Nonlinear Abelian gauge fixing

For an Abelian gauge theory with $A_{\mu }\mapsto A_{\mu }+{\partial }_{\mu }\alpha$, a nonlinear gauge functional $F[A]$ has Faddeev-Popov determinant $\det (\delta F[A+\partial \alpha ]/\delta \alpha )$. This operator can depend on $A$ even though the Adjoint representation of $U(1)$ is trivial. Consequently Faddeev-Popov ghost fields need not decouple in a nonlinear Abelian gauge, unlike a field-independent linear gauge condition.

For $\delta A_{\mu }={\partial }_{\mu }\omega$, the real quadratic gauge functional varies by $(\square +2A\cdot \partial )\omega$. Its Faddeev-Popov operator depends on the gauge field, producing a ghost-gauge vertex even in an Abelian gauge theory. With $\Psi =\int \bar{c}(F[A]+\xi h/2)$, a consistent left BRST symmetry convention gives $s\Psi =hF+\xi h^2/2-\bar{c}(\square +2A\cdot \partial )c$. The auxiliary-field integral at $\xi =0$ imposes the exact condition; finite $\xi$ gives a weighted gauge condition. This real functional is distinct from a complex quadratic Abelian gauge condition.

The nonlinear gauge fixing $\partial \cdot A+\lambda A^2$ produces cubic and quartic photon Feynman vertices. For physical photon polarization vectors, the two-external-leg cubic vertex reduces to a longitudinal internal momentum times the scalar product of the external polarizations. Its exchanged photon propagator cancels that momentum squared. Each of the three exchange graphs contributes $+4i{\lambda }^2({\epsilon }_i\cdot {\epsilon }_j)({\epsilon }_k\cdot {\epsilon }_l)$, cancelling the corresponding term of the quartic vertex. This vanishing physical tree-level Feynman diagram sum does not assert that the off shell four-point function vanishes.

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 $D_{\mu }={\partial }_{\mu }+i{A}_{\mu }$, 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.

A massless complex bosonic charged scalar field with $D_{\mu }={\partial }_{\mu }+i{A}_{\mu }$ 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.

For real $A_{\mu }$ in positive-definite Euclidean signature, $\partial \cdot A$ is real and $A_{\mu }{A}_{\mu }\ge 0$. Therefore $\partial \cdot A+i{A}^2=0$ forces $A^2=0$, and hence $A=0$. Its only real gauge orbit is the zero-field orbit; a field with nonzero electromagnetic curvature cannot be gauge-transformed to zero. Consequently this complex equation is not an admissible gauge fixing for general Hermitian Euclidean U(1) gauge symmetry configurations. A formal complexified treatment needs a specified contour and cannot be justified by a real delta-functional insertion.