Bibliographic record
Abstract
In flat spacetime, the Dirac equation is the “square root” of the Klein–Gordon equation in the sense that, by applying the square of the Dirac operator to the Dirac spinor, one recovers the equation duplicated for each component of the spinor. In the presence of gravity, applying the square of the curved-spacetime Dirac operator to the Dirac spinor does not yield the curved-spacetime Klein–Gordon equation, but instead yields the Schrödinger–Dirac covariant equation. First, we show that the latter equation gives rise to a generalization to spinors of the covariant Gross–Pitaevskii equation. Then, we show that, while the Schrödinger–Dirac equation is not conformally invariant, there exists a generalization of the equation that is conformally invariant but which requires a different conformal transformation of the spinor than that required by the Dirac equation. The new conformal factor acquired by the spinor is found to be a matrix-valued factor obeying a differential equation that involves the Fock–Ivanenko line element. The Schrödinger–Dirac equation coupled to the Maxwell field is then revisited and generalized to particles with higher electric and magnetic moments while respecting gauge symmetry. Finally, Lichnerowicz’s vanishing theorem in the conformal frame is also discussed.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".