Introduction To Quantum Field Theory And The Standard Model

Contents
	Foreword
	Preface
	Frequently cited references
	Index of useful formulae
	A note on the problems
1
	Adding special relativity to quantum mechanics
		1.1Introductory remarks
		1.2Theory of a single free, spinless particle of mass µ
		1.3Determination of the position operator X
2
	The simplest many-particle theory
		2.1First steps in describing a many-particle state
		2.2Occupation number representation
		2.3Operator formalism and the harmonic oscillator
		2.4The operator formalism applied to Fock space
3
	Constructing a scalar quantum field
		3.1Ensuring relativistic causality
		3.2Conditions to be satisfied by a scalar quantum field
		3.3The explicit form of the scalar quantum field
		3.4Turning the argument around: the free scalar field as the fundamental object
		3.5A hint of things to come
	Problems 1
	Solutions 1
4
	The method of the missing box
		4.1Classical particle mechanics
		4.2Quantum particle mechanics
		4.3Classical field theory
		4.4Quantum field theory
		4.5Normal ordering
5
	Symmetries and conservation laws I. Spacetime symmetries
		5.1Symmetries and conservation laws in classical particle mechanics
		5.2Extension to quantum particle mechanics
		5.3Extension to field theory
		5.4Conserved currents are not uniquely defined
		5.5Calculation of currents from spacetime translations
		5.6Lorentz transformations, angular momentum and something else
	Problems 2
	Solutions 2
6
	Symmetries and conservation laws II. Internal symmetries
		6.1Continuous symmetries
		6.2Lorentz transformation properties of the charges
		6.3Discrete symmetries
7
	Introduction to perturbation theory and scattering
		7.1The Schrödinger and Heisenberg pictures
		7.2The interaction picture
		7.3Dyson’s formula
		7.4Scattering and the S-matrix
	Problems 3
	Solutions 3
8
	Perturbation theory I. Wick diagrams
		8.1Three model field theories
		8.2Wick’s theorem
		8.3Dyson’s formula expressed in Wick diagrams
		8.4Connected and disconnected Wick diagrams
		8.5The exact solution of Model 1
	Problems 4
	Solutions 4
9
	Perturbation theory II. Divergences and counterterms
		9.1The need for a counterterm in Model 2
		9.2Evaluating the S matrix in Model 2
		9.3Computing the Model 2 ground state energy
		9.4The ground state wave function in Model 2
		9.5An infrared divergence
	Problem 5
	Solution 5
10
	Mass renormalization and Feynman diagrams
		10.1Mass renormalization in Model 3
			10.2Feynman rules in Model 3
			10.3Feynman diagrams in Model 3 to order g2
			10.4O(g2) nucleon–nucleon scattering in Model 3
11
	Scattering I. Mandelstam variables, CPT and phase space
		11.1Nucleon–antinucleon scattering
		11.2Nucleon–meson scattering and meson pair creation
		11.3Crossing symmetry and CPT invariance
		11.4Phase space and the S matrix
12
	Scattering II. Applications
		12.1Decay processes
		12.2Differential cross-section for a two-particle initial state
		12.3The density of final states for two particles
		12.4The Optical Theorem
		12.5The density of final states for three particles
		12.6A question and a preview
	Problems 6
	Solutions 6
13
	Green’s functions and Heisenberg fields
		13.1The graphical definition of (n)(ki)
		13.2The generating functional Z[ρ] for G(n)(xi)
		13.3Scattering without an adiabatic function
		13.4Green’s functions in the Heisenberg picture
		13.5Constructing in and out states
	Problems 7
	Solutions 7
14
	The LSZ formalism
		14.1Two-particle states
		14.2The proof of the LSZ formula
		14.3Model 3 revisited
		14.4Guessing the Feynman rules for a derivative interaction
	Problems 8
	Solutions 8
15
	Renormalization I. Determination of counterterms
		15.1The perturbative determination of A
		15.2The Källén-Lehmann spectral representation
		15.3The renormalized meson propagator ′
		15.4The meson self-energy to O(g2)
		15.5A table of integrals for one loop
16
	Renormalization II. Generalization and extension
		16.1The meson self-energy to O(g2), completed
		16.2Feynman parametrization for multiloop graphs
		16.3Coupling constant renormalization
		16.4Are all quantum field theories renormalizable?
