Expand around a background without forgetting constraints

Begin with finitely many real coordinates, a stable Euclidean action S, and weight exp[−(S−Jx)/ℏ]. Split x into a background b and a fluctuation. Adjust the source in the Legendre construction so the fluctuation has zero mean. At leading saddle order the source cancels the linear term, while the quadratic fluctuation kernel is S″ evaluated on b. Gaussian integration then produces a half log determinant in ΓE, with a positive coefficient for real bosonic modes.

The one-loop term is an expansion, not a declaration that the interacting integral has become exactly Gaussian. Cubic and higher fluctuation terms generate further loops. Restoring ℏ as a formal loop counter helps state the approximation order even in dimensionless examples. Determinants at two backgrounds can be compared using the same regulator and reference normalization. In continuum theories the resulting trace logarithm may contain divergences requiring counterterms consistent with the theory’s symmetries.

If configurations related by a gauge transformation are physically redundant, an invariant action is flat along the corresponding infinitesimal orbit at a stationary point. Its unreduced Hessian therefore has zero directions. A gauge condition restricts integration to a transverse slice, and the Faddeev–Popov determinant accounts for how that condition changes along the orbit. Simply dropping zero eigenvalues without the orbit measure can yield a gauge-dependent or incorrectly normalized answer.

Locally one may represent the determinant with anticommuting ghost variables, whose Gaussian contribution has a different determinant power from the bosonic fluctuations. This perturbative construction assumes the gauge condition works in the chosen patch. Multiple intersections of a gauge orbit with the slice, residual transformations, and global topology can defeat a naive global one-copy formula. Even a two-dimensional rotation quotient displays the problem if a chosen line intersects every nonzero orbit twice.

In background-field gauge, split transformations of background and fluctuation can preserve a manifest background gauge symmetry. This constrains the effective action and counterterms, making calculations efficient. It does not imply that every off-shell effective potential or truncated field configuration is independent of the quantum gauge-fixing parameter. Physical on-shell observables require the full consistent construction; background covariance is a useful organizing property rather than a universal gauge-independence shortcut.

The matrix lab offers a finite Jacobian check. Reset and set scale to 1 and shear to 0; predict final determinant 1 and singular values 1,1. Reset and set scale to 1 and shear to 1; predict determinant still 1 while singular values become approximately 1.618 and 0.618. Readouts describe final A; Reset geometry is identity and Play or Step interpolates identity↔A. This family demonstrates unchanged area despite stretching, not gauge fixing or an action-Hessian eigensolver.