
1
k 2 − m2
and therefore
∆F (k) =
i
k 2 − m2
.
The propagator of a fermion: We start from
P(x) = i∂/ − m.
Then
(i∂/x − m)
Z
d 4 k −ik·(x−y) −1
e
P̃ (k) =
(2π)4
Z
d4k
(k/ − m) e−ik·(x−y) P̃−1 (k) =
(2π)4
Therefore
(k/ − m) P̃−1 (k) = 1
and
∆F (k) = i
106
k/ + m
.
k 2 − m2
Z
d 4 k −ik·(x−y)
e
.
(2π)4
The propagator of a gauge boson:
In Lorentz gauge the gauge-fixing term is given by
LGF = −
1 µ 2
(∂ Aµ )
2ξ
and therefore
1 µ ν
∂ ∂ .
P(x) = ✷g − 1 −
ξ
µν
Then
P(x)
Z
d 4 k −ik·(x−y) −1
e
P̃ (k) =
(2π)4
Z
d4k 2
1 kµ kν −ik·(x−y) −1
µν
k −g + 1 −
e
P̃ (k).
(2π)4
ξ k2
We have for
1
Mµν = −gµν + 1 −
ξ
kµ kν
kµ kν
and Nµν = −gµν + (1 − ξ) 2
2
k
k
the following relation:
Mµλ N λν = gµν .
Therefore
kµ kν
i
∆F (k) = 2 −gµν + (1 − ξ) 2 .
k
k
9.6.2 Vertices
A general term in Lint (x) has the form
Lint (x) =
Z
d 4 x1 ...d 4 xn αi1 ...in (x, x1 , ..., xn)φi1 (x1 )...φin (xn ).
For the vertex we define
αi1 ...in (x, x1 , ..., xn) =
Z
d 4 k1 d 4 kn −ik1 (x−x1 )−...−ikn (x−xn )
α̃i1 ...in (k1 , ..., kn).
...
e
(2π)4 (2π)4
α̃ contains a factor ik jµ for every derivative ∂/∂x jµ acting on a field with argument x j . The vertex
is then given by
I(k1, ..., kn) = i
∑
permutations
107
(−1)P α̃i1 ...in (k1 , ..., kn).
The summations are over all permutations of indices and momenta. The momenta are taken to
flow inward.
Example: The quark-gluon vertex
i µ
Lint = gψ̄ γ
(T a )i j Aaµ ψ j
=
Z
4
d x1
Z
4
d x2
× gγµ (T a )i j ψ̄i (x1 )ψ j (x2 )Aaµ (x3 ).
Then
Z
d 4 x3 δ4 (x − x1 )δ4 (x − x2 )δ4 (x − x3 )
α̃(k1 , k2 , k3 ) = gγµ (T a )i j
and
I = igγµ (T a )i j .
Example 2: The three-gluon vertex
Lint = −g f abc ∂µ Aaν Abµ Acν
Z
=
Z
=
d 4 x1
d 4 x1
Z
Z
d 4 x2
d 4 x2
Z
Z
d 4 x3 δ4 (x − x1 )δ4 (x − x2 )δ4 (x − x3 ) −g f abc ∂νx1 gµλ Aaµ (x1 )Abν (x2 )Acλ (x3 )
d 4 x3 Aaµ (x1 )Abν (x2 )Acλ (x3 )g f abc gµλ ∂νx1 δ4 (x − x1 )δ4 (x − x2 )δ4 (x − x3 ).
Therefore
α(x, x1 , x2 , x3 ) = g f abc gµλ ∂νx1 δ4 (x − x1 )δ4 (x − x2 )δ4 (x − x3 )
Z
Z
Z
d 4 k2
d 4 k3 −ik1 (x−x1 ) −ik2 (x−x2 ) −ik3 (x−x3 )
d 4 k1
e
e
e
(2π)4 (2π)4 (2π)4
Z
Z
Z
d 4 k1
d 4 k2
d 4 k3 abc µλ ν −ik1 (x−x1 ) −ik2 (x−x2 ) −ik3 (x−x3 )
=
g f g ik1 e
e
e
(2π)4 (2π)4 (2π)4
= g f abc gµλ ∂νx1
and
α̃(k1 , k2 , k3 ) = g f abc gµλ ik1ν .
Then
I = i
∑
permutations
α̃(k1 , k2 , k3 ) = g f abc (k2 − k3 )µ gνλ + (k3 − k1 )ν gλµ + (k1 − k2 )λ gµν .
108
9.6.3 List of Feynman rules
Propagators:
The propagators for the gauge bosons are in the Feynman gauge (ξ = 1).
gauge bosons gluon
−igµν
δab
k2
Aaµ
−igµν
k2
photon Aµ
fermions
quarks
p/+m
ψi i p2 −m2 δi j
i pp2/+m
−m2
leptons ψ
i ab
δ
k2
ca
ghosts
Vertices:
Quark-gluon-vertex:
igγµ Tiaj
3-gluon-vertex:
k1 ,µ,a
k2 ,ν,b
k3 ,λ,c
4-gluon-vertex:
g f abc (k2 − k3 )µ gνλ + (k3 − k1 )ν gλµ + (k1 − k2 )λ gµν
ρ,d
µ,a
λ,c
ν,b
109
h
i
−ig2 f abe f ecd gµλ gνρ − gµρ gνλ + f ace f ebd gµν gλρ − gµρ gλν + f ade f ebc gµν gλρ − gµλ gνρ
Gluon-ghost-vertex:
k,a
µ,b
q,c
−g f abc kµ
Fermion-photon-vertex:
ieQγµ
Additional rules:
An integration
Z
d 4k
(2π)4
for each loop.
A factor (−1) for each closed fermion loop.
