Gauge-Fixing Fermions, Canonical Transformations, and Observables

A gauge-fixing fermion $\Psi$ of ghost number $-1$ selects the graph $\Phi_A^*=\delta\Psi/\delta\Phi^A$, a Lagrangian submanifold of the BV phase space. Restricting a proper master action to that submanifold produces a nondegenerate gauge-fixed action when the gauge condition is transverse. BV observables and integrals are independent of smooth deformations of this choice only when the quantum master equation, measure, support, and boundary hypotheses hold. Singular slices, Gribov horizons, anomalies, and discarded boundary terms invalidate the conclusion.

Required background. The BV complex and classical master equation supply the odd symplectic space; BV fields and antifields fix the physical field content; and the master equation and BV gauge fixing supply the application to gauge theory.

Helpful background. Ghosts, auxiliary fields, and the gauge parameter explain the nonminimal pair, while local slices and Faddeev–Popov geometry state the transversality limitation.

Gauge fixing as a Lagrangian restriction

In Darboux coordinates $(\Phi^A,\Phi_A^*)$ for a degree-$-1$ odd symplectic form, the graph

$$\mathcal L_\Psi: \qquad \Phi_A^*=\frac{\delta\Psi}{\delta\Phi^A}$$

is Lagrangian because the pullback of $\omega_{\mathrm{BV}}=\int\delta\Phi_A^*\,\delta\Phi^A$ is the graded-symmetric second derivative of $\Psi$ contracted with an antisymmetric two-form. Equivalently, a canonical transformation generated by $\Psi(\Phi)$ sends $\Phi_A^*$ to $\Phi_A^*+\delta\Psi/\delta\Phi^A$, after which one restricts to the zero section. The ghost number of $\Psi$ is forced: since $\operatorname{gh}\Phi_A^*=-1-\operatorname{gh}\Phi^A$, the derivative must have that degree.

The minimal BV complex usually has no variables with which to impose a convenient covariant condition. Add a nonminimal contractible pair, the odd antighost $\bar c_a$ of ghost number $-1$ and the even Nakanishi–Lautrup field $b_a$ of ghost number zero, together with the term

$$S_{\mathrm{nm}}=\int d^4x\,\bar c^{*a}b_a .$$

Because $Q\bar c_a=b_a$ and $Qb_a=0$, this pair does not change cohomology. It does change the available Lagrangian submanifolds. The gauge-fixed Hessian must be invertible on the chosen test-function space after residual zero modes are handled; the Lagrangian condition alone does not ensure that analytic requirement. The rank conditions and general gauge-fixing-fermion construction are given in Gomis, París, and Samuel 1995, §§6.1–6.4, pp. 73–89.

Covariant Yang–Mills gauge

For Yang–Mills theory choose

$$\Psi_\xi=\int d^4x\, \bar c_a\left(\partial^\mu A^a_\mu+\frac{\xi}{2}b^a\right).$$

Functional differentiation gives, with integrations by parts and compatible left/right conventions,

$$A^{*\mu}_a=-\partial^\mu\bar c_a,\qquad \bar c^{*a}=\partial^\mu A^a_\mu+\frac{\xi}{2}b^a,\qquad c_a^*=0,\qquad b^{*a}=\frac{\xi}{2}\bar c^a .$$

Substitution into the nonminimal master action produces

$$S_{\Psi_\xi}=S_{\mathrm{YM}}+ \int d^4x\left[ b_a\partial^\mu A^a_\mu+\frac{\xi}{2}b_ab^a -\bar c_a\partial^\mu(D_\mu c)^a \right],$$

up to a correlated overall sign convention for $b$. Eliminating $b$ yields the usual quadratic covariant gauge-fixing term, while the ghost operator is the linearization of the gauge condition along gauge orbits. This is the exact first application developed by the physical BV master-equation page: introduce the nonminimal pair, choose $\Psi_\xi$, and derive the standard covariant gauge-fixed action by canonical restriction.

Three independent checks are available. Every term has ghost number zero. The Abelian limit gives the field-independent Faddeev–Popov operator $-\Box$. Finally, varying with respect to $b$ returns the declared gauge condition. These checks do not establish that the slice intersects every non-Abelian orbit exactly once.

The precise independence theorem

In a finite-dimensional orientable BV manifold with compatible measure, a $\Delta$-closed integrand has equal integrals over closed oriented Lagrangian submanifolds whose bodies are homologous. A $\Delta$-exact integrand integrates to zero. Schwarz proves these BV–Stokes statements in Schwarz 1993, Theorems 1–2, pp. 252–253. Infinitesimally, the variation generated by $\delta\Psi$ becomes a total BV divergence.

For field theory, each noun in that theorem becomes an obligation: the measure and $\Delta$ require regularization, the renormalized action must satisfy the quantum master equation, observables must be quantum-BRST closed, integrations by parts need support or boundary conditions, and the deformation must stay within nonsingular gauge choices. Perturbative renormalization can establish a formal analogue order by order. The finite-dimensional theorem is not a proof that an infinite-dimensional gauge-field measure exists.

The adversarial test is a fermion whose slice crosses a Gribov horizon, where the Faddeev–Popov operator develops a zero mode. A deformation can then acquire a boundary contribution from the restricted region. Likewise, on a manifold with boundary, discarding the term generated while differentiating $\Psi$ changes the action. In either case the strongest surviving statement is local canonical equivalence away from the singularity or boundary; global gauge-fixing independence has not been proved.

Even inside a regular patch, two gauge fermions can produce different propagators and different infrared behavior. Equality of BRST-closed observables is conditional on a common renormalized domain; it does not assert equality of off-shell Green functions.

Exercises

Check the ghost number of $\Psi_\xi$ and its four antifield substitutions.

Solution

$\bar c$ has ghost number $-1$, while $\partial\cdot A$ and $b$ have ghost number zero, so $\Psi_\xi$ has ghost number $-1$. Differentiating with respect to a field of ghost number $g$ gives degree $-1-g$, exactly the ghost number of its antifield.

Integrate out $b$ when $\xi\neq0$.

Solution

The algebraic equation is $b^a=-\xi^{-1}\partial\cdot A^a$ in the displayed convention. Substitution gives $-(2\xi)^{-1}(\partial\cdot A^a)^2$. Reversing the sign chosen for $S_{\mathrm{nm}}$ reverses the correlated $b$ convention but not the ghost determinant or the transversality requirement.

References

• Gomis, Joaquim, Jordi París, and Stuart Samuel. “Antibracket, Antifields and Gauge-Theory Quantization.” Physics Reports 259 (1995): 1–145. DOI; Open PDF.
• Henneaux, Marc, and Claudio Teitelboim. Quantization of Gauge Systems. Princeton, NJ: Princeton University Press, 1992. Publisher.
• Schwarz, Albert. “Geometry of Batalin–Vilkovisky Quantization.” Communications in Mathematical Physics 155 (1993): 249–260. DOI; Open PDF.