QFT: Supplement to Srednicki

A. George

These are solutions to the first ~2/3 of Srednicki’s QFT textbook.

The Real Supplement to Srednicki
I recommend Robert Klauber's book as a supplement to Srednicki.

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Feedback
Found any mistakes? Questions? Comments? If you are willing to update the TeX, send me an e-mail and I will forward you the source. At this point I have little interest in fixing errata myself. Chapter 1: Attempts at a Relativistic Quantum Mechanics

What’s wrong with quantum mechanics?
The Dirac Equation
The Klein-Gordon Equation
Write quantum mechanics as a quantum field theory

Chapter 2: Lorentz Invariance

The Lorentz Group
Propriety and orthochronaity
Unitary operators
Generators

Chapter 3: Canonical Quantization of Scalar Fields

Canonical quantization
Lorentz-Invariant integration measure
Ultraviolet cutoff
Development of a QFT for non-interacting, spin-0 particles

Chapter 4: The Spin-Statistics Theorem

Examples of local, Lorentz-invariant terms
Derivation of spin-statistics theorem
Derivation of the “fundamental object” of the spin-0 QFT (phi)

Chapter 5: The LSZ Formula

The LSZ formula, for transition amplitudes
Multi-particle and bound states
Introduction to renormalization

Chapter 6: Path Integrals in Quantum Mechanics

Path Integrals
Heisenberg and Schrodinger Pictures
Functional Derivatives
A “trick” for ground state to ground state transitions

Chapter 7: The Path Integral for the Harmonic Oscillator

The path integral for the harmonic oscillator
Green’s functions

Chapter 8: The Path Integral for Free Field Theory

The path integral for the free field
The Feynman Propagator

Chapter 9: The Path Integral for Interacting Field Theory

Phi-cubed theory
The path-integral for the interacting field
Writing the path-integral in terms of proto-Feynman diagrams
Diagrams: symmetry factors, connected diagrams, vertex factors
Counterterms, tadpoles

Chapter 10: Scattering Amplitudes and the Feynman Rules

Connected diagrams, again
Calculation of correlation functions and scattering amplitudes
s-, t-, and u-channel diagrams
A general formula for 2 --› 2 scattering, scattering matrix elements
Feynman diagrams

Chapter 11: Cross-Sections and Decay Rates

Cross sections, general and for phi-cubed theory
Decay Rates, general and for phi-cubed theory (prbm. 11.1)
Mandelstam variables

Chapter 12: Dimensional Analysis with ħ = c = 1

Mass dimensionality of various quantities
Preference for dimensionless coupling constants

Chapter 13: The Lehmann-Källén Form of the Exact Propagator

Derivation of an explicit form for the propagator
Spectral density

Chapter 14: Loop Corrections to the Propagator

Derivation for an exact form for the propagator
Self-Energy
1PI diagrams
Feynman’s Formula
Wick Rotations
Pauli-Villars Regularization
Renormalizability of various theories
Dimensional Regularization
Mass Parameter “mu tilde”
Dimensionality parameter epsilon

Chapter 15: The One-Loop Correction in Lehmann-Källén Form

Reconciliation between perturbative and Lehmann-Källén form
Relations between self-energy and spectral density

Chapter 16: Loop Corrections to the Vertex

First-order corrections to the vertex
Normalization condition for Zg.

Chapter 17: Other One-Loop 1PI Vertices

Calculation and convergence of Vn

Chapter 18: Higher-Order Corrections and Renormalizability

Mass dimensionality and renormalizability
Superficial degree of divergence

Chapter 19: Perturbation Theory to all Orders

Combination of all loop-order (and higher) corrections
Skeleton diagrams
Effective quantum action

Chapter 20: Two-Particle Elastic Scattering at One-Loop

Example of perturbation theory at higher orders
Calculation of two-particle scattering amplitude at one-loop order
Summary of all one-loop formulas (in text)

Chapter 21: The Quantum Action

The effective action, which gives proper scattering amplitudes using only tree-level diagrams
Relationship between action and effective action

Chapter 22: Continuous Symmetries and Conserved Currents

Noether Current
Noether Charge
U(1) Transformation for complex fields, and related equations
Noether current/charge from Lorentz symmetry
The stress-energy tensor

