Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 304 / 4

For a short Wilson line, a gauge transformation givesUse $V(x+an)=1+i\alpha (x)+ia{n}^{\mu }{\partial }_{\mu }\alpha (x)+O(a^2,{\alpha }^2)$ and $U=1+iga{n}^{\mu }{A}_{\mu }+O(a^2)$. Keeping terms of order $a$, including $a\alpha$, givesSince the Lie bracket is encoded by the Lie algebra structure constants, $i[\alpha ,A_{\mu }{]}^a=f^{abc}{A}_{\mu }^b{\alpha }^c$, and thereforeThis is the infinitesimal form of the Wilson-line gauge transformation.

The variation found above uses the adjoint covariant derivative:For the Lorenz gauge functional $G^a[A]={\partial }^{\mu }{A}_{\mu }^a$,The field-independent factor $1/g$ may be absorbed into normalization. The Grassmann Gaussian integral exponentiates the Faddeev-Popov determinant with anticommuting Faddeev-Popov ghost fields:A Gaussian average over the gauge condition supplies the covariant gauge-fixing term, so

The quadratic gauge-field action in momentum space isIn terms of the transverse projector of a vector field and longitudinal projector of a vector field,the operator is $k^2(P_T+{\xi }^{-1}{P}_L)$. Its inverse, the gauge-boson propagator, isTherefore $X=1$ and $Y=\xi -1$.

Without gauge fixing, every gauge orbit is integrated infinitely many times and the quadratic gauge-field operator has a zero mode in field theory for every pure-gauge direction. It therefore has no inverse and no propagator. A gauge condition selects one representative per orbit, up to global transformations and the possible nonperturbative Gribov ambiguity, and makes perturbative Gaussian integration well defined.

The anticommuting fields $c$ and $\bar{c}$ represent the field-dependent Faddeev-Popov determinant. They are Lorentz-scalar Grassmann fields, appear only on internal lines, and contribute a minus sign for each closed ghost loop. Their diagrams cancel unphysical gauge-polarization contributions and are required for gauge-independent, unitary amplitudes in a non-Abelian covariant gauge.

For axial gauge, $G^a[A]=n^{\mu }{A}_{\mu }^a$. Its Faddeev-Popov operator isOn the gauge slice $n\cdot A^c=0$, the second term vanishes. HenceThis functional determinant is independent of the gauge field and may be absorbed into the normalization of the path integral. Any introduced ghosts are free and decouple, so no ghost fields are needed in axial gauge.