Introduction to Quantum Field Theory

121 Pages·2003·0.715 MB·English

by  Mulders P.J.

INTRODUCTION TO QUANTUM FIELD
THEORY
P.J. Mulders
Department of Theoretical Physics, Division of Physics and Astronomy,
Faculty of Sciences, Vrije Universiteit,
1081 HV Amsterdam, the Netherlands
E-mail: [email protected]
February 2003 (version 1.1)
Contents
1 Introduction 1
1.1 Quantum (cid:12)eld theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Conventions for three- and four-vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Relativistic wave equations 8
2.1 The Klein-Gordon equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Mode expansion of solutions of the KG equation . . . . . . . . . . . . . . . . . . . . . 9
2.3 Symmetries of the Klein-Gordon equation . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Groups and their representations 12
3.1 The rotation group and SU(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Representations of symmetry groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 The Lorentz group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.4 The generatorsof the Poincar(cid:19)egroup. . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5 The Lorentz group and SL(2;C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.6 Representations of the Poincar(cid:19)egroup . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.6.1 Massive particles: p2 =M2 >0, p0 >0. . . . . . . . . . . . . . . . . . . . . . . 23
3.6.2 Massless particles: p2 =M2 =0, p0 >0. . . . . . . . . . . . . . . . . . . . . . . 24
3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4 The Dirac equation 27
4.1 Spin 1=2 representationsof the Lorentz group . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 General representations of (cid:13) matrices and Dirac spinors . . . . . . . . . . . . . . . . . 29
4.3 Plane wave solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.4 (cid:13) gymnastics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5 Maxwell equations 36
5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6 Classical lagrangian (cid:12)eld theory 38
6.1 Euler-Lagrangeequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.2 Lagrangiansfor spin 0, 1/2 and 1 (cid:12)elds . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.2.1 The scalar (cid:12)eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.2.2 The Dirac (cid:12)eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.2.3 Vector (cid:12)eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.3 Symmetries and conserved (Noether) currents . . . . . . . . . . . . . . . . . . . . . . . 42
1
6.4 Space-time symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.4.1 Translations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.4.2 Lorentz transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.5 (Abelian) gauge theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
7 Quantization of (cid:12)elds 47
7.1 Canonical quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7.2 Field operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7.2.1 The real scalar (cid:12)eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7.2.2 The complex scalar (cid:12)eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
7.2.3 The Dirac (cid:12)eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.2.4 The electromagnetic (cid:12)eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
8 Discrete symmetries 58
8.1 Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
8.2 Charge conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
8.3 Time reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
8.4 Bi-linear combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
8.5 Form factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
8.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
9 Path integrals and quantum mechanics 65
9.1 Time evolution as path integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
9.2 Functional integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
9.3 Time ordered products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
9.4 The interaction picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
9.5 The ground-state to ground-state amplitude . . . . . . . . . . . . . . . . . . . . . . . . 73
9.6 Euclidean formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
9.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
10 Feynman diagrams for scattering amplitudes 77
10.1 Generating functional for scalar (cid:12)elds . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
10.1.1 Generalization to quantum (cid:12)elds . . . . . . . . . . . . . . . . . . . . . . . . . . 77
10.1.2 Z[J] for the free scalar (cid:12)eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
10.1.3 Z[J] in the interacting case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
10.1.4 Connected Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
10.2 Interactions and the S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
10.2.1 The S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
10.2.2 The relation between S and Z[J] . . . . . . . . . . . . . . . . . . . . . . . . . . 85
10.3 Feynman rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
10.3.1 The real scalar (cid:12)eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
10.3.2 Complex scalar (cid:12)elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
10.3.3 Dirac (cid:12)elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
10.3.4 Vector (cid:12)elds and Quantum Electrodynamics. . . . . . . . . . . . . . . . . . . . 91
10.4 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
10.4.1 e(cid:22) scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
10.4.2 e e+ (cid:22) (cid:22)+ scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
(cid:0) (cid:0)
!