Chapter 23: Discrete Symmetries: P, T, C, and Z

Definition of P, T, C, Z
Anti-unitarity of T
Antiparticles (for scalars) defined

Chapter 24: Non-Abelian Symmetries

Generator matrices of SO(N) and SU(N)
Structure coefficients

Chapter 25: Unstable Particles and Resonances

Alternative to the LSZ Formula for unstable particles
Relationship between self-energy and decay constant

Chapter 26: Infrared Divergences

Problem with tree-level amplitude in massless limit
Solution Part 1: amplitude adjusted to account for imperfect detectors

Chapter 27: Other Renormalization Schemes

Solution Part 2: MS bar renormalization
Anomalous dimension of mass parameter
Beta functions
Asymptotic freedom
Infrared freedom

Chapter 28: The Renormalization Group

General derivation of beta function
General derivation of anomalous dimensions

Chapter 29: Effective Field Theory

Wilson's approach: treatment of non-renormalizable theories
Ultraviolet cutoff
Triviality, ultraviolet fixed points and asympototic freedom
Naturalness and the fine-tuning problem
Euclidean space

Chapter 30: Spontaneous Symmetry Breaking

Discrete symmetry breaking, before and after renormalization
Symmetry breaking in the quantum action

Chapter 31: Broken Symmetry and Loop Corrections

Loop corrections to broken symmetries

Chapter 32: Spontaneous Breaking of Continuous Symmetries

Continuous symmetry breaking
Continuous symmetry breaking in the quantum action
Breaking of non-Abelian symmetries
Goldstone's Theorem

Chapter 33: Representations of the Lorentz Group

Tensor transformation under the Lorentz Group
Representations of the Lorentz Group

Chapter 34: Left- and Right-Handed Spinor Fields

Lorent Transformations of Spinor Fields
Dotted and Undotted Indices
Full Decomposition of Fields
The Levi-Cevita Symbol
Self-Dual and Anti-Self Dual Fields

Chapter 35: Manipulating Spinor Indices

Invariant symbols under Lorentz Transformations for spinors
Explicit form for generators of Lorentz Transformations for spinors

Chapter 36: Lagrangians for Spinor Fields

Derivation of a Lagrangian for spinor fields
Interpreting the Dirac Equation
Majorana and Dirac Fields
The charge conjugation operator
Gamma matrices

Chapter 37: Canonic Quantization of Spinor Fields I

Commutation relations for spin-1/2 fields
Feynman Slash Notation
General solution of Dirac Equation

Chapter 38: Spinor Technology

Explicit forms of spinors
Identities involving spinors
Spin sums
Helicity

Chapter 39: Canonical Quantization of Spinor Fields II

Expansion of fermionic creation/annihilation operators
Expansion of free-field Hamiltonian
Conservation of fermion number
Proof of spin-statistics theorem for spin-1/2 particles (problem 39.4)

Chapter 40: Parity, Time Reversal, and Charge Conjugation

Parity, time reversal, and charge conjugation transformations for fermions
Intrisic parity
CPT Theorem

Chapter 41: LSZ Reduction for spin-1/2 particles

The LSZ Formula for fermions
Normalization of the field

Chapter 42: The Free Fermion Propagator

Derivation of free-field correlation functions for fermions

Chapter 43: The Path Integral for Fermion Fields

Path integral for fermion fields
Anticommutation of anticommutating functional derivatives
Review of path integral theory, etc.

Chapter 44: Formal Development of fermionic path integrals

Formal derivation of previous chapter's result
Grassmann Numbers, variables, and calculus

Chapter 45: The Feynman Rules for Dirac Fields

Path integrals, Correlation Functions, and Feynman Rules
Yukawa Theory
Examples of Yukawa scattering amplitudes

Chapter 46: Spin Sums

Squaring the scattering amplitude
Spin-averaging and summing

Chapter 47: Gamma Matrix Technology

Lots of gamma matrix identities
Basis for general 4x4 matrix (problem 47.3)

Chapter 48: Spin-Averaged Cross Sections

Averaging and summing over spins
Examples of matrix elements for real processes

Chapter 49: The Feynman rules for Majorana fields

Feynman Rules for Majorana Fields
Examples of SUSY process (problem 49.1)

Chapter 50: Massless Particles and Spinor Helicity

Scattering amplitudes for massless particles of definite helicity
Twistors
Twistor product, twistor notation

