Towards a more algebraic footing for quantum field theory
Bibliographic record
Abstract
Abstract The predictions of the standard model of particle physics are highly successful in spite of the fact that several parts of the underlying quantum field theoretical framework are analytically problematic. Indeed, it has long been suggested, by Einstein, Schrödinger and others, that analytic problems in the formulation of fundamental laws could be overcome by reformulating these laws without reliance on analytic methods namely, for example, algebraically. With this in mind, we focus here on the analytic ill-definedness of the quantum field theoretic Fourier and Legendre transforms of the generating series of Feynman graphs, including the path integral. To this end, we develop here purely algebraic and combinatorial formulations of the Fourier and Legendre transforms, employing rings of formal power series. These are all-purpose transform methods, i.e. their applicability is not restricted to quantum field theory. When applied in quantum field theory to the generating functionals of Feynman graphs, the new transforms are well-defined and thereby help explain the robustness and success of the predictions of perturbative quantum field theory in spite of analytic difficulties. Technically, we overcome here the problem of the possible divergence of the various generating series of Feynman graphs by constructing Fourier and Legendre transforms of formal power series that operate in a well-defined way on the coefficients of the power series irrespective of whether or not these series converge. Our new methods could, therefore, provide new algebraic and combinatorial perspectives on quantum field theoretic structures that are conventionally thought of as analytic in nature, such as the occurrence of anomalies from the path integral measure. In comparison, the use of formal power series in QFT by Bogolubov, Hepp, Parasiuk and Zimmermann concerned a different kind of divergencies, namely the UV divergencies of loop integrals and their renormalization.
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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 | 0.004 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.011 |
| Scholarly communication | 0.005 | 0.011 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.006 |
| Insufficient payload (model declined to judge) | 0.006 | 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".