10.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
2
11 Scattering theory 95
11.1 kinematics in scattering processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
11.1.1 Phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
11.1.2 Kinematics of 2 2 scattering processes . . . . . . . . . . . . . . . . . . . . . 97
!
11.2 Cross sections and lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
11.2.1 Scattering process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
11.2.2 Decay of particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
11.3 Unitarity condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
11.4 Unstable particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
11.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
12 The standard model 103
12.1 Non-abelian gauge theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
12.1.1 QCD, an example of a nonabelian gauge theory . . . . . . . . . . . . . . . . . . 104
12.1.2 A geometric picture of gauge theories . . . . . . . . . . . . . . . . . . . . . . . 105
12.2 Spontaneous symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
12.2.1 Realization of symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
12.2.2 Chiral symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
12.3 The Higgs mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
12.4 The standard model SU(2) U(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
W Y
(cid:10)
12.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3
Chapter 1
Introduction
1.1 Quantum (cid:12)eld theory
In quantum (cid:12)eld theory the theories of quantum mechanics and special relativity are united. In
quantum mechanics a special role is played by Planck’s constant h, usually given divided by 2(cid:25),
h(cid:22) h=2(cid:25) = 1:054 571 596 (82) 10(cid:0)34 J s
(cid:17) (cid:2)
= 6:582 118 89 (26) 10 22 MeV s: (1.1)
(cid:0)
(cid:2)
In the limit that the action S is much larger than h(cid:22), S h(cid:22), quantum e(cid:11)ects do not play a role
(cid:29)
anymoreandoneisinthe classical domain. Inspecialrelativityaspecialroleisplayedbythe velocity
of light c,
c = 299 792 458 m s(cid:0)1: (1.2)
In the limit that v c one reaches the non-relativistic domain.
(cid:28)
In the framework of quantum mechanics the position of a particle is a well-de(cid:12)ned concept and
the position coordinates can be used as dynamical variables in the description of the particles and
their interactions. The position can in principle be determined at any time with any accuracy, being
an eigenvalue of the position operator. One can talk about states r and the wave function (r) =
j i
r . In this coordinate representation the position operator simply acts as
h jj i
r (r) = r (r): (1.3)
op
The uncertainty principle tells us that in this representation the momentum cannot be fully deter-
mined. The corresponding position and momentum operators do not commute. They satisfy the
well-known (canonical) operator commutation relations
[r ;p ] = ih(cid:22)(cid:14) ; (1.4)
i j ij
where(cid:14) istheKronecker(cid:14)function. Indeed,theactionofthemomentumoperatorisinthecoordinate
ij
representation not as simple. It is given by
p (r) = ih(cid:22)r (r): (1.5)
op (cid:0)
Onecanalsochoosearepresentationinwhichthemomentaoftheparticlesarethedynamicalvariables.
The corresponding states are p and the wave functions ~(p) = p are the Fourier transforms of
j i h jj i
the coordinate space wave functions,
i
~(p) = d3r exp p r (r); (1.6)
(cid:0)h(cid:22) (cid:1)
Z (cid:18) (cid:19)
1
and
d3p i
(r) = exp p r ~(p): (1.7)
(2(cid:25)h(cid:22))3 h(cid:22) (cid:1)
Z (cid:18) (cid:19)
Theexistenceof alimitingvelocity,however,leadstonewfundamentallimitationsonthepossible
measurements of physical quantities. Let us consider the measurement of the position of a particle.
