Quantum theory over dual-complex numbers

Quantum Physics [Submitted on 18 Mar 2026] Quantum theory over dual-complex numbers P. Arrighi, D. Bakircioglu, N. L. Houyet We take quantum theory and replace \mathbb{C} by \mathbb{C}[\varepsilon] where \varepsilon^2=0, i.e. we extend quantum theory to the ring of dual complex numbers. The aim is to develop a common language in which to treat continuous quantum physics and discrete quantum models in a unified manner, including their symmetries. Since quantum theory is linear, introducing \varepsilon is enough to model infinitesimals. A first objection to this programme is that \mathbb{C}[\varepsilon] is not a field, since division by \varepsilon is undefined, while quantum mechanics typically relies on division. A second objection concerns whether unitarity still makes sense given \varepsilon^2 = 0. Hence, the core of this work is dedicated to proving that \dual quantum theory remains fully consistent. In particular, norm is preserved at all times, and renormalization never requires dividing by an infinitesimal. An equivalence with conventional quantum theory is demonstrated: the \dual extension of a parametrized quantum operation automatically provides a linear treatment of its first-order variations. As a first example application, we provide a unified description of both the Dirac equation in the continuum and the Dirac Quantum Walk in the discrete. We establish the discrete Lorentz covariance of the latter. Comments: 19 pages, 3 figures Subjects: Quantum Physics (quant-ph) Cite as: arXiv:2603.17581 [quant-ph]   (or arXiv:2603.17581v1 [quant-ph] for this version)   https://doi.org/10.48550/arXiv.2603.17581 Focus to learn more Submission history From: Nathan Houyet [view email] [v1] Wed, 18 Mar 2026 10:36:17 UTC (75 KB) Access Paper: HTML (experimental) view license Current browse context: quant-ph < prev   |   next > new | recent | 2026-03 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?)