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

The Faddeev-Popov ghost fields $c$ and $\bar{c}$ are spin-zero Grassmann fields in the Adjoint representation of $G$; their statistics are anticommuting, despite their scalar transformation under spacetime rotations. The Nakanishi-Lautrup field $h$ is an auxiliary commuting spin-zero field, also in the Adjoint representation. ThusThese Faddeev-Popov ghost fields are not physical spin-zero fermion states; they represent the gauge-fixing Jacobian, so their scalar spin does not contradict the physical Spin-statistics theorem.

Write $F[A]={\partial }_{\mu }{A}_{\mu }+i{A}_{\mu }{A}_{\mu }$ and choose the U(1) gauge symmetry convention $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }\alpha$. Its infinitesimal variation isA formal BRST symmetry construction uses $s{A}_{\mu }={\partial }_{\mu }c$, $sc=0$, $s\bar{c}=ih$, $sh=0$ and the gauge-fixing fermion $\Psi =\int \bar{c}(F-i\xi h/2)d^{4}x$. Acting with the odd differential givesFor a real admissible gauge functional, integration over $h$ gives $F^2/(2\xi )$ in the Euclidean action, and $\xi \to 0$ imposes the gauge condition. Overall constant phases in the Faddeev-Popov determinant can be absorbed in the measure convention. Here nonlinear Abelian gauge fixing makes ${\mathcal{O}}_A$ depend on $A$. The action for the Faddeev-Popov ghost fields contains the interaction $-2i\bar{c}{A}_{\mu }{\partial }_{\mu }c$, so the Faddeev-Popov ghost fields cannot be discarded as in a linear Abelian gauge, even though the $U(1)$ Adjoint representation is trivial.

There is a genuine reality obstruction for a complex Euclidean gauge condition in the source. For a Hermitian $U(1)$ gauge field, each Euclidean component $A_{\mu }$ is real. The real and imaginary parts of $F=0$ separately require $\partial \cdot A=0$ and ${\sum }_{\mu }{A}_{\mu }^2=0$, hence $A=0$. For example $A_2=B{x}_1$, with other components zero and $B\ne 0$, has curvature $F_{12}=B$; no gauge transformation can put it on this slice because curvature is gauge invariant. Thus the stated condition is not a gauge fixing of general real Euclidean configurations. The boxed action is the intended formal complex-gauge construction. A legitimate complexified contour prescription would be additional data, not an ordinary real delta-functional enforcing this printed condition.

A pair of Faddeev-Popov ghost fields has a Grassmann Gaussian integral, so its functional integral contributes $\det {}^{\prime }{\mathcal{O}}_A$, whereas the effective action is minus the logarithm of the partition weight. Hence its field-dependent contribution isThe 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.

Choose one free, massless complex scalar field with unit charge magnitude and bosonic statistics, minimally coupled to $A$. To fix the sign, use the conventional gauge covariant derivative $D_{\mu }={\partial }_{\mu }-iq{A}_{\mu }$ and take $q=-1$, so $D_{\mu }={\partial }_{\mu }+i{A}_{\mu }$ and $\phi \mapsto e^{-i\alpha }\phi$. Its antiparticle has charge $+1$. The matter data areThere is no scalar self-interaction. For real fields before formal continuation, its action is $S_{\phi }=\int (D_{\mu }\phi {)}^{\ast }(D_{\mu }\phi )d^{4}x=\int \bar{\phi }(-D^2)\phi d^{4}x$, by integration by parts. The Gaussian functional integral gives the inverse functional determinant and thus ${\Gamma }_{\phi }=+{\mathrm{Tr}}^{\prime }\log (-D^2)$, up to normalization.

The key operator identity isOn the formal complex gauge slice, $-D^2=-M_A={\mathcal{O}}_A$. With matching regulators and omitted zero modes, this gives ghost-scalar determinant cancellation in a complex quadratic gauge, leaving the Maxwell action as the background functional:where gauge-fixing terms and field-independent constants are understood separately. For a fixed background the complex scalar field and Faddeev-Popov ghost field integrals are Gaussian, so this determinant cancellation is exact within the formal construction. It is a cancellation of closed bosonic matter loops against ghost loops in that background functional. It does not prove that physical scalar quantum electrodynamics has no radiative corrections: the printed real Euclidean gauge slice contains only $A=0$ and misses every nonzero-curvature orbit. On that literal slice the equality is trivial; a nontrivial version requires the additional complex-contour interpretation.