Thispositioncannotbemeasuredwithin(cid:12)niteprecision. Anydevicethatwantstolocatetheposition
of say a particle within an interval (cid:1)x will contain momentum components p h(cid:22)=(cid:1)x. Therefore if
/
we want (cid:1)x h(cid:22)=mc (where m is the rest mass of the particle), momenta of the order p mc and
(cid:20) /
energies of the order E mc2 are involved. It is then possible to create a particle - antiparticle pair
/
andit isno longerclearof whichparticlewearemeasuringthe position. As aresult,we (cid:12)ndthat the
original particle cannot be located better than within a distance h(cid:22)=mc, its Compton wavelength,
h(cid:22)
(cid:1)x : (1.8)
(cid:21) mc
For a moving particle mc2 E (or by considering the Lorentz contraction of length) one has (cid:1)x
! (cid:21)
h(cid:22)c=E. If the particlemomentumbecomesrelativistic,onehasE pcand(cid:1)x h(cid:22)=p, whichsaysthat
(cid:25) (cid:21)
a particle cannot be located better than its de Broglie wavelength.
Thus the coordinatesof a particle cannot act as dynamical variables (since these must have a precise
meaning).
Some consequences are that only in cases where we restrict ourselves to distances h(cid:22)=mc, the
(cid:29)
conceptofawavefunctionbecomesameaningful(albeitapproximate)concept. Foramasslessparticle
one gets (cid:1)x h(cid:22)=p=(cid:21)=2(cid:25), i.e. the coordinates of a photon only become meaningful in cases where
(cid:29)
the typical dimensions are much larger than the wavelength.
Forthemomentumorenergyofaparticleweknowthatina(cid:12)nitetime(cid:1)t,theenergyuncertainty
is given by (cid:1)E h(cid:22)=(cid:1)t. This implies that the momenta of particles can only be measured exactly
(cid:21)
whenonehasanin(cid:12)nitetimeavailable. Foraparticleininteraction,themomentumchangeswithtime
and a measurement over a long time interval is meaningless. The only case in which the momentum
ofaparticlecanbemeasuredexactlyiswhentheparticleisfreeandstableagainstdecay. Inthiscase
the momentum is conserved and one can let (cid:1)t become in(cid:12)nitely large.
The result thus is that the only observable quantities that can serve as dynamical coordinates are
the momenta (and further the internal degrees of freedom like polarizations, ...) of free particles.
These are the particles in the initial and (cid:12)nal state of a scattering process. The theory will not give
an observable meaning to the time dependence of interaction processes. The description of such a
process as occurring in the course of time is just as unreal as classical paths are in non-relativistic
quantum mechanics.
The main problem in Quantum Field Theory is to determine the probability amplitudes be-
tween well-de(cid:12)ned initial and (cid:12)nal states of a system of free particles. The set of such amplitudes
p ;p ; out p ;p ; in = p ;p ; inS p ;p ; in is the scattering matrix or S-matrix.
h 01 02 jj 1 2 i h 01 02 j j 1 2 i
Anotherpoint that needsto be emphasizedisthe meaning of particlein the abovecontext. Actu-
ally, the better name might be ’degree of freedom’. If the energy is low enough to avoid excitation of
internaldegreesoffreedom,anatomisaperfectexampleofaparticle. Infact,itisthebehaviorunder
Poincar(cid:19)etransformationsorinthelimitv cGallileitransformationsthatdeterminethedescription
(cid:28)
of a particle state, in particular the free particle state.
1.2 Units
It is important to choose an appropriate set of units when one considers a speci(cid:12)c problem, because
physical sizes and magnitudes only acquire a meaning when they are considered in relation to each
other. This is true speci(cid:12)cally for the domain of atomic, nuclear and high energy physics, where
the typical numbers are di(cid:14)cult to conceive on a macroscopic scale. They are governed by a few
fundamental units and constants, which havebeen discussed in the previoussection, namely h(cid:22) and c.
2
Table 1.1: Physical quantities and their canonical dimensions d, determining units (energy)d.
quantity quantity in MKS d
time t t=h(cid:22) -1
length l l=h(cid:22)c -1
energy E E 1
momentum p pc 1
mass m mc2 1
area A A=(h(cid:22)c)2 -2
velocity v v=c 0
force F F h(cid:22)c 2
e2 (cid:11)=e2=4(cid:25)(cid:15) h(cid:22)c 0
0
G G=h(cid:22)c5 -2
It turns out to be convenient to work with units such that h(cid:22) and c are set to one. All length,
time and energy or mass units then can be expressed in one unit and powers thereof, for which one
can use energy (see table 1.1). The elementary unit that is most relevant depends on the domain of
applications,e.g. the eVforatomicphysics,the MeVorGeVfornuclearphysicsandthe GeVorTeV
for high energy physics. To convert to other units of length or time we use appropriate combinations
of h(cid:22) and c, e.g. for lengths
h(cid:22)c = 0:197 326 960 2 (77) GeV fm (1.9)
or (when h(cid:22) =c=1)
1 fm = 10(cid:0)15 m = 5:068 GeV(cid:0)1: (1.10)
For areas, e.g. cross sections, one needs
h(cid:22)2c2 = 0:389 379 292 (30) GeV2 mbarn (1.11)
(1 barn = 10 28 m2 = 102 fm2). For times one needs
(cid:0)
h(cid:22) = 6:582 118 89 (26) 10 22 MeV s; (1.12)
(cid:0)
(cid:2)
or (when h(cid:22) =c=1)
1 s = 1:519 1021 MeV(cid:0)1: (1.13)
(cid:2)
Quantities that do not contain h(cid:22) or c are classical quantities, e.g. the mass of the electron m .
e
Quantities that contain only h(cid:22) are expected to play a role in non-relativistic quantum mechanics,
e.g. the Bohr radius, a =4(cid:25)(cid:15) h(cid:22)2=m e2. Quantities that only contain c occur in classical relativity,
0 e
e.g. the electron rest m1ass m c2 and the classical electron radius r = e2=4(cid:25)(cid:15) m c2. Quantities
e e 0 e
that contain both h(cid:22) and c play a role in relativistic quantum mechanics, e.g. the electron Compton
wavelength (cid:21) = h(cid:22)=m c. One can, however, always make use of h(cid:22) and c to simplify or calculate
(cid:0)e e
quantities,suchasthe dimensionless(cid:12)ne structureconstant(cid:11) e2=4(cid:25)(cid:15) h(cid:22)c= 1/137.03599976(50).
0
(cid:17)
1.3 Conventions for three- and four-vectors
The three components of a vector p in 3-dimensional Euclidean space are indicated by an index i,
thus pi, which can be i = x, y, z, or i = 1, 2, 3. The inner product is indicated
3
p q = piqi =piqi: (1.14)
(cid:1)
i=1
X
3
When arepeated index appears,such ason the righthand side of this equation, summation overthis
indexisassumed(Einsteinsummation convention). The inner product of avectorwith itselfgivesits
length squared. For instance, rotations are those real, linear transformations that do not change the
length of a vector. The following tensors are useful,
1 if i=j
(cid:14)ij = (1.15)
0 if i=j,
(cid:26) 6
1 if ijk is an even permutation of 123
(cid:15)ijk = 1 if ijk is an odd permutation of 123 (1.16)
8 (cid:0)
0 otherwise.
<
They are used for instance in th:e scalar product of two vectors, p q = piqj (cid:14)ij = piqi or the cross
(cid:1)
product of two vectors, (p q)i =(cid:15)ijkpjqk. Useful relations are
(cid:2)
(cid:15)ijk (cid:15)imn = (cid:14)jm(cid:14)kn (cid:14)jn(cid:14)km; (1.17)
(cid:0)
(cid:15)ijk (cid:15)ijl = 2(cid:14)kl: (1.18)
We note that for Euclidean vectors and tensors there exist only one type of indices. No di(cid:11)erence
is made between upper or lower. So we could have used all lower indices in the above equations.
Becauseof ourconventionsin Minkowski space,however,it isconvenientto stickto upper indices for
three-vectors.