Symmetry factor: Multiply the diagram by a factor 1/S, where S is the order of the permutation group of the internal lines and vertices leaving the diagram unchanged when the external
lines are fixed.
External particles:
Outgoing fermion: ū(p)
Outgoing antifermion: v(p)
Incoming fermion: u(p)
Incoming antifermion: v̄(p)
Gauge boson: εµ (k)
110
Polarisation sums:
∑ u(p, λ)ū(p, λ)
= p/ + m,
∑ v(p, λ)v̄(p, λ)
= p/ − m,
λ
λ
∑ ε∗µ(k, λ)εν(k, λ)
= −gµν +
λ
kµ kν
kµ nν + nµ kν
− n2
.
kn
(kn)2
Here nµ is an arbitrary four vector. The dependence on nµ cancels in gauge-invariant quantities.
Using Weyl spinors, a convenient choice of polarisation vectors for the gauge bosons is given by
hq − |γµ |k−i
√
,
2hqki
hq + |γµ |k+i
√
ε−
,
µ (k, q) =
2 [kq]
ε+
µ (k, q) =
where qµ is an arbitrary light-like reference momentum. The dependence on qµ cancels in gaugeinvariant quantities. The polarisation vectors satisfy:
µ
ε±
= 0,
µ (k, q)k
µ
ε±
= 0.
µ (k, q)q
ε+ · ε+
+
ε · ε
∗
− ∗
= ε− · ε−
= 0.
ε+
µ
∗
∗
= ε−
µ.
111
= −1,
10 Spontaneously broken gauge theories
The concept of gauge theories allowed us to describe successfully quantum electrodynamics and
quantum chromodynamcis, the quantum theories of the electromagnetic and the strong force.
Both theories are characterised by the fact, that the particles which mediate the forces (photons
and gluons) are massless particles. This is required by gauge invariance. In fact, a naive mass
term for the gauge bosons in the Lagrangian
Lmass = m2 Aaµ Aa µ
is not invariant under gauge invariance
i
a a
→ U (x) T Aµ (x) + ∂µ U †(x).
g
On the other hand it is an experimental fact, that the W -bosons and the Z-boson have non-zero
masses. As we do not want to abandon the concept of gauge theories, we face the problem on how
to incoporate massive gauge bosons into gauge theories. The solution is provided by the concept
of spontaneously broken gauge theories, also known under the name “Higgs mechanism”. To
start the discussion let us consider a simple physical system with a complex coordinate φ and a
potential
1 2
V (φ) = m2 |φ|2 + λ |φ|2 .
4
2
The potential has a harmonic term m2 |φ|2 and an anharmonic term 41 λ |φ|2 . For m2 > 0 and
λ > 0 has an absolut minimum at φ = 0. In classical mechanics the ground state would therefore
be φ = 0. This is nothing new.
T a Aaµ (x)
Imagine now that the potential is given by
1 2 2
V (φ) = −µ |φ| + λ |φ|
,
4
2
with
2
µ2 > 0,
λ > 0.
Then the potential has the shape of a mexican hat, and φ = 0 corresponds to a local maxima. The
potential has a minima for
2µ2
.
λ
The minimas are described by a circle in the complex plane. The ground state of the physical
system will be one point of this circle, with no preference for any particular point. Without loss
of generality we can choose this point to lie along the positive real axis. Therefore we face the
situation, that the potential has a rotational symmetry around the point φ = 0, while the ground
state has not. This is the concept of a spontaneously broken symmetry. In general one speaks
about a spontaneously broken symmetry, if the Lagrangian of a theory has a certain symmetry,
