OKAZAKI, Junichiro March 8, 2005 - · Cosmological Physics Theoretical Astrophysics Group, Tokyo...
71
Cosmological Physics Theoretical Astrophysics Group, Tokyo Metropolitan University OKAZAKI, Junichiro March 8, 2005 i
Transcript of OKAZAKI, Junichiro March 8, 2005 - · Cosmological Physics Theoretical Astrophysics Group, Tokyo...
Cosmological Physics
Theoretical Astrophysics Group, Tokyo Metropolitan University
OKAZAKI, Junichiro
March 8, 2005
i
CONTENTS iii
Contents
1 Elementary particles and fields 1
I Quantum Field Theory 3
2 Functional analysis 32.1 Banach space & Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Bose-Einstein, Fermi-Dirac statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 PATH INTEGRAL 8
4 PATH INTEGRAL QUANTIZATION 84.1 Generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2 Propagators and gauge consition in QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
II Renormalization 8
5 RENORMALIZATION 95.1 Divergence in φ4 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95.2 Dimensional regularization of φ4 theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.3 Renormalization of φ4 theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.4 Renormalization group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.5 Divergences and dimensional regularization of QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.6 1-loop renormalization of QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.7 Renormalizability of QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.8 Asymptotic freedom of Yang-Mills theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.9 Renormalization of Yang-Mills theories with spontaneous symmetry breakdown . . . . . . . . . . . . . . . . . . . . 20
III Gauge Theory 20
6 Gauge symmetries and conservation laws 206.1 WIGNER PHASE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.2 NOETHER’S THEOREM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.3 THE GAUGE PRINCIPLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.4 FREE FIELDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.5 NON-ABELIAN GAUGE FIELDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.6 GAUGE DIFFICULTIES WITH QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
7 The weak interaction 247.1 FERMI’S THEORY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
8 Lie Group & Lie Algebra 258.1 LIE GROUP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258.2 LIE ALGEBRA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298.3 Connected component including the identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.4 Connected linear Lie group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318.5 REPRESENTATIONS of G & AG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
9 Bundle 329.1 Some definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329.2 Tangent Space ∼ Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349.3 Differential Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
10 Connections & Curvatures 3710.1 Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3710.2 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
iv CONTENTS
11 Non-Abelian gauge symmetries 3711.1 YANG-MILLS THEORIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3711.2 SU(2) AND WEAK INTERACTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
12 Spontaneous symmetry breaking 3812.1 THE ’MEXICAN HAT’ POTENTIAL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3912.2 CLASSICAL AND QUANTUM SYMMETRY BREAKING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3912.3 GOLDSTONE BOSONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
13 The Electroweak Model 3913.1 GAUGE GROUP AND NEUTRAL CURRENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4013.2 MASSES AND UNIFICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
14 Quantum Chromodynamics (QCD) 4114.1 COLOUR AND SU(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4114.2 THE QCD LAGRANGIAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4114.3 CHIRAL SYMMETRY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4114.4 QUARK MIXING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4214.5 VACUUM POLARIZATION AND ASYMPTOTIC FREEDOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4214.6 THE STRONG CP PROBLEM AND THE AXION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
IV BEYOND THE STANDARD MODEL 44
15 Beyond the standard model 4415.1 GRAND UNIFIED THEORIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4415.2 EMPIRICAL EVIDENCE FOR UNIFICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4515.3 SIMPLE GUT MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4515.4 SUCCESSES AND FAILURES OF GUT MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
16 Neutrino masses and mixing 4516.1 Majorana particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4516.2 NEUTRINO OSCILLATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4616.3 SOLAR NEUTRINOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4616.4 MSW EFFECT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4716.5 SUPERNOVA NEUTRINOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
17 Quantum Gravity 4717.1 SEMICLASSICAL QUANTUM GRAVITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4817.2 HAWKING RADIATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4817.3 FIELD THEORY AT NON-ZERO TEMPERATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4917.4 BLACK HOLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5017.5 ALTERNATIVE INSIGHTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
18 Kaluza-Klein models 5018.1 LAWER-DIMENSIONAL GRAVITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5018.2 THE KALUZA-KLEIN ARGUMENT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5018.3 Ashtekar theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
19 Supersymmetries and beyond 5019.1 SUPERSYMMETRY AND THE Λ PROBLEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5019.2 SUPERGRAVITY, SUPERSTRINGS AND THE FUTURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
V CLASSICAL COSMOLOGY 51
20 The isotropic universe 5120.1 THE ROBERTSON-WALKER METRIC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5120.2 DYNAMICS OF THE EXPANSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5320.3 COMMON BUG BANG MISCONCEPTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5420.4 OBSERVATIONS IN COSMOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5420.5 THE ANTHROPIC PRINCIPLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
CONTENTS v
21 Gravitational lensing 5421.1 BASICS OF LIGHT DEFLECTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5421.2 SIMPLE LENS MODELS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5421.3 GENERAL PROPERTIES OF THIN LENSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5421.4 OBSERVATIONS OF GRAVITATIONAL LENSING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5421.5 MICROLENSING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5421.6 DARK-MATTER MAPPING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
22 The age and distance scales 5422.1 THE DISTANCE SCALE AND THE AGE OF THE UNIVERSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5422.2 METHODS FOR AGE DETERMINATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5422.3 LARGE-SCALE DISTANCE MEASUREMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5422.4 THE LOCAL DISTANCE SCALE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5422.5 DIRECT DISTANCE DETERMINATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5422.6 SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
VI THE EARLY UNIVERSE 55
23 The hot big bang 5623.1 THERMODYNAMICS IN THE BIG BANG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5623.2 RELICS OF THE BIG BANG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5623.3 THE PHYSICS OF RECOMBINATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5623.4 THE MICROWAVE BACKGRAOUND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5723.5 PRIMORDIAL NUCLEOSYNTHESIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5723.6 BARYOGENESIS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
24 Topological defects 5724.1 PHASE TRANSITION IN COSMOLOGY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5724.2 CLASSES OF TOPOLOGICAL DEFECT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5724.3 MAGNETIC MONOPOLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5724.4 COSMIC STRINGS AND STRUCTURE FORMATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
25 Inflationary cosmology 5725.1 GENERAL AARGUMENTS FOR INFLATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5725.2 AN OVERVIEW OF INFLATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5725.3 INFLATION FIELD DYNAMICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5725.4 INFLATION MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5725.5 RELLIC FLUCTUATIONS FROM INFLATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5825.6 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
VII OBSERVATIONAL COSMOLOGY 58
26 Matter in the universe 58
A Measure & Integral 59A.1 Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
B QFT 59
C Representation of finite groups 59C.1 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59C.2 Derived Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60C.3 on Finite group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60C.4 Character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61C.5 Orthogonal relation of Character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62C.6 Fine constant & Bohr radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
D PATH INTEGRAL QUANTIZATION 64D.1 Generating function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64D.2 Propagators and gauge consition in QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
1
In this textbook, gµν = diag(+,−,−,−)
RemarkThere exist 3 signs:
ηµν = [S1]× diag(−1,+1,+1,+1)
Rµαβγ = [S2]×(Γµαγ,β − Γµαβ,γ + ΓµσβΓ
σγα − ΓµσγΓ
σβα
)
Gµν = [S3]× 8πGc4
Tµν
then
Rµν = [S2]× [S3]×Rαµαν
[S1] [S2] [S3]Landau − + +MTW + + +
Weinberg + − −Peebles − + +
This Book − + −
1 Elementary particles and fields
All that is needed is a Lagrangian, and then there is a rela-tively standard procedure to follow. 1
→ why certain Lagrangians are found in nature while oth-ers are not.→ the standard model of particle models.The standard model asserts that the building blocks of
physics are a certain set of fundamental particles from whichthe composite particles seen in experiments are constructed;
table 8.1 The particles of the standard model (p.216)
• spin-1/2 fermions
– leptons
– quarks
• spin-1 boson
vector boson
• spin-0 boson
Higgs boson
Degree of freedom;mass, life-time, charge(Q), baryon number(B), lepton
number(L), spin(J), parity(P ), isospin(I) , I3, strangeness(S),hypercharge(Y )
Q = I3 +Y
2, Y = B + S
LEPTONS (B = 0, S = 1/2)particle charge mass
e (electron) -1 0.511 MeVνe (e-neutrino) 0 < 15 eV
µ -1 0.106 GeVνµ 0 < 0.17 MeVτ -1 1.777 GeVντ 0 < 24 MeV
GAUGE BOSONS
particle charge massγ 0 0
g (gluons) 0 0W ±1 80.3 GeVZ 0 91.2 GeV
QUARKS (B = 1/3, J = 1/2)
flavour name charge mass Iz S C B T
u up 2/3 2-8 MeV 1/2 0 0 0 0d down -1/3 5-15MeV -1/2 0 0 0 0c charm 2/3 1.0-1.6 GeV 1974 0 0 1 0 0s strange -1/3 100-300MeV 0 -1 0 0 0t top 2/3 170-190GeV 1994 0 0 0 0 1b bottom -1/3 4.1-4.5 GeV 1976 0 0 0 -1 0
HADRON — LEPTON — LEPTON
HADRONS
1Our task is making a reasonable Lagrangian!
2 1 ELEMENTARY PARTICLES AND FIELDS
MESONS (qq) m(MeV) IG JPC BARYONS (qqq) m(MeV) I JP (L2S,2J)(B = L = 0) (L = 0, J = 1/2)
π 140 1− 0−+ N 939 1/2 (1/2)+(P11)η 547 0+ 0−+ N 1440 1/2 (1/2)+(P11)ρ 770 1+ 1−− N 1520 1/2 (3/2)−(D13)ω 782 0− 1−− N 1535 1/2 (1/2)−(S11)η′ 958 0+ 0−+ ∆ 1232 3/2 (3/2)+(P33)φ 1020 0− 1−− Λ 1116 0 (1/2)+(P01)ηc 2980 0+ 0−+ Λ 1405 0 (1/2)−(S01)J/ψ 3097 0− 1−− Λ 1520 0 (3/2)−(D03)Υ 9460 0− 1−− Σ 1193 1 (1/2)+(P11)K 494 1/2 0− Σ 1385 1 (3/2)+(P13)K∗ 892 1/2 0− Ξ 1318 1/2 (1/2)+(P11)D 1865 1/2 0− Ξ 1530 1/2 (3/2)+(P13)D∗ 2010 1/2 1− Ω 1672 0 (3/2)+(P03)B 5271 1/2 0− Λc 2285 0? (1/2)+?B∗ 5330 1/2 0− Λb 5641 0? (1/2)+?
I; Isospin, G; G-Parity, J ; Spin, P ; Parity, C; Charge-conj. L; Partial Wave
HADRON ( QUARKS + GLUON )
udd uuddds uds, uds uus
dss uss
=
n pΣ− Σ0,Λ Σ+
Ξ− Ξ0
ddd udd uud uuudds uds uus
dss usssss
=
∆− ∆0 ∆+ ∆++
Σ∗− Σ∗0 Σ∗+
Ξ∗− Ξ∗0
Ω−
∆++(= uuu!) → need other freedom ⇒ ’colour’ (RGB)2
2i.e. ∆++ = uuu
3
Part I
Quantum Field Theory
2 Functional analysis
2.1 Banach space & Hilbert space
Definition 2.1.1 NORME; a vector space, ∀ x ∈ E ‖ · ‖ : E −→ R≥0
1. ‖x‖ ≥ 0, ‖x‖ = 0 ⇐⇒ x = 0
2. ‖αx‖ = |α|‖x‖3. ‖x+ y‖ ≤ ‖x‖+ ‖y‖
Then we call ‖ · ‖ norm , and if it does not satisfy ‖x‖ = 0⇔x = 0, we call it semi-norm.The pair (E, ‖ · ‖E) is called a normed space3.
l2; Formulization of Heisenberg (matrix dynamics)L2; Formulization of Schrodinger (wave dynamics)
Definition 2.1.11 BANACH SPACE(E, ‖ · ‖E); Banach space⇐⇒def
E; complete4 w.r.t. T‖·‖E
N.B. Every finite dimensional normed space is Banachspace.
Remark 2.1.12lp, l∞, c0, Lp, L∞, C0 are Banach spaces.
Cc(R), C∞([0, 1]) are not Banach spaces.
Theorem 2.1.13
1. Normed space lp is complete.
2. Normed space Lp is complete. (1907 R.Riesz, E.Ficher)5
Theorem 2.1.14 The completion theorem(X, d); metric space
=⇒ ∃ (X, d); metric space s.t.
1. X ⊂ X
2. ClX = X
3. X; complete
Further, (X1, d1), (X2, d2) satisfy 1,2 and 3=⇒ ∃ φ : X1 → X2 s.t.
1. bijective
2. for x ∈ X, φ(x) = x
3. for x, y ∈ X1, d1(x, y) = d2(φ(x), φ(y))
Why completeness is so important?→ For example, Plancherel theorem
3Example - normed space ⊂ linear sp. ∩ metric sp.
Definition 2.1.2 lp;
lp =defx = (x1, x2, · · · , xn, · · · ) | ‖x‖p ; finite is a vect. sp. and is normed with ‖x‖p =
def(∑∞n=1 |xn|p)
1p
Definition 2.1.3 l∞;l∞ =
defx = (x1, x2, · · · , xn, · · · ) | ‖x‖∞ ; finite is a vect. sp. and is normed with ‖x‖∞ =
defsup1≤n<∞ |xn|
Definition 2.1.4 c0;c0 =
defx = (x1, x2, · · · , xn, · · · ) | xj −−−−→
j−→∞0 is a vect. sp. and is normed with ‖x‖∞ =
defsup1≤n<∞ |xn|
Definition 2.1.5 Lp;
Lp(X,µ) =def
f : X −→ C | ∫
X|f(x)|pdµ(x); finite
f(x) = 0 µ− a.e. is a vect. sp. and is normed with ‖[f ]‖p =def
(∫X|f(x)|pdµ(x)
) 1p
Definition 2.1.6 L∞;
L∞(X,µ) =def
f : essentially boundedf(x) = 0 µ− a.e. is a vect. sp. and is normed with
‖[f ]‖∞ def= ess.sup|f(x)|def= inf
[α
∣∣∣ µ(
x ∈ X∣∣∣ |f(x)| > α
)= 0
]
Definition 2.1.7 C([0, 1]);C([0, 1]) =
def f : [0, 1] −→ C | continuous is a vect. sp. and is normed with ‖f‖∞ = max0≤x≤1 |f(x)|
Definition 2.1.8 C0;C0(R) =
deff : R −→ R | continuous, |f(x)| −−−−−→
|x|−→∞0 is a vect. sp. and is normed with ‖f‖∞ = supR |f |
Definition 2.1.9 Cc;Cc =
deff ∈ C0 | suppf ; compact
Definition 2.1.10 C∞([0, 1]);C∞([0, 1]) =
deff : [0, 1] −→ R | C∞ − class is a vect. sp. and is normed with ‖f‖∞ = sup0≤x≤1 |f(x)|
4i.e. Any Cauchy sequence converges in E5Riemann integral → not complete!!
4 2 FUNCTIONAL ANALYSIS
For f(t) ∈ L2
F (ω) =def
∫ ∞
∞f(t)e−iωt dt
=⇒ F (ω) ∈ L2
Definition 2.1.15 DENSEE; Banach sp.L; subset of E
L is dense ⇐⇒def
Cl(L) = E
Definition 2.1.16 SEPARABLEBanach sp. E is separable ⇐⇒
def∃ L ; subset of E which is
dense and card(L)≤ ℵ0
Example 2.1.17R is a separable Banach space. In fact,∃ Q (⊂ R) is dense, i.e. Cl(Q) = R and card(Q) = ℵ0.
Theorem 2.1.18X; topological space
X; second-countable =⇒ X; separable
Theorem 2.1.19For metric sp. X, the following 3 conditions are equivalent:
1. 2nd-coutable
2. separable
3. Lindelof sp
Definition 2.1.20 INNER PRODUCT
〈·|·〉; inner product6 on Edef⇐⇒ 〈·|·〉 : E × E −→ C satisfying the following condi-
tions;
1. 〈x|x〉 ≥ 0, 〈x|x〉 = 0 ⇔ x = 0
2. 〈αx|y〉 = α〈x|y〉 for α ∈ C
3. 〈x+ y|z〉 = 〈x|z〉+ 〈y|z〉
4. 〈x|y〉 = 〈y|x〉
Example 2.1.22For x, y ∈ l2〈x|y〉 def=
∑∞n=1 xnyn
For f, g ∈ L2
〈f |g〉 =def
∫Xfg∗dµ(x)
2.1.1 Some theorem
Theorem 2.1.23 Hahn-Banach→ mini-max theorem
Theorem 2.1.24 Ascoli-Arzela
Theorem 2.1.25 Hilbert-Schmidt→ integral eq.
1. 1st Fredholm
∫ β
α
K(x, t)y(t) dt = f(x)
6(S, d); metric space⇐⇒def
S; a set
d : S × S −→ R s.t.
1. ∀ x, y ∈ S, d(x, y) ≥ 0
2. ∀ x, y ∈ S, d(x, y) = 0 ⇐⇒ x = y
3. ∀ x, y ∈ S, d(x, y) = d(y, x)
4. ∀ x, y, z ∈ S, d(x, y) + d(y, z) ≥ d(x, z)
(S, n); normed space⇐⇒def
S; a vector space over K
n : S −→ R s.t.
1. ∀ x ∈ S, n(x) ≥ 0
2. ∀ x ∈ S, n(x) = 0 ⇐⇒ x = 0
3. ∀ x ∈ S, ∀ k ∈ K, n(kx) = |k|n(x)
4. ∀ x, y ∈ S, n(x+ y) ≤ n(x) + n(y)
Remark; d(x, y) = n(x− y) satisfies the above 1-4
(S, i); inner product spade⇐⇒def
S; a vector space over Ci : S × S −→ C s.t.
1. ∀ x ∈ S, i(x, x) ≥ 0
2. ∀ x, y ∈ S, i(x, x) = 0 ⇐⇒ x = 0
3. ∀ x, y ∈ S, α ∈ C, i(αx, y) = αi(x, y)
4. ∀ x, y, z ∈ S, i(x+ y, z) = i(x, z) + i(y, z)
5. ∀ x, y ∈ S, i(x, y) = i(y, x)∗
Remark 2.1.21n(x) =
pi(x, x)
d(x, y) = n(x− y) =pi(x− y, x− y)
2.2 Bose-Einstein, Fermi-Dirac statistics 5
2. 2nd Fredholm
y(x)− µ∫ β
α
K(x, t)y(t) dt = f(x)
Theorem 2.1.26 Winner-Tauber→ prime number theorem
Theorem 2.1.27 Stone→ evolution equation，ポッホナーの定理Lemma 2.1.28 Schwartz’s inequality(E, 〈·|·〉E); an inner product space
|〈x|y〉E | ≤ ‖x‖E‖y‖E for any x, y ∈ E
Theorem 2.1.29(E, 〈·|·〉E); an inner product space
Then
1. ‖x‖E def= (〈x|x〉E)12 satisfies the properties of norm.
2. ‖x+ y‖2E + ‖x− y‖2E = 2(‖x‖2E + ‖y‖2E)
Definition 2.1.30 HILBERT SPACE(E, 〈·|·〉E); Hilbert space7
⇐⇒def
(E, ‖ · ‖E); Banach space, where ‖ · ‖E = (〈·|·〉E)12
Remark 2.1.31 NotationH := a Hilbert space
Theorem 2.1.32 von Neumann∃ inner product 〈·|·〉 on E s.t.
√〈x|x〉E = ‖x‖E
⇐⇒ ‖x+y‖2E+‖x−y‖2E = 2(‖x‖2E+‖y‖2E) for ∀x, y ∈ ETheorem 2.1.33H ; an ∞-dim Hilbert sp.
H ; separable =⇒ ∃ complete orthonormal basis which card.≤ ℵ0
2.2 Bose-Einstein, Fermi-Dirac statistics
2.2.1 Bose statistics(of finite dimension)
[aj , a†k] = δjk
[aj , ak] = 0 (j, k = 1, · · · , N)
Qj ≡ 1√2(aj + a†j)
Pj ≡ 1√2i
(aj − a†j)8
=⇒
[Qj , Qk] = [Pj , Pk] = 0[Qk, Pk] = δjk
(♣)
Theorem 2.2.1 von NeumannQj , Pj which satisfy (♣) are determined uniquely exceptunitary-equivalence.
⇓ (infinite dimension)
|r1, r2, · · · 〉 ≡ [a†j1 ]r1 · · · |0〉 (rk = 0, 1, · · · )
=⇒ card( |r1, r2, · · · 〉 ) = ℵℵ00 = ℵ
2.2.2 Fermi statistics (of finite dimension)aj , a†k = δjk
aj , ak = 0 (j, k = 1, · · · , N)γ2j−1 ≡ aj + a†jγ2j ≡ 1
i(aj − a†j)
9
=⇒ γα, γβ = 2δαβ (♥)
10
Theorem 2.2.2 Jordan, Wignerγα which are irreducible and hermitian, and satisfy (♥) aredetermined uniquely except unitary-equivalence. In addition,they are denoted by 2N × 2N matrix.
⇓ (infinite dimension)
|r1, r2, · · · 〉 ≡ [a†j1 ]r1 · · · |0〉 (rk = 0, 1)
=⇒ card( |r1, r2, · · · 〉 ) = 2ℵ0 = ℵ
Dimension of Hilbert sp. is supposed to be countable when wediscuss the quantum mechanics!
(↔ It includes uncountable unitary-nonequivalent Hilbertsp.)
How should we choose the Hilbert sp. much up to ourpurpose?
2.2.3 Asymptotic world
It is necessary to choose what gives an actual particle imageout of it since ∃∞ nonequivalent Hilbert space over the systemof infinite d.o.f. even if it asks for the commutation relationof a field according to the procedure of canonical form fromLagrangian of the system which is carrying out the interaction.
lSince interaction ⇒ ∃ creation and annihilation of a particle,a clear particle image cannot be made to correspond to theoperator of a field.
It is impossible to build Fock sp. and then to set up theHilbert sp. which serves as a foundation of theory.
Although what is necessary is just to be able to choose theHilbert space which is physically meaningful, without callingat Fock sp., such a method is not found out for the moment.
7
Euclidean sp. ⊂ Hilbert sp. ⊂ Banach sp. ⊂ Normed sp. ⊂ Linear(Vect.) sp.
8They are hermitian.9They are hermitian.
10Clifford algebra of 2N -dim
6 2 FUNCTIONAL ANALYSIS
Therefore, it is necessary to suppose the case where it be-comes the group of free particles naturally, without doing ex-ternal influence, and constitute Fock sp. in order to preparethe Hilbert sp.
The group of such free particles is considered to realizeat the time of sufficiently after or sufficiently before the reac-tion between particles. Such the world is called asymptoticworld.
aspt. wld. → interaction → aspt. wld.
Example 2.2.3 2-bodies problem
V (t) =∫d3X
∫d3r |ψ(X, r, t)|V (r)
limt→±∞
〈ψ(t)|V |χ(t)〉 = 0 (♠)
H0; free Hamiltonian for the systemH := H0 + V (r); total Hamiltonian
limt→±∞
〈ψ(t)|H|χ(t)〉 = limt→±∞
〈ψ(t)|H0|χ(t)〉i.e. We may use H0 instead of H in t→ ±∞.
What should be careful of is that; even though, when werewrite (♠) in Heisenberg picture → V (t) = eiHtV e−iHt
limt→±∞
〈ψ|V (t)|χ〉 = 0 ∀ψ, χindependent of time
, we must not consider V (t) → 0 as an operator because ofψ, χ ’s arbitrariness. That is, the norm of state operated byV (t) does not go to 0, but the matrix element goes to 0. =⇒weak limit
Remark 2.2.4the division of ’free part’ and ’interaction part’ is unknown.
=⇒ self-consistent
2.2.4 Commutation relation
FACT Both normal case and anomalous case lead thesame result.11
2.2.5 Fock space
Define |n〉(n =
∑
k
rk
)s.t.
|n〉 =
def
∑r1+r2+···=n c(r1, · · · )|r1, r2, · · · 〉
‖ |n〉 ‖ =def
1
Completeness
And we presume; For λ1, λ2, · · · ∈ C
limN>M→∞
∥∥∥∥∥N∑
n=M
λn|n〉∥∥∥∥∥ = 0
−→We can definea∞∑n=1
λn|n〉 and hence make up a Hilbert
space.
aIn general, Cauchy seq.⇐=6=⇒ convergent seq.
Since |λn| will be without limit small if n is enlarged, thecontribution from a state with n of infinite size is disregarded.For this reason, it is possible to consider this Hilbert space asthe limit which enlarged enough the number of energy quantaof state space with energy quanta of finite #.
We call this Hilbert space Fock space where |0〉 is calledvacuum.
In the case of∑k rk =∞
Considering Fermi statistics, for instance,Define
αj2k−1 =
defaj2k−1 cos τ + a†j2k
sin τ
αj2k=def−a†j2k−1
sin τ + aj2kcos τ
(k = 1, 2, · · · , τ ∈ R)
=⇒ αjk , α†jk′ = δjkjk′ , αjk , αjk′ = 0
Here define ”vacuum” |0〉τ w.r.t. αjk s.t.
αjk |0〉τ = 0 (k = 1, 2, · · · )
Define
Uk(τ) =def
eτ(a†j2k
a†j2k−1−aj2k−1aj2k
)
then
Uk′aj2k−1U†k′ =
αj2k−1 (k = k′)aj2k−1 (k 6= k′)
Uk′aj2kU†k′ =
αj2k
(k = k′)aj2k
(k 6= k′)
Define
U (N)(τ) =def
N∏
k=1
Uk(τ)
then
|0〉τ = U (N)(τ)|0〉
Ak =def
aj2k−1aj2k, Ck =
defa†j2k
aj2k+ a†j2k−1
aj2k−1
⇓
[Ck, Ak] = −2Ck
[Ck, A†k] = 2Ck
[Ak, A†k] = −Ck + 1
A2k = A† 2
k = 0Ak|0〉 = Ck|0〉 = 0
11G.Luders: Z.Naturforsch. 139 (1954) 254, H.Araki: J.Math.Phys. 12 (1971) 1588
2.3 Fields 7
⇒ ∃ fn, gn (A†k −Ak)n|0〉 = (fn + gnA†k)|0〉
where fn, gn satisfy the following conditions;fn+1 = −gngn+1 = fn (f0 = 1, g0 = 0)
→ fn =
(−1)n/2 (n ∈ even)0 (n ∈ odd)
gn =
0 (n ∈ even)(−1)(n−1)/2 (n ∈ odd)
Therefore12
Uk(τ)|0〉 = eτ(A†k−Ak)|0〉
=∞∑n=0
τn
n!(fn + gnA
†k)|0〉
= (cos τ + sin τ ·A†k)|0〉= cos τ(1 + tan τ · a†j2k
a†j2k−1)|0〉
→ |0〉τ = cosN τ
[N∏
k=1
(1 + tan τ · a†j2ka†j2k−1
)
]|0〉
= cosN τ
1 + tan τ
N∑
k=1
a†j2ka†j2k−1
+ · · ·
+tanN τN∏
k=1
a†j2ka†j2k−1
|0〉
=⇒ the expansion of |0〉τ by the bases of the Fock space (sayH0) where |0〉 is the vacuum
Further for vector in Fock sp. (say Hτ ) where |0〉τ→ for example
α†j2lα†j2m−1
|0〉τ =
cosN τ · α†j2lα†j2m−1
[N∏
k=1
(1 + tan τ · a†j2ka†j2k−1
)
]|0〉
Thus any vector in Hτ is expanded by the bases of H0 in casethe finite d.o.f.
2.3 Fields
Quantum Field theory
• for scalar fields
φ =∑
k
√~c
2ωV
(ae−ikµx
µ
+ a†eikµxµ)
• for complex scalar fields
φ =∑
k
√~c
2ωV
(ae−ikµx
µ
+ b†eikµxµ)
and φ†
• for Dirac fields
ψ =∑p,r
√mc2
EV
(bur(p)e−ikµx
µ
+ d†vr(p)eikµxµ)
• for vector fields
A =∑
k, α
√~c
2ωV
(aε(α)e−ikµx
µ
+ a†ε(α)eikµxµ)
2.3.1 Group Theory (← Symmetry )→ Field Theory
1. φ(x) ; scalar, spin-0Schrodinger equation ;
i~∂
∂tψ +
~2
2m∇2ψ = 0
for de Broglie field13 (complex, non-rel).
→ Schrodinger eq.
Klein-Golden equation ;
(∂
∂xµ∂
∂xµ+
(mc~
)2)φ = 0
for Klein-Gordon field (real, relativistic)
2. ψ(x) ; Dirac, spin-1/2Dirac equation ; ( /P −mcI)ψ = 0i.e. (i~γµ∂µ −mcI)ψ = 014
12For simplicity, 0 < τ < π/213NOT probability wave BUT actual wave14Recall gamma matrices;
γ0 =
2664
1 0 0 00 1 0 00 0 −1 00 0 0 −1
3775 , γ1 =
2664
0 0 0 10 0 1 00 −1 0 0−1 0 0 0
3775 , γ2 =
2664
0 0 0 −i0 0 i 00 i 0 0−i 0 0 0
3775 , γ3 =
2664
0 0 1 00 0 0 −1−1 0 0 00 1 0 0
3775 , γ5 =
2664
0 0 1 00 0 0 11 0 0 00 1 0 0
3775
γµ, γν = 2ηµν
operator d.o.f. nonrela-approx
1 1 scalar 1γµ 4 vector γ0 ≈ 0
12(γµγν − γνγµ) 6 tensor σk
γµγ5 4 pseudo-vector σk
γ5 1 pseudo-scalar v · σ/c¿ 1
here γ5 = iγ0γ1γ2γ3
CPT-transformation
Cψ = γ2ψ∗
Pψ = γ0ψ
Tψ = γ1γ3ψ∗
8 4 PATH INTEGRAL QUANTIZATION
3. Aµ ; vector, spin-1
Maxwell equations ;
Fµν,µ = −4πcjµ,
(F[µν,λ] = 0
)
4. ψµα ; gravitino, spin-3/2
Rarita-Schwinger ;
LRS = εµνλρψµγν∂ψρ∂xλ
5. gµν ; gravitons (metric), spin-2
Rµν + gµν
(Λ− 1
2R
)=
8πGc4
Tµν
variables spin Lagrangian density EL equation comment
ψ(x) ∈ C∞(C) 0 i~ψ∗∂
∂tψ − ~2
2m∇ψ∇ψ∗ i~
∂
∂tψ +
~2
2m∇2ψ = 0 de Broglie field (non-rel.)
φ(x) ∈ C∞(R) 012φ,µφ
,µ − 12
(mc~
)2
φ2
¤ +
(mc~
)2φ(x) = 0 Klein-Gordon field
ψα (spinor) 1/2 i~cψγµ∂
∂xµψ −mc2ψψ ( /P −mcI4)ψ(x) = 0 Dirac field
Aµ(vector) 1 −14FµνF
µν Fµν,µ = 0 Maxwell equation(radiation field)
ψαµ 3/2 εµνληψµγν∂ψη∂xλ
εµνληγν∂λψη = 0 Rarita-Schwinger equation
gµν 2 − 116πG
(R+ 2Λ)√−g Rµν − 1
2gµνR = 0 Einstein equation
3 PATH INTEGRAL
4 PATH INTEGRAL QUANTIZA-TION
4.1 Generating function
Z[J ] =def
∫Dφ exp
i
∫d4x
[L(φ) + J(x)φ(x) +
i
2εφ2
]
(4.0)
∝ 〈0,∞|0,−∞〉J
Let us now calculate this for a free particle (field), for which
L0 =12
(∂µφ∂
µφ−m2φ2)
The corresponding vacuum-to-vacuum amplitude is (as N →∞)
Z0[J ] =∫
Dφ exp(i
∫ 12(∂µφ∂µφ− (m2 − iε)φ2 + φJ
d4x
)
(4.1)
Proposition 4.1.1
Z0[J ] =∫
Dφ exp(−i
∫ (12φ(¤ +m2 − iε)φ− φJ
)d4x
)
(4.2)
Proof of 4.1.1
Proposition 4.1.2From the Lagragngian
L =12∂µφ∂µφ− 1
2m2φ2 + Lint
φ obeys the equation of motion
(¤ +m2)φ(x) =∂Lint
∂φ(x)(4.3)
Further, the solution to this equation is
φ(x) = φin(x) +∫dy4ret(x− y) ∂Lint
∂φ(y)(4.4)
φ(x) = φout(x) +∫dy4adv(x− y) ∂Lint
∂φ(y)(4.5)
where 4ret is defined as
4ret = 0 (for x2 > 0, x0 < 0) (4.6)
4adv = 0 (for x2 > 0, x0 > 0) (4.7)
(¤ +m2)4ret(x) = δ4(x) (4.8)
(¤ +m2)4adv(x) = δ4(x) (4.9)
we call them retarded (advanced) Green’s function.
Proof of 4.1.2
4.2 Propagators and gauge consition in QED
Z[J ] =∫
DAµ exp(i
∫(L+ JµAµ)d4x
)(4.10)
where
L = −14FµνFµν (4.11)
9
Part II
Renormalization
5 RENORMALIZATION
Feynman Rules for scalar (or pseudoscalar) and spinortheories
1. To nth order, perturbation theory corresponds to a dia-gram with n vertices.The amplitude for a particular process15 to a particularorder is obtained by adding the amplitudes of all topo-logically nonequivalent connected diagrams.
2. For each incoming spinor particle write u(p)(v(p) for itsantiparticle), and for each outgoing spinor particle u(p)
3. For each vertex write ig (for scalar interaction) or igγ5
(for pseudoscalar), where g is the relevant coupling con-stant read off from the interaction Lagrangian, and mul-tiply by (2π)4δ4 (incoming momenta).
4. For each spinor propagator (internal line) of momentump write
1(2π)4
i
/p−M d4p
5. For each (pseudo-)scalar propagator write
1(2π)4
i
p2 −m2d4p
6. Integrate over internal momenta
5.1 Divergence in φ4 Theory
=⇒ g
∫d4q
(2π)41
q2 −m2(5.0)
=⇒ g2
∫d4q1(2π)4
d4q2(2π)4
δ(q1 + q2 − p1 − p2)(q21 −m2)(q22 −m2)
= g2
∫d4q
(2π)81
(q2 −m2)[(p1 + p2 − q)2 −m2](5.1)
Here there are 4 powers of q in both numerator and denomi-nator, so we get a logarithmic divergence.
Definition 5.1.1 Superficial degree of divergenceD is the superficial degree of divergence.
⇐⇒def
D = dL− 2I (5.2)
where d is spacetime dimension, L is the number of loops,and I is the number of internal lines.
Lemma 5.1.2n; the number of vertices
=⇒ L = I − n+ 1 (5.3)
Lemma 5.1.3E; the number of external lines
=⇒ 4n = E + 2I (5.4)
for φ4 theory.
Proposition 5.1.4
D = d−(d
2− 1
)E + n(d− 4) (5.5)
Example 5.1.5 d = 4We have
D = 4− E (5.6)
which gives the correct result.
Definition 5.1.6 RenormalizableConsider the last term in (5.5). If the coefficient of n ≥ 0, Dincreases with n.=⇒ the complete theory will contain an infinite number ofterms.
However, in φ4 theory in 4 spacetime dimensions, (5.6)shows that D depends on E only. (not on the order in pertur-bation theory)=⇒ we have only a small number of divergent graphs.=⇒ the effects of these can be eliminated by (infinite) renor-malization of various physical quantity.=⇒ If this turn out to be true (which it does) the theory iscalled renormalizable
Remark 5.1.7Of course, we have not yet shown that φ4 theory is renor-malizable − We have shown only that the perturbation seriesdoes not give an infinite number of different types of divergentgraphs, which is obviously a sine qua non for renormalizability.
Proposition 5.1.8For φr theory,
D = d−(d
2− 1
)E + n
[r2(d− 2)− d
](5.9)
Proof of 5.1.8(5.2) and (5.3) → Unchanged
(5.4) → rn = E + 2I
Example 5.1.9 d = 4D = 4− E + n(r − 4)
Example 5.1.10 φ6 theory in d = 4D = 4− E + 2n=⇒ the theory is unrenormalizable
15i.e. with particular ingoing and outgoing external lines
10 5 RENORMALIZATION
Example 5.1.11 φ3 theory in d = 4D = 4− E − n; called super-renormalizableD decreases with n increasing=⇒ ∃ only a finite number of divergent graphs for given E
Example 5.1.12 d = 2D = 2− 2n ← independent of r
Now let us return to equation (5.6), and enquire whetherall graphs with E > 4 are convergent.
Diagrams with 6 external legs in φ4 theory
The left diagram is convergent, as may be seen by writingout the amplitude, but the center diagram contains the 1-loopcontribution to the amplitude, as in (5.0) above, and this isalways divergent. Similarly the right diagram contains, asmarked, two 1-loop contributions to the 4-point function, sois also divergent.This happens with all Feynman diagrams; if they contain hid-den 2- or 4-point functions with 1-loop (or more) =⇒ theywill diverge, despite the formula D = 4−E: This is why D iscalled the superficial degree of divergence.
Theorem 5.1.13 WeinbergA Feynman diagram converges if its degree of divergence D,together with the degree of divergence of all its subgraphs, isnegative.
Proof of 5.1.13Omitted.
The 2 divergent diagrams G(2) and G(4) in (5.0) and (5.1)above are called primitive divergences.
5.1.1 Dimensional analysis
Lemma 5.1.14
S =∫ddxL
is dimensionless (neglecting the 1/~ factor).Hence,
[L] = L−d (L is length)[L] = Λd (Λ is momentum)
(5.17)
Proof of 5.1.14By definition.
Lemma 5.1.15
[φ] = L1−d/2 or Λd/2−1 (5.18)
Proof of 5.1.15erg = s−1 = cm−1 ∵ ~ = c = 1
=⇒ [m] = g = erg = cm−1
=⇒ [m2φ2] = [L] = cm−d
=⇒ [φ2] = cm2−d
=⇒ [φ] = cm1−d/2 or ergd/2−1
Proposition 5.1.16Consider an interaction gφr (g is the coupling constant)
[g] = L−δ =⇒ δ = d+ r − rd
2(5.19)
Proof of 5.1.16[gφr] = [L] = cm−d
=⇒ [g]cm(1−d/2)r = cm−d
=⇒ [g] = cm−d−(1−d/2)r ≡ cm−δ
=⇒ δ = d+ (1− d/2)r
Example 5.1.17
gφ4 : δ = 4− d [g] = Λ4−d (5.20)
gφ3 : δ = 3− d
2[g] = Λ3−d/2 (5.21)
gφ6 : δ = 6− 2d [g] = Λ6−2d (5.22)
Corollary 5.1.18For φr theory
D = d−(δ
2− 1
)E − nδ (5.23)
Proof of 5.1.18
δ = d+ (1− d/2)r =⇒ −d+rd
2= −δ + r
=⇒(??)
D = d−(d
2− 1
)E + n(−δ + r − r)
Remark 5.1.19 Renormalizable theoryRenormalizable =⇒ coupling constant g has a mass dimensionδ ≥ 0
Remark 5.1.20Fermi’s theory of weak interactions:[GF ] = [m]−2 =⇒ unrenormalizable
Example 5.1.21 Dimensions of various Green’s func-tions
1. G(2)(x, y) ≈ ∫ddpeip(x−y)(p2 −m2)−1
has dimension d− 2;and G(4) =
∑G(2)G(2).
2. The momentum-space Green’s function is obtained byFourier transformation of G(n)(xi)
3. By translation invariance, overall momentum conserva-tion allows us to define
G(n)(p1, · · · , pn) = G(n)(p1, · · · , pn−1)δ(P )
and δ(P ) has (mass) dimension
4.∫dzG(x, z)Γ(z, y) = δ(x− y) and δ has mass dimension−d
5.
δ3Γδφ(y)δφ(y′)δφ(y′′)
= −∫dxdx′dx′′Γ(x, y)Γ(x′, y′)Γ(x′′, y′′)
× δ3W
δJ(x)δJ(x′)δJ(x′′)(5.24)
6. Follows by Fourier transformation.
5.2 Dimensional regularization of φ4 theory 11
7. Follows from overall momentum conservation, asG(n)(pi)
Mass dimension in Massd-dimensional dimension
Note Quantity spacetime with d = 4φ d
2 − 1 11 G(n)(x1, · · · , xn) n
(d2 − 1
)n
2 G(n)(p1, · · · , pn) −nd+ n(d2 − 1
) −3n= −n (
d2 + 1
)3 G(n)(p1, · · · , pn−1) d− n (
d2 + 1
)4− 3n
4 Γ(2)(x− y) 2 + d 65 Γ(n)(x1, · · · , xn) n
(d2 + 1
)3n
6 Γ(n)(p1, · · · , pn) −dn+ n(d2 + 1
) −n=
(1− d
2
)7 Γ(n)(p1, · · · , pn−1) d+
(1− d
2
)4− n
Canonical dimensions of various quantities in d-dimensionalspacetime, and in 4-dimensional spacetime
5.2 Dimensional regularization of φ4 theory
We first need to generalize the 4-dim Lagrangian
L =12∂µφ∂µφ− 1
2φ2 − g
4!φ4 (5.25)
to d dimensions.Since
[φ] =d
2− 1, [L] = d
g is dimensionless in 4-dim, but in order to keep it dimension-less in d dimensions, it must be multiplied by µ4−d
L =12∂µφ∂
µφ− 12m2φ2 − 1
4!µ4−dgφ4 (5.26)
where µ is an arbitrary mass parameter
Lemma 5.2.1
The loop integral is
12gµ4−d
∫ddp
(2π)d1
p2 −m2(5.27)
Proposition 5.2.2(5.27) is
− ig
32π2m2
(4πµ2
m2
)2−d/2Γ
(1− d
2
)(5.29)
Proposition 5.2.3Expanding (5.29) about d = 4 gives
igm2
16π2ε+ finite (5.31)
Proof of 5.2.3
Γ(−n+ ε) =(−1)n
n!
[1ε
+ ψ1(n+ 1) +O(ε)]
(5.32)
where
ψ1(n+ 1) =n∑
k=1
1k− γ
and γ = −ψ1(1) = 0.577 is the Euler-Mascheroni const.Put ε = 4− d, then
Γ(1− d/2) = Γ(−1 + ε/2) = −2ε− 1 + γ +O(ε) (5.33)
Therefore
−igm32π2
[−2ε− 1 + γ +O(ε)
] [1 +
ε
2ln
(4πµ2
m2
)]
=igm2
16π2ε+igm2
32π2
[1− γ + ln
(4πµ2
m2
)]+O(ε)
=igm2
16π2ε+ finite (5.34)
¤
Remark 5.2.4The finite part of the correction to the propagator is not veryimportant, but note that it depends on the arbitrary mass µ.
4-point function to order g2:
12g2(µ2)4−d
∫ddp
(2π)d1
p2 −m2
1(p− q)2 −m2
(5.35)
Lemma 5.2.5 The Feynman formula
1ab
=∫ 1
0
dz
[az + b(1− z)]2 (5.36)
Proof of 5.2.5
1ab
=1
b− a(
1a− 1b
)=
1b− a
∫ b
a
dx
x2
Put x = az + b(1− z)Lemma 5.2.6(5.35) =
12g2(µ2)4−d
∫ 1
0
dz
∫ddp
(2π)d1
[p2 −m2 + q2z(1− z)]2 (5.38)
Proof of 5.2.6
Lemma 5.2.7
ig2
2(µ2)4−d
(14π
)d/2 Γ(2− d/2)Γ(2)
×∫ 1
0
dz(q2z(1− z)−m2
)d/2−2
=ig2
32π2(µ2)2−d/2Γ(2− d/2)
∫ 1
0
dz
(q2z(1− z)−m2
4πµ2
)d/2−2
(5.39)
12 5 RENORMALIZATION
Lemma 5.2.8In the limit d→ 4, equation (5.32) gives
Γ(2− d/2) = Γ(ε/2) =1ε− γ +O(ε) (5.41)
Proposition 5.2.9
(5.39) =ig2µε
32π2
(1ε− γ +O(ε)
)
×
1− ε
2
∫ 1
0
dz ln(q2z(1− z)−m2
4πµ2
)
=ig2µε
16π2− ig2µε
32π2
γ +
∫ 1
0
dz ln(q2z(1− z)−m2
4πµ2
)
(5.43)
Proof of 5.2.9
Definition 5.2.10 Mandelstram variables
s =def
(p1 + p2)2, t =def
(p1 + p3)2, u =def
(p1 + p4)2 (5.44)
Corollary 5.2.11Putting
F (s,m, µ) =def
∫ 1
0
dξ ln[sξ(1− ξ)−m2
4πµ2
](5.45)
For (5.35)
ig2µε
16π2− ig2µε
32π2γ + F (s,m, µ) =
ig2µε
16π2ε+ finite (5.47)
Lemma 5.2.12
(5.34) = Σ/i =⇒ to order g, Σ = − gm2
16π2ε+ finite
Lemma 5.2.13
Γ(2)(p) = p2 −m2(1− g
16π2ε
)(5.49)
Remark 5.2.14It is clearly infinite in the limit ε→ 0
Lemma 5.2.15
Γ(4)(pi) = −igµε(
1− 3g16π2ε
)+ finite (5.50)
Remark 5.2.16the corrections we have made are not to the same order in thecoupling g.
5.2.1 Loop expansion
an expansion in L ”=” expansion in ~ ?
Lemma 5.2.17
Z[J(x)] =∫
Dφ expi
~
∫[L+ ~J(x)φ(x)]dx
(5.51)
Splitting up L into ’free’ and ’interacting’ parts L = L0 +Lint
then we obtain
Z[J ] = expi
~Lint
[1i
δ
δJ
]Z0[J ] (5.52)
where
Z0[J ] =∫
Dφ exp[i
~
∫dx(L0 + ~Jφ)
](5.53)
Proof of 5.2.17
Proposition 5.2.18Finally , we have
Z0[J ] = N exp[− i~
2
∫dxdyJ(x)∆F (x− y)J(y)
](5.54)
Proof of 5.2.18
It is clear from (5.52) that each vertex contributes a factor~−1
and from (5.54) that each propagator contributes a factor ~ ,to a general graph in n th order perturbation theory.=⇒ such a graph contributes a factor ~I−n = ~L−1 using (5.3)
Remark 5.2.19Note that we have assumed that our general graph has nopropagators associated with the external legs. What we areexpanding therefore is the vertex function Γ(n) rather than theGreen’s function G(n)
5.3 Renormalization of φ4 theory
Consider the vertex functions Γ(2) and Γ(4), equations (5.49)and (5.50) above. To the (1-loop) approximation we are con-sidering, they should be finite, so we put
Γ(2)(p) = p2 −m21 (5.60)
where m1 is a new parameter, defined by this equation.It is taken to be finite, and to represent the physical mass.
The original mass m is taken to be infinite, and to have nodirect physical significance. It is the mass the particle wouldhave if no interactions were present − and since they are al-ways present, it is an unobservable quantity.
m2 = m21 +
m2g
16π2ε
= m21
(1 +
g
16π2ε
)(5.61)
where substituting m→ m1 in the loop correction.
Remark 5.3.1This is valid since the error involved is of order g2.
Definition 5.3.2 Renormalized massThe above finite (physical) mass, say m1 is called renormal-ized mass, and is given by
m21 = −Γ(2)(0) (5.62)
5.3 Renormalization of φ4 theory 13
We now apply a similar treatment to Γ(4). First let uswrite (5.50) as
iΓ(4)(pi) = gµε − g2µε
32π2
[6ε− 3γ − F (s,m, µ)
− F (t,m, µ)− F (u,m, µ)
](5.64)
Definition 5.3.3 Renormalized coupling constant
g1 =def
gµε − g2µε
32π2
[6ε− 3γ − 3F (0,m, µ)
](5.65)
Lemma 5.3.4Rearranging (5.65) by substituting g1 for g and m1 for m, then
g = g1µ−ε +
3g21µ−2ε
32π2
[2ε− γ − F (0,m1, µ)
](5.67)
Proposition 5.3.5Expressing in terms of g1:
iΓ(4)(pi) = g1 +g21µ−ε
32π2
× (F (s,m1, µ) + F (t,m1, µ) + F (u,m1, µ)− 3F (0,m1, µ))
Corollary 5.3.6It follows immediately that
iΓ(4)(pi = 0) = g1 (5.68)
Proof of 5.3.6pi = 0 =⇒ s = t = u = 0 by definition.
Two loop terms in Γ(2) and Γ(4)
Definition 5.3.7 Renormalized 2-point function & Renormaliza-tion constantDefine the renormalized 2-point function as
Γ(2)r =
defZφ(g1,m1, µ)Γ(2)(p,m1, µ) (5.69)
where Z12φ is called the wave function (or field) renormal-
ization constant
Zφ = 1 + g1Z1 + g21Z2 + · · ·
= 1 + g21Z2 + · · · (5.70)
Definition 5.3.8 Renormalized massThe renormalized vertex function Γ(2)
r now gives a finite massmr:
m2r =
defZφm
21 (5.71)
Γ(4)r = Z2
φΓ(4)(p,m1, µ) (5.72)
iΓ(4)(pi = 0) = gr = Z2φg1 (5.73)
gr = Z2φ (5.74)
the renormalized n-particle vertex function is
Γ(n)r (pi, gr,mr, µ) = Z
n/2φ (gµε)Γ(n)(pi, g,m) (5.75)
Γ(n)(pi, g,m) = Z−n/2φ (gµε)Γ(n)(pi, gr,mr, µ) (5.76)
5.3.1 Counter-terms
=igm2
16π2ε+ finite
which diverges as ε→ 0Adding to L a term
δL1 = − gm2
32π2εφ =
def−δm
2
2φ2 (5.77)
=⇒ this is treated as an interaction, and gives rise to the ad-ditional Feynman rule:
= − igm2
16π2ε= −iδm2 (5.78)
Remark 5.3.9The Lagrangian is now L′ = L + δL1 (δL1 being a counter-term, which is divergent). It may appear strange to introducewhat is clearly a ’mass’ term into the Lagrangian and call itan ’interaction’, but, in fact, there is no contradiction.
The free theory for φ
For L =12(∂µφ)(∂µφ)− 1
2φ2,
regard it as describing a massless field φ, given by the firstterm in L, with an interaction given by the second term.=⇒ The Feynman rules are
=i
p2, = −im2
and the complete propagator is
=i
p2+
i
p2(−im2)
i
p2+
i
p2(−im2)
i
p2(−im2)
i
p2+ · · ·
=i
p2 −m2
which is the usual propagator for a massive field.
A similar treatment may be made to Γ(4).
δL2 = − 14!
3g2µε
16π2εφ4 =
def−Bgµ
ε
4!φ4 (5.79)
δL3 =A
2(∂µφ)2 (5.80)
where 1 +A = Zφ
L =12(∂µφ)2 − 1
2m2φ2 − gµε
4!φ4 (5.81)
LCT =12A(∂µφ)2 − 1
2δm2φ2 − Bgµε
4!φ4 (5.82)
14 5 RENORMALIZATION
Then the total Lagrangian, called the bare Lagrangian LBis
LB = L+ LCT
=(
1 +A
2
)(∂µφ)2 − m2 + δm2
2φ2 − (1 +B)
gµε
4!φ4 (5.83)
Definition 5.3.10 Bare quantities
φB =def
√Zφφ, Zφ = 1 +A
mB =def
Zmm, Z2m =
def
m2 + δm2
1 +A
gB =def
µεZgg, Zg =def
1 +B
(1 +A)2(5.84)
Proposition 5.3.11
LB =12(∂µφB)2 − m2
B
2φ2B −
gB4!φ4B (5.85)
5.4 Renormalization group
Dimensional regularization =⇒ new parameter µ
Lemma 5.4.1The unrenormalized function Γ(n) given by (??) is invariantunder the group of transformations
µ→ esµ (5.86)
Proof of 5.4.1
(??) =⇒ Γ(n) is independent of µ.
Remark 5.4.2These transformations form the renormalization group.
Lemma 5.4.3Introducing the dimensionless differential operator µ(∂/∂µ),we have
µ∂
∂µΓ(n) = 0 (5.87)
or, from (??),
µd
dµ
(Z−n/2φ (gµε)Γ(n)
r (pi, gr,mr, µ))
= 0 (5.88)
Proof of 5.4.3
Lemma 5.4.4(5.88) gives[−nµ ∂
∂µln
√Zφ + µ
∂
∂µ+ µ
∂gr∂µ
∂
∂gr+ µ
∂mr
∂µ
∂
∂mr
]Γ(n)r = 0
(5.90)
Proof of 5.4.4
For simplicity, let us from now on write g, m, Γ(n) forgr, mr, Γ(n)
r
Theorem 5.4.5 Renormalization group equation (RGequation)
γ(g) = µ∂
∂µln
√Zφ
β(g) = µ∂g
∂µ
mγm(g) = µ∂m
∂µ
(5.91)
=⇒ (5.90) becomes
[µ∂
∂µ+ β(g)
∂
∂g− nγ(g) +mγm(g)
∂
∂m
]Γ(n) = 0 (5.93)
Proof of 5.4.5
Remark 5.4.6It expresses the invariance of the renormalized Γ(n) under achange of regularization parameter µ.
Let us now write down a similar equation expressing theinvariance of Γ(n) under a change of scale. Let p → tp, m →tm, µ→ tµ.
Proposition 5.4.7Γ(n) has a mass dimension D given by
D = d+ n
(1− d
2
)
= 4− n+ ε(n
2− 1
)(5.94)
Lemma 5.4.8
−t ∂∂t
+ β∂
∂g− nγ(g) +m (γm(g)− 1)
∂
∂m+D
×Γ(n)(tp, g,m, µ) = 0 (5.95)
Lemma 5.4.9We assume that
Γ(n)(tp,m, g, µ) = f(t)Γ(n)(p,m(t), g(t), µ) (5.96)
then(−t ∂∂t
+t
f
df
dt+ t
∂m
∂t
∂
∂m+ t
∂g
∂t
∂
∂g
)Γ(n)(tp,m, g, µ) = 0
(5.97)
Proof of 5.4.9
∂
∂tΓ(n)(tp,m, g, µ)
=df
dtΓ(n)(p,m(t), g(t), µ)
+ f(t)(∂m
∂t
∂
∂m+∂g
∂t
∂
∂g
)Γ(n)(p,m(t), g(t), µ) (5.98)
5.4 Renormalization group 15
and
t∂
∂tΓ(n)(tp,m, g, µ)
=(tdf
dt+ f(t)t
∂m
∂t
∂
∂m+ f(t)t
∂g
∂t
∂
∂g
)
× Γ(n)(p,m(t), g(t), µ)
=(tdf
dt+ f(t)t
∂m
∂t
∂
∂m+ f(t)t
∂g
∂t
∂
∂g
)1f(t)
× Γ(n)(tp,m, g, µ) (5.99)
Corollary 5.4.10
t∂g(t)∂t
= β(g) (5.100)
g(t) is called a running coupling constant.
