A note on complete gauge-fixing and the constraint algebra
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arXiv
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| Format: | Preprint |
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2026
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| author | Manchanda, Ganga Singh |
| author_facet | Manchanda, Ganga Singh |
| contents | The admissibility of a gauge-fixing is governed by the invertibility of $Δ=\{σ^a,γ_b\}$ where $σ^a$ are gauge-fixing conditions and $γ_b$ are independent first-class constraints. We prove, via the Schur complement, that the determinant of the combined constraint matrix $\mathcal{M}=\{Φ_A, Φ_B\}$ built from all constraints and gauge-fixing conditions factorises as $\det\mathcal{M}\approx\pm(\detΔ)^2\det C$, where $C$ is the second-class constraint matrix, providing an alternative criterion for admissibility. Since $\det C\neq0$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_16990 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A note on complete gauge-fixing and the constraint algebra Manchanda, Ganga Singh General Relativity and Quantum Cosmology The admissibility of a gauge-fixing is governed by the invertibility of $Δ=\{σ^a,γ_b\}$ where $σ^a$ are gauge-fixing conditions and $γ_b$ are independent first-class constraints. We prove, via the Schur complement, that the determinant of the combined constraint matrix $\mathcal{M}=\{Φ_A, Φ_B\}$ built from all constraints and gauge-fixing conditions factorises as $\det\mathcal{M}\approx\pm(\detΔ)^2\det C$, where $C$ is the second-class constraint matrix, providing an alternative criterion for admissibility. Since $\det C\neq0$ 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. |
| title | A note on complete gauge-fixing and the constraint algebra |
| topic | General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2604.16990 |