In special relativity we start with a four-dimensional real vector space E(1,3) with basis n ((cid:22) =
(cid:22)
0,1,2,3). Vectors are denoted x=x(cid:22)n . The length (squared) of a vector is obtained from the scalar
(cid:22)
product,
x2 = x x = x(cid:22)x(cid:23)n n = x(cid:22)x(cid:23)g : (1.19)
(cid:22) (cid:23) (cid:22)(cid:23)
(cid:1) (cid:1)
The quantity g n n is the metric tensor, given by g = g = g = g = 1 (the
(cid:22)(cid:23) (cid:22) (cid:23) 00 11 22 33
(cid:17) (cid:1) (cid:0) (cid:0) (cid:0)
other components arezero). Forfour-vectorsin Minkowski spacewewill use the notationwith upper
indices and write x = (t;x) = (x0;x1;x2;x3), where the coordinate t = x0 is referred to as the time
component, xi are the three space components. Because of the di(cid:11)erent signs occurring in g , it is
(cid:22)(cid:23)
convenient to distinguish lower indices from upper indices. The lower indices are constructed in the
following way, x =g x(cid:23), and are given by (x ;x ;x ;x )=(t; x). One has
(cid:22) (cid:22)(cid:23) 0 1 2 3
(cid:0)
x2 = x(cid:22)x = t2 x2: (1.20)
(cid:22)
(cid:0)
The scalar product of two di(cid:11)erent vectors x and y is denoted
x y = x(cid:22)y(cid:23)g = x(cid:22)y = x y(cid:22) = x0y0 x y: (1.21)
(cid:22)(cid:23) (cid:22) (cid:22)
(cid:1) (cid:0) (cid:1)
Within Minkowski space the real, linear transformations that do not change the length of a four-
vector are called the Lorentz transformations. Therefore it is sensible to give these lengths a special
name, such as eigentime (cid:28) for the position vector (cid:28)2 x2 =t2 x2. The invariant distance between
(cid:17) (cid:0)
two points x and y in Minkowski space is determined from the length ds(cid:22) = (x y)(cid:22). The real,
(cid:0)
linear transformations that leave the length of a vector invariant are called (homogeneous) Lorentz
transformations. The transformationsthat leaveinvariantthe distance ds2 = dt2 (dx2+dy2+dz2)
(cid:0)
between two points are called inhomogeneous Lorentz transformations or Poincar(cid:19)e transformations.
The Poincar(cid:19)etransformations include Lorentz transformations and translations.
Unlike in Euclidean space, the invariant length or distance (squared) is not positive de(cid:12)nite. One
can distinguish:
ds2 >0 (timelike intervals); in this casean inertial system exists in which the two points are at the
(cid:15)
same space point and in that frame ds2 just represents the time di(cid:11)erence ds2 =dt2;
ds2 <0(spacelikeintervals);inthiscaseaninertialsystemexistsinwhichthetwopointsareatthe
(cid:15)
same time and ds2 just represents minus the spatial distance squred ds2 = dx2;
(cid:0)
4
ds2 =0 (lightlike or null intervals); the points lie on the lightcone and they can be connected by a
(cid:15)
light signal.
Many other four vectors can be constructed. Relations independent of a coordinate system are
calledcovariant. Examplesarethescalarproductsabovebutalsorelationslikep(cid:22) =mdx(cid:22)=d(cid:28) forthe
momentum four vector. Note that in this equation one has on left- and righthandside a four vector
because (cid:28) is a scalar quantity! The equation with t =x0 instead of (cid:28) simply would not make sense!
The momentum four vector, explicitly written as (p0;p) = (E;p), is timelike with invariant length
(squared) is
p2 =p p=p(cid:22)p =E2 p2 =m2; (1.22)
(cid:22)
(cid:1) (cid:0)
where m is called the mass of the system.
