Part V: Gauge Field Theories • Chapter 5

Gauge Fixing & Faddeev-Popov Ghosts

Path integral quantization of non-Abelian gauge theories

📺 Video Lectures

For video lectures on Faddeev-Popov quantization and ghost fields, see:

• Tobias Osborne QFT 2016:YouTube Playlist
Path integral approach to gauge fixing with detailed ghost field derivation
• David Tong QFT Lectures:Perimeter Institute (PIRSA)
Covers Faddeev-Popov procedure and BRST symmetry
• David Tong Gauge Theory Notes:Lecture Notes (Cambridge)
Chapter 2: Quantization covers ghost fields in detail

The Gauge Redundancy Problem

In path integral quantization, we sum over all field configurations:

Z = ∫ 𝒟A_μ e^iS[A]

Problem: Gauge-equivalent configurations A_μ and A'_μ = A_μ + D_μα give the same physics but are integrated over separately, leading to an infinite overcounting.

The integral diverges!

We must eliminate gauge redundancy by restricting the integral to one representative from each gauge orbit.

The Faddeev-Popov Procedure

Step 1: Insert clever identity

Insert 1 = Δ_FP[A] ∫𝒟α δ(G[A^α]) into the path integral, where:

• G[A] = 0 is the gauge-fixing condition
• A^α is A gauge-transformed by α
• Δ_FP is the Faddeev-Popov determinant

Step 2: Faddeev-Popov determinant

Δ_FP[A] = det(δG^a[A^α]/δα^b)

This determinant compensates for the volume of the gauge orbit.

Ghost Fields

The Faddeev-Popov determinant can be exponentiated using ghost fields c and c̄:

det(M) = ∫ 𝒟c̄ 𝒟c e^{i∫c̄^aM^abc^b}

Ghost Properties

• Scalars in spacetime (no Lorentz indices)
• Anticommuting (Grassmann-valued, like fermions)
• Transform in adjoint representation of gauge group
• Violate spin-statistics: spin-0 fermions (unphysical!)
• Only appear in loops (internal lines, never external)
• Cancel unphysical gauge boson polarizations

Ghost Lagrangian (Lorenz Gauge)

For Lorenz gauge ∂_μA^{μ a} = 0, the ghost Lagrangian is:

ℒ_ghost = c̄^a ∂_μ D^{μ ab} c^b

where the covariant derivative in the adjoint representation is:

D^{μ ab} = ∂^μδ^ab + gf^abcA^μc

Ghost-Gluon Vertex

The ghost Lagrangian contains a c̄-A-c vertex proportional to gf^abc. Ghosts couple to gluons but not to matter fields (quarks, leptons).

Why Ghosts are Necessary

Without Ghosts

✗ Loop integrals ill-defined
✗ Unitarity violated
✗ Unphysical gauge boson DOF contribute
✗ Ward identities fail

With Ghosts

✓ Well-defined path integral
✓ Unitarity preserved
✓ Unphysical DOF cancel
✓ Ward identities satisfied

Physical Interpretation

Ghosts are purely mathematical artifacts that ensure correct counting of degrees of freedom in gauge theories. They never appear as external particles (no ghost detectors!) but are essential for consistent quantum calculations.

BRST Symmetry

The gauge-fixed Yang-Mills + ghost system possesses a new fermionic symmetry called BRST symmetry(Becchi-Rouet-Stora-Tyutin), generated by a nilpotent operator Q_BRST with Q² = 0.

Key property: Physical states are those annihilated by Q_BRST. This provides a systematic way to identify gauge-invariant observables and proves unitarity and renormalizability.

BRST symmetry is a cornerstone of modern gauge theory and essential for understanding anomalies, cohomology, and string field theory.

Summary

✓ Gauge redundancy: Path integral overcounts gauge-equivalent configurations
✓ Faddeev-Popov procedure: Insert determinant to fix gauge, exponentiate using ghosts
✓ Ghost fields: Anticommuting scalars in adjoint representation, violate spin-statistics
✓ Ghost Lagrangian: ℒ_ghost = c̄^a ∂_μD^μabc^b
✓ Physical role: Cancel unphysical gauge boson polarizations in loops
✓ BRST symmetry: Fermionic symmetry encoding gauge invariance after gauge fixing

Quantum Field Theory Course