Gauge Fixing and Faddeev–Popov Ghosts

Quantizing a gauge theory in the path integral runs into an immediate obstruction: the integral $\int \mathcal{D}A{e}^{iS}$ overcounts, integrating over infinitely many physically identical (gauge-equivalent) configurations. For the abelian photon this is handled by simply adding a gauge-fixing term. For non-abelian Yang–Mills the correct procedure introduces new fields — Faddeev–Popov ghosts — whose necessity is one of the subtlest and most important features of gauge theory.

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

The overcounting problem

The gauge symmetry $A_{\mu }\to A_{\mu }^U$ means the action is constant along gauge orbits — the sets of configurations related by gauge transformations. The path integral $\int \mathcal{D}A{e}^{iS[A]}$ integrates over every point of every orbit, so it contains a divergent factor equal to the (infinite) volume of the gauge group,

$$\int \mathcal{D}A{e}^{iS[A]}=\underset{\text{gauge volume}}{\underbrace{\left(\int \mathcal{D}U\right)}}\times {\int }_{\text{one point per orbit}}\mathcal{D}A{e}^{iS[A]}.$$

Worse, without fixing the gauge the quadratic kinetic operator is non-invertible, so no propagator exists (the vector-field problem). The fix: integrate over only one representative per orbit by imposing a gauge condition $G(A)=0$ (e.g. ${\partial }_{\mu }{A}^{\mu }=0$).

The Faddeev–Popov determinant (Standard machinery)

Faddeev and Popov insert a cleverly written "1" into the path integral:

$$1=\int \mathcal{D}\alpha \delta (G(A^{\alpha }))\det \left(\frac{\delta G(A^{\alpha })}{\delta \alpha }\right),$$

where $A^{\alpha }$ is the gauge transform of $A$ by parameter $\alpha$. Substituting and using gauge invariance of the action and measure factors the gauge volume out cleanly, leaving

$$\int \mathcal{D}A{e}^{iS}\delta (G(A))\underset{\text{Faddeev–Popov determinant}}{\underbrace{\det \left(\frac{\delta G(A^{\alpha })}{\delta \alpha }\right)}}.$$

The Faddeev–Popov determinant is the Jacobian of the gauge condition. For an abelian theory it is field-independent and can be dropped (the photon needs no ghosts). For non-abelian theories it depends on $A$ and cannot be ignored — it encodes real dynamics.

Ghosts: exponentiating the determinant

The determinant is put back into the action by writing it as a Gaussian integral over auxiliary fields. Because a determinant (not its inverse) is needed, those fields must be anticommuting — recall $\int \mathcal{D}\bar{c}\mathcal{D}c{e}^{-\bar{c}Mc}=\det M$:

$$\det \left(\frac{\delta G^a}{\delta {\alpha }^b}\right)=\int \mathcal{D}\bar{c}\mathcal{D}c\exp \left(i\int d^{4}x{\bar{c}}^a{M}^{ab}{c}^b\right).$$

The fields $c^a,{\bar{c}}^a$ are the Faddeev–Popov ghosts: they are

• scalar (spin 0) under the Lorentz group, yet
• anticommuting (Grassmann, fermionic statistics).

They therefore violate the spin–statistics theorem — which is allowed precisely because they are not physical particles. Ghosts never appear as external states; they circulate only in internal loops, where their "wrong" statistics supplies compensating minus signs.

The gauge-fixed Lagrangian

Choosing the covariant $R_{\xi }$ gauge $G^a={\partial }_{\mu }{A}^{a\mu }$, the complete quantized Yang–Mills Lagrangian is

$$\boxed{\mathcal{L}=-\frac{1}{4}{F}_{\mu \nu }^a{F}^{a\mu \nu }-\frac{1}{2\xi }({\partial }_{\mu }{A}^{a\mu }{)}^2+{\bar{c}}^a(-{\partial }_{\mu }{D}^{\mu ab})c^b.}$$

The three terms are the Yang–Mills action, the gauge-fixing term (which supplies the gluon propagator, $\xi$-dependent as for the photon), and the ghost Lagrangian. Because the ghost kinetic operator contains the covariant derivative $D^{\mu ab}={\partial }^{\mu }{\delta }^{ab}+g{f}^{acb}{A}^{c\mu }$, there is a ghost–gluon vertex $\bar{c}Ac$: ghosts couple to gluons. This coupling is where the ghosts do their work.

Why ghosts are necessary: unitarity

The self-interacting gluon propagator propagates unphysical polarizations (timelike and longitudinal), just like the covariant photon. In QED the Ward identity makes these decouple automatically. In non-abelian theory they do not decouple by themselves — a gluon loop alone would violate unitarity (produce negative probabilities). The ghost loops exactly cancel the unphysical gluon contributions, restoring unitarity. This is the concrete content of the statement in QCD/from-postulates that "ghosts do not decouple."

The optical-theorem check — $\mathrm{Im}\mathcal{M}$ equals the physical cross section — works only with the ghost contribution included. The systematic guarantee that this cancellation holds to all orders is BRST symmetry.

Abelian vs. non-abelian

Summary

• The gauge path integral overcounts by the gauge volume; fix by a condition $G(A)=0$.
• The Faddeev–Popov determinant compensates; exponentiating it introduces anticommuting scalar ghosts $c,\bar{c}$.
• Ghosts violate spin–statistics but are unphysical (internal lines only).
• In non-abelian theories ghost loops are necessary for unitarity; abelian ghosts decouple.

Where this leads

• The symmetry guaranteeing consistency: BRST symmetry.
• The physical payoff of the self-coupling: asymptotic freedom.
• The theory in use: QCD.

References

• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 16.2.
• Faddeev & Popov, Phys. Lett. B 25, 29 (1967).
• Weinberg, The Quantum Theory of Fields, Vol. 2, Ch. 15.5–15.7.
• Srednicki, Quantum Field Theory, Ch. 71–72.