The derivative @ is de(cid:12)ned @ =@=@x(cid:22) and we have a four vector @ with components
(cid:22) (cid:22)
@ @ @ @ @
(@ ;@ ;@ ;@ )= ; ; ; = ;r : (1.23)
0 1 2 3
@t @x @y @z @t
(cid:18) (cid:19) (cid:18) (cid:19)
It is easy to convince oneself of the signs in the above equation, because one has
@ x(cid:23) =g(cid:23): (1.24)
(cid:22) (cid:22)
Notethatg(cid:23) withoneupperandlowerindexisconstructedalsoviathemetrictensoritself,g(cid:23) =g g(cid:26)(cid:23)
(cid:22) (cid:22) (cid:22)(cid:26)
and is in essence a Kronecker delta, g0 = g1 = g2 = g3 = 1. The length squared of @ is the
0 1 2 3
d’Alembertian operator, de(cid:12)ned by
@2
2 = @(cid:22)@ = r2: (1.25)
(cid:22) @t2 (cid:0)
The value of the antisymmetric tensor (cid:15)(cid:22)(cid:23)(cid:26)(cid:27) is determined in the same way as for (cid:15)ijk, starting
from
(cid:15)0123 =1: (1.26)
(Note that there are di(cid:11)erent conventions around and sometimes the opposite sign is used). The
product of two tensors is given by
g(cid:22)(cid:22)0 g(cid:22)(cid:23)0 g(cid:22)(cid:26)0 g(cid:22)(cid:27)0
(cid:15)(cid:22)(cid:23)(cid:26)(cid:27)(cid:15)(cid:22)0(cid:23)0(cid:26)0(cid:27)0 = (cid:12) g(cid:23)(cid:22)0 g(cid:23)(cid:23)0 g(cid:23)(cid:26)0 g(cid:23)(cid:27)0 (cid:12); (1.27)
(cid:0)(cid:12) g(cid:26)(cid:22)0 g(cid:26)(cid:23)0 g(cid:26)(cid:26)0 g(cid:26)(cid:27)0 (cid:12)
(cid:12) (cid:12)
(cid:12) g(cid:27)(cid:22)0 g(cid:27)(cid:23)0 g(cid:27)(cid:26)0 g(cid:27)(cid:27)0 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) g(cid:23)(cid:23)0 g(cid:23)(cid:26)0 g(cid:23)(cid:27)0 (cid:12)
(cid:12) (cid:12)
(cid:15)(cid:22)(cid:23)(cid:26)(cid:27)(cid:15) (cid:23)0(cid:26)0(cid:27)0 = g(cid:26)(cid:23)0 g(cid:26)(cid:26)0 g(cid:26)(cid:27)0 ; (1.28)
(cid:22) (cid:0)(cid:12) (cid:12)
(cid:12) g(cid:27)(cid:23)0 g(cid:27)(cid:26)0 g(cid:27)(cid:27)0 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:15)(cid:22)(cid:23)(cid:26)(cid:27)(cid:15) (cid:26)0(cid:27)0 = 2(cid:12) g(cid:26)(cid:26)0g(cid:27)(cid:27)0 g(cid:26)(cid:27)0g(cid:27)(cid:12)(cid:26)0 ; (1.29)
(cid:22)(cid:23) (cid:0)(cid:12) (cid:0) (cid:12)
(cid:15)(cid:22)(cid:23)(cid:26)(cid:27)(cid:15) (cid:27)0 = 6g(cid:16)(cid:27)(cid:27)0; (cid:17) (1.30)
(cid:22)(cid:23)(cid:26) (cid:0)
(cid:15)(cid:22)(cid:23)(cid:26)(cid:27)(cid:15) = 24: (1.31)
(cid:22)(cid:23)(cid:26)(cid:27)
(cid:0)
The (cid:12)rst identity, for instance, is easily proven for (cid:15)0123(cid:15)0123 from which the general case can be
obtainedbymakingpermutationsof indicesonthe lefthandsideandpermutationsofrowsorcolumns
on the righthandside. Each of these permutations leads to a minus sign, but more important has the
samee(cid:11)ectonlefthandsideandrighthandside. Forthecontractionofavectorwiththeantisymmetric
tensor one often uses the shorthand notation
(cid:15)ABCD =(cid:15)(cid:22)(cid:23)(cid:26)(cid:27)A B C D : (1.32)
(cid:22) (cid:23) (cid:26) (cid:27)
5
1.4 References
IntheselecturesIwillfollowforsomepartthe bookofRyder[1]. OthertextbooksofQuantumField
Theory that are useful are given in refs [2-8].
