Skip to main content

Introduction to Theoretical Physics

In other notes, we have explored many topics in physics, from classical mechanics, electromagnetism, optics to quantum mechanics, general relativity, and string theory. We now turn to a self-contained introduction to all the theoretical physics you need to know to understand modern research papers in physics. This includes a review of the necessary mathematics, as well as the physical concepts and theories that underpin modern physics.

To understand the latest research in physics, we must have a solid understanding of classical mechanics. This is often a neglected topic in modern physics education, but in fact classical mechanics is the foundation of all physics. The mathematics of classical mechanics is immensely complicated, covering ideas such as fiber bundles, symplectic geometry, Lie groups, and much more. As such, we will include a dedicated section on the mathematical foundations needed for theoretical physics.

Contents​

Let's take a look at the contents of this set of notes.

Mathematical Methods​

This section deals with fundamental mathematical concepts and methods that are essential for understanding theoretical physics. It includes topics such as:

  • Preliminaries:
    • Set theory --- sets, relations, maps, metrics, cardinalities
    • Topological concepts --- open and closed sets, compactness, connectedness
  • Vector spaces:
    • Vector spaces --- definitions, subspaces, basis, span, linear independence
    • Linear maps --- definitions, kernel, image, rank-nullity theorem
    • Linear functionals and dual spaces
    • Inner product spaces --- definitions, properties, orthogonality, Gram-Schmidt process
    • Multilinear algebra --- tensors
    • Algebras over a field
    • Ideals and quotient algebras
    • Derivation and decomposition of algebras
    • Modules over a ring
    • Operator algebra --- endomorphisms, isomorphisms, automorphisms; idempotents, nilpotents, and involutions; matrix representation of linear operators
    • Change of basis --- similarity transformations, eigenvalues and eigenvectors, diagonalization, Jordan form
  • Infinite-dimensional vector spaces:
    • Hilbert spaces --- L2 spaces, orthonormal basis, projection theorem
    • Linear operators on Hilbert spaces --- bounded and unbounded operators, adjoint operators, self-adjoint operators, compact operators
    • Spectral theory --- spectrum of an operator, spectral decomposition, functional calculus
    • Classic orthogonal polynomials --- Hermite, Laguerre, Legendre, Chebyshev polynomials
    • Fourier analysis --- Fourier series, Fourier transform, Plancherel theorem, convolution theorem
    • Generating functions
  • Complex analysis:
    • Complex functions --- holomorphic functions, Cauchy-Riemann equations, harmonic functions
    • Analytic functions, Taylor and Laurent series
    • Conformal mappings
    • Complex integration --- Cauchy's integral theorem, Cauchy's integral formula, residue theorem
    • Meromorphic functions, Gamma/Beta functions, steepest descent method
  • Differential equations:
    • Separation of variables --- first-order ODEs, second-order ODEs, higher-order ODEs; spherical harmonics
    • Second-order linear ODEs --- Wronskian, adjoint operator, power-series solutions, WKB approximation
    • Complex second-order ODEs --- Fuchsian equations, hypergeometric equations
    • Integral transforms --- Laplace transform, Mellin transform
  • Operations on Hilbert spaces:
    • Operator theory --- spectrum of an operator, spectral theorem, functional calculus
    • Integral equations --- Fredholm and Volterra equations
    • Sturm-Liouville theory
  • Green's functions:
    • In one dimension
    • In three dimensions
  • Group theory and representation theory:
    • Group actions --- definitions, orbits, stabilizers, Burnside's lemma
    • Group representations --- definitions, irreducible representations, Schur's lemma, Maschke's theorem, tensor product of representations
    • Characters --- definitions, orthogonality relations, character tables
  • Tensor analysis:
    • Exterior algebras --- differential forms, wedge product, exterior derivative; symplectic vector spaces, hodge duality
    • Clifford algebras
    • Manifolds --- definitions, charts, atlases, smooth manifolds; tangent and cotangent spaces; vector fields and differential forms
  • Lie theory, applications:
    • Representations of Lie algebras --- definitions, universal enveloping algebra, Poincaré-Birkhoff-Witt theorem
    • Spinors --- Clifford algebras, spin groups, spin representations
    • Calculus of variations --- Euler-Lagrange equations, Noether's theorem
    • Classical field theories --- Lagrangian and Hamiltonian formulations, canonical transformations, Poisson brackets
  • Differential geometry --- Riemannian geometry, connections, curvature, geodesics:
    • Fiber bundles --- definitions, sections, connections, curvature; principal bundles, associated bundles
    • Gauge theories --- Yang-Mills theory, Chern-Simons theory
    • Homology and cohomology --- singular homology, de Rham cohomology, Čech cohomology
    • Morse theory
    • Index theorems --- Atiyah-Singer index theorem

Classical Mechanics and Field Theory​

This section covers the fundamental principles and equations of classical mechanics, including:

(todo)

Quantum Field Theory​

This section introduces the concepts and techniques of quantum field theory, including:

(todo)

String Theory​

This section provides an overview of string theory, including:

(todo)