su(2) spin s representations via CP2s sigma models*
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Bibliographic record
Abstract
Abstract We establish and analyze a new relationship between the matrix functions describing spin fields of a spin s , where <?CDATA $2s\in {{\mathbb{Z}}}^{+}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>2</mml:mn> <mml:mi>s</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> </mml:mrow> </mml:msup> </mml:math> , and <?CDATA ${\mathbb{C}}{P}^{2s}$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="double-struck">C</mml:mi> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>s</mml:mi> </mml:mrow> </mml:msup> </mml:math> two-dimensional Euclidean sigma models. The spin matrices are constructed from the rank-1 Hermitian projectors of the sigma models or from the anti-Hermitian immersion functions of their soliton surfaces in the <?CDATA ${\mathfrak{su}}(2s+1)$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="fraktur">su</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>s</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> algebra. We provide a geometric interpretation of this construction. For the spin fields which can be represented as linear combinations of the generalized Pauli matrices, we find the dynamics equation satisfied by their coefficients. This equation is identical to the stationary equation of a two-dimensional Heisenberg model. We show that the same holds for matrices congruent to the generalized Pauli matrices through any coordinate-independent unitary linear transformation. These properties allow for new interpretations of the spins as compositions of more elementary objects. They also open up the possibility of future applications of the sigma models to the situations which depend on spin behaviour, including spintronics, spin glasses and quantum computing.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.003 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it