1. L.H. Ryder, Quantum Field Theory, Cambridge University Press, 1985.
2. M.E. Peshkin and D.V. Schroeder, An introduction to Quantum Field Theory, Addison-Wesly,
1995.
3. M. Veltman, Diagrammatica, Cambridge University Press, 1994.
4. S. Weinberg, The quantum theory of (cid:12)elds; Vol. I: Foundations, Cambridge University Press,
1995; Vol. II: Modern Applications, Cambridge University Press, 1996.
5. C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, 1980.
1.5 Exercises
Exercise 1.1
(a) The (quantum mechanical) size of the Hydrogen atom is of the order of the Bohr radius, a =
4(cid:25)(cid:15) h(cid:22)2=m e2. Re-expressthisquantityintermsoftheelectronComptonwavelength(cid:21) and1the
0 e (cid:0)e
(cid:12)ne structure constant (cid:11).
(b) Similarly express the (relativistic) classical radius of the electron, r = e2=4(cid:25)(cid:15) m c2 in the
e 0 e
Compton wavelength and the (cid:12)ne structure constant.
(c) Calculate the Compton wavelength of the electron and the quantities under (a) and (b) using
thevalueofh(cid:22)c,(cid:11)andm c2 =0:511MeV.Thisdemonstrateshowacarefuluseofunitscansave
e
a lot of work. One does not need to know h(cid:22), c, (cid:15) , m , e, but only appropriate combinations.
0 e
(d) Use the value of the gravitational constant G=h(cid:22)c5 = 6.71 10 39 GeV 2 to construct a mass
(cid:0) (cid:0)
(cid:2)
M (Planck mass) and a corresponding length r and give their values.
p p
Exercise 1.2
Provethe identity A (B C) = (A C)B - (A B)C using the propertiesof the tensor (cid:15)ijk given
(cid:2) (cid:2) (cid:1) (cid:1)
in section 1.3.
Exercise 1.3
Provethe following relation
(cid:15)(cid:22)(cid:23)(cid:26)(cid:27)g(cid:11)(cid:12) =(cid:15)(cid:11)(cid:23)(cid:26)(cid:27)g(cid:22)(cid:12) +(cid:15)(cid:22)(cid:11)(cid:26)(cid:27)g(cid:23)(cid:12) +(cid:15)(cid:22)(cid:23)(cid:11)(cid:27)g(cid:26)(cid:12) +(cid:15)(cid:22)(cid:23)(cid:26)(cid:11)g(cid:27)(cid:12):
by a simple few-line reasoning[For instance: If (cid:22);(cid:23);(cid:26);(cid:27) is a permutation of 0;1;2;3 the index (cid:11)
f g f g
can only be equal to one of the indices in (cid:15)(cid:22)(cid:23)(cid:26)(cid:27), ...].
Exercise 1.4
Lightcone coordinates for a four vector a, denoted as [a ;a+;a1;a2] or [a ;a+;a ] are de(cid:12)ned as
(cid:0) (cid:0) T
a(cid:6) =(a0 a3)=p2:
(cid:6)
(a) Expressthescalarproducta binlightconecoordinatesandshowinthiswaythatg++ =g =0
(cid:0)(cid:0)
(cid:1)
and g+ =g + =1.
(cid:0) (cid:0)
6