Restricted Weyl invariance in four-dimensional curved spacetime
Bibliographic record
Abstract
We discuss the physics of restricted Weyl invariance, a symmetry of dimensionless actions in four-dimensional curved space time. When we study a scalar field nonminimally coupled to gravity with Weyl(conformal) weight of $\ensuremath{-}1$ (i.e. scalar field with the usual two-derivative kinetic term), we find that dimensionless terms are either fully Weyl invariant or are Weyl invariant if the conformal factor $\mathrm{\ensuremath{\Omega}}(x)$ obeys the condition ${g}^{\ensuremath{\mu}\ensuremath{\nu}}{\ensuremath{\nabla}}_{\ensuremath{\mu}}{\ensuremath{\nabla}}_{\ensuremath{\nu}}\mathrm{\ensuremath{\Omega}}=0$. We refer to the latter as restricted Weyl invariance. We show that all the dimensionless geometric terms such as ${R}^{2}$, ${R}_{\ensuremath{\mu}\ensuremath{\nu}}{R}^{\ensuremath{\mu}\ensuremath{\nu}}$ and ${R}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\sigma}\ensuremath{\tau}}{R}^{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\sigma}\ensuremath{\tau}}$ are restricted Weyl invariant. Restricted Weyl transformations possesses nice mathematical properties such as the existence of a composition and an inverse in four-dimensional space-time. We exemplify the distinction among rigid Weyl invariance, restricted Weyl invariance and the full Weyl invariance in dimensionless actions constructed out of scalar fields and vector fields with Weyl weight zero.
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".