{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T20:57:09Z","timestamp":1775595429449,"version":"3.50.1"},"reference-count":29,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,12,28]],"date-time":"2023-12-28T00:00:00Z","timestamp":1703721600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We consider an alternative to dark matter as a potential solution to various remaining problems in physics: the addition of stochastic perturbations to spacetime to effectively enforce a minimum length and establish a fundamental uncertainty at minimum length (ML) scale. To explore the symmetry of spacetime to such perturbations both in classical and quantum theories, we develop some new tools of stochastic calculus. We derive the generators of rotations and boosts, along with the connection, for stochastically perturbed, minimum length spacetime (\u201cML spacetime\u201d). We find the metric, the directional derivative, and the canonical commutator preserved. ML spacetime follows the Lie algebra of the Poincare group, now expressed in terms of the two-point functions of the stochastic fields (per Ito\u2019s lemma). With the fundamental uncertainty at ML scale a symmetry of spacetime, we require the translational invariance of any classical theory in classical spacetime to also include the stochastic spacetime perturbations. As an application of these ideas, we consider galaxy rotation curves for massive bodies to find that\u2014under the Robertson\u2013Walker minimum length theory\u2014rotational velocity becomes constant as the distance to the center of the galaxy becomes very large. The new tools of stochastic calculus also set the stage to explore new frontiers at the quantum level. We consider a massless scalar field to derive the Ward-like identity for ML currents.<\/jats:p>","DOI":"10.3390\/sym16010036","type":"journal-article","created":{"date-parts":[[2023,12,28]],"date-time":"2023-12-28T03:30:34Z","timestamp":1703734234000},"page":"36","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["The Algebra and Calculus of Stochastically Perturbed Spacetime with Classical and Quantum Applications"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4210-2780","authenticated-orcid":false,"given":"Dragana","family":"Pilipovi\u0107","sequence":"first","affiliation":[{"name":"Department of Physics, University of Illinois Chicago, Chicago, IL 60607, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,12,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1016\/j.physletb.2006.08.021","article-title":"The emergence of background geometry from quantum fluctuations","volume":"641","author":"Janik","year":"2006","journal-title":"Phys. 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