which is not preserved in the ground state of the theory.
|φ|2 =
112
10.1 The Higgs boson
The standard model is based on the gauge group
SU (3) ⊗ SU (2) ⊗U (1),
where SU (3) is the gauge group of the strong interactions, SU (2) the gauge group associated
to the weak isospin and U (1) the gauge group associated to the hypercharge. This group is not
identical to the gauge group of quantum electrodynamics. To avoid confusions, one often writes
UY (1) for the group related to the hypercharge and Uel−magn (1) for the gauge group of QED.
In the standard model, the electroweak sector SU (2) ⊗ UY (1) is spontaneously broken down to
Uel−magn (1). We now study the spontaneously symmetry breakdown in detail.
Within the standard model one assumes an additional complex scalar field, transforming as the
fundamental representation of SU (2) and having hypercharge Y = 1. In the weak isospin space
we can write the field as a two-vector with complex entries:
!
φ+ (x)
φ(x) =
.
√1 (v + H(x) + iχ(x))
2
φ+ (x) is a complex field (two real components). The three components φ(x) and χ(x) are absorbed as the longitudinal modes of Wµ± and Zµ . H(x) is the Higgs field.
The Lagrange density of the Higgs sector
LHiggs =
The covariant derivative is given by
†
Dµ φ (Dµ φ) −V (φ) + LYukawa .
Y
Dµ = ∂µ − igI aWµa − ig′ Bµ ,
2
where I a = 21 σa (σa are the Pauli matrices) and we have Y = 1 for the Higgs doublet. (Note that
our g′ = −g1 (Hollik).)
The Higgs potential is given by
1 † 2
V (φ) = −µ φ φ + λ φ φ .
4
2 †
For µ2 > 0 (and λ > 0) we have spontaneous symmetry breaking. In that case the potential has a
minimum for
φ† φ =
2µ2 v2
= .
λ
2
113
We have
v = 2
r
µ2
.
λ
We write
φ(x) =
√1
2
†
φ (x) =
φ+ (x)
(v + H(x) + iχ(x))
!
,
1
φ (x), √ (v + H(x) − iχ(x)) .
2
−
Then
Dµ φ
†
Dµ φ physical
=
1
1
∂µ H∂µ H + g2 v2 Wµ1Wµ1 +Wµ2Wµ2
2
8
′2
1 2
Bµ
g
−gg′
3
+ v Bµ ,Wµ
.
Wµ3
−gg′ g2
8
The subscript “physical” indicates that we ignored contributions involving the φ+ (x) or χ(x)fields. We define
1
Wµ± = √ Wµ1 ∓ iWµ2
2
and
Aµ
Zµ
=
cos θW sin θW
− sin θW cos θW
Bµ
Wµ3
,
with
cos θW = q
We then obtain
Dµ φ
†
We therefore have
Dµ φ physical
g
g′
sin θW = q
.
2
2
′
g +g
,
g2 + g′ 2
1 vg 2
1
∗
∗
Wµ+ Wµ+ + Wµ− Wµ−
∂µ H∂µ H +
=
2
2 2
q
2
0 0
1 v
Aµ
2
2
′
+
g +g
Aµ , Zµ
.
0 1
Zµ
2 2
v
mW = g,
2
v
mZ =
2
114
q
g2 + g′ 2 .
It is not too difficult to show that the Higgs mass is given by
r
λ
v.
mH =
2
The gauge couplings g and g′ are related to the elementary electric charge by
gg′
e = q
,
g2 + g′ 2
or
g=
e
,
sin θW
g′ =
e
.
cos θW
10.2 Yukawa couplings
The Higgs sector generates also the fermion masses through Yukawa couplings. We discuss this
mechanism first in a simplified model without flavour mixing. The full standard model including
flavour mixing is discussed in the next chapter.
Spin 1/2 particles are described by four-component spinors ψ(x). With the chiral projectors
P± =
1
(1 ± γ5 )
2
we define left- and right-handed spinors:
ψ± (x) = P± ψ(x).
The fermions in the standard model can be grouped into three families:

