Quantum Field Theory

Quantum Field Theory (QFT) is the theoretical framework that merges the principles of quantum mechanics with Einstein's special relativity, providing the foundation for the Standard Model of particle physics. It posits that the fundamental entities of the universe are not particles but continuous fields that permeate all of spacetime. Particles, such as electrons and photons, are understood as localized excitations or "quanta" of these fields, and their interactions, creations, and annihilations are described by the dynamics of these underlying fields, successfully explaining the electromagnetic, weak, and strong nuclear forces.

1. Foundations and Preliminaries
   1. Mathematical Preliminaries
      1. Complex Numbers and Functions
         1. Complex Arithmetic
         2. Complex Exponentials and Logarithms
         3. Analytic Functions
         4. Residue Theorem
      2. Linear Algebra
         1. Vector Spaces
            1. Definition and Properties
            2. Basis and Dimension
            3. Linear Independence
         2. Inner Products
            1. Euclidean Inner Product
            2. Complex Inner Products
            3. Orthogonality and Orthonormalization
         3. Linear Operators
            1. Matrix Representations
            2. Rank and Nullity
         4. Eigenvalues and Eigenvectors
            1. Characteristic Polynomial
            2. Diagonalization
            3. Spectral Theorem
      3. Differential Equations
         1. Ordinary Differential Equations
            1. First-Order Linear Equations
            2. Second-Order Linear Equations
            3. Series Solutions
         2. Partial Differential Equations
            1. Wave Equation
            2. Heat Equation
            3. Laplace Equation
      4. Fourier Analysis
         1. Fourier Series
            1. Periodic Functions
            2. Convergence Properties
         2. Fourier Transforms
            1. Definition and Properties
            2. Convolution Theorem
            3. Parseval's Theorem
         3. Multidimensional Fourier Transforms
      5. Distributions and Delta Functions
         1. Test Functions
         2. Distribution Theory
         3. Delta Function Properties
         4. Derivatives of Distributions
      6. Group Theory Basics
         1. Group Definition and Examples
         2. Subgroups and Cosets
         3. Homomorphisms and Isomorphisms
         4. Lie Groups Introduction
   2. Review of Classical Mechanics
      1. Lagrangian Formalism
         1. Principle of Least Action
            1. Action Functional
            2. Variational Calculus
         2. Euler-Lagrange Equations
            1. Derivation
            2. Applications to Simple Systems
         3. Constraints and Lagrange Multipliers
            1. Holonomic Constraints
            2. Non-holonomic Constraints
      2. Hamiltonian Formalism
         1. Canonical Variables
            1. Generalized Coordinates and Momenta
            2. Legendre Transform
         2. Hamilton's Equations
            1. Derivation from Lagrangian
            2. Phase Space
         3. Poisson Brackets
            1. Definition and Properties
            2. Canonical Transformations
      3. Symmetries and Noether's Theorem
         1. Continuous Symmetries
            1. Translational Symmetry
            2. Rotational Symmetry
            3. Time Translation Symmetry
         2. Conserved Quantities
            1. Energy Conservation
            2. Momentum Conservation
            3. Angular Momentum Conservation
   3. Special Relativity
      1. Spacetime and Minkowski Metric
         1. Events and Worldlines
            1. Spacetime Coordinates
            2. Light Cones
         2. Metric Signature
            1. Minkowski Metric Tensor
            2. Proper Time
      2. Lorentz Transformations
         1. Boosts and Rotations
            1. Standard Configuration
            2. Velocity Addition Formula
         2. Invariant Intervals
            1. Timelike and Spacelike Intervals
            2. Lightlike Intervals
      3. Four-Vectors and Tensors
         1. Four-Position and Four-Momentum
            1. Contravariant and Covariant Vectors
            2. Four-Velocity and Four-Acceleration
         2. Lorentz Covariance
            1. Tensor Transformation Laws
            2. Scalar Products
         3. Index Notation
            1. Einstein Summation Convention
            2. Raising and Lowering Indices
      4. Relativistic Kinematics
         1. Energy-Momentum Relation
            1. Rest Mass and Relativistic Mass
            2. Massless Particles
         2. Center-of-Mass Frame
            1. Invariant Mass
            2. Laboratory Frame Transformations
         3. Mandelstam Variables
            1. Definition for Two-Body Scattering
            2. Physical Regions
   4. Review of Quantum Mechanics
      1. Hilbert Spaces and Operators
         1. State Vectors
            1. Normalization and Probability Interpretation
            2. Superposition Principle
         2. Observables and Hermitian Operators
            1. Eigenvalue Equations
            2. Measurement Postulates
         3. Commutators
            1. Canonical Commutation Relations
            2. Uncertainty Principle
      2. Pictures of Quantum Mechanics
         1. Schrödinger Picture
            1. Time Evolution of States
            2. Time-Independent Hamiltonians
         2. Heisenberg Picture
            1. Time Evolution of Operators
            2. Heisenberg Equations of Motion
         3. Interaction Picture
            1. Separation of Free and Interaction Parts
            2. Time Evolution Operator
      3. The Harmonic Oscillator
         1. Ladder Operators
            1. Creation and Annihilation Operators
            2. Commutation Relations
         2. Energy Eigenstates
            1. Ground State and Excited States
            2. Coherent States
      4. Symmetries in Quantum Mechanics
         1. Unitary Transformations
            1. Wigner's Theorem
            2. Projective Representations
         2. Conservation Laws
            1. Symmetry and Degeneracy
            2. Selection Rules
   5. Classical Field Theory
      1. The Principle of Least Action for Fields
         1. Action Functional
            1. Lagrangian Density
            2. Integration over Spacetime
         2. Field Configurations
            1. Boundary Conditions
            2. Functional Derivatives
      2. Euler-Lagrange Equations for Fields
         1. Derivation for Scalar Fields
            1. Variational Principle for Fields
            2. Field Equations
         2. Generalization to Multiple Fields
            1. Coupled Field Systems
            2. Vector Fields
      3. Spacetime Symmetries and Conserved Currents
         1. Translation Invariance
            1. Energy-Momentum Conservation
            2. Noether's Theorem for Fields
         2. Rotation and Lorentz Invariance
            1. Angular Momentum Conservation
            2. Lorentz Covariance
      4. The Energy-Momentum Tensor
         1. Definition and Properties
            1. Canonical Energy-Momentum Tensor
            2. Symmetric Energy-Momentum Tensor
         2. Conservation Law
            1. Continuity Equation
            2. Physical Interpretation
      5. Specific Classical Fields
         1. The Klein-Gordon Field
            1. Lagrangian Density
            2. Equation of Motion
            3. Plane Wave Solutions
         2. The Dirac Field
            1. Dirac Matrices
               1. Gamma Matrix Algebra
               2. Clifford Algebra
            2. Spinor Solutions
               1. Weyl Spinors
               2. Majorana Spinors
         3. The Electromagnetic Field
            1. Maxwell Lagrangian
               1. Field Strength Tensor
               2. Dual Field Tensor
            2. Gauge Invariance
               1. Gauge Transformations
               2. Physical vs Unphysical Degrees of Freedom