A tutorial on Dirac quantisation by analysing the problem of a ball on an inclined plane as a Hamiltonian system with constraints
Summarize this article with:
Quantum Physics arXiv:2605.28878 (quant-ph) [Submitted on 26 May 2026] Title:A tutorial on Dirac quantisation by analysing the problem of a ball on an inclined plane as a Hamiltonian system with constraints Authors:M. F. Araujo de Resende, Thales Machado F View a PDF of the paper titled A tutorial on Dirac quantisation by analysing the problem of a ball on an inclined plane as a Hamiltonian system with constraints, by M. F. Araujo de Resende and 1 other authors View PDF HTML (experimental) Abstract:In this paper, we present a detailed review/analysis of the Dirac quantisation of Hamiltonian systems with constraints. To this end, we use, as a guide, the physical example provided by the dynamics of a solid ball rolling, without slipping, down an inclined plane under the action of gravity. After all, however simple this physical system may be, it provides a rich framework for this analysis since, in addition to allowing us to discuss scenarios involving holonomic and non-holonomic constraints, it is also a gauge system. Indeed, due to this latter fact, we have carefully detailed how the transition, from classical to quantum mechanics, must be guided by the Dirac-Bergmann algorithm and by the consequent replacement of Dirac brackets with commutators. As a central result, we demonstrate that the restriction of the Hamiltonian operator of this system with constraints to the physical Hilbert subspace (which is identified with the quantisation of these constraints) reproduces the same Schrödinger equation that can be originally obtained in intrinsic terms, a fact that only reinforces the consistency of the Dirac quantisation method. Comments: Subjects: Quantum Physics (quant-ph); High Energy Physics - Theory (hep-th); Classical Physics (physics.class-ph) Cite as: arXiv:2605.28878 [quant-ph] (or arXiv:2605.28878v1 [quant-ph] for this version) https://doi.org/10.48550/arXiv.2605.28878 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Maria Fernanda Araujo de Resende [view email] [v1] Tue, 26 May 2026 07:42:09 UTC (3,612 KB) Full-text links: Access Paper: View a PDF of the paper titled A tutorial on Dirac quantisation by analysing the problem of a ball on an inclined plane as a Hamiltonian system with constraints, by M. F. Araujo de Resende and 1 other authorsView PDFHTML (experimental)TeX Source view license Current browse context: quant-ph new | recent | 2026-05 Change to browse by: hep-th physics physics.class-ph References & Citations INSPIRE HEP NASA ADSGoogle Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: 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?) 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?)