TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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Poincare cotangent moment map and full generator algebra

Statement

Assume ${A C}_{ω}$ . For the group and cotangent conventions of Poincare semidirect group, dual pairing and moment components, the map $μ : T^{*} R^{4} \to g^{*}$ defined by $⟨ μ (x, p), (a, A) ⟩ = ℓ_{(a, A)} (x, p)$ is a smooth equivariant moment map and obeys

${ℓ_{ξ}, ℓ_{η}} = ℓ_{[ξ, η]}, {P^{a}, P^{b}} = 0, {J^{a b}, P^{c}} = η^{a c} P^{b} - η^{b c} P^{a},$

${J^{a b}, J^{c d}} = η^{a c} J^{b d} - η^{a d} J^{b c} - η^{b c} J^{a d} + η^{b d} J^{a c} .$

For shell coordinates write $p = (p_{1}, p_{2}, p_{3})$ and $∣ p ∣^{2} : = p_{1}^{2} + p_{2}^{2} + p_{3}^{2}$ . For each future massive shell with $m > 0$ in kg, $c > 0$ in m/s, $p_{0} = - E / c$ , and $E = \sqrt{m^{2} c^{4} + c^{2} ∣ p ∣^{2}}$ , or nonzero future null shell with $m = 0$ and $∣ p ∣ > 0$ , its Poincare charges descend to the global quotient by complete characteristic lines of the shell constraint. For vectors $u, v \in R^{4}$ , define $(u \land v)^{a b} : = u^{a} v^{b} - u^{b} v^{a}$ . The induced dual action is $(P, J) \mapsto (Λ P, Λ J Λ^{T} + a \land Λ P)$ . On the fixed-time slice $x^{0} = c t$ , with physical position $q^{i} = x^{i}$ , the boost component $B_{i} = J^{0 i} = c t p_{i} - q^{i} E / c$ is explicitly time dependent and satisfies $\partial_{t} B_{i} + {B_{i}, E} = 0$ . These are mathematical identities; conservation is conditional on the stated free-shell evolution.

Facts & Assumptions

Given: The affine Poincaré action, cotangent phase space, and Poisson sign convention of Poincare semidirect group, dual pairing and moment components and the exact conditional suppliers below.

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): assume countable choice ${A C}_{ω}$ only; no full axiom of choice is used.

[F1]

Poincare semidirect group, dual pairing and moment components: $Y_{(a, A)} = a + A x$ , $ℓ_{(a, A)} = p_{b} (a^{b} + A^{b}_{c} x^{c})$ , $P^{a} = η^{a b} p_{b}$ , and $J^{a b} = x^{a} P^{b} - x^{b} P^{a}$ .

[L1]

Canonical Poisson brackets and local Hamiltonian flow: the current H0 draft states ${F, G} = F_{x} G_{p} - F_{p} G_{x}$ , $ι_{X_{H}} ω = d H$ , and local flow only on the open solution domain. Its H25 Step 1 record is ready and hash-current, but the draft carrier has no pipeline_run binding and its ai-altered proof has no independent proof approval. Its coordinate convention matches the proof.

[F2]

The cotangent lift of an action is Hamiltonian with the tautological moment map: for the cotangent lift and the library $\exp (- s ξ)$ convention, $μ^{ξ} = - p (ξ_{Q})$ satisfies $d μ^{ξ} = - ι_{ξ_{T^{*} Q}} ω$ and the cotangent-lift map is coadjoint equivariant.

[L2]

Flat future mass shell has an explicit global orbit quotient: the current H8 draft states the future-shell equation $- p_{0} = P^{0} = \sqrt{m^{2} c^{2} + ∣ p ∣^{2}}$ , complete affine-line flow, global $(a, p)$ quotient, reduced form $d a^{i} \land d p_{i}$ , and nonzero-null restriction. Its statement and dependencies match the H25 manifest. Its Step 1 record is ready/hash-current, but the carrier is still draft and unbound to pipeline_run, so this is not independent proof approval.

[L3]

Free shell in inertial time has the square-root Hamiltonian: the current H8 draft states $p_{0} = - E / c$ , $E = \sqrt{m^{2} c^{4} + c^{2} ∣ p ∣^{2}}$ , $q = a + t E_{p}$ , and $\dot{q} = c^{2} p / E$ , $\dot{p} = 0$ on $x^{0} = c t$ . The supplier is a mathematical deduction within its stated chart, positive-root and nonzero-null hypotheses; it introduces no physical action adoption.