 
 

t
c
u
 d   s   b 

 
 

 νe  ,  νµ  ,  ντ  .
τ
µ
e
The families differ only by the masses of their members.
The quantum numbers of the fermions in the electro-weak sector:
The left-handed components (uL , dL ) and (νL , eL ) transform as the fundamental representation
under the SU (2) group. The right-handed components uR , dR , νR and eR transform as a singlet
under the SU (2) group.
In detail one has:
115
I3
Y
Q
uL
1
2
1
3
2
3
dL
− 12
1
3
νL
1
2
eL
− 12
I3
Y
Q
uR
0
4
3
2
3
− 31
dR
0 − 23
− 13
−1
0
νR
0
0
0
−1
−1
eR
0
−2
−1
The electric charge is given by the Gell-Mann-Nishijima formula:
Q = I3 +
Y
2
Remark: The table contains a right-handed neutrino, which does not interact with any other particle.
The Yukawa couplings are given by
n
o
LYukawa = ∑ −λd ūL , d¯L φdR − λu ūL , d¯L φC uR − λe (ν̄L , ēL ) φeR − λν (ν̄L , ēL ) φC νR + h.c.
families
where the charge-conjugate Higgs field is given by
φC = 2iI 2 φ∗ = iσ2 φ∗ =
0 1
−1 0
√1
2
φ− (x)
(v + H(x) − iχ(x))
!
=
√1 (v + H(x) − iχ(x))
2
−φ− (x)
!
Example:
−λd
!
φ+ (x)
ūL , d¯L φdR + h.c. = −λd ūL , d¯L
dR + h.c.
√1 (v + H(x) + iχ(x))
2
0
vλd
¯
dR + interaction terms + h.c.
= − √ ūL , dL
1
2
vλd
= − √ d¯L dR + interaction terms + h.c.
2
Therefore the Yukawa couplings generate the masses of the fermions. From the above example
we obtain
1
md = √ vλd .
2
116
10.3 Feynman rules in the electroweak sector
As with any gauge theory, we also have to fix the gauge for the electroweak sector. A useful
gauge fixing condition is given in the electroweak sector by the ’t Hooft gauge (also called Rξ gauge):
LGF =
1 µ
1
1 µ 2
(∂ Zµ − mZ ξZ χ)2 −
(∂ Aµ )
− (∂µWµ+ − imW ξW φ+ )(∂µWµ− + imW ξW φ− ) −
ξW
2ξZ
2ξγ
|
{z
} |
{z
}
SU(2)
U(1)
ξ = 0 corresponds to Landau gauge, ξ = 1 to the Feynman gauge. φ+ , φ− and χ are called the
would-be Goldstone fields.
We now list the most important Feynman rules for the electroweak sector.
The propagators for the W - and Z-bosons are given in ’t Hoofts Rξ=1 gauge.
gauge bosons photon
Higgs sector
Aµ
−igµν
k2
W-boson Wµ±
−igµν
2
k2 −mW
Z-boson
Zµ
−igµν
k2 −m2Z
Higgs
H
i
k2 −m2H
Fermion-Z-vertex:
where
ie
γµ v f − a f γ5 ,
2 sin θW cos θW
v f = I3 − 2Q sin2 θW ,
a f = I3 .
Fermion-W-vertex:
ie
γµ (1 − γ5 )V jk
2 2 sin θw
√
Fermion-Higgs-vertex:
ie m f
2 sin θw mW
117
11 Flavour mixing
We now come to the final ingredient of the standard model: flavour mixing. Let us consider the
electroweak sector, in particular the coupling of quarks to the electroweak gauge bosons. Recall
the Lagrange density for the quark sector:
µ
uL
µ
′ µ
′
′
¯
¯
+ ūR iγ Dµ uR + dR iγ Dµ dR
Lfermions = ∑
ūL , dL iγ Dµ
dL′
families
For reasons, which will become clear later, we put a prime on all d-type quark fields. The
Lagrange density is obtained by replacing the ordinary derivative ∂µ with the covariant derivative
Y
Dµ = ∂µ − igI aWµa − ig′ Bµ .
2
This is required by gauge invariance. This Lagrange density does not allow for mixing between
the various quark flavours.
On the other hand the Yukawa couplings are given by
n
o
LYukawa = ∑ −λd ūL , d¯L φdR − λu ūL , d¯L φC uR + h.c.
families
where the charge-conjugate Higgs field is given by
φC = iσ2 φ∗ = 2iI2 φ∗ .
Note that now the prime is missing on the d-type quark fields. We have seen that the Yukawa
terms lead to mass terms for the fermions:
−λd ūL , d¯L φdR − λu ūL , d¯L φC uR + h.c. =
vλu
vλd
− √ d¯L dR − √ ūL uR + h.c. + interaction terms
2
2
However, the Yukawa couplings are not constrained by any gauge symmetry and we could allow
for flavour mixing in the Yukawa terms. In fact, nature has chosen this possibility. We therefore
consider a general mass term of the form
Lmass =
∑
d¯L′′ Md dR′′ + ū′′L Mu u′′R + h.c.
families
where Md and Mu are (arbitrary) complex 3 × 3 matrices in family space. A matrix M can be
diagonalised by a biunitary transformation
V −1 MW = M̃,
118
where M̃ is a diagonal matrix.
Proof: Using the polar decomposition, M can be written as
M = HU,
where H is hermitian and U is a unitary matrix. H can be diagonalised by a unitary matrix V :
V −1 HV = M̃,
therefore W = U −1V .
The gauge part of the Lagrange density
µ
u
L
µ
′
µ
′
′
+ ūR iγ Dµ uR + d¯R iγ Dµ dR
Lfermions = ∑
ūL , d¯L iγ Dµ
dL′
families
is invariant under the rotations with respect to the family index:
uL
uL
→ SL
dL′
dL′
uR → SR,u uR
dR → SR,d dR
Using this freedom we have with
Mu = Vu M̃uWu−1 ,
∑
families
Md = Vd M̃dWd−1 ,
d¯L′′Vd M̃d Wd−1 dR′′ + ū′′LVu M̃u Wu−1 u′′R + h.c. =
|{z}
| {z } |{z} | {z }
d¯L′ Vu−1Vd
dR′
ū′L
u′R
d¯L′ Vu−1Vd M̃d dR′ + ū′L M̃u u′R + h.c.
|
{z }
families
∑
d¯L
Vu−1Vd describes the quark mixing and is a unitary 3 × 3 matrix:
VCKM = Vu−1Vd .
Note that
¯
d ′ = VCKM d and d¯′VCKM = d.
A unitary n × n matrix is decribed by n2 real parameters, out of these
n(n + 1)
2
119
are phases. For 3 × 3 matrices we have three angles and six phases. We still have the freedom to
redefine our fields by a unitary diagonal matrix:
 iφ

e 1 0
0
ψ →  0 eiφ2 0  ψ
0
0 eiφ3
This can be used to eliminate 2n − 1 phases, e.g. five out of the six phases for three generations.
This leaves one “physical” phase in the CKM-matrix.
Standard parameterisations:
The CKM-matrix connects the weak eigenstates (d ′ , s′ , b′ ) with the mass eigenstates (d, s, b):

 ′ 
 
Vud Vus Vub
d
d
 s′  =  Vcd Vcs Vcb   s 
b′
Vtd Vts Vtb
b
Standard parametrisation:

VCKM

c12 c13
s12 c13
s13 e−iδ
=  −s12 c23 − c12 s23 s13 eiδ c12 c23 − s12 s23 s13 eiδ s23 c13  ,
s12 s23 − c12 c23 s13 eiδ −s23 c12 − s12 c23 s13 eiδ c23 c13
with ci j = cos θi j and si j = sin θi j .
Wolfenstein parametrisation:

VCKM
Neutrino mixing:

2
λ
Aλ3 (ρ − iη)
1 − λ2


2
4
= 
+O λ .
Aλ2
−λ
1 − λ2
Aλ3 (1 − ρ − iη) −Aλ2
1
In the lepton sector one uses for Dirac neutrinos the lepton mixing matrix



