What Neural Network Field Theory Can and Cannot Realise on a Computer

[2608.21523] What Neural Network Field Theory Can and Cannot Realise on a Computer Skip to main content Search Submit Donate Log in High Energy Physics - Theory arXiv:2608.21523 (hep-th) [Submitted on 21 Aug 2026 ( v1 ), last revised 27 Aug 2026 (this version, v2)] Title: What Neural Network Field Theory Can and Cannot Realise on a Computer Authors: Thomas R. Harvey View PDF HTML (experimental) Abstract: One aim of neural network field theory is to put a quantum or effective field theory on a computer, with the network ensemble itself as the theory. We ask how far that aim can be pushed for a function class regular enough to be computed with. Our main result is a no-go theorem with assumptions that hold for standard network architectures. We use it to separate four versions of neural network field theory, according to whether the defining object is the finite width ensemble or its infinite width limit, and whether the target we want to compute is a quantum or an effective field theory. Neither finite width interpretation is straightforwardly consistent. For finite width ensembles with finite variance at each point, the QFT interpretation fails reflection positivity, while the EFT interpretation establishes no scale separation by which the positivity violation can be placed outside its domain of validity. Of the two limit versions, one can be simulated in full and the other only in part, as only its smeared correlators are computable with a controlled error. As such, at the level of a controlled numerical computation, the QFT and EFT versions cannot be distinguished. One dimension escapes the obstruction, yet reflection positivity is shown to still fail there at every finite width for the cosine network. Two escapes from the theorem remain, giving up either finite variance at a point or exact rotation invariance, and we discuss both of these possibilities. Subjects: High Energy Physics - Theory (hep-th) ; Machine Learning (cs.LG); High Energy Physics - Lattice (hep-lat); High Energy Physics - Phenomenology (hep-ph); Mathematical Physics (math-ph) Cite as: arXiv:2608.21523 [hep-th] (or arXiv:2608.21523v2 [hep-th] for this version) https://doi.org/10.48550/arXiv.2608.21523 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Thomas Harvey [ view email ] [v1] Fri, 21 Aug 2026 18:00:10 UTC (34 KB) [v2] Thu, 27 Aug 2026 20:37:50 UTC (34 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: hep-th < prev | next > new | recent | 2026-08 Change to browse by: cs cs.LG hep-lat hep-ph math math-ph math.MP References & Citations INSPIRE HEP SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) IArxiv recommender toggle IArxiv Recommender ( What is IArxiv? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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