Lemma 5.4.11
Γ(n)(tp,m, g, µ) = t4−n exp[−n
∫ t
0
dtγ (g(t))
t
]
×Γ(n)(p,m(t), g(t), µ) (5.101)
is the solution to the RG equation (5.95)
Proof of 5.4.11
Remark 5.4.12This is the solution to the RG equation (5.95), in terms of therunning coupling constant g(t) and running mass m(t). Theexponential term is the anomalous term.
Remark 5.4.13The physical at large momentum is governed by m(t) and g(t),and particular use of the renormalization group is to study thelarge (or even the small) momentum behavior of QFT.
Let us examine some possible behavior of g(t) as t → ∞,i.e. at large momentum, and assume that (5.100) is still validthere:
t∂g(t)∂t
= β(g(t)) (5.100)
First, suppose that β(g) has the form shown in below:
g s.t. β(g = 0, g0) = 0 are called fixed points.We may see that as t → ∞, a value of g near to g0 tendstowards g0.For if g < g0 =⇒ 0 < β =⇒ g increases with increasing t,and is driven towards g0.If g0 < g =⇒ β < 0 =⇒ g decreases with increasing t =⇒ gis driven back towards g0Hence g(∞) = g0 called an ultra-violet stable fixed point.
By analogous argument, if g is small =⇒ g → 0 as t→ 0,and g(0) = 0 is called an infra-red stable fixed point.
Second, suppose that β(g) has the form shown below
Again ∃2 fixed points, but the sign of β is reversed.=⇒ g = g0 is an infra-red stable fixed point, and g = 0 isan ultra-violet stable fixed point. =⇒ For the latter behavior,perturbation theory gets better and better at higher energies,and infinite momentum limit, the coupling constant vanishes.This is known as asymptotic freedom.
5.4.1 The asymptotic behavior of φ4 theory
Assume that 1-loop expression (5.65) for the renormalized cou-pling constant is a reliable pointer to the asymptotic regime.
Lemma 5.4.14Ignoring the finite corrections, we then have
g1 = gµε(
1 +3g
16π2ε
)(5.107)
so that
µ∂g1∂µ
= εgµε +3g2
16π2µε
Lemma 5.4.15By (5.91)
β(g) = limε→0
µ∂g1∂µ
=3g2
16π2(5.109)
Therefore
∂
∂sg(s) = β(g(s)) (5.110)
Proof of 5.4.15
Remark 5.4.16We see that the effective (running) coupling const. increaseswith s, i.e. with increasing momentum.=⇒ φ4 theory is not asymptotic free.
Proposition 5.4.17The solution to (5.109) is
g =g0
1− ag0 ln(µ/µ0)(5.112)
where a = 3/16π2
Proof of 5.4.17
Remark 5.4.18In fact, g increases with increasing µ.
16 5 RENORMALIZATION
5.5 Divergences and dimensional regulariza-tion of QED
The only particle in QED are photons and electrons.Divergences occur in several types of Feynman diagram16.=⇒ We shall treat the diagrams in the same systematic wayin which we analyzed φ4 theory
Lemma 5.5.1 The general formula for DL; the number of loopsPi; the number of internal photon linesEi; the number of internal electron linesd; dimension of space-time
=⇒ D = dL− 2Pi − Ei (5.113)
Lemma 5.5.2In addition, letn; the number of verticesPe; the number of external photon linesEe; the number of external electron lines
=⇒
L = Ei + Pi − n+ 12n = Ee + 2Ein = Pe + 2Pi
(5.114)
Proposition 5.5.3The above relations give
D = (d− 1)Ei + (d− 2)Pi − d(n− 1)
D = d+ n
(d
2− 2
)−
(d− 1
2
)Ee −
(d− 2
2
)Pe
(5.115)
Proof of 5.5.3We shall omit.
Example 5.5.4 d = 4
D = 4− 32Ee − Pe (5.116)
showing that D is independent of n, the sine qua non forrenormalizability.
5.5.1 Electron self-energy diagram
Proposition 5.5.5The Feynman rules give
−iΣ(p) = (−ie)2∫
d4k
(2π)4γµ
i
/p− /k −m−igµνk2
γν (5.117)
17 The electron self-energy diagramhas Ee = 2, Pe = 0 =⇒ D = 2
5.5.2 Photon self-energy diagram
Proposition 5.5.6The photon self-energy is denoted Πµν and is also called vac-uum polarization. Unlike electron self-energy, it has no clas-sical counterpart.
The Feynman rules give
iΠµν(k) = −(−ie)2∫
d4
(2π)4tr
(γµ
i
/p−mγνi
/p− /k −m)
(5.118)
Remark 5.5.7It is clear that this integral is quadratically divergent, as an-ticipated.
The photon self-energy diagram hasEe = 0, Pe = 2 =⇒ D = 2
Remark 5.5.8Note that this graph gives a modified photon propagator, sothat, in the Feynman gauge, to one loop
iD′µν(k) = −igµνk2
+(−igµλ
k2
)iΠλζ(k)
(−igζνk2
)(5.119)
5.5.3 Primitively divergent graphs
These self-energy graphs are primitively divergent graphs.There are 3 more primitively divergent graphs in QED.
The first one of them is the vertex graph having Ee =2, Pe = 1 =⇒ D = 0, a logarithmic divergence.
Proposition 5.5.9The Feynman rules give
− ieΛµ(p, q, p+ q)
= (−ie)3∫
d4k
(2π)4−igρσ
(k + p)2γρ
i
/k − q/−mγµi
/k −mγσ (5.120)
Remark 5.5.10This vertex graph, and the 2 self-energy graphs above, all havethe property that the removal of their infinities results in a re-definition of various physical quantities. i.e. no extra termsin the Lagrangian are required of a type which are not therealready.But the 2 remaining primitively divergent graphs could posea serious threat to the renormalizability of QED.
The first of these is the 3-photon coupling:
It has Ee = 0, Pe = 3 =⇒ D = 1It turns out, however, that this graph is cancelled by a similar
16e.g. the electron and photon self-energy diagrams17The dashed lines denote photons, and the solid lines denote electrons
5.6 1-loop renormalization of QED 17
graph with the electron arrows reversed.=⇒ It may be ignored. (Furry’s theorem)
The 2nd diagram which could cause trouble for renormal-izability:
It has Ee = 0, Pe = 4 =⇒ D = 0; super-ficially logarithmically divergentIt turns out, however, that due to gauge invariance, this graphis actually convergent.=⇒ It causes no trouble
⇓Our task = to calculate 3 primitively divergent graphs usingdimensional regularization
Recall the Lagrangian for Nakanishi-Lautrup form withFeynman gauge:
L = iψ /Dψ −mψψ − 14F 2µν −
12(∂µAµ)2
= iψγµ∂µψ −mψψ− eAµψγµψ − 1
4(∂µAν − ∂νAµ)2 − 1
2(∂µAµ)2
Remark 5.5.11In d-dim spacetime,
[L] = d, [ψ] =d− 1
2, [Aµ] =
d− 22
Proposition 5.5.12
Σ(p) =e2
8π2ε(−/p+ 4m) + finite (5.121)
Proof of 5.5.12
Proposition 5.5.13
Πµν(k) =e2
6π2ε(kµkν − gµνk2) + finite (5.122)
Proof of 5.5.13
Proposition 5.5.14
Λµ = Λ(1)µ + Λ(2)
µ (5.123)
where
Λ(1)µ (p, q, p′) =
e2
8π2εγµ + finite (5.124)
Λ(2)µ (p, q, p′)
=e2
16π2
∫ 1
0
dx
∫ 1−x
0
dy
× γν ( /p′(1− y)− /px+m) γµ (/p(1− x)− /p′y +m) γν
m2(x+ y)− p2x(1− x)− p′2y(1− y) + 2p · p′xy(5.125)
Proof of 5.5.14
divergent part
Σ(p) =e2
8π2ε(−/p+ 4m) + finite
Πµν(k) =e2
6π2ε(kµkν − gµνk2) + finite
Λ(1)µ (p, q, p′) =
e2
8π2εγµ + finite
Remark 5.5.15The divergent parts of Σ and Λ above satisfy the Ward iden-tity.
5.6 1-loop renormalization of QED
Now let us consider what counter-terms it is necessary to addto L to make the above quantities finite.
5.6.1 Anomalous magnetic moment of the electron
5.6.2 Asymptotic behavior
Proposition 5.6.1
eB = eµε/2(
1 +e2
12π2ε
)(5.126)
Corollary 5.6.2
µ∂e
∂µ= − ε
2e+
e3
12π2+
e5
96π4ε(5.127)
Corollary 5.6.3
µ∂e
∂µ=
e3
12π2(5.128)
Corollary 5.6.4The solution to (5.128) is
e2(µ) =e2(µ0)
1− e2(µ0)6π2
lnµ
µ0
(5.130)
Remark 5.6.5We explicitly the increase of e with µ.
Definition 5.6.6 Landau singularityThe µ which makes the denominator of (5.130) be 0 is calledLandau singularity
µ = µ0 exp(
6π2
e2(µ0)
)(5.132)
The increasing of e with µ, i.e. with decreasing distance,has an analogy in macroscopic electrostatics in a dielectric
18 5 RENORMALIZATION
medium. Here, the presence of an electric charge polarizes themedium.
If the free charge density is ρf and the polarization chargedensity is ρp. we have the equations
∇ ·E = ρf in vacuo (5.133)∇ ·E = ρf + ρp in medium (5.134)
Putting ∇ · P = −ρp gives
∇ · (E + P ) = ρf (5.135)or ∇ ·D = ρf (5.136)
where D =def
E + P .
If P = αE, then
D = (1 + α)E (5.137)
=⇒ ∇ ·E =(
11 + α
)ρf (5.138)
=⇒ we see that the effect if the medium is to ’screen’ the origi-nal charge. From a distance, the charge (as measured by E) ismeasured to be less than it is nearby. At a distance much lessthan molecular or atomic dimensions, the ’bare’, unscreenedcharge is seen.
The same situation holds in QED. Here the photon self-
energy graph may be interpreted by say-ing the vacuum is ’filled’ with virtual e+e− pairs which screena bare change, in the manner drawn in Figure below.So the charge e increases with decreasing distance, i.e. withan increasing momentum (or mass) scale.
We shall see below that the situation in this regard is verydifferent in QCD.
5.7 Renormalizability of QED
5.8 Asymptotic freedom of Yang-Mills theo-ries
We shall perform calculations analogous to those in the sec-tion 5.6, with the aim of showing that at high energies therunning coupling constant in Yang-Mills theories approacheszero, a property known as asymptotic freedom.
Remark 5.8.1It turns out that asymptotic freedom is a property possessedby all non-Abelian gauge theories, and, so far as is known,only by non-Abelian gauge theories.
The key to asymptotic behavior in the quantity β(g) de-fined by (5.91) and g is the Yang-Mills coupling constant.
β(g) = µ
∂g
∂µ
gB = gµε/2Z1Z−12 Z
− 12
3
(5.140)
Here,Z1; the renormalization constant for the quark-gluon-gluonvertexZ2; for quark wave functionZ3; for gluon wave function (self-energy)
Lemma 5.8.2Applying the Feynman rules with Feynman gauge gives
− iΣab(p) = −g2µ4−d∫
ddk
(2π)dγµ
1/p− /k −m
× γν gµν
k2(T c)ad(T c)db (5.141)
where T c = λc/2.
Proof of 5.8.2
Remark 5.8.3It is seen immediately that this is simply (T cT c)ab multipliedby the corresponding self-energy expression in QED:
Σab(p) = (T cT c)abΣ(QED)
= (T cT c)abg2
8π2ε(−/p+ 4m) (5.142)
Lemma 5.8.4
T cT c =14(λ2
1 + · · ·+ λ28)
=431 (5.143)
((T cT c)ab =43δab)
≡ C2(F )δab
Remark 5.8.5
C2(F ) =N2 − 1
2Nfor SU(N)
Lemma 5.8.6From (5.142)
Σab =g2
6π2ε(−/p+ 4m)δab (5.145)
Proposition 5.8.7The fermion (quarks) wave function is renormalized by
√Z2,
where
Z2 = 1− g2
6π2ε(5.146)
Next, we turn to vacuum polarization in QCD, or gluonself-energy.
LEFT: Gluonloop contribution to vacuum polarization in QCD.
CENTER: Ghost loop contribution to vacuum polarization.RIGHT: Quark loop contribution to vacuum polarization in
QCD
5.8 Asymptotic freedom of Yang-Mills theories 19
Define these contributions respectively Πabµν(1), Πab
µν(2), andΠabµν(3).
Proposition 5.8.8The gluon contribution; Πab
µν(1)
Πabµν(1) =
−g2
16π2εfacdf bcd
(113pµpν − 19
6gµνp
2
)(5.147)
Proposition 5.8.9The ghost contribution; Πab
µν(2)
Πabµν(2) =
g2
16π2εfacdf bcd
(13pµpν +
16gµνp
2
)(5.148)
Proposition 5.8.10The quark contribution; Πab
µν(3)
Πabµν(3) = tr(T aT b)
g2
6π2ε
(pµpν − gµνp2
)(5.149)
Lemma 5.8.11
facdf bcd = 3δab (5.150)
=def
= δabC2(G)C2(G) = N for SU(N) (5.151)
Corollary 5.8.12Gathering these results,
Πabµν(1 + 2 + 3) =
g2
8π2ε
(gµνp
2 − pµpν) (
53C2(G)− 2nF
3
)δab
=g2
8π2ε
(gµνp
2 − pµpν) (
5− 2nF3
)δab
(5.152)
Proposition 5.8.13
Z3 = 1 +g2
8π2ε
(5− 2nF
3
)(5.153)
Proof of 5.8.13Comparing equations (??) and (??) andRequiring Z3 to cancel this divergence in a counter-term.
Proposition 5.8.14
Z1 = 1− g
8π2ε
133
(5.154)
Corollary 5.8.15Bringing together equations (5.140), (5.146), (5.153), and(5.154)
gB = gµε/2(
1− g2
8π2ε
133
)
×(
1 +g2
8π2ε
)[1− g2
8π2ε
(52− nF
3
)]
= gµε/2(
1 +g2
16π2ε
(−11 +
2nF3
))(5.159)
Corollary 5.8.16
β(g) = µ∂g
∂µ=
g3
16π2
(−11 +
2nF3
)(5.160)
Remark 5.8.17If the number of quark flavors is nF ≤ 16, =⇒ β < 0 and g de-creases with increasing mass scale µ, so the theory is asymp-totic free.It seems likely that in nature nF = 6 < 16, so asymptoticfreedom is a property possessed by QCD, and is the justifica-tion of the parton model, according to which partons behavealmost like free particles when interacting at high momentumtransfer with photons, inside a hadron.
Proposition 5.8.18Writing
β(g) =dg
dt
where t =def
lnµ, equation (5.159) may be written as
dg
dt= −bg3
b =def
11− 2nF /316π2
(5.162)
and
d
dt(g−2) = 2b
Proof of 5.8.18
Proposition 5.8.19The solution is1g2
=1g2
+ 2bt
=⇒ g2 =g2
1 + 2btg2
Here define αs =def
g2/4π
=⇒ αs(t) =α0
1 + 8πbtα0Now t = lnµ, which in deep inelastic scattering xp. we mayrepresent as 1
2 ln(Q2/Λ2)
=⇒ αs(Q2) =1
1α0
+ 4πb ln(Q2
Λ2
) (5.163)
and ignoring the 1/α0, we get
αs(Q2) =4π(
11− 2nF3
)ln
(Q2
Λ2
) (5.164)
where Λ is a scale ’chosen’ by the world in which we live.
20 6 GAUGE SYMMETRIES AND CONSERVATION LAWS
5.9 Renormalization of Yang-Mills theorieswith spontaneous symmetry breakdown
Part III
Gauge Theory
6 Gauge symmetries and conserva-tion laws
symmetry → keep the action invariant (NOT sufficient condi-tion)
Symmetry for Quantum Field Theory
↑
It doesn’t suffice it to say that ∃ ”Noether invariance”18
Even if Noether inv. exists, the physical result with samequantity of the field before transformation and quantity19 ofthe field after the transformation corresponding to this is notnecessarily drawn.
lIn order two theories before and behind conversion are com-pletely equal and to realize symmetry, unitary op. which con-nects two on this Hilbert space must exist.
The spontaneous breaking of symmetry depends on such aunitary operator not existing.
gauge = a hidden freedom
We already have known the gauge invariance in electro-magnetics. What do this fact show us?
Invariance give us the conservative quantities supported bythe Noether theorem.
6.1 WIGNER PHASE
H ; Hilbert sp. for some system|A〉, |B〉, · · · ∈H
|A〉 → |A′〉|B〉 → |B′〉
|A′〉 /∈H (generally)
When |A′〉 /∈H , symmetry is said to be broken spontaneously,and we say that such a system is in broken phase.When |A′〉 ∈H , we say that the system is in Wigner phase.
6.2 NOETHER’S THEOREM
Theorem 6.2.1 Noether theorem∃ global symmetry ⇐⇒ ∃ conserved current
CM; conservation of energy and momentumL: independent of all the position coordinates
=⇒ conservation of momentumL: independent of time=⇒ conservation of energy
classical
discrete dynamics −→ a constant of the motionfield dynamics −→ a conserved current
Lagrangian to EL equation
S =∫Ldt =
∫∫LdV dt =
1c
∫LdΩ
L = L(φ, φ,µ)
=⇒ δS =∫d4x δL
=∫d4x
(∂L∂φ
δφ+∂L∂φ,µ
δφ,µ
)
=∫d4x δφ
(∂L∂φ− d
dxµ
(∂L∂φ,µ
))
+∫dSµ
∂L∂φ,µ
δφ
= 0
∴ d
dxµ
(∂L∂φ,µ
)− ∂L∂φ
= 0 Euler− Lagrange eq.
Conserved quantities;
dLdxµ
=∂L∂φ
∂φ
∂xµ+
∂L∂φ,ν
∂φ,ν∂xµ
=d
dxν
(∂L∂φ,ν
)∂φ
∂xµ+
∂L∂φ,ν
∂φ,µ∂xν
=d
dxν
(∂L∂φν
∂φ
∂xµ
)
= gνµdLdxν
∴ 0 =d
dxν
(∂L∂φ,ν
∂φ
∂xµ− gνµL
)
0 =d
dxν
(∂L∂φ,ν
∂φ
∂xµ− gµνL
)
∂L∂α
=∂L∂φ
∂φ
∂α+
∂L∂φ,µ
∂φ,µ∂α
= 0 (independent of α explicitly)
⇐⇒ d
dxµ
(∂L∂φ,µ
∂φ
∂α
)= 0
Hence, we define the conserved quantities
Tµν ≡ ∂L∂φ,ν
∂φ
∂xµ− gµνL
Jµ ≡ ∂L∂φ,µ
∂φ
∂α
18In classical theory, it is N.S.C.19It is the operator defined in the same Hilbert space.
6.2 NOETHER’S THEOREM 21
How to treat Lagrangian
1. The order of the products of ψ(x), ψ†(x)A is freely un-changeable.
2. Both the infinitesimal change
δψ(x, t), δψ†(x, t), δψ(x, t), δψ†(x, t)
at the time t and
ψ(x′, t), ψ†(x′, t), ψ(x′, t), ψ†(x′, t)
at the same time t
are commutative [anticommutative] in Bose [Fermi]statistics.
6.2.1 Noether currents
transformation for ψα(x) under infinitesimal transformation ofa group G;
ψα(x)→ ψ′α(x) = ψα + δψα(x) (♦)
where δψα(x) =∑A εAfA,α(x)
δL(x) =∑α
(δψα
∂
∂ψα+ δψα,µ
∂
∂ψα,µ
)L
Since
δψα,µ∂L∂ψα,µ
=def
∂
∂xµ
([δψα
∂
∂ψα,µ
]L
)+Rα
=⇒ Rα = −δψα ∂
∂xµ
[∂L∂ψα,µ
]
then if L is supposed to be invariant under (♦) i.e. δL ≡ 0
0 = −∑α
δψα(x)(
∂
∂xµ∂L∂ψα,µ
− ∂L∂ψα
)
+∑α
∂
∂xµ
([δψα
∂
∂ψα,µ
]L
)
=∑α
(∂L∂ψα
− ∂
∂xµ
[∂L∂ψα,µ
])δψα +
∑α
∂
∂xµ
[∂L∂ψα,µ
δψα
]
= −∑α
∑
A
εAfA,α(x)(
∂
∂xµ
[∂L∂ψα,µ
]− ∂L∂ψα
)
+∑α
∑
A
∂
∂xµ
([εαfA,α(x)
∂
∂ψα,µ
]L
)
Because εA are arbitrary infinitesimal parameter,
∑α
fA,α(x)(
∂
∂xµ
[∂L∂ψα,µ
]− ∂L∂ψα
)
−∑α
∂
∂xµ
[∂L∂ψα,µ
fA,α(x)]
= 0
This equation is identity w.r.t. ψα(x) as and when δψα(x)satisfy the normal commutation relation.
If ψα are on-shell,
∂
∂xµ
[∂L∂ψα,µ
]− ∂L∂ψα
≡ 0
here we define Noether current
−∑α
[∂L∂ψα,µ
fA,α(x)]
=def
JµA(x)
and then∂µJ
µA = 0
Since ψα in the Lagrangian are not necessarily on-shelland undefined as operator, it is not sufficient to suppose onlycommutation relation between ψα and field. In addition tothis, fA,α(x) should satisfy the same commutation relationas δψα(x). If this property is ensured when ψα are on-shell,Noether currents are conserved.
However, fields must obey canonical commutation relationas operator when on-shell. The above condition are not alwayssatisfied. We should make certain that Noether currents areconserved by using Euler-Lagrange equation.
Furthermore, under some transformation, law of conser-vation itself loses its meaning because the definition of JµA isincomplete even if we show that Noether currents are con-served.
Even if it is going to calculate the matrix element whichsandwiched JµA by two state vector at this time (with for ex-ample, perturbation theory), expression may be imperfect andmay be unable to draw an answer.
So, it corrects so that this may be defined clearly, and itis necessary to make it fill a physical demand (for example,gauge invariance).
But redefining the currents in this way, the law of conser-vation will often break. → (quantum) anomaly
It needs to be cautious only of the quantum field theorynecessarily not being completely defined by having given La-grangian.
6.2.2 Noether charges
∂µJµA(x) = 0 ∴ ∂J0
A
∂t+
∂
∂xkJkA(x) = 0
0 =∫d3x
∂
∂xµJµA(x)
=∫d3x
∂J0A
∂t+
∫d3x
∂
∂xkJkA(x)
=d
dt
∫d3x J0
A(x) +∫dSkJ
kA(x)
=d
dt
∫d3x J0
A(x)
IA =def
∫d3x J0
A(x)
; Noether charges
claim.1 When we apply the normal commutation relation,IA satisfy Lie alg. of G
[IA, IB ] = i∑
C
f CAB IC
claim.2 δψα = i∑A εA[ψα(x), IA]
22 6 GAUGE SYMMETRIES AND CONSERVATION LAWS
6.3 THE GAUGE PRINCIPLE
global symmetry vs. local symmetryWhile ”global” means that the transformation all space
and time, ”local” means that the transformation at somepoint. i.e. difference between constant and function of x.
Example 6.3.1 For EM, gauge transformation
Aµ = Aµ +∂χ
∂xµ
makes no differece.
Fµν → Fµν
= ∂µAν − ∂νAµ= ∂µ(Aν + ∂νχ)− ∂ν(Aµ + ∂µχ)= ∂µAν − ∂νAµ
Example 6.3.2 For QM, the phase transformation
ψ 7→ eiθψ
has no influence on all observers.As is usual, any such transformation that leaves the system
invariant is called a symmetry.
Example 6.3.3 The (free) Dirac Lagrangian
L = iψγµ∂
∂xµψ −mψψ
(L = i~cψγµ
∂
∂xµψ −mc2ψψ
)
If we transform ψ into ψeiθ(xµ), then
ψ → ψ′ = ψ′†γ0
= (ψeiθ(x))†γ0
= ψ†e−iθ(x)γ0
= e−iθ(x)ψ†γ0
= e−iθ(x)ψ
So
L → iψ†eiθ(x)γ0γµ∂
∂xµψeiθ(x) −mψ†e−iθ(x)γ0ψeiθ(x)
= L − ψγµ ∂θ(x)∂xµ
ψ
−→ depending on∂θ(x)∂xµ
Consider the effect on∂ψ
∂xµ
∂ψ
∂xµ→ ∂
∂xµ
(ψeiθ(x)
)= eiθ(x)(ψ,µ + iθ,µψ)
the 2nd term spoils gauge invariance. We introduce the co-variant derivative Dµ (in order to compensate this failure) s.t.
Dµ ≡ ∂
∂xµ− Vµ
and Vµ follows the next transformation;
Vµ → Vµ + i∂θ(x)∂xµ
Then,
Dµψ = ψ,µ − Vµψ
→ eiθ(x)(ψ,µ + iθ,µψ)−(Vµ + i
∂θ
∂xµ
)ψeiθ(x)
= eiθ(x)Dµψ
Therefore the (new) Dirac Lagrangian
L = iψγµDµψ −mψψ
is gauge invariant.
(∵)
L = iψγµDµψ −mψψ→ ie−iθψγµeiθDµψ −mψψ= iψγµDµψ −mψψ
phase in QM should be locally unobservable→ introducinga new field, whose interaction and kinetic energy terms can berecognized as exactly those of electromagnetism. + telling usthat the photon have zero mass.