Chapter 51: Loop Corrections in Yukawa Theory

Lehmann-Kaellen and 1PI form of the exact propagator for fermions
Loop corrections to the propagator for fermions
Loop corrections to the vertex for fermions
Renormalization factors for fermions
"Bag of tricks" for evaluating diagrams, revisited
Yukawa Interactions

Chapter 52: Beta Functions in Yukawa Theory

Calculation of beta functions and anomalous dimensions, again
Ultraviolet and infrared stable fixed points (problem 52.3)

Chapter 53: Functional Determinants

Meaning and example of functional determinant
Derivation of minus sign for fermion loops in Feynman rules

Chapter 54: Maxwell's Equations

Maxwell's equations in Heaviside-Lorentz units
Gauge invariance
Four-current and field-strength tensor
Lagrangian for electromagnetism

Chapter 55: Electrodynamics in Coulomb Gauge

Gauges in electromagnetism; Coulomb Gauge
Polarization vectors
Field operator in spin-1
Canonical quantization of spin-1 fields
Hamiltonian for Coulombic potentials

Chapter 56: LSZ Reduction for Photons

Path integral for photons
Renormalization of electromagnetic Lagrangian
Propagator for free-field photon
Feynman gauge

Chapter 57: The path integral for photons

Path integral for photons, again
Non-invertible matrices, an example

Chapter 58: Spinor Electrodynamics

Conserved Noether current as electric charge
Electrodynamic Lagrangian
Gauge covariant derivatives
Feynman rules for spinor electrodynamics

Chapter 59: Scattering in Spinor Electrodynamics

Calculation of matrix elements for spinors and photons
Completeness relation for polarization vectors
Electron-positron annihilation

Chapter 60: Spinor helicity in Spinor Electrodynamics

Spinor electrodynamics for massless particles
Twistor products, again

Chapter 61: Scalar Electrodynamics

Calculation of matrix elements for scalars and photons
Global and local symmetries

Chapter 62: Loop Corrections in Spinor Electrodynamics

R-xi gauge
Symmetric integration, again
Example of loop corrections with spinors and photons

Chapter 63: The Vertex Function in Spinor Electrodynamics

Normalization condition for vertex functions
Form functions

Chapter 64: The Magnetic Moment of the Electron

Magnetic Moment of the Electron
Lande g-factor for electron
Electron Spin Resonance (slides)

Chapter 65: Loop Corrections in Scalar Electrodynamics

One-loop corrections to scalar-photon interactions
Example of gauge invariance (problem 65.2)

Chapter 66: Beta Functions in Quantum Electrodynamics

Beta functions and anomalous dimensions for scalar and spinor electrodynamics
Modified decoupling subtraction renomalization scheme (problem 66.4)
The corrected coupling constant for electromagnetism (problem 66.4)

Chapter 67: Ward Identities in Quantum Electrodynamics I

Ward Identity, revisited
Contact Terms
Schwinger-Dyson Equations, revisited
Transverse polarization of photons (slides)

Chapter 68: Ward Identities in Quantum Electrodynamics II

Ward Identity, revisited
Relationships between renormalization factors in quantum electrodynamics

Chapter 69: Nonabelian Gauge Theory

Quantum Chromodynamics: particles and fields
Transfromation of nonabelian (Yang-Mills) Gauge Theories
Langrians in QCD
Nonabelian field strength tensor
Representation-independent results

Chapter 70: Group Representations

Generator Matrices in SU(3), SU(2), etc.
Fundamental and adjoint representations
Index, quadratic Casimir of a group
Invariants of a group, singlets
Anomaly coefficients

Chapter 71: The Path Integral for Nonabelian Gauge Theory

Fadeev-Popov Theory, "Ghosts"
Gauge fixing term
Application of Grassmann variables, functional determinants

Chapter 72: The Feynman Rules for non-Abelian Gauge Theory

Vertex factors and propagators for non-Abelian theories

Representation Theory: Very Basic Notes

Other QFT Textbooks
I am familiar with Zee's book as well as Peskin and Schroeder. I personally find Peskin-Schroeder
to be terrible and Zee to be fairly good, but if I were to teach this course, I would not use either;
I would use only Srednicki and Klauber. There are several other books too, but I am not familiar with
them. My understanding is that almost all books use phi^4 theory as an example, which Srednicki covers
only in the problems (Srednicki uses phi^3).

I've also seen Ramond's book; I do not pretend to be familiar with it, but one of my professors
used it to present some simple applications to condensed matter; you may find it useful for that purpose.

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