Gauge Fixing, BRST, and BV

Gauge fixing chooses a workable description of redundant variables; BRST and BV encode the associated identities in graded algebra. This chapter develops those constructions locally and perturbatively while keeping global copies, anomalies, and the physical interpretation of cohomology distinct.

Enter this chapter

Begin with The Faddeev–Popov Construction and its finite-dimensional orbit model before introducing ghosts.

Helpful background. Regulated Bosonic Field Integrals and Chains, Homology, Cohomology, and Exact Sequences supply regulated integration and complexes modulo exact terms.

Topics

1. The Faddeev–Popov Construction — Derives the determinant associated with a locally transverse gauge slice.
2. Ghosts, Auxiliary Fields, and Gauge-Parameter Dependence — Represents determinants with Grassmann variables and tracks gauge parameters.
3. Gribov Copies and the Limits of Local Gauge Fixing — Explains why local transversality does not imply a global unique slice.
4. The BRST Differential and Gauge-Fixed Complex — Encodes infinitesimal gauge transformations in a nilpotent graded differential.
5. BRST Cohomology and Physical Observables — Tests when closed quantities modulo exact ones represent physical information.
6. Slavnov–Taylor and Zinn-Justin Identities — Derives identities for nonlinear BRST variations and their sources.
7. BV Fields, Antifields, and the Odd Symplectic Structure — Introduces antifields and the odd symplectic pairing.
8. Master Equations and BV Gauge Fixing — Separates classical master equations from regulated quantum consistency.

How the ideas fit together

A nonzero Faddeev–Popov determinant tests one local intersection with a gauge orbit; it does not exclude another intersection elsewhere. BRST instead uses a nilpotent differential to organize infinitesimal redundancy, and BV extends the construction to more general closure or reducibility structures. See Fuster, Henneaux, and Maas 2005, §§ 2–5.

The BRST complex and its limits distinguish algebraic nilpotency from global gauge fixing and quantum anomaly freedom. A cohomology calculation becomes a claim about physical states only after its domain, grading, and relevant positivity or quotient conditions have been supplied.

Review the chapter

Why is nilpotency necessary before defining BRST cohomology as closed quantities modulo exact ones?

Solution

If $s^2=0$, every exact quantity $sX$ is closed because $s(sX)=0$. Thus the image is contained in the kernel and the quotient is defined. Without that containment, identifying closed representatives modulo arbitrary exact terms is not a cohomology construction.

Where to go next

Continue to Anomalies, Inflow, and Matching to test whether quantum effects obstruct the identities used here.

References

• Andrea Fuster, Marc Henneaux, and Axel Maas, “BRST-Antifield Quantization: A Short Review,” International Journal of Geometric Methods in Modern Physics 2 (2005), 939–964. DOI. Open PDF.