Bibliographic record
Abstract
At present, our notion of space is a classical concept. Taking the point of view that quantum theory is more fundamental than classical physics, and that space should be given a purely quantum definition, we revisit the notion of Euclidean space from the point of view of quantum mechanics. Since space appears in physics in the form of labels on relativistic fields or Schr\"odinger wave functionals, we propose to define Euclidean quantum space as a choice of polarization for the Heisenberg algebra of quantum theory. We show, following Mackey, that generically, such polarizations contain a fundamental length scale and that contrary to what is implied by the Schr\"odinger polarization, they possess topologically distinct spectra. These are the modular spaces. We show that they naturally come equipped with additional geometrical structures usually encountered in the context of string theory or generalized geometry. Moreover, we show how modular space reconciles the presence of a fundamental scale with translation and rotation invariance. We also discuss how the usual classical notion of space comes out as a form of thermodynamical limit of modular space while the Schr\"odinger space is a singular limit.
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How this classification was reachedexpand
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) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.004 |
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; both teacher heads agree on what is shown here.
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".