L = iψγµDµψ −mψψ − 14FµνFµν (13)
where
Dµ ≡ ∂µ − ieAµ (14)
(L = i~cψγµ
(∂µ − i e~cAµ
)ψ −mc2ψψ − 1
16πFµνFµν
)
6.4 FREE FIELDS
Definition 6.4.1 free fieldsfree fields ∼= linear equation of motionsLfree
∼= quadratic
Ltotal(φ, φ,µ) = Lfree + gLinteraction + · · ·
Example 6.4.2 real scalar φ
S =∫d4x
[12(φ,µ)2 − 1
2m2φ2 + λ1φ
3 + λ2φ4 + · · ·
]
Example 6.4.3 complex scalar φ 6= φ∗
Lfree =∂φ
∂xµ∂φ∗
∂xµ−m2φφ∗
global U(1)
φ −→ eiαφ
φ∗ −→ e−iαφ∗
↓local U(1); ∂µ −→ Dµ = ∂µ − ieAµ
here Aµ → A′µ = Aµ +1e
∂α(x)∂xµ
6.5 NON-ABELIAN GAUGE FIELDS 23
Lfree → DµφDµφ∗ −m2φφ∗
= Lfree(φ, φ∗) + Lint(φ, φ∗, Aµ)
Example 6.4.4 QED = Quantum Electrodynamics∼ Abelian U(1) gauge theory
Lfree = −14FµνF
µν + iψγµ∂ψ
∂xµ−mψψ
global U(1)
ψ(x) −→ eiαψ(x)ψ(x) −→ e−iαψ(x)
↓local U(1); ∂µ −→ Dµ = ∂µ − ieAµ
LQED = Lfree + Lint
Lint = e(ψγµψ)Aµ = eJµAµ
Example 6.4.5 ”many” complex φi(x) (i = 1, 2, · · · , n);same mass m
L = ∂µφi∂µφ∗i −m2φiφ
∗i
global gauge transformation;
φ −→ φ′i = ωijφj(x) ω ∈ SU(n)
φ∗iφi −→ φ′∗i φ′i = φ∗kω
∗kiωijφj = φ∗jφj
φ =
φ1
φ2
...φn
, φ† =
(φ∗1, φ∗2, · · · , φ∗n
)
−→φ′ = ωφ
L = ∂µφ†∂µφ−m2φ†φ
globally U(n) invariant
6.5 NON-ABELIAN GAUGE FIELDS
Example 6.5.1 SU(2)
φ =(φ1
φ2
), ω ∈ SU(2)
φ(x) −→ ωφ(x)
vector representation (= adjoint) of SU(2)
ξa(x) (a = 1, 2, 3), ξ −→ ξ′ = ωξω−1
L = ∂µφ†∂µφ+ ∂µξ
a∂µξa
− λ1(φ†φ)2 − λ2(ξaξa)2 − λ3φ(τaξa)φ
ξij = τaijξa(x)
φ†ξφ→ φ′†ξ′φ′
= φ†ω†(ωξω−1)ωφ
= φ†ξφ
Example 6.5.2 φiα(x)
i = 1, 2, · · · , n in fundamental of SU(n)α = 1, 2, · · · ,m in fundamental of SU(m)
φiα → φ′iα = ωijΩαβφjβ
L = ∂µφ∗iα∂
µφiα −m2φ∗iαφiα − λ(φ∗iαφiα)2
= ∂µφ†∂µφ−m2φ†φ− λ(φ†φ)2
describing free quarks20
Example 6.5.3 G′ = SU(2)⊗ U(1), ω ∈ SU(2), α ∈ U(1)φ, χ; doublets = fundamental representξ; singlet = trivial representation
φ → ωφ
χ → ωχ
ξ → ξ
φ → eiqφαφ
χ → eiqχαχ
ξ → eiqξαξ
Lint = λ[(φ†ξ)χ+ h.c.]
charge conservation; qχ + qξ = qφ
6.6 GAUGE DIFFICULTIES WITH QED
Taking the action S = − 116πc
∫FµνFµνdΩ, then by definition
we get the canonical conjugate momentum, say πµ, but if wequantize these quantities then we confuse the outcome.
πµ =∂L∂Aµ
= ∂0Aµ − ∂µA0
In particular,
π0 = 0 !!!?
we cannot quantize.This is because a massless field like EM has fewer d.o.s.
than a massive vector field.
⇒ Modifying the Lagrangian :L = −14FµνFµν
massive d.o.f. → the gauge invariance21
→ Adding a gauge-fixing term to the L:
L = −14FµνFµν − 1
2
(∂
∂xµAµ
)2
(16)
20quarks = spinors · · · in fundamental of SU(3)⊗ SU(6)
ψµiα;µ = Lorentz 0, 1, 2, 3, i = colour 1, 2, 3, α = flavour 1, 2, 3, 4, 5, 6
SU(3); (R, G, B) - ”colour” - ”local”SU(6); (s, c, t, b, u, d) - ”flavour” - ”global”
21Thus, the m→ 0 limit of the propagator for a massive vector particle is not the propagator for the photon.
24 7 THE WEAK INTERACTION
Nakanishi-Lautrup form
L = −14FµνF
µν +B(x)∂Aµ
∂xµ+
12αB(x)2
where α; non-dim const. ∈ R, and B(x); scalar
α = 1↔ Feynman gauge
α = 0↔ Landau gauge
Euler-Lagrange equation
¤Aµ − ∂
∂xµAν,ν −
∂
∂xµB(x) = 0
∂
∂xµAµ + αB(x) = 0
πk(x) = ∂0Ak + ∂kA0
π0(x) = B(x)πB = 0
−→ Aµ, πµ; canonical variable
α = 1; Feynman gaugeE-L eq.
¤Aµ − ∂
∂xµAν,ν −
∂
∂xµB(x) = 0
B(x) = − ∂
∂xµAµ
=⇒
¤Aµ = 0
B(x) = − ∂
∂xµAµ
Then,
LFey = −14FµνFµν −
(∂Aµ
∂xµ
)2
+12
(∂Aµ
∂xµ
)2
= −14FµνFµν − 1
2
(∂Aµ
∂xµ
)2
← corresponding to (16)
7 The weak interaction
xp
• Hadron interactionK+ → π+ + π+ + π−, K+ → π+ + π0
• Lepton interactionµ− → e− + νe + νµ
• Hadron-Lepton interactionn→ p+ e− + νe, π
+ → µ− + νµ
Chirality and helicityfor massive particles, its helicity is depending on the observer.→ mixed state
for massless particle, its chirality is either left-handed or right-handed.
=⇒ chiral symmetry
neutrinos are regarded as massless particle =⇒ system isnot invariant with C, P, CT, PT transf. and, in fact, CP, Tsymmetry are also broken (← Yukawa coupling const.).
7.1 FERMI’S THEORY
Empirically22,
LV−A = −GF√2
[ψpγµ(1− gγ5)ψn][ψeγµ(1− γ5)ψν ]
where23 the Fermi constant takes the numerical value
GF(~c)3
= 1.166× 10−5 GeV−2
and, g ' 1.26; protons and neutrons are composite particles
7.1.1 PARITY VIOLATION
the weak interactions violate parityCh.6 → chirality = helicity (for massless particles)
7.1.2 THE NEED FOR VECTOR BOSONS
the Fermi Lagrangian cannot be an exact theory of theweak interaction because it’s not renormalizable.
Ch.7: renormalizable theories must involve dimension-less coupling constants.
HoweverGF
(~c)3(Gev)−2
LV−A is peculiar in describing a zero-range interaction:there is no field for a particle to mediate the weak interaction
−→←−
Natural way;
intermediate vector bosonlow energy−−−−−−−→
limitFermi’s L
Wµ ← must be charged (∵ We need to efficiently convertbetween leptons & neutrinos)
and a complex field must be involved scalar field (spin-0)A sensible candidate theory
Anzats
L = gJµW †µ + hermitian conjugate
where g would be dimensionless, and
Jµ = ψeγµ(1− γ5)ψνe
claim; g2 =GF√
2M2W
To obtain this relation and to show that the vector-boson the-ory reduces to Fermi’s theory at low energy, we need to calcu-late matrix elements in 2 theories; Fermi case(1), vector-bosoncase(2).
22Classical Maxwell theory −→ QED23γ5 ≡ iγ0γ1γ2γ3
25
1. Fermi case - 1st order
M∝ GF√2aγµ(1− γ5)bcγµ(1− γ5)d (21)
2. Vector-boson case - 2nd order
M∝ g2aγµ(1− γ5)b(gµν − kµkν/M2
W
k2 −M2W
)cγν(1− γ5)d
(22)
the Feynman Rule (how to make the propagator)
1. For each vertex, write ieγα
2. For each internal photon line, of 4-momentum k, write−igαβk2 + iε
3. For each internal fermion line, of 4-momentum q,
writei
q/−m+ iε
4. For each external line, write a factor depending ontype:
(a) ur(p)[ur(p)] for an input [output] electron
(b) vr(p)[vr(p)] for an input [output] positron
(c) εrα(k) for an input or output photon
5. For each closed fermion loop, multiply by (−1) andtake the trace over spinor indices24
The Fourier-space equivalent of the real-space propa-gator
1. For a massless vector fields,
1i~〈0|T [Aµ(x)Aν(x′)]|0〉
→ −gµνGF =−gµνk2 + iε
(7.83)
2. For a massive vector fields,
1i~〈0|T [Bµ(x)Bν(x′)]|0〉
→ −gµν +kµkνm2
GF =−gµν + kµkν/m
2
k2 −m2 + iε(7.82)
3. For a spinor field
1i~〈0|T [ψµ(x)ψβ(x′)]|0〉
→ (q/+m)αβGF =(q/+m)αβk2 −m2 + iε
(7.81)
———————————-
kµ ¿MW =⇒ 1k2 −M2
W
= − 1M2W
(1 +
k2
M2W
)
=⇒ gµν − kµkν/M2W
k2 −M2W
' − gµν
M2W
(21) & (22) =⇒ g2
M2W
∼ GF√2
The vector-boson theory only be renormalizable in the casewhere MW = 0 =⇒ the weak interaction would be a force ofinfinite range. To cure this problem involves making the Wfield a gauge field, analogous to the position of the photon inEM.
8 Lie Group & Lie Algebra
Sophus Lie 1842-1899
8.1 LIE GROUP
Definition 8.1.1 LIE GROUPG is a Lie group if G satisfying the following 3 conditions;
1. G is a group25,24which corresponds to summing over spin states of the virtual e± pair25DISCRETE GROUPS
What are groups ?
• symmetry（19, 20c ）• invariance
• the structure of Math.
axiom1; associativityx(yz) = (xy)z for any x, y, z ∈ G
G is called a semi-group, if satisfying AXIOM 1.
axiom2; identity element∃ e ∈ G s.t. xe = ex = x for any x ∈ G
G is called a monoid, if satisfying axiom 1 & 2.
axiom3; inverse elementFor any x ∈ G ∃ x′ ∈ G s.t. xx′ = x′x = e
G is called a group, if satisfying axiom1, 2, & 3.
GroupA set G is a group
def⇐⇒G×G −→ Gis defined on G satisfying the following conditions;
(a) associativity; (xy)z = x(yz)
(b) an identity element; ∃e ∈ G s.t. ∀x ∈ G, xe = ex = x
26 8 LIE GROUP & LIE ALGEBRA
2. G is a manifold,
3. ’inner operation’ is C∞ mapping.
i.e.
1) G 3 g 7−→ g−1 ∈ G is C∞
and
2)G×G 3 (x, y) 7−→ xy ∈ G is C∞
N.B. 3. ⇔ (x, y) 7−→ xy−1 が C∞
Definition 8.1.2 LINEAR LIE GROUPG is called linear Lie group iff G is Lie subgroup ofGL(n,K);
1. G is a submanifold of GL(n,K)
2. G is a subgroup of GL(n,K)
(c) the inverse element; ∀x ∈ G, ∃x′ ∈ G s.t. xx′ = x′x = e
Since the algebraic structure is determined by the inner operation,we should write groups as (G, ·)
And, ifxy = yx ∀ x, y ∈ G
, then G is called an abelian group, or commutative group.
Proposition ”the uniqueness of identity & inverse” ;The identity & inverse element exist uniquely.
NOTATION;Denote inverse element by x−1 ∗26
Sufficient Condition of Being a GroupG; a setH; a subset of G
(a) An inner operation is defined on G satisfying associativity & the following conditions;
i. ∃e ∈ G s.t. ∀x ∈ G, ex = x ; called left-identity
ii. ∀x ∈ G, ∃x′ ∈ G s.t. x′x = e ; called left-inverse
=⇒ G is a group.
(b) An inner operation is defined on G( 6= ∅) satisfying associativity & the following condition;
∀g, g′ ∈ G, the next 2 equations¡gx = g′fg = g′ has a root x, y ∈ G.
=⇒ G is a group.
(c) H is a non-empty set and¡x, y ∈ H =⇒ xy ∈ Hx ∈ H =⇒ x−1 ∈ H
=⇒ H is a subgroup of G.27
(d) H is a non-empty set and
x, y ∈ H =⇒ x−1y ∈ H
=⇒ H is a subgroup of G
Example; subgroups of GL(n,R) GL(n,C)
SL(n,R) A ∈ GL(n,R) | det(A) = 1 (real) special linearSL(n,C) A ∈ GL(n,C) | det(A) = 1 (complex) special linearO(n) A ∈ GL(n,R) | tAA = I orthogonal
SO(n) = SL(n,R) ∩O(n) A ∈ SL(n,R) | tAA = I special orthonormalU(n) A ∈ GL(n,C) | AA† = I unitary
SU(n) = SL(n,C) ∩ U(n) A ∈ SL(n,C) | AA† = I special unitary
These groups are called classical groups.
Examples
GL(1,R) = R\0 = R×
GL(1,C) = C\0 = C×
O(1) = −1, 1 = S0 (0− dim sphere)
SO(1) = 1U(1) = z ∈ C | zz = 1 = z ∈ C | |z|2 = 1
= z ∈ C | |z| = 1 ∼= S1
SU(1) = 1
8.1 LIE GROUP 27
Differentiable Manifold of class Cr & Mappingof class Cr
Definition 8.1.3 TOPOLOGICAL MANIFOLDM ; topological space28
M is called a topological manifold , if satisfying thefollowing conditions;
1. M is a Hausdorff space.29
2. ∀ P ∈ M, ∃ (U,ϕ); m-dim coordinate neighborhood30
s.t.
P ∈ (U,ϕ), here ϕ is homeomorphism31 from U ∈ Oonto U ′ ⊂ Rm.
The problem whether metrization is possible for topologi-cal space is one theme of a topological space theory.
Axioms of separation
X; top. sp.
Definition 8.1.4 Frechet, T1
∀a, x(6= a) ∈ X, ∃O ∈ O; a ∈ O, x /∈ ODefinition 8.1.5 Hausdorff, T2
∀a, x(6= a) ∈ X, ∃O,O′ ∈ O; a ∈ O, y ∈ O′, O ∩O′ = ∅
Definition 8.1.6 Vietoris, T3
∀x, x /∈ F (closed), ∃O,O′ ∈ O; x ∈ O, F ⊂ O′, O ∩O′ = ∅
Definition 8.1.7 Tychonoff, T3 12∀x0 ∈ X,F (closed) 63 x0, ∃f : X → [0, 1] s.t.
f(x0) = 0, x ∈ F =⇒ f(x) = 1
Definition 8.1.8 Tietze, T4
∀F,Gclosed, F ∩G = ∅ =⇒ ∃O,O′ ∈ O s.t.
F ⊂ O, G ⊂ O′, O ∩O′ = ∅
Definition 8.1.9 DIFFERENTIAL MANIFOLD OF CLASS Cr
M ; topological space
M is called a differential manifolds of class Cr , ifsatisfying the following condition;
1. M is a Hausdorff space.
2. ∃ a family of m-dim coordinate neighborhood,(Uα, ϕα)α∈A32, s.t. M =
⋃α∈A Uα
3. Uα∩Uβ 6= ∅ =⇒ ϕβ ϕ−1α : ϕα(Uα∩Uβ)→ ϕβ(Uα∩Uβ)
is Cr mapping33. ( r ∈ N ∪ ∞)
Remark 8.1.10Topological manifolds is C0 manifolds.
i.e. if M satisfies the conditions 1, 2 (in case r = 0 ) ,then M is a topological manifolds.
Caution;Since Cr-manifold is determined by topological space M
and Cr-atlas S, therefore we should write Cr-manifold as(M,S).
Regular Point & Critical Point
Definition 8.1.11 REGULAR POINTP is called a regular point, when
(df)P : TP (M) −→ Tf(P )(N)
is linear mapping from TP (M) onto Tf(P )(N).
Definition 8.1.12 CRITICAL POINTP is called a critical point, when
(df)P : TP (M) −→ Tf(P )(N)
is not linear mapping from TP (M) onto Tf(P )(N).
Remark 8.1.13 NOTATIONCf =
defcritical value of f ⊂M
28M ; topological spacedef⇐⇒M is a set and O is a family of subsets of M .We call O topology of M , if satisfying the following conditions;
1. M ∈ O and ∅ ∈ O
2. Oi, Oj ∈ O =⇒ Oi ∩Oj ∈ O
3. Oλλ∈Λ ∈ O =⇒ Sλ∈ΛOλ ∈ O
The pair (M,O) is called topological space, and Oi is called open set in M .
29∀ P,Q ∈M, ∃U, V ∈ O s.t. P ∈ U, Q ∈ V, U ∩ V = ∅30The pair U and ϕ s.t. U ⊂
openM, ϕ : U
∼→ U ′ ⊂open
Rm , and we call ϕ a local coordinate system on U .
In addition, ϕ(P ) = (x1, x2, · · · , xm) is called a local coordinate at P w.r.t. (U,ϕ).
31ϕ is homeomorphismdef⇔ ϕ is bijective, continuous and open mapping from U onto U ′
Then U and U ′ is called homeomorphic, denoted by U ∼= U ′
32(Uα, ϕα)α∈A is called a system of coordinate neighborhoods, or atlas,denoted by S = (Uα, ϕα)α∈A.
S which satisfies the condition 3 is called Cr-atlas.33ϕβ ϕ−1
α , which is homeomorphism , is called coordinate transformation.
28 8 LIE GROUP & LIE ALGEBRA
Definition 8.1.14 CRITICAL VALUEQ is called a critical value, when
Q ∈ f(Cf ) ⊂ N
Definition 8.1.15 REGULAR VALUEQ is called a regular value, when
Q /∈ f(Cf ) ⊂ N
Theorem 8.1.16Q ∈ N ; regular value of f , Cr mapping from M to N
f−1(Q) 6= ∅
=⇒ f−1(Q) is a Cr submanifold of M of (m− n)-dim.
Immersion & Embedding
In this section, we setM ; m-dim Cr manifoldN ; n-dim Cr manifoldf : M → N ; Cr mapping−→ rank(Jf)P ≤ mindimM, dimN 34
Definition 8.1.17 IMMERSIONf : M → N ; immersion
def⇐⇒ ∀P ∈ M, (df)P : TP (M) −→ Tf(P )(N) is linear andinjective.⇐⇒ rank(Jf)P = dimM = m
Definition 8.1.18 SUBMERSIONf : M → N ; submersion
def⇐⇒ ∀P ∈ M, (df)P : TP (M) −→ Tf(P )(N) is linear andsurjective.⇐⇒ rank(Jf)P = dimN = n⇐⇒ Every point in M is regular point35 w.r.t. f .
Definition 8.1.19 EMBEDDINGf : M → N ; embedding
def⇐⇒1. f is immersion.
2. f : M → f(M)(⊂ N) is homeomorphism36.
Theorem 8.1.20 Embedding theoremFor arbitrary class-Cr37 compact manifold of m-dim, say M ,∃ g ∈ Cr : M → Rn; embedding38
Theorem 8.1.21 Whitney theoremFor arbitrary class-Cr39 σ-compact manifold of m-dim, sayM ,
∃ g(∈ Cr) : M → R2m+1; embedding and g(M) is closedset in R2m+1
Example 8.1.22 GL(n,R) = A ∈M(n,R) | det(A) 6= 0
Let us check for GL(n,R) to be a Lie group.
・to be a group
identity element∃I ∈ GL(n,R)
inverse elementA ∈ GL(n,R)⇒ ∃A−1 ∈ GL(n,R)
associativityA,B,C ∈ GL(n,R)⇒ (AB)C = A(BC)
・to be a manifold
GL(2,R) ⊂ M(2,R) =(
a bc d
)| a, b, c, d ∈ R
∼=
R4 = (a, b, c, d) | a, b, c, d ∈ R
det : M(2,R) −→ R
R ⊃open
R×
GL(2,R) = det−1(R×)
GL(2,R) ⊂open
M(2,R) ∼= R4
Since GL(2, R) is an open subset of R4 which is a classC∞ manifold, then it is C∞ manifold.
・for the operation w.r.t. grp to be C∞
1. inverse element
GL(2,R) 3 A =(a bc d
)
7−→ A−1
=1
ad− bc(
d −b−c a
)∈ GL(2,R)
R4 3 (a, b, c, d)
7−→(
d
ad− bc ,−b
ad− bc ,−c
ad− bc ,a
ad− bc)∈ R4
2. multiplication
Theorem 8.1.23 Cartan theoremA subgroup of Lie group is also Lie group.
34
(Jf)P =
2664
∂f1
∂x1 · · · ∂f1
∂xm
.... . .
...∂fn
∂x1 · · · ∂fn
∂xm
3775
35By definition m ≥ n, and every points in N is regular value of f36f is homeomorphism
def⇔ f is bijective, continuous and open mapping from U onto V
Then U and V is called homeomorphic, denoted by U ∼= V
371 ≤ r ≤ ∞38n is a sufficiently large integer.391 ≤ r ≤ ∞
8.2 LIE ALGEBRA 29
8.2 LIE ALGEBRA
Definition 8.2.1 LIE RINGR is called a Lie ring40 if Addition(+) andmultiplication([∗, ∗], say commutator) are defined on R andsatisfy the followings:
1. abelian w.r.t. Addition
(a) associativity; r1 + (r2 + r3) = (r1 + r2) + r3
(b) identity; 0R ∈ R s.t. r + 0R = 0R + r = r
(c) inverse; −r ∈ R s.t. r + (−r) = (−r) + r = 0R
2. multiplication [∗, ∗](a) Jacobi iden.
[r1, [r2, r3]] + [r2, [r3, r1]] + [r3, [r1, r2]] = 0R(b) [r, r] = 0R
3. distributive principle
(a) [r1, r2 + r3] = [r1, r2] + [r1, r3]
(b) [r1 + r2, r3] = [r1, r3] + [r2, r3]
r, r1, r2, r3 ∈ R
Proposition 8.2.2
• [r1, r2] = −[r2, r1]
• [r, 0R] = 0R
Definition 8.2.3 LIE ALGEBRAR is called a Lie algebra over K if
1. R; Lie ring
2. R is vect. sp.41 over K (field)
multiplication [r1, r2] ∈ R, say commutator, is definedon R s.t.
for any r1, r2, r3 ∈ R[kr1, r2] = [r1, kr2] = k[r1, r2] k ∈ K
Example 8.2.4Let U be an associative algebra and define
[a, b] = a · b− b · a a, b ∈ U .where a · b is the multiplication on U
Then, U is Lie alg. w.r.t. this new multiplication [a, b].
Definition 8.2.5 LINEAR LIE ALGBRAR is called a linear Lie algebra over K if
1. R; subset of M(n,C)
2. R is vect. sp. over K (field)40RINGR: a ringR is a (commutative) ring if
1. R is a commutative group w.r.t. addition.
2. ∀r1, r2, r3 ∈ R, (r1r2)r3 = r1(r2r3)
3. ∃1 ∈ R s.t. ∀r ∈ R, 1r = r1 = r
4. ∀r1, r2, r3 ∈ R, (r1 + r2)r3 = r1r3 + r2r3
5. ∀r1, r2 ∈ R, r1r2 = r2r1
N.B. We don’t assume 1 6= 0 , when defining a ring.41
R: K-moduledef⇐⇒
(1) R is an abelian w.r.t. addition.i.e. (R,+):an abelian group ⇔ Addition is defined on R, and satisfies the following conditions;
(a) associativity; r + (r′ + r′′) = (r + r′) + r′′
(b) an identity element; ∃0 ∈ R s.t. 0 + r = r
(c) the inverse element; ∀r ∈ R, ∃ − r ∈ R s.t. r + (−r) = 0
(d) commutativity; r + r′ = r′ + r
(2) K ×R −→ R : scalar multiplication s.t.for any k, k′ ∈ K, r, r′ ∈ R(a) (k + k′)r = kr + k′r
(b) (kk′)r = k(k′r)
(c) k(r + r′) = kr + kr′
(d) 1K · r = r
FIELDF: a setF is a field if
(a) F is a commutative ring,
(b) ∀x ∈ F, x 6= 0 =⇒ ∃ x−1 ∈ F s.t. x−1x = xx−1 = 1,
(c) 0 6= 1.
Example1. F= 0, 1 ; field of 2 elements2. Q = rational number3. Q(
√2) = a+ b
√2| a, b ∈ Q
30 8 LIE GROUP & LIE ALGEBRA
3. ∀r1, r2, [r1, r2] ≡ r1r2 − r2r1 ∈ R
Definition 8.2.6 LIE ALG. OF LIE GRP.Lie algebra of Lie group G def= T1G
G ≡ AGProposition
1. AG is a vector space. (→ ∃ basis, dima )
2. AG is closed under commutation [∗,∗].aTheoremLet V be vector sp. over K ⇒ V has basis.
G 3 g(t) = 1 + tA+O(t2) for through 1A ≡ AG
Example 8.2.7
1. U(n) 3 g(t) = 1 +At+O(t2)
UU† = 1 =⇒ [1 +At+O(t2)][1 +A†t+O(t2)] = 1
=⇒ A+A† = 0
=⇒ A = −A† (antihermitian)
Let A be hermitian, then A = A†
−→ A = iA, U = eA = eiA
2. O(n)
OOT = 1 =⇒ A = −AT (antisymmetric)
−→ O = eA
3. SU(n) = U ∈ U(n) | detU = 1
g(t) ∈ U(n)⇒ A† = −A
det g(t) = det(1 + tA+O(t2))
= 1 + (trA)t+O(t2)−→ trA = 0
4. SO(n)−→ AT = −A, trA = 0
Remark 8.2.8 Notation; BasisAG; (finitely generated) vect. sp. −→ ∃ basisbasis; Tk ∈ AG (k = 1, 2, · · · ,dimAG)
Definition 8.2.9 Structure constantLet T1, · · ·Tn be basis of AG.Since [Ti, Tj ] ∈ AG,
[Ti, Tj ] = c kij Tk
c kij is called structure constant.
FACT structure constant determine the structure of Lie al-gebra.
Proposition 8.2.10
1. Not depending upon T
2. c kij = −c k
ji
3. c lij c
mlk + c l
jk cm
li + c lki c
mlj = 0
Example 8.2.11 SU(n)
Lemma 8.2.12
Tr(TiTj) =δij2
(normalization)
Tr(Ti[Tj , Tk]) = icijk2
Proposition 8.2.13
cijk = cjki = ckij
= −cikj = −ckji = −cjik
Theorem 8.2.14
dimG = n =⇒ dimAG = n
Definition 8.2.15 SubalgebraS is called a subalgebra of Lie algebra R if
1. S is subspace of Q
2. S is closed w.r.t. [∗, ∗] i.e.
[s1, s2] ∈ S ∀s1, s2 ∈ S
Definition 8.2.16 Invariant subalgebraS is called an invariant subalgebra of Lie alg. R if S is anideal of R w.r.t. [ , ].
i.e. ∀r ∈ R, ∀s ∈ S, [r, s] ∈ SDefinition 8.2.17 Commutative Lie algebra
R is a commutative Lie algebra if
∀r, r′, [r, r′] = 0
Definition 8.2.18 Semi-simple Lie algebras | [s, r] = 0 ∀r ∈ R is commutative invariant subalgebra,and called a center of R. It is clear that the center of R is anideal of R
Lie alg. R which has no commutative invariant subalgebra,is called semi-simple Lie algebra.
Lie alg. which has no invariant subalgebra is called simpleLie algebra.
Theorem 8.2.19semi-simple Lie alg. is a direct sum of simple Lie algebra.
Proposition 8.2.20
dimSU(n) = dim su(n) = n2 − 1
Example 8.2.21 basis in su(2) → Pauli matrices
σ1 =(
0 11 0
), σ2 =
(0 −ii 0
), σ3 =
(1 00 −1
)
8.3 Connected component including the identity 31
Example 8.2.22 basis in su(3) → Gell-Mann matrices
λ1 =
0 1 01 0 00 0 0
, λ2 =
0 −i 0i 0 00 0 0
λ3 =
1 0 00 −1 00 0 0
, λ4 =
0 0 10 0 01 0 0
λ5 =
0 0 −i0 0 0i 0 0
, λ6 =
0 0 00 0 10 1 0
λ7 =
0 0 00 0 −i0 i 0
, λ8 =
1√3
0 00 1√
30
0 0 −2√3
Definition 8.2.23 Homomorphism and IsomorphismR,S; Lie alg. over K
1. σ ∈ Hom42(R,S) ⇐⇒def
(a) σ is linear mapping,(b) For r1, r2 ∈ R, σ([r1, r2]) = [σ(r1), σ(r2)]
2. In particular σ ∈Hom(R,S) is bijective, σ is called aisomorphism. Then σ−1 : S → R is also isomorphism.We say that R and S is isomorphic (as Lie alg.) when ∃isomorphism.
