Symmetries & Conserved Currents
Symmetry is the organizing principle of all modern physics. Noether's theorem guarantees that every continuous symmetry of the action produces a conserved current and a conserved charge. Promoted to the quantum level, these conservation laws become Ward-Takahashi identities — exact operator relations that hold to all orders in perturbation theory, constrain radiative corrections, and are the backbone of the proof that gauge theories are renormalizable.
Key Concepts
- Noether current: jμ = (∂ℒ/∂(∂μφᵢ)) δφᵢ — conserved when equations of motion hold
- Conserved charge: Q = ∫d³x j⁰(x), dQ/dt = 0, generates the symmetry: [Q,φ] = iδφ
- U(1) vector current: jμ = ψ̄γμψ, Q = electron number (lepton number conservation)
- Ward identity (QED): kμMμ = 0 — longitudinal photons decouple from physical amplitudes
- Ward-Takahashi identity: kμΓμ(p,p+k) = S_F⁻¹(p+k) − S_F⁻¹(p) — exact at all orders
- Non-Abelian current algebra: [Q^a, Q^b] = if^{abc}Q^c — Lie algebra of the symmetry group
Key Equations
Example Problem
The Dirac Lagrangian is invariant under ψ → e^{iα}ψ. Derive the Noether current and verify it is conserved on-shell.
δψ = iαψ for global U(1). jμ = (∂ℒ/∂(∂μψ̄))·0 + (∂ℒ/∂(∂μψ))·(iψ) = (iψ̄γμ)(iψ) = −ψ̄γμψ → with sign convention jμ = ψ̄γμψ. Conservation: ∂μjμ = (∂μψ̄)γμψ + ψ̄γμ∂μψ. On-shell: iγμ∂μψ = mψ and i(∂μψ̄)γμ = −mψ̄, so ∂μjμ = −im·ψ̄ψ + im·ψ̄ψ = 0. ✓
Key Takeaways
- Noether's theorem: every differentiable symmetry of the action → a conserved current jμ with ∂μjμ = 0 and a time-independent charge Q
- The Ward identity kμMμ = 0 (QED) follows from current conservation and ensures only transverse photon polarizations contribute to S-matrix elements
- The Ward-Takahashi identity kμΓμ = S⁻¹_F(p+k) − S⁻¹_F(p) is exact at all loop orders — it relates vertex corrections and self-energies
- Non-Abelian current algebra [Q^a, Q^b] = if^{abc}Q^c encodes the group structure and constrains soft-particle emission amplitudes