Universality from Quadratic Angular-Momentum Operations: Exact Dynamical Lie Algebras and the Role of the Casimir Obstruction

• URL: http://arxiv.org/abs/2609.15302v1
• Date: Mon, 14 Sep 2026 09:52:25 GMT
• Title: Universality from Quadratic Angular-Momentum Operations: Exact Dynamical Lie Algebras and the Role of the Casimir Obstruction
• Abstract summary: We show that the Lie algebra generated by all degree-at-most-two elements does not exhaust the universal enveloping algebra of $mathfraksu_mathbbR(2)$.<n>We derive a practical degree-two criterion and discuss finite-dimensional SSRC.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Universality relies on combining operators with different algebraic properties, which depend on the physical structure of the Hilbert space encoding the information. Systems of identical symmetric particles---including bosons and symmetric collections of spin-$1/2$ particles---can be mapped to angular-momentum systems, where rotations supplemented by quadratic, spin-squeezing interactions provide universal control. We revisit universality in this framework and confirm that such linear and quadratic operations provide universal control on every fixed-particle-number subspace. At the abstract level, however, we show that the dynamical Lie algebra generated by all degree-at-most-two elements does not exhaust the full universal enveloping algebra of $\mathfrak{su}_{\mathbb{R}}(2)$, contrary to a stronger claim made previously: higher powers of the quadratic Casimir element cannot be generated by commutators and linear combinations. This obstruction does not conflict with fixed-sector universality because, under a fixed irreducible spin-$n/2$ representation, all powers of the Casimir element act as scalar multiples of the identity. We determine the resulting abstract dynamical Lie algebras exactly and show that the central-free and full degree-at-most-two algebras map, respectively, onto $\mathfrak{su}_{\mathbb{R}}(n+1)$ and $\mathfrak{u}_{\mathbb{R}}(n+1)$, establishing special-unitary and full-unitary universality. We further derive a practical degree-two universality criterion and discuss finite-dimensional SSRC universality.

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