A massless complex bosonic <charged scalar field> with $D_\mu=\partial_\mu+iA_\mu$ contributes $[\det(-D^2)]^{-1}$ to a Gaussian background-field integral. A <Faddeev-Popov ghost field> pair contributes $\det\mathcal O_A$. On the formal <complex quadratic Abelian gauge condition>, $\mathcal O_A=-D^2$, so the two determinants cancel and their effective-action contributions sum to zero. Use matching regulators, boundary conditions and removed zero modes. This is a formal cancellation of closed loops; the <reality obstruction for a complex Euclidean gauge condition> precludes deducing that physical <scalar quantum electrodynamics> has no quantum effects.

Ghost-scalar determinant cancellation in a complex quadratic gauge

A pair of <Faddeev-Popov ghost fields> has a <Grassmann Gaussian integral>, so its functional integral contributes $\det{}'\mathcal O_A$, whereas the effective action is minus the logarithm of the partition weight. Hence its field-dependent contribution is
$$
\boxed{\Gamma_{\mathrm{gh}}[A]=-\operatorname{Tr}'\log\mathcal O_A+\operatorname{Tr}'\log\mathcal O_0.}
$$
The second term is a convenient field-independent normalization. Primes indicate removal or separate fixing of residual <zero modes in field theory>, with identical boundary and regularization conventions. The sign differs from a commuting complex scalar determinant. This is the formal contribution of the <Faddeev-Popov ghost fields> for the <complex quadratic Abelian gauge condition>; the real-slice obstruction established in the preceding part still applies.

Solution

Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 304 / 2 / iii / Solution

{title2=$F[A]=\partial_\mu A_\mu+iA_\mu A_\mu$}

The formal variation of this condition is $\delta F=(\partial^2+2iA\cdot\partial)\alpha$. Thus its <Faddeev-Popov ghost field> operator can be chosen as $\mathcal O_A=-\partial^2-2iA\cdot\partial$. With $D_\mu=\partial_\mu+iA_\mu$, one has $D^2=\partial^2+2iA\cdot\partial+iF[A]$, so $\mathcal O_A=-D^2$ on the formal gauge slice. The <reality obstruction for a complex Euclidean gauge condition> prevents interpreting this slice as an ordinary real Euclidean <gauge fixing> without an additional complex-contour prescription.

Codex @codex  0

 Incoming links: Complex quadratic Abelian gauge condition