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Positive classical spin Poincare orbit and stabilizer jump

Statement

Assume ${A C}_{ω}$ . Let $G = R^{4} ⋊ S O^{+} (1, 3)$ act coadjointly on the dual charge space of Poincare semidirect group, dual pairing and moment components. At a charge point $(P^{a}, J^{a b})$ , define $P_{b} : = η_{b a} P^{a}$ , $J_{c d} : = η_{c e} η_{d f} J^{e f}$ , and $P^{2} : = η_{a b} P^{a} P^{b}$ . Let $ε_{a b c d}$ be the totally antisymmetric Minkowski volume tensor with $ε_{0123} = + 1$ ; raising all indices with $η = diag (- 1, 1, 1, 1)$ gives $ε^{0123} = - 1$ . Define the algebraic vector $W^{a} : = \frac{1}{2} ε^{a b c d} P_{b} J_{c d}$ , and write $W^{2} : = η_{a b} W^{a} W^{b}$ and $W \cdot P : = η_{a b} W^{a} P^{b}$ . For fixed mass $m > 0$ in kg, angular-momentum parameter $s > 0$ in kg m $^{2}$ /s, and $c > 0$ in m/s, take the rest charge point with $P^{a} = (m c, 0, 0, 0)$ and $J^{12} = s$ , all other independent components of $J$ being zero. Its coadjoint orbit is an eight-dimensional smooth symplectic manifold with its KKS form. On this orbit, $P^{2} = - m^{2} c^{2}$ , $P$ is future, $W \cdot P = 0$ , and $W^{2} = m^{2} c^{2} s^{2}$ . The exact joint invariant set with these values is this one orbit. At $s = 0$ the rest stabilizer has dimension four and the spinless orbit has dimension six; no smoothness claim is made for a larger union of Casimir level sets.

Facts & Assumptions

Given: The proper orthochronous Poincaré group and charge pairing, $m > 0$ , $s > 0$ , and the rest representative in the statement.

[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: the Poincaré group is a ten-dimensional Lie group with semidirect bracket and dual translation/antisymmetric Lorentz charges $(P, J)$ ; here $P_{b} = η_{b a} P^{a}$ and $J_{c d} = η_{c e} η_{d f} J^{e f}$ are lowered with $η$ .

[L1]

Poincare cotangent moment map and full generator algebra: the current assigned A draft states the affine coadjoint action (P,J)->(Lambda P,Lambda J Lambda^T+a wedge Lambda P) and the charge map with the exp(-s xi) convention. The carrier exists as this batch-26 draft, but its H0/H8 inputs and proof review remain unresolved.

[F2]

Coadjoint orbits are symplectic manifolds: under ${A C}_{ω}$ , every coadjoint orbit has the canonical smooth homogeneous-space structure and the KKS form is smooth, closed, nondegenerate, with $ω_{β} (ξ_{O}, η_{O}) = β ([ξ, η])$ .

Proof

technique · rest-frame classification and stabilizer calculation

Given: The rest covector and exact coadjoint action conditional on [L1].

1.1F1L1algebra

In the rest frame $P = (m c, 0)$ , the spatial components of $W$ are the three independent duals of $J^{i j}$ , multiplied by the nonzero factor $m c$ . Thus $∣ W ∣ = m c s$ and $W^{0} = 0$ , so $W \cdot P = 0$ and $W^{2} = m^{2} c^{2} s^{2}$ . Under translations, $J$ changes by $a \land P$ ; inserting this into $W$ adds an epsilon contraction with two copies of $P$ , which vanishes by antisymmetry. Lorentz covariance of $ε$ under $S O^{+}$ and the charge action [L1] preserve these scalar identities over the orbit.

1.2F1L1algebra

A stabilizer element must fix $P = (m c, 0)$ , so its Lorentz part is a spatial rotation. Preserving the nonzero spin vector leaves exactly the $S O (2)$ rotations about that axis, of dimension one. In the coadjoint action, preserving $J^{0 i} = 0$ then forces the spatial translation part to vanish, while translations parallel to $P$ remain free, giving one further dimension. The stabilizer has dimension two, and since $\dim G = 10$ by [F1], the orbit dimension is eight.

2.1F1L1step 1.1algebra

Any future timelike $P$ with $P^{2} = - m^{2} c^{2}$ can be sent to rest by a proper orthochronous Lorentz transformation. In that frame, choose the translation part $a$ to remove the three $J^{0 i}$ components; the remaining spatial spin vector has norm $s$ by step 1.1, and a spatial rotation sends it to the specified $J^{12} = s$ axis. Therefore every pair with the stated invariants lies in the rest representative's orbit; the converse follows from step 1.1.

3.1F1F2L1A1∎

Applying [F2] to this orbit gives its canonical smooth homogeneous-space structure and nondegenerate closed KKS two-form. If $s = 0$ , the stabilizer instead contains all three spatial rotations and the one translation parallel to $P$ , hence has dimension four and its orbit dimension is $10 - 4 = 6$ . This establishes the stabilizer jump without asserting regularity of any ambient Casimir locus.

Positive classical spin Poincare orbit and stabilizer jump

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources