PU-402 · Lagrangian Field Theory
Starting from the field action and the variational principle, the unit builds classical field theory as a machine that turns Lagrangian densities and their symmetries into equations of motion and conserved currents, unifying scalar, spinor, and gauge fields under Noether's theorem. It then canonically quantizes these fields — deriving particles, propagators, and the spin-statistics link — and closes with spontaneous symmetry breaking and the Higgs mechanism, the templates on which the Standard Model is built.
Lectures
| L01 | From Particles to Fields: The Continuum Limit — |
| L02 | The Field Action and Functional Derivatives — |
| L03 | Euler-Lagrange Equations for Fields |
| L04 | The Klein-Gordon Field and Relativistic Dispersion |
| L05 | Symmetries of the Action and Noether's First Theorem |
| L06 | Spacetime Translations and the Stress-Energy Tensor |
| L07 | Lorentz Invariance and the Belinfante Improvement |
| L08 | Internal Symmetries and Conserved Charges |
| L09 | The Lorentz Group and Its Representations |
| L10 | Spinors and the Dirac Lagrangian |
| L11 | Electromagnetism as a Field Theory |
| L12 | Local Gauge Invariance and Minimal Coupling |
| L13 | Massive Vector Fields: The Proca Theory |
| L14 | Hamiltonian Formulation of Classical Fields |
| L15 | Canonical Quantization: Fields Become Operators |
| L16 | Mode Expansion and the Particle Interpretation |
| L17 | The Vacuum, Fock Space, and Normal Ordering |
| L18 | The Feynman Propagator |
| L19 | Charges as Generators of Symmetry |
| L20 | Microcausality and the Spin-Statistics Theorem |
| L21 | Spontaneous Symmetry Breaking |
| L22 | Goldstone's Theorem and Massless Modes |
| L23 | The Abelian Higgs Mechanism |
| L24 | Interacting Fields and Perturbation Theory — |
| L25 | Outlook: Toward Non-Abelian Gauge Theory and the Standard Model — |
Derivations homed in this unit
Euler-Lagrange Equations for Fields
Stationarity of the action functional S=∫ℒ(φ,∂μφ)d⁴x under δφ vanishing on the boundary yields ∂μ(∂ℒ/∂(∂μφ)) − ∂ℒ/∂φ = 0.
Noether's First Theorem
Every continuous symmetry of the action that leaves the equations of motion invariant yields a locally conserved current ∂μj^μ=0 and a conserved charge.
Canonical Stress-Energy Tensor
Applying Noether's theorem to spacetime translation invariance gives the canonical energy-momentum tensor T^μν whose conservation encodes energy and momentum of the field.
Belinfante-Rosenfeld Improvement
Adding a superpotential fixed by Lorentz (rotation-boost) invariance improves the canonical tensor into the symmetric, gauge-invariant Belinfante tensor that couples to gravity.
Klein-Gordon Field from Its Lagrangian
The real scalar Lagrangian ½(∂μφ)²−½m²φ² yields the Klein-Gordon equation (□+m²)φ=0 and the relativistic dispersion ω²=k²+m².
Representations of the Lorentz Group
Decomposing so(1,3)≅su(2)⊕su(2) classifies fields as (j₁,j₂) representations, delivering scalars, Weyl/Dirac spinors, and vectors as the building blocks of Lorentz-covariant Lagrangians.
Dirac Equation from the Spinor Lagrangian
The Lagrangian ψ̄(iγ^μ∂μ−m)ψ with the Clifford algebra {γ^μ,γ^ν}=2η^μν gives the Dirac equation and a positive-definite conserved probability current.
Maxwell's Equations from −¼F²
Varying ℒ=−¼F_μνF^μν−j^μA_μ recovers the inhomogeneous Maxwell equations, while the homogeneous pair follows identically from F=dA (the Bianchi identity).
Local U(1) Gauge Invariance and Minimal Coupling
Demanding invariance under local phase rotations forces the covariant derivative D_μ=∂_μ−ieA_μ, generates minimal coupling, and ties current conservation to gauge symmetry.
Proca Field and Its Degrees of Freedom
Adding ½m²A_μA^μ breaks gauge invariance, forces ∂_μA^μ=0 as a constraint, and leaves a massive vector with exactly three physical polarizations.
Hamiltonian Formulation for Fields
The Legendre transform π=∂ℒ/∂φ̇ defines the conjugate momentum and Hamiltonian density ℒ, giving the field-theoretic Poisson-bracket dynamics.
Canonical Quantization of a Field
Promoting fields to operators and Poisson brackets to commutators imposes equal-time [φ(x),π(y)]=iδ³(x−y), turning a classical field into a quantum operator field.
Mode Expansion and Fock Space
Expanding the quantized Klein-Gordon field in plane waves yields creation/annihilation operators whose algebra builds a Fock space of relativistic particle states.
The Feynman Propagator
The time-ordered vacuum two-point function ⟨0|Tφ(x)φ(y)|0⟩ is a Green's function of the Klein-Gordon operator, fixed by the iε contour prescription.
Conserved Charge as Symmetry Generator
In the quantized theory the Noether charge Q satisfies [Q,φ]=−iδφ, so the conserved charge is precisely the generator of the symmetry transformation.
The Spin-Statistics Theorem
Microcausality (vanishing of field commutators at spacelike separation) and positivity of energy force integer-spin fields to be quantized with commutators and half-integer with anticommutators.
Goldstone's Theorem
Spontaneous breaking of a continuous global symmetry produces one massless scalar (Goldstone) boson per broken generator, seen as flat directions of the potential.
The Abelian Higgs Mechanism
Gauging a spontaneously broken U(1) lets the would-be Goldstone boson become the longitudinal mode of the gauge field, giving it a mass while conserving total degrees of freedom.