**QFT: Supplement to Srednicki**

A. George

These are solutions to the first ~2/3 of Srednicki’s QFT textbook.I recommend Robert Klauber's book as a supplement to Srednicki.

In your terminal (Mac or Linux) type this command:

for ((i = 1; i < 98; i++)); do wget http://hep.ucsb.edu/people/cag/qft/QFT_Notes_${i}.pdf;

wget http://hep.ucsb.edu/people/cag/qft/QFT-${i}.pdf; done

If it complains that wget is not installed, find instructions here.

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.

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

- The Lorentz Group
- Propriety and orthochronaity
- Unitary operators
- Generators

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

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

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

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

- The path integral for the harmonic oscillator
- Green’s functions

- The path integral for the free field
- The Feynman Propagator

- 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

- 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

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

- Mass dimensionality of various quantities
- Preference for dimensionless coupling constants

- Derivation of an explicit form for the propagator
- Spectral density

- 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

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

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

- Calculation and convergence of Vn

- Mass dimensionality and renormalizability
- Superficial degree of divergence

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

- 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)

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

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

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

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

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

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

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

- General derivation of beta function
- General derivation of anomalous dimensions

- 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

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

- Loop corrections to broken symmetries

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

- Tensor transformation under the Lorentz Group
- Representations of the Lorentz Group

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

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

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

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

- Explicit forms of spinors
- Identities involving spinors
- Spin sums
- Helicity

- 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)

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

- The LSZ Formula for fermions
- Normalization of the field

- Derivation of free-field correlation functions for fermions

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

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

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

- Squaring the scattering amplitude
- Spin-averaging and summing

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

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

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

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

- 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

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

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

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

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

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

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

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

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

- Spinor electrodynamics for massless particles
- Twistor products, again

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

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

- Normalization condition for vertex functions
- Form functions

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

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

- 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)

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

- Ward Identity, revisited
- Relationships between renormalization factors in quantum electrodynamics

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

- 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

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

- Vertex factors and propagators for non-Abelian theories

Representation Theory: Very Basic Notes

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.