	Problems 9
	Solutions 9
17
	Unstable particles
		17.1Calculating the propagator  for µ > 2m
		17.2The Breit–Wigner formula
		17.3A first look at the exponential decay law
		17.4Obtaining the decay law by stationary phase approximation
18
	Representations of the Lorentz Group
		18.1Defining the problem: Lorentz transformations in general
		18.2Irreducible representations of the rotation group
		18.3Irreducible representations of the Lorentz group
		18.4Properties of the SO(3) representations D(s)
		18.5Properties of the SO(3, 1) representations D(s+, s–)
	Problems 10
	Solutions 10
19
	The Dirac Equation I. Constructing a Lagrangian
		19.1Building vectors out of spinors
		19.2A Lagrangian for Weyl spinors
		19.3The Weyl equation
		19.4The Dirac equation
20
	The Dirac Equation II. Solutions
		20.1The Dirac basis
		20.2Plane wave solutions
		20.3Pauli’s theorem
		20.4The γ matrices
		20.5Bilinear spinor products
		20.6Orthogonality and completeness
	Problems 11
	Solutions 11
21
	The Dirac Equation III. Quantization and Feynman Rules
		21.1Canonical quantization of the Dirac field
		21.2Wick’s theorem for Fermi fields
		21.3Calculating the Dirac propagator
		21.4An example: Nucleon–meson scattering
		21.5The Feynman rules for theories involving fermions
		21.6Summing and averaging over spin states
	Problems 12
	Solutions 12
22
	CPT and Fermi fields
		22.1Parity and Fermi fields
		22.2The Majorana representation
		22.3Charge conjugation and Fermi fields
		22.4PT invariance and Fermi fields
		22.5The CPT theorem and Fermi fields
23
	Renormalization of spin- theories
		23.1Lessons from Model 3
			A digression on theories that do not conserve parity
		23.2The renormalized Dirac propagator ′
		23.3The spectral representation of ′
		23.4The nucleon self-energy ′
		23.5The renormalized coupling constant
	Problems 13
	Solutions 13
24
	Isospin
		24.1Field theoretic constraints on coupling constants
		24.2The nucleon and pion as isospin multiplets
		24.3Experimental consequences of isospin conservation
		24.4Hypercharge and G-parity
25
	Coping with infinities: regularization and renormalization
		25.1Regularization
		25.2The BPHZ algorithm
		25.3Applying the algorithm
		25.4Survey of renormalizable theories for spin 0 and spin ½
	Problems 14
	Solutions 14
26
	Vector fields
		26.1The free real vector field
		26.2The Proca equation and its solutions
		26.3Canonical quantization of the Proca field
		26.4The limit μ → 0: a simple physical consequence
		26.5Feynman rules for a real massive vector field
27
	Electromagnetic interactions and minimal coupling
		27.1Gauge invariance and conserved currents
		27.2The minimal coupling prescription
		27.3Technical problems
	Problems 15
	Solutions 15
28
	Functional integration and Feynman rules
		28.1First steps with functional integrals
		28.2Functional integrals in field theory
		28.3The Euclidean Z0[J] for a free theory
		28.4The Euclidean Z[J] for an interacting field theory
		28.5Feynman rules from functional integrals
		28.6The functional integral for massive vector mesons
29
	Extending the methods of functional integrals
		29.1Functional integration for Fermi fields
		29.2Derivative interactions via functional integrals
		29.3Ghost fields
		29.4The Hamiltonian form of the generating functional
		29.5How to eliminate constrained variables
		29.6Functional integrals for QED with massive photons
	Problems 16
	Solutions 16
30
	Electrodynamics with a massive photon
		30.1Obtaining the Feynman rules for scalar electrodynamics
		30.2The Feynman rules for massive photon electrodynamics
			Scalar electrodynamics with a massive photon
			Spinor electrodynamics with a massive photon
		30.3Some low order computations in spinor electrodynamics
		30.4Quantizing massless electrodynamics with functional integrals
31
	The Faddeev–Popov prescription
		31.1The prescription in a finite number of dimensions
		31.2Extending the prescription to a gauge field theory
		31.3Applying the prescription to QED
		31.4Equivalence of the Faddeev–Popov prescription and canonical quantization
		31.5Revisiting the massive vector theory
		31.6A first look at renormalization in QED
	Problems 17
	Solutions 17
32
	Generating functionals and Green’s functions
		32.1The loop expansion
		32.2The generating functional for 1PI Green’s functions
		32.3Connecting statistical mechanics with quantum field theory
		32.4Quantum electrodynamics in a covariant gauge