Proposition 8.2.24R,S; Lie algs. over K
σ ∈ Hom(R,S)Then
1. σ(R) is a subalgebra of S.
2. Ker(σ) is an ideal of R
Proposition 8.2.25
R; solvable Lie alg. ⇐⇒ S has s−chain
Proposition 8.2.26R; Lie alg.
The following 3 condition are equivalent;
1. R is nilpotent
2. R has an ascending central chain of finite length.
3. ∃ n-chain
8.3 Connected component including theidentity
In this section, G, G0 denote the linear Lie grp. and the con-nected component including 1∈ G respectively.
Proposition 8.3.1x ∈ C; some connected component of G
Then
C = xG0 = xg| g ∈ G0= G0x = gx| g ∈ G0
Theorem 8.3.2
1. G0 C G
2. G0 is open and closed in G.
Therefore G0 is closed in GL(n,C) i.e. G0 is a linearLie group.
Corollary 8.3.3
AG = AG0
8.4 Connected linear Lie group
In this section, H, G denotes a connected linear Lie group.
Theorem 8.4.1For any b > 0, U := g ∈ G| ‖g − 1G‖ < b
Then G is generated by U .
Corollary 8.4.2Every G’s element =
∐i exp(ri) ri ∈ AG
Proposition 8.4.3
H ⊂ G ⇐⇒ AH ⊂ AG
Corollary 8.4.4
AG = AH =⇒ G = H
Definition 8.4.5 INNER AUTOMORPHISMIg ∈Inn(G)43
⇐⇒def
Ig(g′) = gg′g−1 for g′ ∈ GProposition 8.4.6Ig ∈Inn(G)
Then,dIg : AG → AG
are inner automorphism.
∀r ∈ AG, dIg(r) = grg−1
8.5 REPRESENTATIONS of G & AG
Definition 8.5.1 Representation of G
D ∈ Hom(G,GL(n,C))
is called a representation of G.
Definition 8.5.2 Representation of Ag
ρ ∈ Hom(AG, GL(n,C))
is called a representation of AG.
Remark 8.5.3 homomorphism for ring
1. ρ(A+B) = ρ(A) + ρ(B) A,B ∈ AG
2. ρ(AB) = ρ(A)ρ(B)
3. ρ(1AG) = 1
42Homomorphism43inner automorphism
32 9 BUNDLE
homomorphism44 for R-module45
1. ρ(r1m1+r2m2) = r1ρ(m1)+r2ρ(m2) r1,2 ∈ R, v1,2 ∈M
Remark 8.5.4
ρ ([A, B]) = [ρ(A), ρ(B)]
D(g(t)) = D(1 + εA) = 1 + εD(A)
For basis,[ρ(Ti), ρ(Tj)] = f k
ij ρ(Tk)
For compact connected Lie group,
D(g) = etiρ(Ti)
Definition 8.5.5 IrreducibleD(G) in V is irreducible if@ non-trivial invariant subspace in V
Remark 8.5.6∅, V is trivially invariant.
Definition 8.5.7 EquivalentD(G), D′(G′) are equivalent⇐⇒def
∃ irreducible S ∈ V s.t.
∀g, D′(g) = SD(g)S−1
Definition 8.5.8 Invariant subspaceD(G) in V
W is invariant subspace (⊂ V )
∀ψ ∈W, D(g)ψ ∈W i.e. ideal
Example 8.5.9 Fundamental representation
G; n× n matrices ∀‖gij‖ ∈ G
ψ =
ψ1
...ψn
∈ V
(T (g) · ψ)i =def
gijψi
For basis ei in VT (g)ei = gijej
Example 8.5.10 G = SU(n)
fundamental repr. is irreducible, and complex. (n > 2)
Example 8.5.11 G = SO(n)
fundamental repr. is irreducible, and real.
Example 8.5.12 Adjoint representation of G
∀g ∈ G, Ad(g)Ad(g)A =
defgAg−1
−→ Ad(g) ∈ AG
9 Bundle
9.1 Some definitions
Definition 9.1.1 COMPACT MANIFOLDSM ; Cr manifold
M is called compact Cr manifolds, if, however any opencovering of M , Uαα∈A, may be given, we can take so finitea pair from the open covering Uαα∈A that M =
⋃ni=1 Ui.
i.e.
∃I ⊂ A s.t.
#I <∞, M =⋃
i∈IUi
Remark 9.1.2noncompact means;
∃ open covering Uαα∈A s.t.
M 6⊂ 46
n⋃
i=1
Ui ⊂⋃
α∈AUα
Definition 9.1.3 σ-COMPACTM ; topological space
M is called σ-compact manifold if
∃Ki ⊂M ; compact subsets of M s.t.
M =∞⋃
i=1
Ki
i.e.
∃Ki ⊂M s.t.
Ki =⋃
j∈J⊂AUj , #J <∞
M =∞⋃
i=1
Ki
Theorem 9.1.4 Whitney TheoremFor arbitrary class-Cr47 σ-compact manifold M of m-dim,
∃ N s.t. class-C∞ m.f.d. and Cr-diffeomorphic to M
Definition 9.1.5 PARACOMPACTM ; topological space
M is called paracompact48 manifold if
For any opencovering Uαα∈A,44=linear mapping for vect. sp.45Ring R is R-module.46whatever we may choose471 ≤ r ≤ ∞48compact ⊂ σ-compact ⊂ paracompact
9.1 Some definitions 33
∃ Vββ∈B ; locally finite49 open covering s.t. Vββ∈B isrefinement50(or subdivision) of Uαα∈A
FACTParacompact manifold of countable union of simple con-
nected component is σ-compact.
Definition 9.1.6 Cs-FUNCTIONM ; Cr-manifold of dim m
f : M −→ R is called a function of class Cs on M , iffor every (Uα, ϕα) ∈ S, f ϕ−1
α : U ′α → R ∗51 is a functionof class Cs on U ′ ⊂ RmDefinition 9.1.7 Cs-MAPPINGM ; Cr-manifold of dim m
N ; Cr-manifold of dim nf : M −→ N ; continuous mapping52
f is called a mapping of class Cs at P , if satisfying thefollowing conditions;∃(U,ϕ) 3 P, (V, ψ) 3 f(P )
1. f(U) ⊂ V2. ψ f ϕ−1 is class Cs.
f is called a mapping of class Cs, if class Cs at eachpoint of M .
Definition 9.1.8 Cs DIFFEOMORPHISMM ; Cr-manifold
N ; Cr-manifold
f : M −→ N is called a diffeomorphism of class Cs, ifsatisfying the following conditions;
1. f is bijective.
2. Both f and f−1 are Cs-mapping.
In addition, we call M and N Cs diffeomorphic, denoted
by MCs
∼= N, or M ∼= N
Definition 9.1.9 Cr-SUBMANIFOLDL ; a subset of M
L is called a Cr-submanifold of dim l, if satisfying thefollowing condition;
1. In case m = l, then L is open set in M
2. In case m > l, then ∀P ∈ L, ∃(U,ϕ) 3 P s.t.
L ∩ U = (x1, x2, · · · , xm) ∈ U | xl+1 = · · · = xm = 0
COLUMN ”∼morphism”
homomorphism
G, G′; groups
g1, g2 ∈ GHom(G,G′)
=deff : G→ G′ | f(g1 G g2) = f(g1) G′ f(g2)
FACT Hom(G,G′) is an abelian group if G′ is anabelian group and we define addition by (f1 + f2)(g) =f1(g) + f2(g) g ∈ G.
endomorphism
G; group
g1, g2 ∈ G
Hom(G,G)=deff : G→ G | f(g1g2) = f(g1)f(g2)
≡ End(G)
FACT End(G) is a non-abelian ring if G is an abeliangroup and we define multiplication by the compositionof mapping.
isomorphism
f ; isomorphism from G to G′
⇐⇒def
f ∈ Hom(G,G′) & bijective
automorphism
f ; automorphism from G to G
⇐⇒def
f ∈ End(G) & bijective
Notation
Aut(G) ≡ f | automorphismFACT Aut(G) is a group since f, g ∈ Aut(G)⇒ f g ∈
Aut(G), and f−1 ∈ Aut(G).
inner automorphism
Inn(G) =deffg|f(g1) = gg1g
−1, g ∈ G
FACT Inn(G) C Aut(G)
homeomorphism
diffeomorphism49Vββ∈B is locally finite coveringdef⇐⇒ ∀P ∈M, ∀U ∈ U(P ), ∃β; finite, s.t. U ∩ Vβ 6= ∅
50Vββ∈B is refinement of Uαα∈Adef⇐⇒ ∀Vβ , ∃Uα s.t. Vβ ⊂ Uα
51ϕα : U∼→ U ′ ⊂ R ; homeo.
52f is continuousdef⇐⇒ ∀ V ⊂
openN, f−1(V ) ⊂
openM
34 9 BUNDLE
9.2 Tangent Space ∼ Bundle
Definition 9.2.1 DERIVATIVEf, g ∈ C∞(U), a, b ∈ R
Z : C∞(U) → C∞(U) is called a derivative of C∞(U) ifsatisfying the following condition;
1. Z(af + bg) = aZf + bZg; linear mapping
2. Z(fg) = f · Zg + Zf · g; Leibniz rule
Here (f · Zg)(x) = f(x)Zg(x)
Definition 9.2.2 FIBER BUNDLEFiber bundle (E, π, M, F, G) is consisted of the followings;
1. E; C∞-manifold, called a total space
2. M ; C∞-manifold, called a base
3. F ; C∞-manifold, called a fiber
4. E π−→M ; surjection, called a projection
π−1(p) =def
Fp ' F ;
is called a fiber at p.
5. G; Lie group acting on F from left-hand-side, called astructure group
6. local trivialization; open covering of M , say Uαα∈A,and φα s.t.
φα : Uα × F → π−1(Uα)
π φα(p, f) = p
7. transition function;
φα(p, f) ≡ φα,p(f) s.t.
φα,p : F ∼−−−−→diffeo.
Fp;
diffeomorphic, and
tαβ(p) ≡ φ−1α,p φβ,p : F → F
should be in G (imperative; φ ∈ G).
Then, φα and φβ are associated on Uα ∩ Uβ 6= ∅ by thesmooth fn. tαβ in such a way that
φβ(p, f) = φα(p, tαβ(p)f)
tαβ is called a transition function.
Proposition 9.2.3 Some properties
(a) tαβ tβγ tγα = idG
(b) tβα = t−1αβ
(c) tαα = idG
Definition 9.2.4 TRIVIAL BUNDLEFiber bundle where any transition function is identity map iscalled a trivial bundle.
trivial bundle = M × FDefinition 9.2.5 PRINCIPAL FIBER BUNDLE
(E, π,M,F,G)→ (P, π,M,G,G) ≡ (P, π,M,G)
is called a G-bundle, or principal (fiber) bundle.
Local trivializationFor u ∈ π−1(Ui), p = π(u), put
φi : Ui ×G→ π−1(Ui)
local trivialization given by φ−1i (u) = (p, gi)
P ×G 3 (u, g) 7→ u · g ∈ Pand the followings are satisfied;
1. (ug)g′ = u(gg′) for u ∈ P, g, g′ ∈ G2. u · 1G = u (1G; identity in G)
Furthermore, the next conditions are satisfied;
1. π(ug) = π(u)
2. π(u) = π(u′) ⇒ ∃1 g ∈ G s.t.
u′ = ug
Definition 9.2.6 VECTOR BUNDLEVector bundle is a fiber bundle whose fiber is a vector space.⇒ F ' RkLet M be a C∞-m.f.d. of n-dim. Then, E is of (n+k)-dim.
Since transition fn. tij : F ' Rk → F ′ ' Rk, tij ∈GL(k, R). (If F ' Ck, then tij ∈ GL(k, C))
Definition 9.2.7 SECTIONM ; C∞-m.f.d.
(E, π, M); fiber bundles is called a section s.t. π s = id ;conti.
Remark 9.2.8 NotationΓ(E) ≡ s : M → E | π s = idM53
Obviously, s is homeo. from M onto s(M). In particular,we call s0 a zero section s.t.
for each P ∈M, s0 : P → 0EP∈ EP
As is often case, we identify M with s0(M), and considerM as a subtopological sp. of E.
Proposition 9.2.9π : E →M
E satisfies the local triviality.⇐⇒ ∀P ∈M, ∃U ∈ U(P) s.t.
∀Q ∈ U, ∃s1(Q), · · · , sr(Q) ∈ EQThus we pick them out as basis in EQ, and define rank (ordim) of E = r
For vector bundle, we define
(s+ s′)(p) = s(p) + s′(p)(fs)(p) = f(p)s(p) f ∈ F(M)
53Γ(E) is vect. sp. of ∞-dim,and is A0(M)-module
9.2 Tangent Space ∼ Bundle 35
Definition 9.2.10 TANGENT SPACE
TPM ⊂subvect. sp.
⟨(∂
∂x1
)
P
, · · · ,(
∂
∂xn
)
P
⟩
is called a tangent (vector) space.
T ∗PM ⊂subvect. sp.
⟨(dx1, · · · , dxn)
P
⟩ ≡ Ω1
is called a cotangent (vector) space.
c.f.for X ∈ TPM , X = ai
∂
∂xifor ω ∈ T ∗PM, ω = aidx
i
Definition 9.2.11 TANGENT BUNDLE
TM =∐
P∈MTPM
is called a tangent bundle (of 2n-dim).
T ∗M =∐
P∈MT ∗PM
is called a cotangent bundle (of 2n-dim).
Definition 9.2.12 TENSOR BUNDLE OF TYPE (r, s)
T rsM =
(r⊗TM
)⊗
(s⊗T ∗M
)
is called a tensor bundle of type (r, s)
Definition 9.2.13 VECTOR FIELDSΓ(TM) is called a vector field ≡ X.Γ(T rs ) is called a tensor field of type (r, s).
Definition 9.2.14 one-para. loc. transf. grpfλ : M ∼→M : diffeomorphic s.t.
• f0 = id
• fλ1+λ2 = fλ1 fλ2
fλ induces the curve → introducing the vector field
Definition 9.2.15 COMPLETE VECTOR FIELDS
LIE DERIVATIVE
for some func. f ,
£Xf =def
lim∆t→0
f(xµ(t+ ∆t))− f(xµ(t))∆t
(†)
where t is a parameter of the curve induced by the vector fieldX
Definition 9.2.16 Lie derivativeX ∈ TxM , fλ; one-para, loc. transf. grp. induced by X
=⇒ (£XT )x =def
limλ→0
Tx − ((fλ)∗T )xλ
which corresponds to (†) and, in particular, for Y ∈ TxM
£XY = [X,Y ]
claim. 1; [X,Y ] is also a vector field.claim. 2; [X,Y ] is a derivative, say Lie derivative.cf. covariant derivative vs. Lie derivative
∇XY = ∇Xµ∂µ(Y ν∂ν)
= Xµ∇µ(Y ν∂ν) (∇µ := ∇∂µ)
= Xµ(∂µY ν + ΓνµλYλ)∂ν (∇µ∂ν = Γλµν∂λ)
£XY = [X,Y ] = [Xµ∂µ, Yν∂ν ]
= Xµ∂µ (Y ν∂ν)− Y ν∂ν (Xµ∂µ)=
Definition 9.2.17 LIE ALGEBRAG; Lie grp.
T1GG ≡ AG
is called a Lie algebra.
i.e.aisi(1G) | s : G 3 1G 7→ T1G
G
54
Theorem 9.2.18 FrobeniusM ; m-dim C∞-m.f.d.N ; n-dim sub-m.f.d. of MX1, · · · , Xn ∈ Γ(TM); lin. independent
Then
∃(m−n) yp s.t. XIy
p = 0 (p = n+ 1, · · · ,m)
m∃ fKIJ(x) s.t. [XI , XJ ] = fKIJXK
where I, J,K = 1, 2, · · · , nProof of 9.2.18
(⇓); trivial(⇑);
Let xµ be l.c.s.54RINGA ring A is a set, with multiplication and addition satisfying the following conditions.
1. A is an abelian group with respect to addition.
2. the multiplication is associative and has an identity element 1.
3. x, y, z ∈ A⇒ x(y + z) = xy + xz, (x+ y)z = xz + yz (distributive)
Notation0: the identity element with respect to addition.1: the identity element with respect to multiplication.−a: the inverse for a with respect to addition.
General linear group := matrices which is ”regular”
36 9 BUNDLE
By assumption XI = XµI (x)
∂
∂xµand Xi
I is non-singular ma-trix after permutating the bases appropriately.
⇒ ∃ XIi
Define Yi =def
XIi XI
⇒Yi =∂
∂xi+ Y pi
∂
∂xp(p = n+ 1, · · · , n+m)
⇒[Yi, Yj ] = (YiYpj − YjY pi )
∂
∂xp
However Yi; closed w.r.t. [∗, ∗]∴ [Yi, Yj ] = 0
⇒∃ (xµ) = (xi, yp) s.t. Yi =∂
∂xi
⇒XIyp = Xi
I(Yiyp) = 0
¤
Definition 9.2.19 BUNDLE ISOMORPHISM(P, π,M,G), (Q, π,N,G); G-principal fiber bundle
f is called a bundle isomorphism if satisfying the fol-lowing conditions;
1. f is diffeo.
2. For u ∈ P, g ∈ G, f(ug) = f(u)g55
In particular, Q = P ⇒ f is called a bundle automorphism.
Remark 9.2.20 Notation
1. Aut(P )≡ bundle automorphism with (P,M,G)2. Aut0(P ) ≡ f ∈ Aut(P )| f = idM ⊂ Aut(P ) which is
called a
Definition 9.2.21 ASSOCIATED BUNDLEP (M,G); G-principal bundle
g ∈ G acts on F from left-hand-side s.t.
(u, f)→ (ug, g−1f) u ∈ P, f ∈ FThen a associated (fiber) bundle (E, π,M,G, F, P ) is de-fined as a quotient space identified (u, f) with (ug, g−1f) s.t.
P × F/G.
9.3 Differential Form
Definition 9.3.1 DIFFERENTIAL k-FORM
s ∈ Γ
(k∧T ∗M
) ∣∣∣ smooth
⇐⇒ ω :k⊗TPM → R ∀P ∈M
s.t.
1. linear
2. skew-symmetric
3. differentiable
notation
Definition 9.3.2 EXTERIOR DERIVATIVE
d : Ap(M)→ Ap+1(M) linear
Lemma 9.3.3
1. d(ω ∧ ω′) = dω ∧ ω′ + (−1)pω ∧ ω′ for ω ∈ Ap(M)
2. dd = 0
3. for f ∈ C∞(M), df = f,idxi
Definition 9.3.4 PULLBACKFor each p, linear mapping ϕ∗ : ΩpV → ΩpU s.t.
1. p = 0; ϕ∗f ≡ f · ϕfor f ∈ Ω0V
2. p > 0; (ϕ∗ω)(x) = (Λpϕ∗x)(ω(y))
for ω ∈ ΩpV, y = ϕ(x)
is called pullback.
properties of pullback For ∀ ω, η, ρ ∈ Ω
1. ϕ∗(ω + η) = ϕ∗ω + ϕ∗η
2. ϕ∗(η ∧ ρ) = ϕ∗η ∧ ϕ∗ρ3. In particular, for f ∈ C∞(V ),
ϕ∗(fρ) = (f · ϕ)ϕ∗ρ
, andd(ϕ∗ω) = ϕ∗(dω)
Definition 9.3.5 RIEMANNIAN METRICSection 〈, 〉, which is symmetric and positive, (E⊗E)∗ is calleda Riemannian metric of E.
Theorem 9.3.6M ; paracompact top. sp.
π : E →M ; vector bundleThen metric of E always do exists!
Definition 9.3.7 De Rham cohomologyDe Rham cohomology
for R or open set U ⊂ R,
HjDR =
def
ker d : Ωk(U)→ Ωk+1(U)Im d : Ωk−1(U)→ Ωk(U)
Theorem 9.3.8 Poincare duality theoremM ; oriented, ∃ good finite covering
⇓Hj(M) ' Hom(Hj
c (M),R) = (Hn−jc (M))∗
Hjc (M) ' (Hn−j(M))∗
55This condition shows that f inducesf : M → N
37
10 Connections & Curvatures
connections; the relation between TxM ↔ T ′xMcurvatures; the relation between ∂µ ↔ ∇µ
10.1 Connection
Principal bundles P 56 are also C∞-mfd.
Definition 10.1.1 Connection formfor u ∈ P, π(u) = x ∈M
TuP = VuP ⊕HuP
set M × G 3 (x, g) as the loc. coord. in P and alg. g ascorresponding to G.
[λa2i,λb2i
]= fabc
λc2i
where fabc are structure constants. Then we can define thefollowing quantities:
A =def
Aaµλa2idxµ (connection form)
ω =def
g−1dg + g−1Ag
Remark 10.1.2connection Aaµ = gauge potentialcoord. transf. for fiber in P = gauge transf.
10.2 Curvature
Definition 10.2.1 curvature
[Dµ, Dν ] =def−F aµνRa
Proposition 10.2.2 exterior covariant derivative
11 Non-Abelian gauge symmetries
Historically, the first manifestation of this idea was isospin:
2-state function ψ =[np
]
11.1 YANG-MILLS THEORIES
Yang-Mills (1954) ; the extension of Gauge field for non-Abelian group57
11.2 SU(2) AND WEAK INTERACTIONS
2-state system; spin-1/2( → 2S+1 = 2) = Fermion ↔ Diracfield
multiplet structureleptons: (
eνe
) (µνµ
) (τντ
)
quarks58:
(ud
) (cs
) (tb
)→
strong interactionweak interaction
pair ↔ 2 possible states we can flipcharge-exchange interaction; intermediate vector-
boson, W±(80.3 GeV/c2)question arising: the observability of changes in phase of
the wave fn. and of the analogues of rotations in the internalsp.
↓ gauge transformation
ψ → eiθMψ, M ∈Mat(2)
Renormalizability→ want; ψ†ψ unchanged
ψ†ψ → (eiθMψ)†(eiθMψ)
= ψ†e−iθM†+iθMψ
∴ M† = M.
i.e. generator M must be Hermitian
56(P, π, M, F = G, G)57
PHYSICS MATHEMATICS
Gauge type principal fiber bundleGauge local trivial mapping
Gauge group structure groupGauge transformation bundle automorphism (transition fn.)――Gauge fields――
Gauge potential connectionField strength curvature (form)
YM field YM connectionWave fn. section of vector bundle
phase factor parallel transformationinstanton autodual YM connection over R4 or S4
integrability Bianchi identitytopological quantum number index of principal fiber bundle
EM field connection over U(1)-bundleisospin gauge field connection over SU(2)-bundle
action integral Yang-Mills functional
58Upper quarks have 2/3 e, lower ones -1/3 e
38 12 SPONTANEOUS SYMMETRY BREAKING
the pure phase transformations59 (M = I2)→ Hermitian & traceless generator
Anzats
weak interaction should be invariant under local SU(2)transformation.
dimSU(2) =3→ i = 1, 2, 3; We will need to introduce 3 gaugefields.−→ the gauge-covariant derivative
Dµ ≡ ∂µ − ig∑i=1,2,3W
(i)µ M (i)
By analogy60 with U(1), to ensure gauge invariance
W (j)µ →W (j)
µ +1g∂µθ
(j)
But, this doesn’t work−NON-ABELIAN;
Dµeiθ(j)M(j) 6= eiθ
(j)M(j)Dµ
⇓
W (j)µ →W ′(j)
µ = W (j)µ +
1g∂µθ
(j) + εjklW(k)µ θ(l) (37)
Lkin = −14F (j)µν F
(j)µν (38.1)
F (j)µν = ∂µW
(j)ν − ∂νW (j)
µ + gεjklW(k)µ W (l)
ν (38.2)
→ the gauge bosons interact amongst themselves.
12 Spontaneous symmetry breaking
Gauge bosons are massless particles. ←Experimentally NO!
critical problem
‖
the gauge invariance → all gauge bosons to be massless
for EM, OK.for otherwise, which have a finite range fixed by the mass ofthe mediating particles.
Example time-independent sol. of KG eq.
(−∇2 + µ2)ψ = 0 → ψ ∝ 1re−µr
; this corresponds to an interaction that cuts off at a range
µ−1 =~mc
.−→ This is Yukawa’s original argument constraining the massof the particle that mediates the strong interaction.
The uncertainty principle saying that
∆E ·∆t & ~⇐⇒ mc2 ·∆t & ~
⇐⇒ c∆t & ~mc
= µ−1 = λ −−−→m→0
∞The particle can travel no further than c∆t; limiting the rangeof interaction
Analogy to classical theory
φ; Nambu-Goldstone particles
φ(x)→ φ(x) + α(x)
Aµ → Aµ +1e∂µα(x)
Then,
L(x) = −18Fµν , Fµν − 1
4∂µφ− eAµ, ∂µφ− eAµ
= −18Fµν , Fµν − e2
4Vµ, V µ
where Vµ(x) =def
1e∂µφ(x)−Aµ(x)
DefineFµν =
def∂µVν − ∂νVµ
=⇒ Fµν , Fµν = Fµν , Fµν
∴ L = −18Fµν , Fµν − e2
4Vµ, V µ
= −18Fµν , Fµν − e2
4Vµ, V µ
This is exactly the Lagrangian for mass=e2 particles.Euler-Lagrange eq.
