Quantum Simulation of Hyperbolic Equations and the Nonexistence of a Dirac Path Measure

Quantum Physics [Submitted on 9 Apr 2026] Quantum Simulation of Hyperbolic Equations and the Nonexistence of a Dirac Path Measure Sumita Datta We revisit the longstanding issue of why no well defined probability measure exists corresponding to a classical (Kolmogorov) path integral representation of the Dirac equation in Minkowski space. Two complementary perspectives are compared: (i) Zastawniak's observation that the distributional character of the Dirac propagator (presence of derivatives of the delta distribution) obstructs the construction of a nonnegative transition kernel, and (ii) the indefinite signature of the Minkowski metric which prevents positivity of the action and yields oscillatory integrals. We show how these viewpoints can be unified as different manifestations of a single mathematical obstruction from measure theoretical point of view, and we discuss consequences for stochastic representations of relativistic first-order equations. Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2604.07847 [quant-ph]   (or arXiv:2604.07847v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2604.07847 Focus to learn more Submission history From: Sumita Datta Dr [view email] [v1] Thu, 9 Apr 2026 05:59:17 UTC (1,294 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-04 References & Citations INSPIRE HEP NASA ADS 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 Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)