Ghosts that Connect

Abstract

What is a Faddeev--Popov ghost? The textbook answer is that it is a trick: a way of writing a functional determinant as a term in the action, and nothing more. Dougherty 2021 and Dougherty and Read 2026 have argued that this undersells the ghost, which on their account encodes classical structure of the configuration space, viz. the degree-one generators of the Chevalley--Eilenberg complex of the action Lie algebroid. I agree with them that the ghost encodes classical structure. But I think the structure is bigger than they say. The ghost, I argue, is a principal connection $\varpi$ on the bundle $F\to F/G$ of gauge potentials over physical configurations; the BRST operator is the exterior derivative along the orbits; and the Chevalley--Eilenberg ghost coincides with $\varpi$ restricted to the orbits (and every $\varpi$ has the same restriction). The restriction drops the horizontal complement, and I show that gauge-fixed quantisation needs that complement. A gauge-fixing installs it as a pairing of configurations across orbits (a counterpart relation, in Lewis's sense); dressed observables and local counterfactuals rely on this pairing, but it is discarded by the quotient $\F/\G$. In the non-Abelian case Singer's theorem forbids a flat $\varpi$, and there the pairing exists only as parallel transport. Finally, BRST survives gauge-fixing because it is vertical and rigid, and so leaves any such pairing unchanged.

arXiv comment31p, 3 figures