Matrix logarithms and range of the exponential maps for the symmetry groups SL(2,R),SL(2,C) , and the Lorentz group
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Bibliographic record
Abstract
Abstract Physicists know that covering the continuously connected component <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mrow> <mml:mi mathvariant="italic"></mml:mi> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>↑</mml:mo> </mml:mrow> </mml:msubsup> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>ln</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:munder accentunder="true"> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="true">̲</mml:mo> </mml:mrow> </mml:munder> <mml:mo stretchy="false">)</mml:mo> </mml:math> of a general matrix <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:munder accentunder="true"> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="true">̲</mml:mo> </mml:mrow> </mml:munder> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>ln</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:munder accentunder="true"> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="true">̲</mml:mo> </mml:mrow> </mml:munder> <mml:mo stretchy="false">)</mml:mo> </mml:math> for all matrices <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:munder accentunder="true"> <mml:mrow> <mml:mi>M</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="true">̲</mml:mo> </mml:mrow> </mml:munder> </mml:math> in the fundamental representations of the non-compact groups <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic">SL</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi mathvariant="italic">SL</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="double-struck">C</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> , and SO (1, 2). The construction for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic">SL</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="double-struck">C</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic">SL</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="italic">SL</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="double-struck">C</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> . Therefore we revisit the maximal neighbourhoods <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="italic"></mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>⊂</mml:mo> <mml:mi mathvariant="italic">SL</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="double-struck">R</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| 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