33
	The renormalization of QED
		33.1Counterterms and gauge invariance
		33.2Counterterms in QED with a massive photon
		33.3Gauge-invariant cutoffs
		33.4The Ward identity and Green’s functions
		33.5The Ward identity and counterterms
	Problems 18
	Solutions 18
34
	Two famous results in QED
		34.1Coulomb’s Law
		34.2The electron’s anomalous magnetic moment in quantum mechanics
		34.3The electron’s anomalous magnetic moment in QED
35
	Confronting experiment with QED
		35.1Higher order contributions to the electron’s magnetic moment
		35.2The anomalous magnetic moment of the muon
		35.3A low-energy theorem
		35.4Photon-induced corrections to strong interaction processes (via symmetries)
	Problems 19
	Solutions 19
36
	Introducing SU(3)
		36.1Decays of the η
		36.2An informal historical introduction to SU(3)
		36.3Tensor methods for SU(n)
		36.4Applying tensor methods in SU(2)
		36.5Tensor representations of SU(3)
37
	Irreducible multiplets in SU(3)
		37.1The irreducible representations q and
		37.2Matrix tricks with SU(3)
		37.3Isospin and hypercharge decomposition
		37.4Direct products in SU(3)
		37.5Symmetry and antisymmetry in the Clebsch–Gordan coefficients
	Problems 20
	Solutions 20
38
	SU(3): Proofs and applications
		38.1Irreducibility, inequivalence, and completeness of the IR’s
		38.2The operators I, Y and Q in SU(3)
		38.3Electromagnetic form factors of the baryon octet
		38.4Electromagnetic mass splittings of the baryon octet
39
	Broken SU(3) and the naive quark model
		39.1The Gell-Mann–Okubo mass formula derived
		39.2The Gell-Mann–Okubo mass formula applied
		39.3The Gell-Mann–Okubo mass formula challenged
		39.4The naive quark model (and how it grew)
		39.5What can you build out of three quarks?
		39.6A sketch of quantum chromodynamics
	Problems 21
	Solutions 21
40
	Weak interactions and their currents
		40.1The weak interactions circa 1965
		40.2The conserved vector current hypothesis
		40.3The Cabibbo angle
		40.4The Goldberger–Treiman relation
41
	Current algebra and PCAC
		41.1The PCAC hypothesis and its interpretation
		41.2Two isotriplet currents
		41.3The gradient-coupling model
		41.4Adler’s Rule for the emission of a soft pion
		41.5Equal-time current commutators
	Problems 22
	Solutions 22
42
	Current algebra and pion scattering
		42.1Pion–hadron scattering without current algebra
		42.2Pion–hadron scattering and current algebra
		42.3Pion–pion scattering
		42.4Some operators and their eigenvalues
43
	A first look at spontaneous symmetry breaking
		43.1The man in a ferromagnet
		43.2Spontaneous symmetry breaking in field theory: Examples
		43.3Spontaneous symmetry breaking in field theory: The general case
		43.4Goldstone’s Theorem
	Problems 23
	Solutions 23
44
	Perturbative spontaneous symmetry breaking
		44.1One vacuum or many?
		44.2Perturbative spontaneous symmetry breaking in the general case
		44.3Calculating the effective potential
		44.4The physical meaning of the effective potential
45
	Topics in spontaneous symmetry breaking
		45.1Three heuristic aspects of the effective potential
		45.2Fermions and the effective potential
		45.3Spontaneous symmetry breaking and soft pions: the sigma model
		45.4The physics of the sigma model
	Problems 24
	Solutions 24
46
	The Higgs mechanism and non-Abelian gauge fields
		46.1The Abelian Higgs model
		46.2Non-Abelian gauge field theories
		46.3Yang–Mills fields and spontaneous symmetry breaking
47
	Quantizing non-Abelian gauge fields
		47.1Quantization of gauge fields by the Faddeev–Popov method
		47.2Feynman rules for a non-Abelian gauge theory
		47.3Renormalization of pure gauge field theories
		47.4The effective potential for a gauge theory
	Problems 25
	Solutions 25
48
	The Glashow–Salam–Weinberg Model I. A theory of leptons
		48.1Putting the pieces together
		48.2The electron-neutrino weak interactions
		48.3Electromagnetic interactions of the electron and neutrino
		48.4Adding in the other leptons
		48.5Summary and outlook
49
	The Glashow–Salam–Weinberg Model II. Adding quarks
		49.1A simplified quark model
		49.2Charm and the GIM mechanism
		49.3Lower bounds on scalar boson masses
50
	The Renormalization Group
		50.1The renormalization group for ϕ4 theory
		50.2The renormalization group equation
		50.3The solution to the renormalization group equation
		50.4Applications of the renormalization group equation
	Concordance of videos and chapters
	Index