Faddeev-Popov determinant (${\Delta }_{\mathrm{FP}}$)

The Faddeev-Popov determinant is the functional Jacobianthat compensates for the change from integration along a gauge orbit to a gauge-fixing condition $G(A)=0$. It can be represented by a path integral over a Faddeev-Popov ghost field pair.

Suppose a gauge fixing condition intersects each gauge orbit once in the local region under consideration, and its linearized Faddeev-Popov operator has no residual zero modes after boundary conditions are imposed. The functional Jacobian for changing from the gauge transformation parameter to the gauge condition gives the displayed identity. Inserting it in a gauge-invariant functional integral allows the integration along the gauge orbit to factor out and cancel the formal gauge-group volume. In a perturbative neighborhood the Jacobian's sign or phase is fixed; a field-independent determinant factor is absorbed into normalization. The Gribov ambiguity prevents treating this local identity as a globally unique gauge choice without further qualifications.

Assume an invariant volume measure, an invariant integrand and one regular gauge slice representative per orbit. At that representative, let $M$ be the Jacobian matrix of $F$ along the group coordinates, whose Haar measure is normalized to ordinary coordinate measure at the identity. Changing coordinates from the group parameter to $F$ gives ${\int }_{G}d\mu (g)\delta (F(x_g))=1/|\det M(x_0)|$. Multiply by its reciprocal and insert this identity into the original integral. Invariance permits a change of variables $x\mapsto x_g$ for each $g$, factoring out the group volume. The absolute determinant is required for ordinary unoriented real integration; a consistently positive orientation permits writing $\det M$. Infinite group volume is divided out formally or treated with a regulator.

For a gauge condition $\chi [A]=0$, the linearization of $\chi [A+D\omega ]$ in the gauge parameter is the Faddeev-Popov operator. Its determinant is represented by Faddeev-Popov ghost fields.

A Faddeev-Popov ghost field is a Grassmann-valued scalar field whose Gaussian functional integral represents the Faddeev-Popov determinant. Ghosts occur only on internal lines and cancel unphysical gauge-field contributions.

A ghost propagator is the two-point function $⟨Tc(x)\bar{c}(y)⟩$ of a Faddeev-Popov ghost field and its independent Faddeev-Popov antighost field. For quadratic action $\int \bar{c}Mc$ and source ordering consistent with that field ordering, the Grassmann integral gives $G_c=i{M}_F^{-1}$. The subscript specifies the causal pole prescription. For the negative derivative kinetic term $-\int \partial \bar{c}\cdot \partial c$, the Fourier kernel is $M=-p^2$. With signature $(-,+,+,+)$ this gives $G_c=-i/(p^2-i0)$. Field normalization multiplies the inverse kernel accordingly, while an antighost sign redefinition changes propagator and vertex signs together. Anticommuting statistics and loop signs do not turn these auxiliary fields into physical asymptotic particles.

With signature $(+---)$ and action $-\int {\partial }^{\mu }\bar{c}{\partial }_{\mu }c$, integration by parts gives the quadratic kernel $K_c=\square$, or $-p^2$ in Fourier space. The Grassmann Gaussian integral with odd sources gives $Z=\exp [-i\bar{\eta }{K}_{c,F}^{-1}\eta ]$. Left and right source derivatives, including their insertion factors, yield $⟨Tc\bar{c}⟩=i{K}_{c,F}^{-1}$ and hence the displayed sign. Replacing $\bar{c}$ by $-\bar{c}$ changes this propagator and the ghost-gluon vertex together; it is a convention change, not a physical discrepancy.

Expanding the ghost covariant derivative in Yang-Mills theory produces an interaction between the Faddeev-Popov ghost field, Faddeev-Popov antighost field, and gauge boson. For the displayed kinetic convention, Fourier factors $e^{-ipx}$ and all momenta incoming give vertex $-g{f}^{abc}{k}_{\mu }$, where $k$ is the antighost momentum. The structure constants encode the Adjoint representation. Propagator, vertex, Fourier, and ghost-order conventions must be chosen consistently.

For canonically normalized fields with action $-\int [\frac{1}{4}{F}^2+\frac{1}{2}(\partial \cdot A{)}^2+\partial \bar{c}\cdot Dc]$, use signature $(-,+,+,+)$ and incoming Fourier modes $e^{ipx}$. The ghost-gluon vertex is $g{p}_{\mu }{f}_{abc}$ and the ghost propagator is $-i/(p^2-i0)$. The open ghost-line one-particle-irreducible two-point vertex at one loop is $i{g}^{2}C{I}_d(p)$, whereA counterterm ${\mathcal{L}}_{\mathrm{ct}}=-\delta Z_c\partial \bar{c}\cdot \partial c$ inserts $-i\delta Z_c{p}^2$ and cancels this pole. Equivalently, writing $d=4-2ϵ$ gives $\delta Z_c=g^{2}C/(32{\pi }^2ϵ)$. There is no closed ghost-loop sign for this open-line graph. The Feynman i-epsilon prescription, Fourier convention and antighost sign must be kept consistent throughout.

The Grassmann-odd partner of the Faddeev-Popov ghost field in the gauge-fixing determinant representation. In an auxiliary-field BRST symmetry description it obeys $sb=h$, where $h$ is the Nakanishi-Lautrup field. Its ordinary spacetime derivative remains odd, which matters when moving an anticommuting BRST parameter past it.

The additive grading assigns ghost number $+1$ to $c$ and $-1$ to $\bar{c}$. Physical observables have ghost number zero. This grading is distinct from spin or the Adjoint representation.

For ghost term $-({\partial }^{\mu }\bar{c})\cdot D_{\mu }c$ and the current convention $\delta S=-\int ({\partial }_{\mu }\theta )j_G^{\mu }$, localize $\delta c=\theta c$, $\delta \bar{c}=-\theta \bar{c}$. The two differentiated parameters give this current. Its Noether charge grades ghosts by $+1$ and antighosts by $-1$, up to the corresponding overall generator normalization.

A ghost loop is a closed cycle of ghost propagators in a Feynman diagram. Its Grassmann statistics contribute an additional minus sign.