Notes
In this section, I post a collection of texts on miscellaneous topics that I have written over the years since I was a graduate student. While the early texts are mostly straightforward compilations of useful formulas, the more recent ones offer introductions to a range of topics. All these texts were written just for my own record, any use is at your own risk! In each category, the files are ordered chronologically, that is, the most recent additions are at the top.
Pure mathematics background
(pdf) Expansion of Riemannian metric in geodesic normal coordinates (explicit expression up to the fourth order).
(pdf) Introduction to the basic method for finding continuous symmetries of differential equations.
(pdf) Pedestrian introduction to the basic concepts of differential geometry (differential and integral calculus on manifolds).
(pdf) Introduction to the theory of orthogonal polynomials.
(pdf) Some basic notes on differential geometry of surfaces in Euclidean space.
(pdf) Derivation of the Haar measure for unitary groups.
Mathematical methods in quantum field/many-body theory
(pdf) Overview of technical details of the coset construction of effective Lagrangians for internal symmetries.
(pdf) Tutorial on advanced quantum field theory: calculation of the two-loop self-energy in a scalar field theory.
(pdf) Asymptotic expansion of singular integrals; includes expansion of selected multiloop Feynman diagrams in spacetime dimension, and expansion of loop integrals in a mass or gap parameter.
(pdf) Introductory notes on the Källén-Lehmann spectral representation of the propagator.
(pdf) How to calculate traces of differential operators: method of (covariant) symbols.
(pdf) Details of construction of the low-energy effective Lagrangian for ferromagnets.
(pdf) Notes on the construction of effective Lagrangians for Nambu-Goldstone bosons in nonrelativistic systems.
(pdf) Some more advanced loop integrals as well as finite-temperature sum-integrals using dimensional regularization. Only integrals that I checked and/or used myself are included; much more complete lists are available in literature.
(pdf) Algebra of block matrices and the basic formulas of the Nambu-Gorkov formalism; derivation of the Grassmann integral on the Nambu space.
(pdf) General theory of Fierz transformations with explicit expressions for the Fierz coefficients of the Lorentz algebra in both the particle-antiparticle and the particle-particle channel (including Lorentz-violating, rotationally-covariant fermion bilinears).
(pdf) Loop integrals using Feynman parameterization; explicit expressions for the divergent and finite parts of the integrals in dimensional as well as cutoff regularization.
Quantum many-body physics
(pdf) Notes on the LOFF phase in imbalanced superconductors.
(pdf) Details on fitting parameters in the two-flavor NJL model in various regularization schemes. (Just for practitioners; if you don't know what NJL model is, ignore this.)
(pdf) Notes on the BCS theory and a derivation of the Ginzburg-Landau functional.
Miscellaneous
(pdf) Mechanics of systems whose configuration space is a Lie group. Examples illustrating the general theory include rotations of a rigid body and the hydrodynamics of ideal incompressible fluids.
(pdf) List of useful formulas for manipulation of matrix exponentials.
(arXiv) Discussion of aspects of classical relativistic field theory with nonzero chemical potential.
(arXiv) Overview of aspects of Noether theorem in classical field theory.
(pdf) Cross-sections and decay rates; basic formulas.
(pdf) Algebra of Dirac gamma matrices; most standard and some less standard formulas.