−→ ¤V µ − ∂µ∂νV ν − e2V µ(x) = 0
Operate ∂µ −→ ∂µVµ = 0
Therefore EL eq. reduces to
(¤− e2)V µ(x) = 0, ∂µV µ(x) = 0
· · · Proca eq. for vector field59
eiθM =Xn
1
n!(iθM)n ∈Mat(2)
put M = 12
eiθ12 =
ůeiθ 00 eiθ
ÿ
60cf. for EM
Dµ ≡ ∂µ − ieAµ
Aµ → Aµ +1
e∂µθ
12.1 THE ’MEXICAN HAT’ POTENTIAL 39
12.1 THE ’MEXICAN HAT’ POTENTIAL
the Lagrangian for a charged scalar field:
L = ∂µφ∂µφ− V (φ) (43)
where the potential V (φ) is just a mass term V = m|φ|2→ adding in a term that looks like a self-interaction:
V (φ) = m2|φ|2 + λ|φ|4
V (φ) = −|µ2|(φ∗φ) + λ(φ∗φ)2 (44)
= µ2|φ|2 + λ|φ|4
= λ
(|φ|2 +
µ2
2λ
)2
− µ4
4λ
ssb
-1-0.5
00.5
1 -1
-0.5
0
0.5
1
-0.4-0.35-0.3
-0.25-0.2
-0.15-0.1
-0.050
→ Vmin is given by
|φ|GS = u ≡√−µ2
2λ(45)
and
Vmin = −µ4
4λ(46)
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
-1.5 -1 -0.5 0 0.5 1 1.5
-x**2+x**4
12.2 CLASSICAL AND QUANTUM SYM-METRY BREAKING
The Mexican hat potential generates equations of motion thatare rotationally symmetric, and yet the solution (=the posi-tion of the ball) violates this symmetry. → we always need toask whether the symmetric solution is stable.
the vacuum expectation value(vev) of φ:
〈0|φ|0〉 = ueiθ (47)
The angle θ is arbitrary; the original U(1) symmetry will nolonger be obeyed.(spontaneous symmetry breaking)
In the simple Abelian U(1) case, the gauge-invariant La-grangian is
L = (∂µ + ieAµ)φ(∂µ − ieAµ)φ∗ − V (φ)− 14FµνF
µν (48)
Assume
φ = (u+R)eiθ (49)|φ| = u+R
then
L = (∂µ + ieAµ)φ(∂µ − ieAµ)φ∗ − V (φ)− 14FµνF
µν
= [∂µ((u+R)eiθ) + ieAµ(u+R)eiθ]
× [∂µ((u+R)e−iθ)− ieAµ(u+R)e−iθ]− V (φ)− 14F 2
= [R,µ + i(θ,µ + eAµ)(u+R)]eiθ
× [(R,µ − i(θ,µ + eAµ)(u+R)]e−iθ − V (φ)− 14F 2
= R,µR,µ + (θ,µ + eAµ)(θ,µ + eAµ)(u+R)2 − V (φ)− 1
4F 2
= R,µR,µ + (θ,µθ,µ + 2eθ,µAµ + e2AµAµ)(u2 + 2uR+R2)
− µ2(u+R)2 − λ(u+R)4 − 14FµνFµν
= R,µR,µ + u2θ,µθ,µ + u2e2AµAµ + 2eu2θ,µA
µ + · · ·− µ2(u+R)2 − λ(u+R)4 +
14F 2
= R,µR,µ − (µ2 + 6λu2)R2 + u2θ,µθ,µ + u2e2AµAµ + · · ·
− µ2u2 − λ(u4 + 4uR3 +R4) +14F 2
= R,µR,µ + 2µ2R2 + u2θ,µθ,µ + e2u2AµAµ + · · ·+ 1
4F 2
Here, take θ equal to zero, then
L = R,µR,µ + 2µ2R2 + e2u2AµAµ − V (R)− 14FµνFµν
+ interaction terms (50)
;Higgs boson
12.3 GOLDSTONE BOSONS
Instead, take Aµ = 0, then
L = R,µR,µ + 2µ2R2 + u2θ,µθ,µ − V (R) (51)
→ the new massless field: Goldstone bosonthe gauge field eats the Goldstone boson and acquires a massin the process.
13 The Electroweak Model
weak interaction;Wµ
electroweakpredict−−−−→ Zµ
; done by S.L.Glashow
+
Renormalizability ; done by S.Weinberg and A.Salam
mass> 0, renormalizable fieldson gauge theory + spontaneous sym. breaking
⇒ Higgs mechanism
40 13 THE ELECTROWEAK MODEL
13.1 GAUGE GROUP AND NEUTRALCURRENTS
the weak interaction involves doublets like (e, νe) → the nat-ural gauge group is one that will deal with rotations in this2D space: SU(2)
Empirically the theory must violate paritydefining left- and right-handed components via
ψL =def
12(1− γ5)ψ = Π−ψ
ψR =def
12(1 + γ5)ψ = Π+ψ
assume; only the left-handed multiplets like
Le =def
(e−
νe
)
L
obey the SU(2) transformation.→ we talk of invariance under SU(2)L ⊗ U(1).SU(2) has 3 generators, and U(1) has 1⇒ the theory contains4 gauge fieldsQED and weak interactions → 3: photon, W±
→ unsuspected ingredient; single degree of freedom, which isthe Z0 boson.=⇒ ∃ neutral currents: weak interactions (that are not ac-companied by charge exchange, such as lepton-neutrino scat-tering) (detect CERN, 1973)
13.2 MASSES AND UNIFICATION
Higgs mechanism → introducing gauge fields and giving themmass while maintaining gauge invariance↔ the photon appears to be massless→ for a massless photon by taking a doublet of Higgs fields ofthe form
φ =(φ1
φ2
)
in terms of which the Mexican hat potential becomes
V (φ) = −µ2φ†φ+ λ(φ†φ)2
The symmetry breaking → a vacuum expectation value(vev) for φ
〈0|φ|0〉 =
(0v√2
)(56)
where v =
õ2
λHow to change under gauge transformations involving spa-
tially varying phase angles θ and θ for the U(1) and SU(2)For SU(2)
L→ e−igσ·θ/2L
R→ R
φ→ e−igσ·θ/2φ
For U(1)
L→ eig′θ/2L
R→ eig′θR
φ→ e−ig′θ/2φ
→ This defines the 2 coupling constants, g, g′ of the 2 groupsinvolved.
The Lagrangian invariant under these transformations cannow be written down:the fermions, the gauge boson, the Higgs field, and interactionterm
L = Lψ + LAµ + Lφ + Linteraction (59)
Aµi ; the 3 SU(2) gauge fieldsBµ; the U(1) field
1. The fermion part;
Lψ = ψiγµDµψ
the covariant derivative for left-handed
Dµ =def
∂µ − ig
2σ ·Aµ +
ig′
2Bµ
for right-handed lepton fields (right-handed neutrinosare not included)
Dµ =def
∂µ + ig′Bµ
2. The gauge part (just the kinetic energy);
LAµ = −14FiµνF
µνi −
14GµνG
µν
where
Gµν =def
∂µBν − ∂νBµF iµν =
def∂µA
iν − ∂νAiµ − gεijkAjµAkν
3. The Higgs part;
Lφ = (Dµφ)†(Dµφ)− V (φ)
where
Dµ =def
∂µ − ig
2σ ·Aµ − ig′
2Bµ
Having specified the Lagrangian, all that remains is tobreak the symmetry by the transformation
φ→
0v + η√
2
(67)
where η is the field corresponding to the Higgs boson. Themost important feature of the result is the term that comesfrom the vector boson coupling in Lφ:
L =v2
8(g2Aµ ·Aµ + g′2BµBµ + 2gg′BµA
µ3 ) (68)
[ZµA′µ
]=
[cos θW − sin θWsin θW cos θW
] [A
(3)µ
Bµ
](70)
41
tan θW =g′
g(71)
MW =gv
2(72.1)
MZ =MW
cos θW(72.2)
e = g sin θW (74)
GF√2
=g2
8M2W
(75)
sin2 θW ' 0.23 (θW ' 29) (73)
There are chosen so that the A′µ field is massless and corre-sponds to the photon;
θW is called the Weinberg angle, given by (71)
The relation to e (73, 74) gives g ' 0.65, g′ ' 0.36, and alsotells us a number of great importance, which is the energyscale of the symmetry breaking:
v = 246 GeV (76)
→ MW ' 80GeV and MZ ' 90GeV
the mass of the Higgs boson
MH =√
2µ =√
2λv (77)
14 Quantum Chromodynamics(QCD)
hadrons← quarks (spin-1/2)
u(2/3), c(2/3), t(2/3)d(−1/3), s(−1/3), b(−1/3)
quarks ↔ fundamental repr.gluons ↔ adjoint repr.
14.1 COLOUR AND SU(3)
ψ =
1√6εijkq
i1qj2qk3 (baryon)
1√2qi1q
i2 (meson)
(79)
the list of ingredients needed for the standard model. Theoverall gauge group that includes the electroweak model andQCD is
U(1)⊗ SU(2)L ⊗ SU(3)C (80)
L; left-handed, C; coloured multipletsThe direct-product nature of the group means that the 3
parts of the fundamental interactions inhabit rather differentworlds, so that QCD can do some extent be considered in iso-lation. This is particularly clear in the electroweak symmetrybreaking, which leaves the SU(3) part of the group unchanged:
U(1)⊗ SU(2)L ⊗ SU(3)C → U(2)⊗ SU(3)C (81)
14.2 THE QCD LAGRANGIAN
LQCD =∑
flavours
ψ(iγµDµ +m)ψ − 12Tr(FµνFµν) (82)
where the colour degree of freedom is hidden in the notation;we have used ψ to denote a column vector of 3 Dirac spinors:
ψ ≡
ψRψGψB
(83)
The covariant derivative must be a 3× 3 matrix
Dµ = ∂µ − igGµ (84)
Gµ ≡ 12GiµΛ
i (i = 1, 2, · · · , 8) (85)
is a combination of vector gauge fields called gluons.As before, the gauge-invariant field tensor is given by
Fµν =1ig
[Dµ, Dν ]= ∂µGν − ∂νGµ + ig[Gµ, Gν ] (86)
[Λa, Λb] = 2ifabcΛc (87.1)
Tr(ΛaΛb) = 2δab (87.2)
14.3 CHIRAL SYMMETRY
Ignoring the quark masses =⇒ chiral symmetry
qL =
def
12(1− γ5)q =
defΠ−q
qR =def
12(1 + γ5)q =
defΠ+q
(88)
N.B.
γ5qL =12(γ5 − 1)q (∵ γ5γ5 = 1)
= −12(1− γ5)q = −qL
γ5qR =12(γ5 + 1)q (∵ γ5γ5 = 1)
=12(1 + γ5)q = qR
i.e. qL and qR are eigen vect. of γ5 belonging to -1 and 1respectively.Fields with zero quark masses, the quark part of Lagrangian:
L = qLiγµDµqL + qRiγ
µDµqR (89)
the form of the mass term that has been neglected is
L = muuu+mddd+msss (90)
→ dynamical breaking of chiral symmetry
42 14 QUANTUM CHROMODYNAMICS (QCD)
14.4 QUARK MIXING
Fact K → π + π, Λ→ p+ π (s→ u)@ reason why the states that are recognized as distinct
’flavour’ of quark by the strong interaction should also be thosethat are distinguished by the weak interaction
Consider the low-energy Fermi effective Lagrangian
L =GF√
2JµJ†µ (91)
where
Jµ = u[γµ(1− γ5)]d′ + c[γµ(1− γ5)]s′ (92)
which is of the expected form except for a rotation(d′
s′
)=
(cos θc sin θc− sin θc cos θc
)(ds
)(93)
where θc ' 13 is the Cabbibo angle.With more quarks, there are more mixing possibilities.
→ the general form of the current has to be a combination ofall possible ψLγµψL terms:
Jµ = (u c t)LγµV
dsb
L
where V is the Kobayashi-Maskawa matrix.61
The CP eigenstates of the theory 6= mass eigenstates=⇒ different time evolution=⇒ given initial state → evolve towards a mixture.
14.5 VACUUM POLARIZATION ANDASYMPTOTIC FREEDOM
Recall; Renormalization(Appendix 5, p.9)
The second-order diagrams and their higher-order coun-terpart: the effect of modifying the apparent strength of theinteraction from its bare value, and moreover in a way that inprinciple depends on energy. (vacuum polarization62)
The vacuum polarization diagrams affect directly the mea-sure strength of an interaction, as quantified by its dimension-less coupling constant g.
The lowest-order correction to the coupling constant willthen look like
g(k2) = g0
[1 + g2
0
∫I(k, q)d4q
](95)
where g(k2) is empirical value of coupling constant, and g0 thebare coupling constant.
g(k22)− g(k2
1) =(g20
)3∫
(I(k2, q)− I(k1, q)) d4q
' (g(k2
1))3
∫(I(k2, q)− I(k1, q)) d4q (96)
Arguing by dimensions once again, the change with energymust satisfy the renormalization group equation (p.209)
dg(k2)d ln k2
= β(g(k2)
)(97)
⇐⇒ the quantum effects operating around a given energyknow only about the effective strength of the interaction atthat energy, there being no other dimensionless numbers inthe problem.
The perturbation approach above would suggest thatβ(g) ∝ g3 in lowest order, in which case the equation inte-grates to
g(k2) =±1
ln(k2/λ2)(98)
=⇒ The apparent interaction strength changes logarithmicallywith energy.For the Abelian U(1) theory, β > 0; For the Cain, β < 0−− the former result is in accord with our experience of EM→ we might expect that a vacuum of virtual pairs would acta little like a plasma in shielding a charge=⇒ the effective g would increase as the energy increased =⇒allowing a smaller de Broglie wavelength to resolve distancescloser to an electron.−− the latter case, intuition offers no such assistance
Expressing the coupling constant in terms of a critical en-ergy scale λ is not very sensible in the case of QED, sincepractical energies are always ¿ λ. For QCD, however,
λQCD ' 200MeV (99)
is a scale of critical importance. → asymptotic freedom.
At extremely low energy: the effective strength of the interac-tion binding the quarks increases without limit as the distancebetween them increases, so that free single quarks are effec-tively forbidden.
14.6 THE STRONG CP PROBLEM ANDTHE AXION
QCD → ignoring one additional possible gauge-invariant con-tribution to the Lagrangian, which takes the form
L =g2
16π2θ TrFµνFµν ∀θ (100)
61
V =
0@
cos θ12 cos θ13 sin θ12 cos θ13 sin θ13e−iδ13
− sin θ12 cos θ23 − cos θ12 sin θ23 sin θ13eiδ13 cos θ12 cos θ23 − sin θ12 sin θ23 sin θ13eiδ13 sin θ23 cos θ13sin θ12 sin θ23 − cos θ12 cos θ23 sin θ13eiδ13 − cos θ12 sin θ23 − sin θ12 cos θ23 sin θ13eiδ13 cos θ23 cos θ13
1A
=
0@
0.9745 ∼ 0.9760 0.217 ∼ 0.224 0.0018 ∼ 0.00450.217 ∼ 0.224 0.9737 ∼ 0.9753 0.036 ∼ 0.0420.004 ∼ 0.013 0.035 ∼ 0.042 0.9991 ∼ 0.9994
1A
62because of the analogy with a charge placed in a dielectic medium, for which the electric field is reduced to1
εtimes its value in a vacuum, owing
to the effect the electric field has on the polarization molecules.
14.6 THE STRONG CP PROBLEM AND THE AXION 43
where the dual field tensor is Fµν =12εµναβF
αβ . Euler-Lagrange eq. is invariant even though we add to this term.However, this term is not invariant under P, CP transf.→ ∃ some quantity breaking P, CP, T symmetry←→ Fact P, C, T are conserved ! (magnetic moment of neu-trons =⇒ |θ| ≤ 10−9 )→ ∴ Mechanism leading θ to 0 should exist (strong CP prob-lem)
Peccei-Quinn symmetry · · · an axial version of a simplephase rotation, and so is denoted U(1)PQ
The most commonly accepted idea for solving the strongCP problem was proposed by Peccei & Quinn(1977): θ isdriven to zero through the breaking of a new symmetry. Ifthis symmetry is global, its breaking will as usual produce aGoldstone boson, the axion63
Quarks & scalar fields behave as follows:
q → eiβγ5q (103.1)
φ→ e−2iβφ (103.2)
V (φ) =12λ(|φ|2 − v2
PQ)2 +m2av
2PQ(1− cos θ) (104)
' 12λ(|φ|2 − v2
PQ)2 +m2av
2PQ
2!θ2
Here θ is the phase of φ, choosing the origin so that θ = 0corresponds to the value desired for solving the strong CPproblem.
(x**2+y**2 - 1)**2 +1 - x/(sqrt(x**2+y**2)) 25 20 15 10 5
-1.5-1
-0.50
0.51
1.5 -2-1.5
-1-0.5
00.5
11.5
2
0.5
1
1.5
2
QCD can generate the potential automatically via instantoneffects, which correspond to quantum tunnelling between dif-ferent vacua in a way that cannot be treated by perturbationtheory.→ arising the quark-hadron phase transition, at energy ofλQCD ' 200 MeV.
m2av
2PQ(cos θ − 1) ∼ λ4
QCD(cos θ − 1) (105)
→ the axion mass can be very small if vPQ is assumed tocorrespond to new physics as À λQCD
ma ∼λ2
QCD
vPQ(106)
”maxion is very small =⇒ unimportant in Cosmology ??”
← this is not so.
The initial energy density is V (φ) ∼ m2av
2PQ (if the initial an-
gle is of order unity, the transverse field is ∼ vPQ). And theproper number density: na =
def
V (φ)ma
na ∼ mav2PQ (107)
whether this corresponds to a classical field, we have toask whether there are many quanta per cubic wavelength. Ifinflation did happen, the wavelength ∼ ∞, so this is certainlysatisfied.
The worse case would be for an axion momentum pa ∼ mac→ the wave length in natural units is m−1
a
and there are N ∼ (vPQ/ma)2 quanta per cubic wave-length;the axion field therefore produces a Bose condensate, whichwill acts as cold dark matter.
Before using the above number density with ma ∼λ2
QCD
vPQ
in order to get the density of relic axions, we need to take intoaccount one subtlety.
At T ∼ λQCD; the mass term switches on (but not in-stantaneously) =⇒ the main effect arises by arguing that therelevant effective axion mass must be such that th e period ofthe oscillations is of order the age of the universe at T ∼ λQCD
In natural units, the radiation-dominated time-temperature relation is
t/GeV−1 = g−1/2∗ (T/109.283GeV)−2 (108)
where g∗ is the ] of equivalent relativistic boson species.64
t(λQCD) ← for the period of oscillation of the axion field=⇒ giving meff
a ' 10−10.1 eVscaling this as a(t)−3 to the present, and multiplying by
the low-temperature axion mass gives the present-day massdensity:
ρ0 ∼ (2.73K/ΛQCD)3mameffa v
2PQ (109)
∼(
2.73KΛQCD
)3 λ4QCD
mameffa ∝
1ma
which scales as m−1a .
Putting numbers into this equation to deduce Ωh2 gives a crit-ical ma of ∼ 10−3eV for Ωh2 = 1.65
Exact calculations:
Ωh2 '( ma
10−5eV
)−1
(110)
=⇒ Cosmologically interesting axion mass requires a ratherspecific energy scale for symmetry breaking:vPQ ' 4 ×1012GeV.66
63one of the most commonly cited candidates for dark matter in cosmology64about 60 just prior to λQCD → Chapter 965
ρ0 = 1.88× 10−29Ωh2g cm−3
66It is worth nothing that the scale cannot be much less than this, since astrophysical arguments such as the possibility that axions are emitted bystars limit the axion mass to ≤ 0.01eV.
44 15 BEYOND THE STANDARD MODEL
Inflationthe axion → reasonably well-motivated model; isocurva-
ture density fluctuations.
Remark 14.6.1 Dynamics of structure formationorigin and growth of inhomogeneities; to understand the pro-cesses that cause the universe to depart from uniform density.Ch.10, 11 → discussion at some length the 2 most promisingexisting ideas:Either
• through the effect of topological defects formed in a cos-mological phase transition
• through the amplification of quantum zero-point fluctu-ations during an inflationary era
→ Neither of these ideas can yet be regarded as established,but it is astonishing that we are able to contemplate the ob-servational consequences.
isocurvature perturbation; the total density remainshomogeneous and so there is no perturbation to the spatialcurvature
δρmatter = −δρrad (111)
It is proved that all quantum fields emerge from inflationbearing a spectrum of irregularities that are a frozen-in formof the quantum fluctuations that exist on small scales duringthe early parts of the inflationary phase.(→ Chapter 11)
δφ =H
2π(112)
where H is the Hubble parameter during inflation.=⇒ the axion field starts moving towards θ = 0 from differentvalues of θ.
The fractional fluctuation in axion number density is thesame as this energy fluctuation:
φ⊥ = |φ|θ ' vPQθ
∴ δφ⊥ = vPQδθ
=⇒ δθ =δφ⊥vPQ
=H
2πvPQ(113)
δnana
=H
2πvPQ(114)
In natural units, H =√VIEp
with the Ωh2 = 1 where VI
is the inflationary energy density ∼ E4I , and Ep is the Planck
energy.sensible density fluctuations require EI ∼ 5× 1013GeV.
Part IV
BEYOND THESTANDARD MODELIn the standard model there exist some fact we cannot explain.
15 Beyond the standard model
1. Gravity
2. Elegance
15.1 GRAND UNIFIED THEORIES
1. 3 coupling consts. Why 3?
2. 3 generations. why 3??
3. left -right asymmetric ???
4. charge is quantized, why ????
5. too many parameters67, why ?????
The main idea behind grand unification = to work by anal-ogy with the success of the standard modelWe postulate that the law s governing particle physics obeysome as yet undiscovered symmetry, which is hidden in thepresent universe through the Higgs mechanism in which ascalar field acquires a non-zero vev.
In the SU(2) part of the standard model, we write wavefunctions as multiplet such as
ψ =(e−
νe
)(115)
← which corresponds to the assumption that electron and itsneutrino are merely 2 different states of the same underlyingentity, in the same way that electrons of spin up↑ and spindown↓ are related; similar arguments apply to multiplets ofquarks.
However, although the standard model consists of 3 fami-lies each containing 1 lepton multiplet and 1 quark multiplet,the leptons and hadrons are distinct, and only the hadron mul-tiplets appear in the part of the Lagrangian devoted to SU(3)add the colour symmetry. → In conventional particle physics,this is dignified by the conservation of baryon number: leptonsand quarks cannot be transmuted into each other.
Nevertheless, this conservation law has never had the samestatus as conservation of charge or mass, both of which aregenerated by simple global symmetries of the Lagrangian andrelate to the production of long-range forces.=⇒ there has been an increasing willingness to breach whatmay be an artificial barrier and contemplate the possibilitythat quarks and leptons may be related.
The basic idea of grand unification is then to write, sym-bolically,
ψ =(
quarksleptons
)(116)
The idea now is to find some way of accommodating all thevarious particles of the standard model into a set of multipletsand to write down a Lagrangian that is invariant under theoperation of some group G
G→ SU(3)⊗ SU(2)⊗ U(1)
To avoid a conflict with the existing body of acceleratordata, any such symmetry-breaking scale must lie well abovethe maximum energy studied to date:EGUT ÀMW ∼ 100GeV
67coupling consts.: 3 Higgs: 2 Yukawa: 9 KM: 4 → 18 parameters
15.2 EMPIRICAL EVIDENCE FOR UNIFICATION 45
15.2 EMPIRICAL EVIDENCE FOR UNI-FICATION
the form of the variation with energy of the coupling constantsin the standard model→ logarithmically varying functions described by renor-
malization group equations(p.209) of the form(q.v. 5.163)
1α(E1)
=1
α(E2)+
b
2πlog
(E2
E1
)(118)
where α ≡ g2
4π(119)
The b coefficients are
b[U(1)] = −43Ng (120.1)
b[SU(N)] = −113N − 4
3Ng (120.2)
where Ng is the ] of distinct families.Given 3 coupling constant values αi(MW ) and extending
them linearly to higher energies, we would not in general ex-pect the lines to intersect, but they do!=⇒ This fact strongly suggests that ∃ something special aboutthe energy of the intersection(MX):At this point, the strengths of all 3 fundamental interactionsbecome equal:
αi(MX) ' 140, MX ' 1014−15 GeV (121)
Here, almost every decade of energy from the mass of theelectron to the mass of the Z boson contains new particles.=⇒ One might expect that TeV energies would open up newphysical phenomena, and so on indefinitely.
The prediction of the GUT idea is that, instead, we nextencounter a desert of at least 12 powers of 10 in energy inwhich nothing interesting happens.
The other suspicious aspect of the GUT hypothesis = theunification scale is so close to the Planck scale.
15.3 SIMPLE GUT MODELS
Cartan → 4 distinct categories of simple groups68
1. special orthogonal group (SO(n))
2. special unitary group (SU(n))
3. symplectic group (Sp(n))
→ leaving invariant the skew-product of two 2n-dimensional real vectors:
y · S · x = invariant
S = diag([
0 1−1 0
] [0 1−1 0
]· · ·
)(8.122)
4. exceptional groups; G2, F4, E6, E7, E8
ψ ≡
dc1dc2dc3e−
−νe
L
, χ ≡
0 uc3 −uc2 −u1 −d1
−uc3 0 uc1 −u2 −d2
uc2 −uc1 0 −u3 −d3
u1 u2 u3 0 −e+d1 d2 d3 e+ 0
L
The main feature is the Higgs sector, in which φ is a 5×5matrix that acquires the vev
φ ∝ diag(
1, 1, 1,−34,−3
2
)(8.124)
in order to break SU(5) but leave the residual symmetry ofthe standard model.
15.4 SUCCESSES AND FAILURES OFGUT MODELS
The SU(2) model makes many predictions, most of which aregeneric to GUTs. GUTs are theories that explicitly violateconservation of baryon number, allowing the interesting newpossibility of proton decay:
p −→ e+π0 −→ e+γγ (8.125)
analogous to the weak decay of the neutron, where tn in nat-ural units is ∼ g−4
weakM4W /m
5e (Ch.9)
τp ∼ ~/c2
α2GUT
M4GUT
m5p
∼ 1030 yr. (126)
→← xp · · · the Cerenkov radiation from the photons gener-ated by the final pion decay. =⇒ limit on the proton lifetimeof τp ≥ 1032 yr.
The other (greatest) defect of GUTs: No natural explana-tion for the violation of parity
SU(5) GUTs do not correspond to the real world.
16 Neutrino masses and mixing
16.1 Majorana particle
electrons & quarks obey Dirac eq.→ ∃ antiparticle (i.e. ∃ quantum number ”charge”)Neutrinos have no charge (← weak interaction & gravity) andare extremely light (← implying that neutrinos’ mass origi-nates otherwise).
At first, since neutrinos have no charge, their antiparticleis unapparent. In general, for spinor we can operate the chargeconjugate operator:
C(ψ) = γ2(ψ)∗ = γ2(ψL + ψR)∗
= γ2(Π−ψ∗ + Π+ψ∗) = Π+γ2ψ∗ + Π−γ2ψ∗
= (C(ψ))R + (C(ψ))L
C(ψL) = (C(ψ))RC(ψR) = (C(ψ))L
Another consequence of GUTs → neutrinos have the non-zero mass that are forbidden in the standard model.=⇒ neutrinos are mixed state with left and right helicity.=⇒ For a single neutrino species, the mass term in the La-grangian is
L = −mνν68those that can not be expressed as a direct product of other groups
46 16 NEUTRINO MASSES AND MIXING
where ν is the neutrino wave fn. In terms of the chiral fields
νR,L =def
1± γ5
2ν (127)
=⇒ L = −m(νLνR + νRνL)Here νLνL and νRνR vanish.
∵ νL = ν1 + γ5
2=⇒ νLνL ∝ (1 + γ5)(1− γ5) (γ5γ5 = 1)
If we don’t distinguish particle from antiparticle, the fol-lowing mixed state
νM =def
νL + C(ν)R = νL + C(νL)
also satisfies Dirac equation. Calculating its antiparticle
C(νM ) = C(νL + C(νL)) = C(νL) + CC(νL)= C(νL) + νL = νM (∵ CC = 1)
∴ C(νM ) = νM
Particles like this is called Majorana particle.Mass term of Majorana particle, say LM , is
LM = −MνMνM
= −M(νLC(νL) + C(νL)νL)
=⇒ In general, We get the mass term
−(Lm + LM ) = m(νLνR + νRνL) +M(νLνcR + νcRνL)
=(νL νL
) M
m
2−m
20
(νcRνR
)
+(νcR νR
)M −m
2m
20
(νLνcL
)
≡ −LmMThen m¿M =⇒
LmM = M(ν1Lν1R + ν1Rν1L) +m2
4M(ν2Lν2R + ν2Rν2L)
whereν1 =
defνcR + νL − m
2M(νcL − νR)
ν2 =def
νR + νcL −m
2M(νcR − νL)
=⇒ generalization; Dirac mass term
L = −νL ·M · νR + Hermitian conjugate (128)
where now
ν =
νeνµντ
(129)
and the 3× 3 Hermitian matrix M is the mass matrix.For Majorana neutrinos, νR → CνL and the Lagrangian is multiplied
by a factor 1/2 (C is the charge-conjugate operator)
16.2 NEUTRINO OSCILLATIONS
Since the mass matrix is not necessarily non-diagonal, themass eigenstates involved may not be the states that we rec-ognize in xp as the distinct neutrino types. This phenomenonhas already been seen with quark mixing, where the weakand strong interactions effectively have different definitions ofquarks.
the neutrinos are massless =⇒ any mixing can be definedaway by making new linear combinations of fields.
the neutrinos have mass =⇒ the mixing will have observ-able consequence
(νeνµ
)=
(cos θ sin θ− sin θ cos θ
)(ν1ν2
)(130)
where ν1, ν2 are the mass eigenstates
νi = νi(0)e−iEit/~ (131)
ψ(t) = ν1(0) cos θe−iE1t/~ + ν2(0) sin θe−iE2t/~ (132)
〈νµ(0)|ψ(t)〉 = sin θ cos θ[−e−iE1t/~ + e−iE2t/~] (133)
P (νe → νµ) =12
sin2(2θ)1− cos[(E1 − E2)t/~] (134)
oscillation is improbable in the limit of small mixing.
