Matrix logarithms and range of the exponential maps for the symmetry groups SL(2,R),SL(2,C) , and the Lorentz group
Bibliographic record
Abstract
Abstract Physicists know that covering the continuously connected component + ↑ of the Lorentz group can be achieved through two Lie algebra exponentials, whereas one exponential is sufficient for compact symmetry groups like SU(N) or SO(N). On the other hand, both the general Baker-Campbell-Hausdorff formula for the combination of matrix exponentials in a series of higher order commutators, and the possibility to define the logarithm ln ( M ̲ ) of a general matrix M ̲ through the Jordan normal form, seem to naively suggest that even for non-compact groups a single exponential should be sufficient. We provide explicit constructions of ln ( M ̲ ) for all matrices M ̲ in the fundamental representations of the non-compact groups SL ( 2 , R ) , SL ( 2 , C ) , and SO(1, 2). The construction for SL ( 2 , C ) also yields logarithms for SO(1, 3) through the spinor representations. However, it is well known that single Lie algebra exponentials are not sufficient to cover the Lie groups SL ( 2 , R ) and SL ( 2 , C ) . Therefore we revisit the maximal neighbourhoods 1 ⊂ SL ( 2 , R ) and
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.005 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.009 | 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".