Proof

technique · direct cotangent moment map and canonical differentiation

Given: The library sign conventions, [F1], and the exact conditional shell statements [L1]–[L3].

1.1F1F2L1A1algebra

For $ξ = (a, A)$ the cotangent lift has positive infinitesimal field $(a + A x, - A^{T} p) = - ξ_{T^{*} Q}$ because the library fundamental vector is generated by $\exp (- s ξ)$ . With $ω = d x^{a} \land d p_{a}$ , $ι_{X_{ℓ_{ξ}}} ω = d ℓ_{ξ}$ by [L1], as $\partial_{p} ℓ = a + A x$ and $- \partial_{x} ℓ = - A^{T} p$ . Therefore $d ℓ_{ξ} = - ι_{ξ_{T^{*} Q}} ω$ . The cotangent lift is symplectic and equivariant by [F2], so $μ$ is the claimed equivariant moment map.

1.2F1L1algebra

Substituting $P^{a} = η^{a b} p_{b}$ and $J^{a b} = x^{a} P^{b} - x^{b} P^{a}$ into [L1]'s canonical bracket, Leibniz gives ${P^{a}, P^{b}} = 0$ and ${J^{a b}, P^{c}} = η^{a c} P^{b} - η^{b c} P^{a}$ . Applying Leibniz to the four products in ${J^{a b}, J^{c d}}$ gives the four displayed metric terms, with signs fixed by ${x^{a}, p_{b}} = δ_{b}^{a}$ .

1.3F1algebra

The full dual action follows directly from the semidirect group law, without restricting to the cotangent moment-map image. For $g = (a, Λ)$ , multiplication and inversion give ${Ad}_{g} (b, B) = (Λ b - Λ B Λ^{- 1} a, Λ B Λ^{- 1})$ , hence ${Ad}_{g^{- 1}} (b, B) = (Λ^{- 1} (b + B a), Λ^{- 1} B Λ)$ . For an arbitrary dual charge $(P, J)$ use the pairing in [F1], $⟨ (P, J), (b, B) ⟩ = P_{b} b^{b} - \frac{1}{2} J_{b c} B^{b c}$ , and define $g \cdot (P, J)$ by pairing with ${Ad}_{g^{- 1}} (b, B)$ . Lorentz invariance of $η$ transforms the first term to $P_{b}^{'} b^{b} + P_{b}^{'} B^{b}_{c} a^{c}$ , where $P^{'} = Λ P$ , and the second term to $- \frac{1}{2} (Λ J Λ^{T})_{b c} B^{b c}$ . Antisymmetry gives $P_{b}^{'} B^{b}_{c} a^{c} = - \frac{1}{2} (a \land P^{'})_{b c} B^{b c}$ . Thus for every $(b, B)$ the pairing equals that of $(Λ P, Λ J Λ^{T} + a \land Λ P)$ ; uniqueness of the dual coordinates proves the formula on all of $g^{*}$ . For cotangent-image charges it also agrees with $x^{'} = Λ x + a$ , $p^{'} = Λ^{- T} p$ , and $J^{'} = x^{'} \land P^{'}$ , as required by equivariance.

1.4F1L2L1algebra

On either future shell, the characteristic shift changes $J$ by $α P \land P = 0$ and leaves $P$ fixed, so each charge is constant on quotient fibres. In H8 coordinates $(a, p)$ the descended $P$ and $J$ are explicit smooth functions, so they descend smoothly through the quotient of [L2]. Lorentz invariance of $p^{2}$ gives ${ℓ_{ξ}, C_{m}} = 0$ for Lorentz generators; translations are constants in flat spacetime, so their brackets with $C_{m}$ also vanish.

2.1F1L1step 1.1algebra

For $η = (b, B)$ , the canonical bracket gives ${ℓ_{ξ}, ℓ_{η}} = p \cdot A (b + B x) - p \cdot B (a + A x) = p \cdot (A b - B a + [A, B] x) = ℓ_{[(a, A), (b, B)]},$ using the semidirect Lie bracket from [F1]. This proves the full dual generator identity.

3.1F1L2L3algebra∎

At $x^{0} = c t$ , $P^{0} = E / c$ , hence $B_{i} = J^{0 i} = c t p_{i} - q^{i} E / c$ . With the reduced canonical bracket from [L2]–[L3], ${B_{i}, E} = - c p_{i}$ and $\partial_{t} B_{i} = c p_{i}$ . Their sum is zero, so $B_{i}$ is conserved along the free reduced motion, with the partial-time term essential.

Poincare cotangent moment map and full generator algebra

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources