2026/07/29 by Henrique Gomes
Physics and Astronomy · #physics.hist-ph #hep-th
arxiv created 2026/07/29 · arxiv updated 2026/07/31
The Faddeev--Popov procedure poses two conceptual puzzles. Puzzle~(1): if gauge-equivalent configurations represent the same physics, the quotient \F/\G should suffice to compute physical amplitudes --- yet the procedure requires anti-commuting auxiliary fields, the ghosts, with no analogue on the quotient. What structure of \F do they encode? Puzzle~(2): gauge-fixing was supposed to eliminate local gauge symmetry, yet the gauge-fixed theory retains BRST --- a residual symmetry that acts on the gauge potential as an infinitesimal gauge transformation. Why does it survive? Both puzzles dissolve together. Following \textciteDougherty2021 and \textciteDoughertyRead2026, I take ghosts to encode classical content of \F → \F/\G, but identify a different structure: a principal connection \varpi on this bundle. The ghost is \varpi; the BRST operator is the vertical exterior derivative on field space; the Maurer--Cartan equation is its vertical Cartan structure equation. The Faddeev--Popov calculus draws only on \varpi's vertical content, which the algebraic reading also captures; \varpi's horizontal content, on which the Vilkovisky--DeWitt programme rests, supplies the cross-orbit pairing that gauge-fixing, dressing-based quantisation, and counterfactual comparison require and the quotient discards. BRST is the rigid, vertical symmetry that preserves this pairing; this is why it survives gauge-fixing.