Part I: Classical Field Theory
Classical field theory provides the foundation for quantum field theory. We begin with the Lagrangian formalism for fields, derive conservation laws via Noether's theorem, and study the classical dynamics of scalar, electromagnetic, and Dirac fields.
Part Overview
Before quantization, we must understand classical field theory. Fields are functions of spacetime Ο(xΞΌ) with infinitely many degrees of freedom. The Lagrangian formalism extends from discrete mechanics to continuous systems, and Noether's theorem connects symmetries to conserved currents.
Key Topics
- β’ Transition from discrete to continuous systems
- β’ Lagrangian density and Euler-Lagrange equations for fields
- β’ Noether's theorem: symmetries β conserved currents
- β’ Real and complex scalar fields (Klein-Gordon equation)
- β’ Electromagnetic field and gauge invariance
- β’ Dirac field and spinor representations
- β’ Energy-momentum tensor and conserved charges
50+ pages | 7 chapters | Foundation for all QFT
Chapters
Chapter 1: Lagrangian Field Theory
From particle mechanics to field theory. Lagrangian density, action principle, Euler-Lagrange equations for fields, Hamilton's equations, and canonical quantization.
Chapter 2: Noether's Theorem
Continuous symmetries and conserved currents. Spacetime translations (energy-momentum), Lorentz transformations (angular momentum), internal symmetries, and the Noether procedure.
Chapter 3: Real & Complex Scalar Fields
Klein-Gordon equation, plane wave solutions, conserved currents for complex scalars, U(1) symmetry, and interpretation of field equations.
Chapter 4: Electromagnetic Field
Maxwell's equations, gauge invariance, field strength tensor, Lagrangian for electromagnetism, coupling to charged fields, and gauge fixing.
Chapter 5: Dirac Field
Dirac equation, gamma matrices, spinor representations, Lorentz covariance, plane wave solutions, Dirac Lagrangian, and conserved current.
Chapter 6: Internal & Spacetime Symmetries
PoincarΓ© group, internal symmetries (U(1), SU(N)), discrete symmetries (C, P, T), CPT theorem, and Coleman-Mandula theorem.
Chapter 7: Energy-Momentum Tensor
Canonical and symmetric energy-momentum tensors, Belinfante-Rosenfeld procedure, conservation laws, and applications to scalar, electromagnetic, and Dirac fields.
Prerequisites
Required Background
- β’ Classical mechanics (Lagrangian/Hamiltonian)
- β’ Special relativity and 4-vector notation
- β’ Variational calculus
- β’ Linear algebra and group theory basics
Helpful but Not Required
- β’ Quantum mechanics
- β’ Electromagnetism (Maxwell theory)
- β’ Differential geometry
- β’ Representation theory