The Cohesive Path Integral and the Osterwalder-Schrader Axioms for Quantum Yang-Mills
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2025
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| _version_ | 1866902073528811520 |
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| author | Janik, John |
| author_facet | Janik, John |
| contents | <p>We establish a rigorous mathematical proof strategy for quantum Yang-Mills theory</p> <p>through the framework of cohesive (∞,1)-topoi, demonstrating that the path integral</p> <p>may be formulated as a well-defined derived pushforward π<span>!</span>(e^<span>−S</span>). Building on the an-</p> <p>alytic properties of the Yang-Mills moduli stack established in our companion work, we</p> <p>prove that this cohesive path integral converges, preserves reflection positivity, and sat-</p> <p>isfies the Osterwalder-Schrader axioms—conditional on three fundamental hypotheses:</p> <p>nuclear Fr´echet structure of the moduli stack, constructible stratification, and weighted</p> <p>compactness of strata. Our approach replaces the ill-defined functional measure DA</p> <p>with the categorically rigorous pushforward π<span>!</span>, automatically implementing gauge in-</p> <p>variance without ghost fields or gauge fixing. We demonstrate how cohesive modalities</p> <p>(Π ⊣♭ ⊣♯ ⊣♭<span>dR</span>) provide natural frameworks for renormalization and the semiclas-</p> <p>sical limit. For the vacuum sector, we derive confinement through non-perturbative</p> <p>mechanisms, establishing a mass gap via modal spectral analysis. While our results are</p> <p>necessarily conditional on deep analytic properties that remain open problems, we pro-</p> <p>vide a complete proof strategy and identify precisely which mathematical foundations</p> <p>must be established for unconditional Yang-Mills existence. This work demonstrates</p> <p>that the apparent mathematical difficulties of quantum field theory stem from inade-</p> <p>quate frameworks rather than fundamental obstacles, with the cohesive topos structure</p> <p>revealing path integration as a natural consequence of homotopy-theoretic principles.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_15662141 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Cohesive Path Integral and the Osterwalder-Schrader Axioms for Quantum Yang-Mills Janik, John <p>We establish a rigorous mathematical proof strategy for quantum Yang-Mills theory</p> <p>through the framework of cohesive (∞,1)-topoi, demonstrating that the path integral</p> <p>may be formulated as a well-defined derived pushforward π<span>!</span>(e^<span>−S</span>). Building on the an-</p> <p>alytic properties of the Yang-Mills moduli stack established in our companion work, we</p> <p>prove that this cohesive path integral converges, preserves reflection positivity, and sat-</p> <p>isfies the Osterwalder-Schrader axioms—conditional on three fundamental hypotheses:</p> <p>nuclear Fr´echet structure of the moduli stack, constructible stratification, and weighted</p> <p>compactness of strata. Our approach replaces the ill-defined functional measure DA</p> <p>with the categorically rigorous pushforward π<span>!</span>, automatically implementing gauge in-</p> <p>variance without ghost fields or gauge fixing. We demonstrate how cohesive modalities</p> <p>(Π ⊣♭ ⊣♯ ⊣♭<span>dR</span>) provide natural frameworks for renormalization and the semiclas-</p> <p>sical limit. For the vacuum sector, we derive confinement through non-perturbative</p> <p>mechanisms, establishing a mass gap via modal spectral analysis. While our results are</p> <p>necessarily conditional on deep analytic properties that remain open problems, we pro-</p> <p>vide a complete proof strategy and identify precisely which mathematical foundations</p> <p>must be established for unconditional Yang-Mills existence. This work demonstrates</p> <p>that the apparent mathematical difficulties of quantum field theory stem from inade-</p> <p>quate frameworks rather than fundamental obstacles, with the cohesive topos structure</p> <p>revealing path integration as a natural consequence of homotopy-theoretic principles.</p> |
| title | The Cohesive Path Integral and the Osterwalder-Schrader Axioms for Quantum Yang-Mills |
| url | https://doi.org/10.5281/zenodo.15662141 |