A note on complete gauge-fixing and the constraint algebra

Abstract

The admissibility of a gauge-fixing is governed by the invertibility of $\hat{I}”=\{\ddot{I}{ƒ}^a,{\hat{I}}_b^3\}$ where $\ddot{I}{ƒ}^a$ are gauge-fixing conditions and ${\hat{I}}_b^3$ are independent first-class constraints. We prove, via the Schur complement, that the determinant of the combined constraint matrix $\mathcal{M}=\{\hat{I}{¦}_A,\hat{I}{¦}_B\}$ built from all constraints and gauge-fixing conditions factorises as $\det \mathcal{M}\approx \pm (\det \hat{I}”{)}^2\det C$, where $C$ is the second-class constraint matrix, providing an alternative criterion for admissibility. Since $\det C\ne 0$ by definition, the second-class sector decouples entirely from the gauge-fixing sector. In the algebraic case, this factorisation identifies the Hamiltonian admissibility criterion of Henneaux and Teitelboim with the Lagrangian completeness criterion of Motohashi, Suyama, and Takahashi. We identify a metric ansatz as gauge-fixing at the action level and analyse completeness in the context of spherically symmetric spacetime. The factorisation ensures that completeness is robust to the second-class sector that arises in modified theories of gravity.

3 pages, 0 figures