BRST Symmetry and Unitarity

The Faddeev–Popov procedure gauge-fixes Yang–Mills theory and introduces ghosts, but the gauge-fixed Lagrangian is no longer gauge-invariant — the original symmetry that guaranteed consistency is gone. Remarkably, a residual global fermionic symmetry survives, mixing gauge fields and ghosts: BRST symmetry (Becchi–Rouet–Stora–Tyutin). It is the organizing principle that guarantees gauge independence, unitarity, and renormalizability of non-abelian gauge theories to all orders.

Conventions: $\hslash =c=1$, ${\eta }_{\mu \nu }=\mathrm{diag}(+1,-1,-1,-1)$.

The BRST transformation

The gauge-fixed Lagrangian $\mathcal{L}={\mathcal{L}}_{\text{YM}}+{\mathcal{L}}_{\text{gf}}+{\mathcal{L}}_{\text{ghost}}$ is invariant under a global transformation with a single anticommuting constant parameter $\theta$, in which the ghost $c^a$ plays the role of the gauge parameter:

$$\boxed{\begin{array}{rlrl}\delta A_{\mu }^a& =\theta (D_{\mu }c{)}^a,& \delta c^a& =-\frac{1}{2}\theta g{f}^{abc}{c}^b{c}^c,\\ \delta {\bar{c}}^a& =\theta B^a,& \delta B^a& =0,\end{array}}$$

with $B^a$ the (bosonic) Nakanishi–Lautrup auxiliary field enforcing the gauge condition. The gauge-field variation is just a gauge transformation with parameter $c^a$ — so BRST is "gauge invariance with the parameter promoted to the ghost."

Nilpotency

The defining property of the BRST operator $s$ (defined by $\delta X=\theta sX$) is that it is nilpotent:

$$\boxed{s^2=0.}$$

This follows from the Jacobi identity for the structure constants and is the algebraic heart of the formalism. Nilpotency has a cohomological consequence that pins down the physical states.

The physical Hilbert space as BRST cohomology

Nilpotency $s^2=0$ means states split by the action of the conserved BRST charge $Q_B$ (with $Q_B^2=0$). Define physical states as those annihilated by $Q_B$, modulo those that are $Q_B$ of something:

$${\mathcal{H}}_{\text{phys}}=\frac{\ker Q_B}{\mathrm{im}{Q}_B}(\text{BRST cohomology}).$$

The mechanism, unpacked:

• States with the wrong sign of norm (timelike/longitudinal gluons, ghosts, antighosts) are either not BRST-closed (excluded from $\ker Q_B$) or are BRST-exact (quotiented out as zero-norm).
• What survives in the cohomology are exactly the two transverse physical polarizations of each gauge boson — the same physical content as the Gupta–Bleuler condition, now valid non-abelian and to all orders.

This is the rigorous, all-orders replacement for the case-by-case ghost cancellations of Faddeev–Popov: unphysical states pair into BRST doublets that decouple from all physical amplitudes, guaranteeing unitarity of the S-matrix on ${\mathcal{H}}_{\text{phys}}$.

Slavnov–Taylor identities

BRST invariance of the generating functional produces exact relations among Green's functions — the Slavnov–Taylor identities, the non-abelian generalization of the Ward–Takahashi identities. They are the functional statement $s\Gamma =0$ on the effective action (in the Zinn-Justin form). Their consequences:

• Gauge independence of physical S-matrix elements (the $\xi$-dependence cancels).
• Constraints on renormalization that force the many renormalization constants of Yang–Mills (gluon, ghost, quark, and each vertex) to be related, leaving a single renormalized coupling — the non-abelian analogue of $Z_1=Z_2$.
• The backbone of 't Hooft's proof that non-abelian gauge theories (and the Higgs-broken ones) are renormalizable, which made the electroweak theory a credible quantum theory and earned the 1999 Nobel Prize.

Why BRST matters

BRST turns the ad hoc ghost bookkeeping into a symmetry principle, and is the modern foundation for quantizing any theory with a gauge redundancy — including gravity and string theory.

Summary

• BRST symmetry is a global fermionic symmetry of the gauge-fixed action, with the ghost as the gauge parameter.
• Its charge is nilpotent, $Q_B^2=0$; physical states are the BRST cohomology.
• Unphysical modes decouple ⇒ unitarity; Slavnov–Taylor identities give gauge independence and renormalizability.

Where this leads

• The self-coupling's UV payoff: asymptotic freedom.
• Renormalizable broken gauge theory: the Higgs mechanism.
• The abelian ancestor of these identities: Ward–Takahashi identities.

References

• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 16.4.
• Becchi, Rouet & Stora, Ann. Phys. 98, 287 (1976).
• Weinberg, The Quantum Theory of Fields, Vol. 2, Ch. 15.7.
• Srednicki, Quantum Field Theory, Ch. 74.