E = (p2c2 +m2c4)1/2 (16.0)
' pc+m2c3
2p' pc+
m2c4
2E(135)
L =4πE~
c3∆(m2)' 2.5m
E/MeV∆(m2)/(eV)2
(136)
For sin2 2θ = 1, the limits for νe, νµ mixing are roughly∆(m2) ≤ 0.02eV2 and ∆(m2) ≤ 0.1eV2,increasing by approximately a factor of ten at the smallestangles probed (sin2 2θ ' 0.01).
16.3 SOLAR NEUTRINOS
Standard Solar Model(SSM) predicts 7.9± 2.9 SNU69
16.3.1 Homestake xp. (USA)
R. Davis Jr. at South Dakota37Cl : target
37Cl + νe → 37Ar + e− (Eν ≥ 0.814MeV)
→ 2.1± 0.3 SNU
16.3.2 KAMIOKANDE xp.
e: target
νe + e− → νe + e−
16.3.3 GALLEX(EU), SAGE(Rus)
71Ga + νe → 71Ge + e− (Eν ≥ 0.233MeV)69SNU(Solar Neutrino Unit) means the number of reaction per unit time for the 1036 targets
16.4 MSW EFFECT 47
Reaction E/Mev 37Cl/SNU 71Ga/SNUp+ p→ D + e+ + νe < 0.42 0.0 707Be +e− →7Li+νe 0.86 1.2 368B→ 24He+e+ + νe < 14 6.2 14
16.4 MSW EFFECT
for matter ← L.Wolfensteinfor solar neutrinos ← S.P.Mikheyev & A.Yu.Smirnov→ MSW effect70
The weak interaction gives a phase change that can be readoff immediately from the Fermi Lagrangian:
~dφ
dt= 〈|HF |〉 =
√2GFne (137)
→ being incorporated into the equation of motion for the neu-trino mixture
|ν(t)〉 = ae(t)|νe〉+ aµ(t)|νµ〉 (138)
where the coefficients ai obey the oscillation equation of mo-tion
i~(aeaµ
)=π~cL
(cos 2θ − sin 2θ− sin 2θ − cos 2θ
)(aeaµ
)(139)
L; the oscillation length defined earlier.The effect of the matter-induced MSW phase rotation is to
add to the matrix on the right-hand-side an additional term
that equalsπ~cLM
diag(1,−1), choosing to eliminate an unmea-
surable overall phase. The matter oscillation length is, fromabove,
LM =√
2π~cGFne
(140)
which is of order 107cm in the core of the Sun (so that matteroscillation effects can be important).
What does it mean, when we take ∆(m2) ' 10−5eV2?To justify the assumption that the neutrino masses are un-
likely to be all nearly equal, we can appeal to an illustrativemeans of generating neutrino masses, known as the seesawmechanism71, which can be realized in GUT models.
Suppose that the neutrino mass (for a single generation) isa combination of the Dirac and Majorana forms:
L = (νL νcL)(
0 mm M
)(νcRνR
)(141)
The mass M must be large to explain why right-handedneutrinos are not seen, and could easily be of order the GUTmass.About this mass matrix, its eigenvalues have dynamic range,
which are approximately M and −m2
M72 The small mass can
easily be a small fraction of an eV for quark masses in theGeV range and M ∼ 1015GeV. More precisely, the seesawmechanism predicts the scaling
m(νe) : m(νµ) : m(ντ ) = m2u : m2
c : m2t (142)
Using this with the above mass for the ντ=⇒ a τ neutrino mass of around 10 eV would be indicated.73
Analogy to Yukawa interactionmassive → νRSU(2)-singlet Higgs field Φ
=⇒ −Lmν = G2
16.5 SUPERNOVA NEUTRINOS
observing neutrino masses by supernova 1987A; in this case,neutrinos are emitted in a sharp pulse and travel over ex-tremely long paths=⇒ it is worth considering the small deviation of their veloc-ities from c:
v
c=
(1− 1
γ2
)1/2
' 1− 12γ2
= 1− m2c4
2E2(143)
Over a path L, the difference in arrival times is therefore
∆t =L
2c∆
(m2c4
E2
)(144)
which is again sensitive to differences in squared mass for agiven energy.
17 Quantum Gravity
Planck units:
mpl ≡√~cG∼ 1019GeV→ 1
lpl ≡√~Gc3∼ 10−33cm→ 1 (145)
tpl ≡√~Gc5∼ 10−43s→ 1
Heisenberg uncertainty:
∆E ·∆t = ∆mc2 ·∆t ' ~∆x ·∆p = ∆x ·∆mc ' ~ (17.0)
Although the general theory of relativity is made as a the-ory which describes a macroscopic gravity phenomenon, sup-posing it has described natural fundamental law, it is possiblethat the gravity interaction in the microscopic world is alsodescribed. However, when it is going to extend the generaltheory of relativity to the microscopic world, in order that asubstance may show quantum theory-behavior, the problemof it becoming impossible for space-time structure to describethe space-time structure itself classically in the general theoryof relativity determined with a substance arises.
70Mikheyev-Smirnov-Wolfenstein71The first step was made by Yanagida.72The unexpected − sign in the second case can easily be amended by inserting a factor i in the definition of the corresponding eigenfield.73As discussed at length in Chapter 9, 12, 15, such a mass would be of the greatest importance for cosmology, as neutrinos of this mass would
contribute roughly a closure density in the form of hot dark matter.
48 17 QUANTUM GRAVITY
In order to describe the gravity phenomenon in the micro-scopic world conformably, it is necessary to unite the frame-work and the general theory of relativity of the quantum the-ory, and such a theory is called quantum gravity theory. Al-though the quantum nature of space-time does not becomeimportant on the scale of an atom or a nucleus, becomingimportant for the first time on the scale (?10−33cm) calledPlanck length so that it may understand by easy calculationis expected. For example, although an elementary particle isusually treated as a point particle, by the general theory of rel-ativity, a point particle positive in mass cannot exist in a strictmeaning, but if a naked singular point will not be accepted, itwill serve as a black hole. Although an actual elementary par-ticle is not served as a black hole, this depends on the quantumeffect of gravity.
Schwarzschild radius rg expected from a classic theory willbecome smaller than Planck length to an elementary particlewith mass smaller than Planck mass, and describing space-time structure classically on the scale of rg will lose a meaning.This result means simultaneously that the black hole of masssmaller than Planck mass does not exist.
Although the quantum nature of gravity can be disregardedin practice in the usual world, when discussing the early stagesof the universe, situations differ. Since the curvature of space-time will become large infinite if an initial singular point isapproached although the universe surely has an initial singu-lar point if space-time structure is described classically, thequantum fluctuation of space-time cannot be disregarded inthe near. However, since the structures near the initial sin-gular point of the universe are the initial conditions over timedevelopment, in order to solve the origin of the structure ofthe present universe, the knowledge about quantum theory-behavior of space-time structure is needed.
Although construction of quantum gravity theory has in-quired for a long time, it has not yet resulted in the success. Al-though there is approach based on the path integration whichhas as close as what attracted attention by the application toa quantum universe theory in recent years a relation as Diracquantization and it although
1. Semiclassical theory
2. Perturbation theory
3. Canonical quantization
(a) Gauge fixing
i. ADM74 theoryii. BRS theory
(b) Dirac quantization / Wheeler-De Witt equation
4. Path integral
5. alternative theory of gravity
are mentioned to the composition method, the problem is heldtoo.
In the usual quantum mechanics, a wave function is inter-preted as what gives the probable prediction to the observationvalue of the amount of physics – having (Copenhagen inter-pretation) – it is simply inapplicable to a quantum gravitysystem Although the equipment which measures the amountof physics of the quantum system expressed with operator is
outside and it is assumed in this interpretation that an ob-servation result is expressed by the classic quantity which de-scribes this equipment, the distinction with such an objectsystem and observation equipment does not exist in the gen-eral theory of relativity. When aimed at the whole universe, itcannot consider that observation equipment is external. Fur-thermore, supposing the wave motion function has describedthe state of the whole universe, the state will change withobservation in un-unfortunate (contraction of wave packet).First of all, an objective definition is not clear in what obser-vation means. Although various ideas, such as extension ofquantum mechanics using the interpretation based on WKBapproximation, a multi-world interpretation, and path integra-tion, are proposed in order to solve the interpretation problemof the above canonical quantum gravity theory, the answerwith still satisfying present is not given.
L =√−g
12κR+
[φ,µφ∗,µ − (m2 + ξR2)|φ|2]
(146)
we shall use the gravitational coupling constant
κ =def
8πGc4
(147)
17.1 SEMICLASSICAL QUANTUMGRAVITY
gµν = ηµν +√κhµν ,
√κhµν À 1 (148)
By
hµν = hµν − 12ηµνh (149)
h =def
hµµ = −hµµ
and the General Relativity equivalent of the Lorentz gauge,hµν,µ = 0, the lower-order Lagrangian becomes
L = hµν,λhµν,λ + (φ,µφ∗,µ −m2|φ|2)+κ(2hµνφ,µφ∗,ν − hm2|φ|2) (150)
→ the quantum of the field must have spin-2 (the graviton).
17.2 HAWKING RADIATION
the vacuum state |0〉 can be defined to satisfy
〈0|Tµν |0〉 = 0 (152)
LI = gD(τ)φ(xµ(τ)) (153)
where τ is the detector’s proper time.固有状態はディテクターとスカラー場の直積空間上の元と
する。
A(a→ b) = ig
⟨b∣∣∣∫Dφ dτ
∣∣∣a⟩
(154)
74Arnowitt, Deser, Misner
17.3 FIELD THEORY AT NON-ZERO TEMPERATURE 49
in the Heisenberg picture with a time-dependent detector op-erator D(τ) = eiHτD(0)e−iHτ
A(a→ b) = ig〈b|D(0)|a〉∫ei(Eb−Ea)τ 〈b|φ(τ)|a〉 dτ (155)
φ =∑
k
(akuk(xµ) + complex conjugate) (156)
where
uk =1√2ωV
e−ikµxµ
(157)
Let the φ part of the initial |a〉 state be |b〉（これも，固有状態の φ成分に演算すると考える）, then
〈b|φ|a〉 =1√2ωV
e−ikµx
µ√n+ 1 (emis.)
eikµxµ√
n (abs.)(158)
A(a→ b) = ig〈b|D(0)|a〉 1√2ωV
∫ei(Eb−Ea±ω)τ dτ
×√
n+ 1 (emission.)√n (absorption)
(159)
the integral → 2π · δ(Eb − Ea ± ω)If the detector is placed in a thermal bath with n =1
e~ω/kBT−1, the detection rate becomes
dp(a→ b)dt
= g2|〈b|D(0)|a〉|2(ωV
π
)−1δ(Ea − Eb − ω)e~ω/kBT − 1
(160)
Next we consider the detector in gravitational field.シュバルツ半径に十分近い所にいるときの議論ではリンド
ラー座標を用いるとよい。
x =1a[cosh(aτ)− 1]
t =1a
sinh(aτ)
observers comoving with the detector inhabit Rindler space.こうすることでフラットな空間の中での量子論を考えればよい事になる。
p(a→ b) = g2|〈b|D(0)|a〉|2
×∫∫
dτdτ ′ G+(x[τ ], x[τ ′])ei(Ea−Eb)(τ−τ ′)
(162)
Here, G+ is the Green function of the φ field, and is a functionof separation: G(x, x′) = G(x− x′)
dp(a→ b)dt
= g2|〈b|D(0)|a〉|2∫d∆τG+(∆τ)ei(Ea−Eb)∆τ
(163)
G+(x, x′) = 〈0|φ(x)φ(x′)|0〉 (164)
dp(a→ b) =g2
2π|〈b|D(0)|a〉|2 ∆E
e2π∆E/a − 1(165)
Taccel =(~ck
)a
2π(166)
加速している粒子が感じる「温度」
T
K' 3.7× 10−7γ2
R/km(167)
重力場にあるということは上の温度をもつ輻射場にあると思ってよいという事
17.3 FIELD THEORY AT NON-ZEROTEMPERATURE
Planck 分布（黒体輻射場）の中におけるカノニカル平均を考える。
〈O〉 =1Z
∑
i
〈i|O|i〉e−Ei/kBT (168)
where Z is the partition function. Compactify this in terms ofthe density matrix ρ:
〈O〉 =Tr(ρO)Tr(ρ)
(169.1)
ρ = e−H/kBT (169.2)
時間発展ぶんの eiHt と Boltzmann factorを対応させてかんがえる。
Euclidean time; t→ it′
〈G+(x, y)〉 = 〈φ(x)φ(y)〉= Tr[e−H/kBTφ(x)φ(y)]/Tr(ρ)
= Tr[e−H/kBTφ(x)eH/kBT e−H/kBTφ(y)]/Tr(ρ)= 〈φ(y)φ(x; t→ t+ i/kBT )〉= 〈G−(x; t→ t+ i/kBT, y)〉 (170)
→ the Green function G = G+ +G− is periodic.Gが周期的になることはうえの式から演繹出来る。3行目の前半の式はHeisenbergの時間発展を表している事に
相当し，Boltzmann factorに対応している事が分かる。この事は，Green函数が周期的になっているということを表している。
Minkowski空間において座標変換を施すと
ds2 = a2r2dτ2 − dr2 − dy2 − dz2 (171)
さらに，Wick回転により
−ds2 = (dr2 + a2r2dτ2) + dy2 + dz2 (172)
が得られる。Similarly, for de-Sitter spacetime, the metric is given by
ds2 = dt2 − cosh2(Ht)[dr2 + dS2k(r)dψ
2] (173)
under t → it, coshHt → cosHt =⇒ the period in Euclidean= 2π/H. Therefore
Tde Sitter =(~kB
)H
2π(174)
という温度（この温度における輻射場があると「感じる」）があると思ってよい。
50 19 SUPERSYMMETRIES AND BEYOND
17.4 BLACK HOLES
a =GM
r2
(1− 2GM
c21r2
)−1/2
(175)
TH =(~c3
k
)1
8πGM(176)
LBH = Γ4π(
2GMc2
)2
σT 4BH
= Γ~c2
15360π
(GM
c2
)−2
(177)
where Γ is a fudge factor to be found by reference to an exactcalculation
tBH =5120πc2
Γ~G
(GM
c2
)3
= 1.50× 1066
(M
M¯
)3
yr (178)
17.5 ALTERNATIVE INSIGHTS
the role of event horizon;∆t & ~/E → ∆x & ~c/EF∆x ∼ E: E2 & c~F→ F ∼ ∆xGM(E/c2)/r3 & (~/c)GM/r3
Emax ∼ (~c3)(GM)−1 (179)
which is E ∼ kBTBH
18 Kaluza-Klein models
G. Nordstrøm, Th. Kaluza, O. KleinIn an ideal 5-dimensional space, its metric ds2 is supposed
to be given
ds2 = gµνdxµdxν + φ2(x)(dx4 + κ2Aµ(x)dxµ)2
Supposing it has a form independent of 5th coordinate x4, thefive dimensions over this metric Einstein equation is equivalentto the Maxwell-Einstein theory to four-dimension an electro-magnetic field and a gravity field.
The assumption to the 5-dim metric in Kaluza-Klein theoryis equivalent to the 5-dim spacetime having isometric trans-formation group of 1-dim, and as for an electromagnetic fieldappearing automatically, it has that it is the gauge theory ofthe Abel group U(1) which is one dimension exactly, and aclose relation.
It is expected that consider the space-time of a still higherdimension and the 4-dimensional gravity theory it carries outan interaction to the gauge field corresponding to Group G isobtained from this. This anticipation is actually right. (R.Kerner) It is the direct motive which resurfaced as a unifica-tion theory containing gravity.
18.1 LAWER-DIMENSIONAL GRAVITY
Rαβγδ =1
n− 2(gαγRβδ + gβδRαγ − gαδRβγ − gβγRαδ)
− R
(n− 1)(n− 2)(gαγgβδ − gαδgβγ) + Cαβγδ (180)
The tensor Cαβγδ is the Weyl tensor75 For n = 2, the equa-tion is
Rαβγδ =R
2(gαδgβγ − gαγgβδ)
18.2 THE KALUZA-KLEIN ARGUMENT
g(5) = φ−1/3
(gµν + φAµAν φAµ
φAν φ
)(181)
S = − 116πG(5)
∫R(5)
√g(5) d4xµdy (182)
S = − 116πG
∫ (R+
14φFµνFµν +
16φ2
∂µφ∂µφ
)√g d4xµ
(183)
18.3 Ashtekar theory
Kaluza-Klein theory was the trial in which unification theorywould be built, by extending Einstein theory to the space-time of a dimension higher than four dimensions, and takingin gauge theory in gravity theory.
On the other hand, there is theory which makes gravitytheory and gauge theory the method of connecting by rewrit-ing the 4-dimensional Einstein theory itself in the form neargauge theory conversely.
19 Supersymmetries and beyond
Q|boson〉 = |fermion〉Q|fermion〉 = |boson〉
Each known particle must have a superpartnergauge hierarchy problemunnaturallarge hadron collider
19.1 SUPERSYMMETRY AND THE ΛPROBLEM
For bosonic fields ([a, a†] = 1)
E =∑ (
n+12
)~ω ⇒ Evac =
∑ 12~ω (185)
For fermionic fields (a, a† = 1 )76
E =∑ (
n− 12
)~ω (186)
75vanishes for n ≤ 376If supersymmetry were to be an exact symmetry, there would have to exist a fermionic field for each bosonic field.
19.2 SUPERGRAVITY, SUPERSTRINGS AND THE FUTURE 51
19.2 SUPERGRAVITY, SUPERSTRINGSAND THE FUTURE
L = ∂µφ∗∂µφ+i
2ψγµ∂µψ (187)
write the scalar field as φ = φ1 + iφ2 =⇒ the followinginfinitesimal supersymmetry transformation changes the La-grangian only by a term that is a total derivative, which is asusual ignored by setting a suitable boundary at infinity:
δφ1 = εψ
δφ2 = iεγ5ψ (188)
δφ3 = −iγµ[∂µ(φ1 + iγ5ψ2)]ε
where ε is a spinor generating the transformation.a new gauge field χµ whose transformation eats the deriva-
tive term:
δχ ∝ ∂µε (189)
→ Rarita-Schwinger spin-3/2 field supersymmetry=⇒ generating a superpartner for the χ particle whose spin is1/2 greater=⇒ introducing a 2-spin field; supergravity
extended supergravity
[φ(x), π(x′)] = i~δ(x− x′) (190)
superstrings
Part V
CLASSICAL COSMOLOGY
20 The isotropic universe
Considering the simplest possible mass distribution:one whose properties are homogeneous(constant density)and isotropic(the same in all directions).
20.1 THE ROBERTSON-WALKER MET-RIC
For
v = Σ · r + Ω ∧ r (20.0)
isotropy =⇒ Ω = 0 and all the principal value of the sheertensor Σ must be equal.
Postulate 20.1.1 Hubble lawThe only allowed velocity field on a local scale is expansion (orcontraction) with velocity proportional to distance:
v = Hr (20.1)
Definition 20.1.2 TransitiveG; isometric Lie group of Mf ∈ G : M −→ M ; diffeo.Acting transitively on M⇐⇒def
∀x, x′ ∈M, ∃f ∈ G : x 7−→ x′
Definition 20.1.3 HomogeneousM is homogeneous ifisometric mapping acts on M transitively.
Definition 20.1.4 Isotropic groupG; isometric Lie group of MHx =
deff ∈ G| f(x) = x 3 e
Hx ⊂ G (subgroup)
Remark 20.1.5f ∈ Hx induces the transformation s.t.
TxM 3 Xµ ∂
∂xµ7−→ X ′µ ∂
∂dxµ
We call this group linear isotropic group.
Definition 20.1.6 Isotropicx ∈M is called to be isotropic if Hx = SO(m)We call M to be isotropic if every point of M is isotropic.
Example 20.1.7isotropic universe has the same form for the comoving spatialpart of its metric as the surface of a sphere:
a 3-sphere embedded in 4-dim Euclidean sp. would bedefined by the coordinate relation
x2 + y2 + z2 + w2 = R2
Now define the equivalent of spherical polers and write
x = R sinα sinβ sin γy = R sinα sinβ cos γz = R sinα cosβw = R cosα
=⇒ |(dx, dy, dz, dw)|2= R2[dα2 + sin2 α(dβ2 + sin2 βdγ2)] (20.2)
Lemma 20.1.8With the Robertson-Walker metric, we obtain
Γ000 = Γi00 = Γ0
i0 = Γ0ij = Γij0 = 0 (i 6= j = 1, 2, 3) (20.3)
Γii0 =a
a(not contracting) (20.4)
Γ011 =
aa
1− kr2 (20.5)
Γ022 = −r(1− kr2) (20.6)
Γ033 = −r sin2 θ(1− kr2) (20.7)
Γ112 = Γ1
13 = Γ123 = 0 (20.8)
Γ111 =
kr
1− kr2 , Γ122 = Γ1
33 = −1r
(20.9)
Γ211 = Γ2
22 = Γ213 = Γ2
23 = 0 (20.10)
Γ212 =
1r, Γ2
33 = − sin θ cos θ (20.11)
Γ311 = Γ3
12 = Γ322 = Γ3
33 = 0 (20.12)
Γ313 =
1r, Γ3
23 = cot θ (20.13)
52 20 THE ISOTROPIC UNIVERSE
Proof of 20.1.8We shall omit
Corollary 20.1.9From above Christoffel symbol, we obtain Ricci tensor:
R00 = −3aa
a2(20.14)
R11 =1
1− kr2(a2 − aa+ 2k
)(20.15)
R22 = r2(a2 − aa+ 2k
)(20.16)
R33 = r2 sin2 θ(a2 − aa+ 2k
)(20.17)
Other Rµν = 0
Proof of 20.1.9We shall omit.
Corollary 20.1.10From Ricci tensor, we gain Ricci scalar:
R =def
gµνRµν = −6
((a
a
)2
+a
a+k
a
)(20.18)
20.2 DYNAMICS OF THE EXPANSION 53
Proposition 20.1.11The Robertson-Walker metric satisfies Einstein’s equationsand obtain the Friedmann equations.
Proof of 20.1.11The Robertson-Walker metric is
gµν =
−1 0 0 0
0R2(t)
1− kr2 0 0
0 0 R2(t)r2 00 0 0 R2(t)r2 sin2 θ
(20.19)
and the energy-momentum tensor is
Tµν = (ρ+ p)uµuν − pgµν (20.20)
Einstein’s field gravitational field equations;
Gµν = −8πGc4
Tµν (20.21)
=⇒
G00 =
3(R+ k)R2
= 8πGρ
G11 =
2RR+ R2 + k
R2= 8πGp
(20.22)
Remark 20.1.12We define the very useful function:
Sk(r) =def
sin r (k = 1)sinh r (k = −1)r (k = 0)
(20.23)
and its cosine-like analogue, which will be useful later:
Ck(r) =def
√1− kS2
k(r) (20.24)
=
cos r (k = 1)cosh r (k = −1)1 (k = 0)
(20.25)
Definition 20.1.13 Dimensionless scale factor
a(t) =def
R(t)R0
(20.26)
Definition 20.1.14 Conformal time
η =∫ t
0
cdt′
R(t′)(20.27)
Remark 20.1.15This transformation induces a conformal transformation.
Definition 20.1.16 Redshift
νemit
νobs=def
1 + z (20.28)
is called redshift parameter.77
Remark 20.1.17
1 + z ' 1 +v
c
Proposition 20.1.18
1 + z =R(tobs)R(temit)
(20.29)
Corollary 20.1.19In terms of the normalized scale factor a(t) we have
a(t) =1
1 + z(20.30)
Corollary 20.1.20
1 + z =
√1 + β
1− β (20.31)
20.2 DYNAMICS OF THE EXPANSION
Definition 20.2.1
Lemma 20.2.2Friedmann’s equation that expresses conservation of energy:
12(Rr)2 − GM
Rr= constant (20.32)
Proposition 20.2.3GR becomes even more vital in giving us the constant of inte-gration in Friedmann’s equation:
R2 − 8πG3
ρR2 = −kc2 (20.33)
Remark 20.2.4This equation covers all contributions to ρ. i.e. those frommatter, radiation, and vacuum
Definition 20.2.5 Density parameter
77
z =def
νemit − νobs
νobs
54 22 THE AGE AND DISTANCE SCALES
Ω =def
ρ
ρc=
8πGρ3H2
(20.34)
Lemma 20.2.6For pressureless matter; ρ ∝ R−3
For radiation; ρ ∝ R−4
For vacuum; ρ = const.
