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Topic 18 of 22

Quantizing Gauge Theories

Quantizing non-Abelian gauge theories is subtle: the path integral over gauge fields overcounts physically equivalent configurations related by gauge transformations. Faddeev and Popov showed how to fix a gauge by inserting a delta function and a compensating Jacobian — which introduces anticommuting scalar "ghost" fields. The resulting theory has a residual fermionic symmetry called BRST invariance, which replaces gauge invariance and ensures the theory is unitary and renormalizable.

Key Concepts

  • Gauge orbit: all Aμ related by gauge transformations are physically equivalent — path integral overcounts
  • Faddeev-Popov: insert 1 = ∫Dα δ(f[A^α]) det(δf/δα), fix gauge with gauge condition f[A]=0
  • Ghost Lagrangian: ℒ_ghost = c̄^a(−∂μDμ)^{ab}c^b from Faddeev-Popov determinant
  • Ghost fields c^a, c̄^a are Grassmann scalars (anticommuting, spin 0) — they violate spin-statistics by design
  • BRST: residual fermionic symmetry sA^a_μ = Dμc^a, sc^a = −½gf^{abc}c^bc^c, etc.
  • Physical states: BRST-closed and not BRST-exact (cohomology); ghosts decouple from physical amplitudes

Key Equations

Gauge-fixed Lagrangian
L=LYM−12ξ(∂μAμa)2+cˉa(−∂μDμab)cb\mathcal{L}=\mathcal{L}_{\rm YM}-\frac{1}{2\xi}(\partial^\mu A^a_\mu)^2+\bar{c}^a(-\partial^\mu D_\mu^{ab})c^b
Ghost propagator
⟨ca(x) cˉb(y)⟩=δabΔF(x−y)\langle c^a(x)\,\bar c^b(y)\rangle=\delta^{ab}\Delta_F(x-y)
BRST variation
sAμa=Dμabcb,sca=−g2fabccbcc,scˉa=BasA^a_\mu=D^{ab}_\mu c^b,\quad sc^a=-\tfrac{g}{2}f^{abc}c^bc^c,\quad s\bar c^a=B^a
BRST nilpotency
s2=0  ⇒  physical states=ker⁡s/im ss^2=0\;\Rightarrow\;\text{physical states}=\ker s/\mathrm{im}\,s
Worked Example

Example Problem

Problem

Why do ghost fields not appear in external states despite being part of the Lagrangian?

Solution

Physical states are defined by the BRST condition: Q_BRST|phys⟩ = 0. Ghost fields c^a have ghost number +1 and c̄^a have −1; physical states must have ghost number 0. So while ghosts propagate in loops (canceling unphysical gauge boson polarizations), they never appear as incoming or outgoing particles in physical scattering amplitudes.

Key Takeaways

  • The Faddeev-Popov procedure fixes gauge redundancy by restricting the path integral to one field configuration per gauge orbit
  • Ghost fields c^a, c̄^a are anticommuting scalars whose loop contributions cancel unphysical longitudinal/temporal gauge boson polarizations
  • BRST symmetry s is the quantum-level replacement for gauge invariance: s² = 0 (nilpotent) and physical states form the BRST cohomology
  • In QED, ghosts decouple (Abelian → f^{abc}=0 → free ghosts); in QCD they are essential at loop level for unitarity
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