Notes

lattice

Chiral Lagrangian in QCD, spurions, lattice discretizations of QCD, Ginsparg-Wilson, chiral Lagrangians in LQCD, mixed actions. These notes were originally written as a…

Colliders

part3

Collider observables, important processes, muon decay and semi-leptonic decays from 4-Fermi theory, and a cheat sheet. These notes were originally written when I was…

Hadron Structure

Deep inelastic scattering, structure functions, parton model, sum rules, parton distribution functions and DGLAP, the Skyrme model, LQCD approaches.

Instantons

part3

Instantons in QM and QFT, homotopy, solitons, fiber bundles and characteristic classes, strong CP problem, U(1) problem. These notes were originally written when I was…

Local Operator Renormalization

lattice

Renormalization of local composite operators, mixing between operators, mixing on the lattice. These notes were originally written as a final project for MIT’s 8.325…

Local Operator Renormalization on the Lattice

lattice

Lattice specific variant of my local operator renormalization notes (much less polished, as it is not a class project). RI/MOM and variants, quark EMT, momentum sources…

Representation Theory

rep-theory

Lie groups and algebras, exponential map, representation of Lie groups, irreps, Schur’s lemma, combining irreps, addition of angular momentum, representations of SU(3)…

The Lorentz and Poincaré Groups

rep-theory

The Lorentz algebra and its finite-dimensional representations, Dirac and Majorana spinors, spinor index conventions, and discrete symmetries.

The Standard Model

part3

Introduction to the Standard Model of Particle Physics. Particles and gauge groups, spontaneous symmetry breaking and the Higgs boson, Goldstone equivalence, electroweak…

Math

Differential Forms

Smooth manifolds, tangent bundle, cotangent bundle, differential forms, sections, wedge products, exterior derivative, Lie derivative, interior products, integration of…

Galois Theory

algebra

Galois extensions, roots of unity, cyclic extensions, Galois groups, Galois correspondence, Galois cohomology. These notes are currently my recap notes from Math 250A at UC…

Group Theory

algebra

Groups, subgroups, homomorphisms, symmetric and dihedral groups, isomorphism theorems, quotients, actions,

Homotopy

alg-top

Homotopy, homotopy equivalence, fundamental group, calculations, Brower fixed point theorem. These notes are based on Math 142 at UC Berkeley, taught by Dr. Jamie Conway.

Linear Algebra

algebra

Vector spaces, linear maps, isomorphisms, quotient spaces, rank-nullity theorem.

Measure Theory

Measures and their construction, measurable functions, integration from simple functions through the convergence theorems, and an introduction to normed vector spaces…

Metric Spaces

topology

Metrics and metric spaces, open and closed sets, limits, continuity, sequences, isometry, completion, p-norms.

Point Set Topology

topology

Introduction to topology. Convergence, continuity, base, quotients, products, compactness, Bolzano-Weierstrauss, Heine-Borel, Tychonoff’s Theorem and the axiom of choice…

Probability

Probability from a measure theoretic point of view. Measure theory, sigma algebras, premeasures and construction of measures, Lebesgue measure, independence of events…

Set Theory

Sets, quantifiers, number systems, maps, equivalence relations, partitions, cosets, countability, construction of number systems, (for the future) ordinals, cardinals.

Simplicial Homology

alg-top

Simplices, chain complexes, chain maps, simplicial homology, some category theory, homotopy and its relation to simplicial homology, chain homotopy, exact sequences, cross…

The Moment Problem

Moment problem and its formulation as a functional, Riesz functional, Jacobi operator, semidefinite programming.

Topological Manifolds

alg-top

Topological manifolds, classification of 1- and 2-manifolds, free producs and abelianization, connect sums. These notes are based on Math 142 at UC Berkeley, taught by Dr.…

QFT I (8.323) Recitation Notes

When I was a graduate student at MIT, I TA’d for Relativistic Quantum Field Theory I (8.323) two years in a row. I got a lot of freedom in shaping the recitations how I wanted to, and doing it two years in a row let me iterate on my recitations and improve them. QFT requires an immense about of mathematical machinery; so much so that the first semester’s course is usually just raw calculation and hard to remember that you’re in a physics class. I tried to make the recitations complement the lectures and also give the students a taste of the physics to come in QFT II and III. It was a lot of fun, and definitely one of the most rewarding experiences of my graduate school career.

Recitation 1: Relativity and Fields

QFT resources, natural units, index notation, the Lorentz group, classical field theory.

Recitation 2: Canonical Quantization

Heisenberg and Schrodinger pictures, quantum vs classical mechanics, canonical quantization.

Recitation 3: Symmetries

Symmetries, Lie groups, generators, conserved quantities, and complex fields. I left out 2 pages of handwritten notes on contour integration because of the file size; email…

Recitation 4: The Path Integral and Functionals

Path integral in QM, stationary phase, discrete vs continuous degrees of freedom, functional calculus, operators, Gaussian integrals and Wick contractions.

Recitation 5: Correlation Functions

Correlation functions, generating functionals, Wick’s theorem again, free theories, the vacuum state, Wick rotation, anecdotes from my research, scattering overview.

Recitation 6: Perturbation Theory

Green’s functions, perturbation theory from Hamiltonian, Feynman diagrams, example calculation in phi^3.

Recitation 7: Effective Field Theory

Momentum space Feynman rules, introduction to effective field theory, matching to EFTs, integrating out degrees of freedom, examples from electroweak theory (4-Fermi) and…

Recitation 8: Dirac Theory

Gamma matrices and the Dirac algebra, Dirac vs Weyl representations, generators and an example, representation theory of the Dirac group, irreps of SU(2), representations of…

Recitation 9: Chirality and Propagators

Gamma5 and chirality, vector and axial transformations, the chiral anomaly, spinor index notation, propagators.

Recitation 10: Grassmann Numbers and Electroweak Theory

Grassmann numbers, Grassmann integral, fermion path integral, exterior algebra, chirality in electroweak theory, toy model of the Higgs boson with global symmetries.

Recitation 11: The Geometry of Gauge Invariance