 ′ 
νe
Ue1 Ue2 Ue3
ν1
 ν′µ  =  Uµ1 Uµ2 Uµ3   ν2 
ν3
Uτ1 Uτ2 Uτ3
ν′τ
ν′e , ν′µ and ν′τ are the weak eigenstates, whereas ν1 , ν2 and ν3 are the mass eigenstates.
120
12 Summary: The Standard Model
To conclude these lectures we summarise the Lagrange density of the standard model. The
Lagrange density for the standard model is split into three parts:
LSM = Lgauge + Lfermions + LHiggs .
The Lagrange density for the gauge bosons:
1
| 4 {z
1
} | 4 {z
1
} | 4 {z
a aµν
a
− Wµν
Lgauge = − Fµν
F
W µνa − Bµν Bµν +LGF + LFP ,
SU(3)
where
SU(2)
U(1)
}
a
abc
Fµν
= ∂µ Aaν − ∂ν Aaµ + g3 fSU(3)
Abµ Acν ,
a
abc
WµbWνc ,
Wµν
= ∂µWνa − ∂νWµa + g fSU(2)
Bµν = ∂µ Bν − ∂ν Bµ .
For SU (3), the indices a, b and c label the generators of SU (3) and run from 1 to 8. For SU (2),
they label the generators of SU (2) and run from 1 to 3. The gauge fixing part (’t Hooft gauge):
1 µ a 2
(∂ Aµ )
2ξg
|
{z
}
LGF = −
SU(3)
1 µ
1
1 µ 2
(∂ Zµ − mZ ξZ χ)2 −
(∂ Aµ )
− (∂µWµ+ − imW ξW φ+ )(∂µWµ− + imW ξW φ− ) −
ξ
2ξZ
2ξ
| W
{z
} | γ {z
}
SU(2)
U(1)
ξ = 0 corresponds to Landau gauge, ξ = 1 to the Feynman gauge. φ+ , φ− and χ are called the
would-be Goldstone fields and have their origin in the Higgs sector. The fields Wµa and Bµ are
related to the Wµ± , Zµ and Aµ fields as follows:
1
Wµ± = √ Wµ1 ∓ iWµ2 ,
2
Bµ
cos θW sin θW
Aµ
.
=
Wµ3
− sin θW cos θW
Zµ
The Faddeev-Popov term for QCD reads:
a
µ
LFP = c̄ −∂
Dab
µ
cb
The covariant derivative in the fundamental representation reads
abc
Dab
= δab ∂µ − g3 fSU(3)
Acµ .
µ
121
In the electroweak sector we have the ghost fields d ± (x), d Z (x) and d γ (x). The Faddeev-Popov
term has the form
LFP = d¯α K αβ d β .
The Lagrange density for the fermion sector:
µ
uL
′
¯
+ ūR iγµ Dµ uR + d¯R′ iγµ Dµ dR′
Lfermions = ∑
ūL , dL iγ Dµ
′
d
L
families
′ µ
νL
′ µ
′
µ
′
+ ν̄R iγ Dµ νR + ēR iγ Dµ eR ,
+ ν̄L , ēL iγ Dµ
eL
∂µ − igT a Aaµ − igI aWµa − ig′ Y2 Bµ , quarks,
Dµ =
∂µ − igI aWµa − ig′ Y2 Bµ ,
leptons.
Note that a right-handed neutrino (with no interactions through Dµ ) has been added.
The Lagrange density of the Higgs sector
†
1 2
LHiggs = Dµ φ (Dµ φ) + µ2 φ† φ − λ φ† φ + LYukawa ,
4
The covariant derivative is given as before by
Y
Dµ = ∂µ − igI aWµa − ig′ Bµ .
2
The Higgs doublet is parameterised as follows:
!
φ+ (x)
,
φ(x) =
√1 (v + H(x) + iχ(x))
2
1
†
−
φ (x) =
φ (x), √ (v + H(x) − iχ(x)) .
2
The Higgs doublet has Y = 1.
The Yukawa couplings are given by
n
o
LYukawa = ∑ −λd ūL , d¯L φdR − λu ūL , d¯L φC uR − λe (ν̄L , ēL ) φeR − λν (ν̄L , ēL ) φC νR + h.c.
families
The CKM-matrix connects the weak eigenstates (d ′ , s′ , b′ ) with the mass eigenstates (d, s, b):

 ′ 
 
Vud Vus Vub
d
d
 s′  =  Vcd Vcs Vcb   s 
b′
Vtd Vts Vtb
b
In the lepton sector one uses for Dirac neutrinos the lepton mixing matrix
 ′ 



νe
Ue1 Ue2 Ue3
ν1
 ν′µ  =  Uµ1 Uµ2 Uµ3   ν2 
Uτ1 Uτ2 Uτ3
ν3
ν′τ
ν′e , ν′µ and ν′τ are the weak eigenstates, whereas ν1 , ν2 and ν3 are the mass eigenstates.
122

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