Proof of 20.2.6
Definition 20.2.7 Deceleration parameter
q =def− RRR2
(20.35)
Proposition 20.2.8
20.3 COMMON BUG BANG MISCON-CEPTION
Definition 20.3.1
20.4 OBSERVATIONS IN COSMOLOGY
Definition 20.4.1
20.5 THE ANTHROPIC PRINCIPLE
Definition 20.5.1
21 Gravitational lensing
Definition 21.0.2
21.1 BASICS OF LIGHT DEFLECTION
Definition 21.1.1
21.2 SIMPLE LENS MODELS
Definition 21.2.1
21.3 GENERAL PROPERTIES OF THINLENSES
Definition 21.3.1
21.4 OBSERVATIONS OF GRAVITA-TIONAL LENSING
Definition 21.4.1
21.5 MICROLENSING
Definition 21.5.1
21.6 DARK-MATTER MAPPING
Definition 21.6.1
22 The age and distance scales
Definition 22.0.2
22.1 THE DISTANCE SCALE AND THEAGE OF THE UNIVERSE
Definition 22.1.1
22.2 METHODS FOR AGE DETERMINA-TION
Definition 22.2.1
22.3 LARGE-SCALE DISTANCE MEA-SUREMENTS
Definition 22.3.1
22.4 THE LOCAL DISTANCE SCALE
Definition 22.4.1
22.5 DIRECT DISTANCE DETERMINA-TIONS
Definition 22.5.1
22.6 SUMMARY
Definition 22.6.1
55
Part VI
THE EARLY UNIVERSE
56 23 THE HOT BIG BANG
23 The hot big bang
23.1 THERMODYNAMICS IN THE BIGBANG
23.1.1 ADIABATIC EXPANSION
Lemma 23.1.1For a ratio of specific heats Γ
T ∝ R−3(Γ−1) (23.1)
23.1.2 QUANTUM GRAVITY LIMIT
mP ≡√~cG' 1019 GeV
`P ≡√~Gc3' 10−33 cm
tP ≡√~Gc5' 10−43 s
(23.2)
23.1.3 COLLISIONLESS EQUILIBRIUM BACK-GRAOUNDS
The study of matter under the extremes of pressure and tem-perature expected in the early phases of the expanding uni-verse might be expected to be a difficult task.→ simplifying in 2 ways
¤ Thermal equilibrium
¤ The perfect gas approximation
→We need the thermodynamics of a possibly relativistic per-fect gas.
dN = gV
(2π)3d3k (23.3)
〈f〉 =[e(ε−µ)/kBT ± 1
]−1
(23.4)
n = g1
(2π~)3
∫ ∞
0
dp4πp2
eε(p)/kBT ± 1(23.5)
where ε√p2c2 +m2c4
♣ Ultrarelativistic limit: For kBT À mc2 → as if massless
n =(kBT
c
)3 4πg(2π~)3
∫ ∞
0
dyy2
ey ± 1(23.6)
♣ Nonrelativistic limit:
n = e−mc2/kBT (2mkBT )3/2
4πg(2π~)3
∫ ∞
0
dy e−y2y2
(23.7)
23.1.4 ENTROPY OF THE BACK GRAOUND
S =4πgV(2π~)3
∫ ∞
0
dpp2
eε/kBT ± 1
(ε
T+p2c2
3εT
)(23.8)
S
M= 1.09× 1016
(Ωh2
)−1erg deg−1 g−1 (23.9)
S = g4πkBV(2π~)3
∫dp p2 ln fF,B +
∫dp p2 (ε− µ)/kBT
e(ε−µ)/kBT ± 1
(23.10)
23.1.5 FORMULAE FOR ULTRARELATIVISTICBACKGROUNDS
u = gπ2
30kBT
(kBT
~c
)3
= 3P (23.11)
s
kB= g
2π2
45
(kBT
~c
)3
= 3.602n (23.12)
78
Lemma 23.1.2
nF =34gFgBnB , uF =
78gFgBuB , sF =
78gFgBsB (23.13)
Definition 23.1.3 Effective degeneracy factor for u os s
g∗ =def
∑
bosons
gi +78
∑
fermions
gj (23.14)
23.1.6 NEUTRINO DECOUPLING
23.1.7 MASSIVE NEUTRINOS
23.2 RELICS OF THE BIG BANG
Proposition 23.2.1 The general relativistic form of theBoltzmann equationThe general relativistic form of the Boltzmann equation forparticles affected by the gravitational forces and collisions is
[pµ
∂
∂xµ− Γµνλp
νpλ∂
∂pµ
]f = C (23.15)
Proof of 23.2.1
df(~x, ~p)dλ
= C (23.16)
where C is the evolution of phase-space density under colli-sions.
df
dλ=
∂f
∂xµdxµ
dλ+
∂f
∂pµdpµ
dλ= C (23.17)
23.3 THE PHYSICS OF RECOMBINA-TION
How should the ionization of a plasma change in ther-mal equilirium?
78Remember that g = 2 for photons.
23.4 THE MICROWAVE BACKGRAOUND 57
23.3.1 EQUILIBRIUM (SAHA) THEORY
the ratio of partition functions:
Pfree
Pbound=
∑free gie
−Ei/kBT
∑boundgie−Ei/kBT
=Zfree
Zbound(23.18)
Proposition 23.3.1 SAHA equationSaha equation
x2
1− x =(2πmekBT )3/2
n(2π~)3e−χ/kBT (23.19)
23.3.2 NON-EQUILIBRIUM REALITY
23.3.3 THE LAST-SCATTERING SHELL
23.4 THE MICROWAVE BACKGRAOUND
23.4.1 OBSERVATIONS
23.4.2 THE DIPOLE ANISOTROPY
23.5 PRIMORDIAL NUCLEOSYNTHESIS
23.5.1 NEUTRON FREEZE-OUT
23.5.2 NEUTRINO FREEZE-OUT
23.5.3 CONSTRUCTION OF NUCLEONS
23.5.4 THE PRIMODIAL HELIUM ABUNDANCE
23.5.5 THE NUMBER OF PARTICLE GENERA-TIONS
23.5.6 OTHER LIGHT-ELEMENT ABUN-DANCES
23.6 BARYOGENESIS
Lemma 23.6.1 Sakharov conditionsfor baryosynthesis,
1. ∆B 6= 0 reactions
2. CP violation
3. non-equilibrium conditions
23.6.1 OTHER MECHANISMS
23.6.2 LEPTON ASYMMETRY
Proposition 23.6.2For the freeze-out neutron-to-proton ratio, which assumedµ = 0:
nnnp
= exp(− Q
kBT− µνkBT
)(23.20)
24 Topological defects
Definition 24.0.3
Definition 24.0.4
24.1 PHASE TRANSITION IN COSMOL-OGY
24.2 CLASSES OF TOPOLOGICAL DE-FECT
24.2.1 CHARACTERISTIC MASSES AND DENSI-TIES
24.2.2 GAUGE VS GLOBAL DEFECTS
24.3 MAGNETIC MONOPOLES
24.4 COSMIC STRINGS AND STRUC-TURE FORMATION
Definition 24.4.1
24.4.1 STRING SPACETIME
24.4.2 SCALING OF STRING NETWORKS
24.4.3 GRAVITATIONAL DENSING BYSTRINGS
24.4.4 HUNTING STRINGS
24.4.5 CMB ANISOTROPIES WITH STRINGS
25 Inflationary cosmology
25.1 GENERAL AARGUMENTS FOR IN-FLATION
25.1.1 THE HORIZON PROBLEM
25.1.2 THE FLATNESS PROBLEM
25.1.3 THE ANTIMATTER PROBLEM
25.1.4 THE STRUCTURE PROBLEM
25.1.5 THE EXPANSION PROBLEM
Definition 25.1.1
25.2 AN OVERVIEW OF INFLATION
25.2.1 EOS FOR INFLATION
25.2.2 DE SITTER SPACE AND INFLATION
25.2.3 REHEATING FROM INFLATION
25.2.4 QUANTUM FLUCTUATIONS
25.3 INFLATION FIELD DYNAMICS
25.3.1 QUANTUM FIELDS AT HIGH TEMPERA-TURE
25.3.2 DYNAMICS OF THE INFLATION FIELD
25.3.3 ENDING INFLATION
25.4 INFLATION MODEL
Definition 25.4.1
58 26 MATTER IN THE UNIVERSE
25.4.1 EARLY INFLATION MODELS
25.4.2 CHAOTIC INFLATION MODELS
25.4.3 CRITERIA FOR INFLATION
Definition 25.4.2
25.5 RELLIC FLUCTUATIONS FROM IN-FLATION
25.5.1 MOTIVATION
Definition 25.5.1
Definition 25.5.2
Definition 25.5.3
25.5.2 THE FLUCTUATION SPECTRUM
25.5.3 INFLATION COUPLING
25.5.4 GRAVITY WAVES AND TILT
Definition 25.5.4 TiltDefine the tilt of the fluctuation spectrum as follows:
tilt =def
1− n =def−d ln δ2Hd ln k
(25.0)
25.6 CONCLUSIONS
25.6.1 THE TESTABILITY OF INFLATION
25.6.2 THE Λ PROBLEM
Part VII
OBSERVATIONALCOSMOLOGY
Definition 25.6.1
Definition 25.6.2
26 Matter in the universe
Definition 26.0.3
Definition 26.0.4
59
A Measure & Integral
A.1 Measure
Definition A.1.1 Measurable space(X,S) is called a measurable space if satisfying the followingconditions:
1. emptyset, X ∈ S2. A ∈ S =⇒ Ac ∈ S3. Ai ∈ S =⇒ ⋃∞
n=1An ∈ S
Definition A.1.2 Measureµ which is R-value function on S is called measure on S
1. 0 ≤ µ(E) ≤ ∞, µ(∅) = 0
2. A1, · · · ∈ S, Ai ∩Aj = ∅=⇒ µ (
⋃∞n=1An) =
∑∞n=1 µ(An)
Definition A.1.3 Measure space(X,S, µ) is called a measure space.
Definition A.1.4 σ-finite(X,S, µ) is finite. ⇐⇒
defµ(X) <∞
(X,S, µ) is σ-finite.⇐⇒def
∃Xnn∈N; seq. of measurable set s.t.
X =∞⋃n=1
Xn, µ(Xn) <∞
Definition A.1.5 Measurable functionf which is R-value function defined on X is S-measurable⇐⇒def
∀α, x|x ∈ X, f(x) > α ∈ S
B QFT
C Representation of finite groups
group · · · Galoisx2 − 2 = 0 → x = ±√2
permutation↓ generalizationautomorphismGL(V )Diff(M)↓ abstraction(abstract) group [the axioms of group]=⇒ Inversely, for an abstract group, what its representationsare?
C.1 Representations
Definition C.1.1 Linear RepresentationG; grpV ; K-vect. sp.GL(V ); the set of nonsingular linear transformations on VThen
ρ : G −→ GL(V ); group hom. (C.2)
is called (linear) representation of G on V
Example C.1.3 G = 1, ω, ω2 ∼= Z/3Zω =
−1±√32
Let’ consider the following permutation for X = A, B, C
1; id (C.4)ω : A 7→ B 7→ C 7→ A (C.5)
ω2 : A 7→ C 7→ B 7→ A (C.6)
G 7−→isom.
G3 = permutation on XCorresponding to this, the following representation (permu-tation repr.) is obtained.
For V = CvA ⊕ CvB ⊕ CvC (C.7)= 〈vA, vB , vC〉 (C.8)
ρ(1) =
1 0 00 1 00 0 1
(C.9)
ρ(ω) =
0 0 11 0 00 1 0
(C.10)
ρ(ω2) =
0 1 00 0 11 0 0
(C.11)
In general, when G acts on X as a permutation (= bijec-tion), the representation defined
ρ(g)(x) =def
vg(x) (C.12)
on V = 〈vx|x ∈ X〉 (C.13)
is called permutation representation.When corresponding to the matrix, ρ : G→ GL(n,C), we
call matrix representation.
Example C.1.14 zero representation
ρ : G −→ GL(1,K) = K× =def
K\0g 7→ 1When n = 0, V = 0 formally.(zero repr.)
Definition C.1.15 Regular Representationgroup G acts on itself by left-hand-side
G 3 g : h 7→ g · h;G→ G(permutation!) (C.16)
Corresponding to this, permutation repr. is defined such as
G 3 g :∑
h∈G, ah∈Kahvh 7−→
∑
h∈Gahvgh (C.17)
We call this repr. regular representation.And deg ρ = |G| = dimV
Remark C.1.18 Notation
V = KG = K[G]
i.e. to make K-coefficient polynomial: vg · vh =def
vg·h→ In this sense, it is a ring, say group-ring
For simplicity, we omit ρ. e.g. ρ(g)v := gv
60 C REPRESENTATION OF FINITE GROUPS
Definition C.1.19 Invariant subspaceV ; repr. of G
W ⊂ V ; linear sub sp. of VBy definition (W is ”linear sub”) it closed w.r.t. addition
and scalar multiplication. However, we don’ t know whetherit closed w.r.t. action of G or not. So the next definition ismeaningful.When ∀g ∈ G, g(W ) ⊂W ,W is called an invariant subspace. In fact, g(W ) = W
Definition C.1.20 Subrepresentation
ρ|W : G→ GL(W ), g 7→ ρ(g) ∈ GL(V ) (C.21)
is called a sub representation.
Definition C.1.22 Quotient representation
V/W 3 v +W 7−→ gv + g(W ) = gv +W (C.23)ρ : G −→ GL(V/W ) (C.24)
is called a quotient representation.
Definition C.1.25 Irreducible RepresentationV is called an irreducible representation when V has noinvariant subspace except 0&V .
C.2 Derived Representation
Set V, W, Vλ; representations
Definition C.2.1 Direct sum RepresentationV ⊕W, g(v ⊕ w) =
defgv ⊕ gw
;multiply matrix A & B on V & W respectively.
=⇒ on V ⊕W ,(A 00 B
)which called a direct sum repre-
sentation.
Definition C.2.2 Symmetrize operator⊗nV = V ⊗ V ⊗ · · · ⊗ V
Definition C.2.3 Alternate operator
Definition C.2.4 Tensor product Representation
Definition C.2.5 Dual Representation
Definition C.2.6
Definition C.2.7 G-morphV, W ; repr.ϕ : V →W is G-morph.
⇐⇒def
ϕ(gv) = gϕ(v)
(commutative to G-operating) Hom(V,W )∪
HomG(V,W )‖ commutative with G-acting
Hom(V,W )G the fixed point of G
Definition C.2.8 Isomorphism
Proposition C.2.9kerϕ, imϕ are subrepresentation of V, W
Proof of C.2.9
ϕ(v) = 0, ϕ(gv) = gϕ(v) = 0
Lemma C.2.10 SchurV,W ; irreducible repr. of Gϕ ∈ HomG(V,W )=⇒
1. ϕ is either isom. or 0-map.
2. V = W ; finite-dim. K; alg. closed (= K)=⇒ ∃λ ∈ K, ϕ = λidV
Proof of C.2.10
1. kerϕ, Imϕ; G-invariant subsp. of V,W=⇒ kerϕ Imϕ = 0, V (W )
=⇒
kerϕ = 0 =⇒ Imϕ = W (=⇒ isom)kerϕ = V =⇒ Imϕ = 0 (=⇒ 0−map)
2. ∃ λ; e-val. of ϕ (K; alg. closed)⇐⇒ ker(ϕ− λid) 6= 0=⇒ ker(ϕ− λid) = V=⇒ ϕ = λid
Remark C.2.11V
'−−−−→ϕ0(fix)
W ; isom
=⇒ ϕ = λϕ0 (∃λ ∈ K)
Corollary C.2.12G; abelianV ; finite-dim irreducible repr. over K = K=⇒ dimV = 1
Proof of C.2.12
C.3 on Finite group
Remark C.3.1V ; repr. 3 v 6= 0Consider some linear subsp. expanded by gv|g ∈ G, whichis of finite-dim and G-inv.dimV =∞ =⇒ V ; reducible
∴ V ; irreducible =⇒ V = 〈gv |g ∈ G〉C; finite-dim
Theorem C.3.2 Maschkechar(K) 6 ||G|V ; repr. of finite grp. GV ⊃W ; subrepr. of W
=⇒ ∃W ′, V = W ⊕W ′, W ′; G−inv.
Proof of C.3.2V = W ⊕∃ U ; It’s possible to decompose as vect. sp.
Define π0 : V →W s.t.π0|W = id, π0|U = 0and define
π =def
1|G|
∑
g∈Gg
(π0(g−1v)
)(∈W ) ∀v ∈ V
C.4 Character 61
claim.1 π : V →W is G-linear (π ∈ HomG(V,W ))∵∀f ∈ G
π(hv) =1|G|h
∑
g∈Gh−1g
(π0(g−1hv)
)(C.3)
=h
|G|∑
g∈Gg′
(π0(g′−1v)
)(g′ = h−1g) (C.4)
= hπ(v) (C.5)
claim.2 Imπ = W∵∀v ∈W
π(v) =1|G|
∑
g∈Gg
(π0(g−1v)
)(C.6)
=1|G|
∑
g∈Gv = v (C.7)
In particular, π = id on W .W ′ =
defkerπ ; G-inv.
=⇒ V = W ⊕W ′
Corollary C.3.8 Completely irreducibleFinite-dim repr. of finite grp is a direct sum of irreduciblereprs.
Remark C.3.9similarly for compact group
Proposition C.3.10
V '⊕
Va '⊕
Wb
=⇒ ∀a, ∃b s.t. Va'−→Wb
Remark C.3.11The decomposition such as V =
⊕Va is not always unique.(
1 00 1
)
↔ 1⊕ 1 = Cv1 ⊕ Cv2Remark C.3.12
R+ 3 a 7−→(
1 a0 1
)∈ GL(2,R)
a+ b 7−→(
1 a+ b0 1
)=
(1 a0 1
)(1 b0 1
)
Though the invariance of R(
10
)leads to irreducible, it is
not completely irreducible.
Example C.3.13Permutation repr. of S3 for 1, 2, 3.C3 = 〈e1, e2, e3〉CS→ GL(3,C); permutation repr.C3 ⊃ C(e1 + e2 + e3); S-inv.
By Maschke,C3 = U ⊕∃ V (← 2dim)We may choose, for example, the following basis.
V = 〈e2 − e1, e3 − e1〉C
S3 3 e↔(
1 00 1
)(C.14)
(1, 2)↔(−1 −1
0 1
)(C.15)
(1, 3)↔(
1 0−1 −1
)(C.16)
(2, 3)↔(
0 11 0
)(C.17)
(1, 2, 3)↔(−1 −1
1 0
)(C.18)
(1, 3, 2)↔(
0 1−1 −1
)(C.19)
C.4 Character
Definition C.4.1 CharacterFor ρ : G→ GL(V ) repr. of finite-dim
χρ(g) =def
trρ(g) (C.2)
χ : G −→ K (C.3)
is called the character of ρ.
Remark C.4.4G 3 g : ρ(g) : finiteorder=⇒ ρ(g)n = id=⇒ e-vals. are nilpotent of 1, diagonalable (∵ x2 − 1 is sepa-rable.)
Remark C.4.5 symmetric expression
χ(g) =∑
i
λi (C.6)
χ(g2) =∑
i
λ2i (C.7)
...
χ(gn) =∑
i
λni (C.8)
leads the symmetric expressions of λi.i.e.
λ1 + · · ·+ λn = a1 (C.9)λ1λ2 + · · ·+ λn−1λn = a2 (C.10)
...λ1λ2 · · ·λn = an (C.11)
→ tn − a1tn−1 + · · ·+ (−1)nan = 0 (C.12)
C.4.1 properties of character
1. χ1(g) ≡ 1
2. χ(e) = dimV (= deg ρ)
3. χ(h−1gh) = χ(g)
4. χV⊕W (g) = χV (g) + χW (g)
5. χV⊗W (g) = χV (g) · χW6. χV ∗(g) = χV (g−1)
7. χSym2V (g) =12χ(g2) + χ(g)2
62 C REPRESENTATION OF FINITE GROUPS
8. χV2 V (g) =12−χ(g2) + χ(g)2
Remark C.4.13
V ⊗ V = Sym2V ⊕2∧V (C.14)
χV⊗W = χSym2V + χV2 V (C.15)
Remark C.4.16(ρ, V ) ' (σ,W ) equivalence
⇐⇒def
V∼−→∃ϕ W
ρ(g) ↓ ↓ σ(g)V −→
gϕ=ϕW
gϕ := ρ(g)−1ϕσ(g)
trσ(g) = tr(ϕ ρ(g) ϕ−1)= trρ(g)
∴ χρ = χσ
i.e. For each equivalent repr. sp., the character is the samevalue.
ρ ' σ =⇒ χρ = χσ
C.5 Orthogonal relation of Character
C.5.1 Fixed point set
V ; repr. of G
Definition C.5.1 Fixed point in V of G
V G =defv ∈ V | ∀g ∈ Gρ(g)v(= gv) = v (C.2)
Remark C.5.3V G is G-invariant subsp. ' 1⊕ dimV G
(1; unit repr.)In the following discussion, we assume G; finite group,
charK = 0.
Definition C.5.4 average outDefine the average out function:
ϕ =def
1|G|
∑
g∈Gg : V → V (C.5)
Proposition C.5.6
ϕ is the projector from V onto V G
Proof of C.5.6
Lemma C.5.7
ϕ ∈ homG(V, V )
Proof of C.5.7
∀h ∈ G
ϕ(hv) =1|G|
∑
g∈G(C.8)
=1|G|
∑
g∈G(gh)v (C.9)
=h
|G|∑
g∈G(h−1gh)v (C.10)
= hϕ(v) (C.11)
¥
Lemma C.5.12
Imϕ ⊂ V G
Proof of C.5.12∀h ∈ G
hϕ(v) =h
|G|∑
g∈Ggv (C.13)
=1|G|
∑
g∈G(hg)v = ϕ(v) (C.14)
¥
Lemma C.5.15
Imϕ ⊃ V G
Proof of C.5.15Assume that∀h ∈ G, hv = v. Then
ϕ(v) =1|G|
∑
g∈Ggv =
1|G|
∑
g∈Gv (C.16)
=v
|G|∑
g∈G1 = v (C.17)
¥
Lemma C.5.18
ϕ ϕ = ϕ
Proof of C.5.18
ϕ ϕ(v) = ϕ(v) (∈ V G) (C.19)
¥
Therefore ϕ : V ³ V G is the projector.i.e. Imϕ = V G, ϕ|V G = id
Remark C.5.20
dimV G = trϕ (C.21)
=1|G|
∑
g∈Gtr(g) =
1|G|
∑
g∈Gχ(g) (C.22)
In particular, when V is not nontrivial irreducible repr.∑
g∈Gχ(g) = 0
C.5 Orthogonal relation of Character 63
C.5.2 Orthogonal relation
V,W ; repr. sp. of finite dimensional over CG; finite group
homC(V,W ) ' V ∗ ⊗W (C.23)⋃
homG(V,W ) = hom(V,W )G (C.24)
∴ dimhomG(V,W ) =1|G|
∑
g∈Gχhom(V,W )(g) (C.25)
=1|G|
∑
g∈GχV ∗(g)χW (g) (C.26)
=
1|G|
∑
g∈GχV (g)χW (g)
(C.27)
∴ 1|G|
∑
g∈GχV (g)χW (g) = dim homG(V,W ) (C.28)
=
0 (V W )1 (V ∼= W )
(C.29)
want to write it down like (χV , χW ).
Remark C.5.30the 2nd ”=” in the above equation is satisfied if V,W ; irred.by Schur lemma.
Definition C.5.31 Set of class function
Cclass(G) =defϕ : G→ C; class fn.
Remark C.5.32
1. Cclass(G) is a C-linear sp.
(ϕ+ ψ)(g) =def
ϕ(g) + ψ(g)(aϕ)(g) =def
aϕ(g) (a ∈ C)
2. χ ∈ Cclass(G)In this sense, Cclass(G) includes the concept of the char-acter.
Definition C.5.33For α, β ∈ Cclass(G), define
〈α, β〉 =def
1|G|
∑
g∈Gα(g)β(g) (C.34)
Proposition C.5.35〈∗|∗〉 is an Hermite innerproduct over Cclass(G)i.e.
• 〈α|β〉 is semilinear w.r.t. α and linear w.r.t. β
〈aα|β〉 = a〈α|β〉
• 〈α|β〉 = 〈β|α〉• 〈α|α〉 ≥ 0, 〈α|α〉 ⇐⇒ α = 0
Proof of C.5.35We shall omit.
Theorem C.5.36
χV | V ; irred. is ONB of Cclass(G)
Corollary C.5.37
]irred.repr./ ∼= ≤ dimCclass(G)
Corollary C.5.38 finite dimensional repr. of G → character of G is injective.
Corollary C.5.39
V ; irred. ⇐⇒ 〈χV |χV 〉 (C.40)
Remark C.5.41
〈χV |χV 〉 ⇐⇒ V ∼= W ⊕W ′
〈χV |χV 〉 ⇐⇒ V ∼= W ⊕W ′ ⊕W ′′
〈χV |χV 〉 ⇐⇒ V ∼= W ⊕W ′ ⊕W ′′ ⊕W ′′′ or W⊕2
Corollary C.5.42Vi; irred. repr.〈χV |χVi
〉 = ](Vi included in V as component of direct sum )
Proof of C.5.42
〈χV |χVi〉 =
⟨∑
j
ajχVj |χVi
⟩= ai (C.43)
Remark C.5.44When V = R; regular repr., dimR = |G|
χR(g) =
|G| (g = e)0 (g 6= e)
(C.45)
Therefore, for Vi; irred. repr., R ∼= ⊕iV ⊕aii
ai = 〈χR|χVi〉
=1|G|
∑
g∈GχR(g)χVi
(g)
=onlyg=e
1|G|
∑
g∈G|G|χVi
(e) = dimVi (C.46)
Lemma C.5.47For α ∈ C(G) = fn. from G to C and V ; complex repr., wedefine
ϕα,V =def
∑
g∈G: V → V (linear) (C.48)
64 D PATH INTEGRAL QUANTIZATION
Then,∀ V ; finite dimensional repr.
ϕα,V ∈ homG(V, V ) (C.49)
m
α ∈ Cclass(G) (C.50)
Proof of C.5.50(⇓): ∀h ∈ G
ϕα,R(hve) = hϕα,R(ve) = h∑
g′∈Gα(g′)vg′
=∑
g′∈Gα(g′)vhg′
=∑
g∈Gα(h−1gh)vgh (g′ = h−1gh)
=∑
g∈Gα(g)vgh
Since vgh is basis,
∀g, α(g) = α(h−1gh)
∴ α ∈ Cclass(G)
¥
(⇑): ∀h ∈ G
ϕα,V (hv) =∑
g∈Gα(g)ghv
= h∑
g∈Gα(g)h−1ghv
= h∑
g∈Gα(h−1gh)h−1ghv
= hϕα,V (v)
¥
C.6 Fine constant & Bohr radius
α =def
e2
~c' 1
137; fine structure const.
a0 =~2
me2; 1st Bohr radius
re =e2
mc2; classical electron radius
σT =8π3r2e ; Thomson cross section
λe =~mc
; Compton wave length
µB =e~
2mc; Borh magneton
then
λeα
=~ce2
~mc
=~2
me2= a0
reα
=~ce2
e2
mc2=
~mc
= λe
=⇒ re = αλe, λe = αa0
re < λe < a0
D PATH INTEGRAL QUANTIZA-TION
D.1 Generating function
Z[J ] =def
∫Dφ exp
i
∫d4x
[L(φ) + J(x)φ(x) +
i
2εφ2
]
(D.1)
∝ 〈0,∞|0,−∞〉J
Let us now calculate this for a free particle (field), for which
L0 =12
(∂µφ∂
µφ−m2φ2)
The corresponding vacuum-to-vacuum amplitude is (as N →∞)
Z0[J ] =∫
Dφ exp(i
∫ 12(∂µφ∂µφ− (m2 − iε)φ2 + φJ
d4x
)
(D.2)
Proposition D.1.3
Z0[J ] =∫
Dφ exp(−i
∫ (12φ(¤ +m2 − iε)φ− φJ
)d4x
)
(D.4)
Proof of D.1.4
Proposition D.1.5From the Lagragngian
L =12∂µφ∂µφ− 1
2m2φ2 + Lint
φ obeys the equation of motion
(¤ +m2)φ(x) =∂Lint
∂φ(x)(D.6)
Further, the solution to this equation is
φ(x) = φin(x) +∫dy4ret(x− y) ∂Lint
∂φ(y)(D.7)
φ(x) = φout(x) +∫dy4adv(x− y) ∂Lint
∂φ(y)(D.8)
where 4ret is defined as
4ret = 0 (for x2 > 0, x0 < 0) (D.9)
4adv = 0 (for x2 > 0, x0 > 0) (D.10)
(¤ +m2)4ret(x) = δ4(x) (D.11)
(¤ +m2)4adv(x) = δ4(x) (D.12)
we call them retarded (advanced) Green’s function.
Proof of D.1.12
D.2 Propagators and gauge consition in QED 65
D.2 Propagators and gauge consition inQED
Z[J ] =∫
DAµ exp(i
∫(L+ JµAµ)d4x
)(D.1)
where
L = −14FµνFµν (D.2)
References
[1] 藤川和男, ゲージ場の理論，岩波書店，1988
[2] 大貫義郎，場の量子論，岩波書店，1978
[3] 佐藤隆文・小玉英雄，一般相対性理論，岩波書店，1973
[4] 中西襄，場の量子論，培風館，1975
[5] 戸塚洋二，素粒子物理，岩波書店，
[6] 原・稲見・青木，素粒子物理学，朝倉書店
