Physical and Mathematical properties of some Deformations of Heisenberg Algebra
| dc.contributor.advisor | Reyes Villamil, Milton Armando | |
| dc.contributor.author | Jaramillo Quiceno, Julio César | |
| dc.contributor.orcid | Jaramillo Qiuceno, Julio César [0000-0002-3518-6680] | |
| dc.contributor.researchgate | https://www.researchgate.net/profile/Julio-Jaramillo?ev=hdr_xprf | |
| dc.date.accessioned | 2025-09-02T15:56:22Z | |
| dc.date.available | 2025-09-02T15:56:22Z | |
| dc.date.issued | 2025-04 | |
| dc.description | ilustraciones, diagramas, fotografías | |
| dc.description.abstract | En este trabajo se analizan las propiedades físicas y matemáticas de algunas deformaciones del álgebra de Heisenberg. Inicialmente, se definen las álgebras libres, seguidas de la versión clásica del álgebra de Heisenberg y algunas de sus versiones deformadas. A continuación, se abordan las propiedades algebraicas y combinatorias de estas versiones deformadas. Además, se presenta la conexión con el operador de Dirac y su relación con las versiones deformadas de algunas álgebras de Heisenberg, se exploran también las relaciones con el espacio de tensores ⊗2 , se propone un nuevo tipo de álgebra de Weyl y se mencionan otras posibles conexiones con diversas álgebras. Finalmente, se discuten algunas aplicaciones en física y se dejan abiertas las posibilidades para investigaciones futuras basadas en este trabajo. (Texto tomado de la fuente) | spa |
| dc.description.abstract | In this work, the physical and mathematical properties of some deformations of the Heisenberg algebra are analyzed. Initially, free algebras are defined, followed by the classical version of the Heisenberg algebra and some of its deformed versions. Next, the algebraic and combinatorial properties of these deformed versions are addressed. Additionally, the connection with the Dirac operator and its relationship with the deformed versions of certain Heisenberg algebras is presented. The relationships with the tensor space ⊗2 are also explored, a new type of Weyl algebra is proposed, and other possible connections with various algebras are mentioned. Finally, some applications in physics are discussed, and possibilities for future research based on this work are left open | eng |
| dc.description.curriculararea | Física.Sede Bogotá | |
| dc.description.degreelevel | Maestría | |
| dc.description.degreename | Magister en Ciencias Físicas | |
| dc.format.extent | viii, 72 páginas | |
| dc.format.mimetype | application/pdf | |
| dc.identifier.instname | Universidad Nacional de Colombia | spa |
| dc.identifier.reponame | Repositorio Institucional Universidad Nacional de Colombia | spa |
| dc.identifier.repourl | https://repositorio.unal.edu.co/ | spa |
| dc.identifier.uri | https://repositorio.unal.edu.co/handle/unal/88539 | |
| dc.language.iso | eng | |
| dc.publisher | Universidad Nacional de Colombia | |
| dc.publisher.branch | Universidad Nacional de Colombia - Sede Bogotá | |
| dc.publisher.faculty | Facultad de Ciencias | |
| dc.publisher.place | Bogotá, Colombia | |
| dc.publisher.program | Bogotá - Ciencias - Maestría en Ciencias - Física | |
| dc.relation.references | H. Ahmed, U. Bekbaev, and I. Rakhimov. Classification of two-dimensional left and right unital algebras over algebraically closed fields and R. Journal of Physics: Conferences Series, 1489, 2020. V | |
| dc.relation.references | M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, 1964 | |
| dc.relation.references | A. Barut and A. Bracken. Exact Solutions of the Heisenberg Equations and Zitterbewegung of the Electron in a Constant Uniform Magnetic Field. Aust. J. Phys, (35):353–370, 1982. 51 | |
| dc.relation.references | K. Brown and K. R. Goodearl. Lectures on Algebraic Quantum Groups. Birkäuser-Verlag, 2002. 3 | |
| dc.relation.references | J. Bueso, J. Gómez-Torrecillas, and A. Verschoren. Algorithmic Methods in Non-Commutative Algebra: Applications to Quantum Groups. Mathematical Modelling: Theory and Applications, Springer Dordrecht, 2003. 2, 3 | |
| dc.relation.references | V. Bardek and S. Meljanac. Deformed Heisenberg algebras, a Fock space representation and the Calogero model. European Physical Journal C, (17):539– 547, 2000 | |
| dc.relation.references | J. Bagger and H. Nicolai. Julius Erich Wess. Physics Today, 1(52), 2009 | |
| dc.relation.references | M. Chaichian, R. Gonzalez, and C. Montonen. Statistics of q-oscillators, quons and relations to fractional statistics. Journal of Physics A: Mathematical General, 26:417–4034„ 1993 | |
| dc.relation.references | B. Cerchiai, R. Hinterding, J. Madore, and J. Wess. A calculus based on a q-deformed Heisenberg algebra. European Physical Journal C, (3):547– 558, 1999. 50, 51 | |
| dc.relation.references | X. Chen and M. Lu. Varieties of modules over the quantum plane. Journal of Algebra, 580:158–198, 2021. V, 9, 41 | |
| dc.relation.references | P. M. Cohn. Free Rings and Their Relations, volume 2. Academic Press, London, 1985. 5 | |
| dc.relation.references | B. Cooperstein. Advanced Linear Algebra. CRC Press, second edition, 2015. V, 11 | |
| dc.relation.references | S. C. Coutinho. A Primer of Algebraic D-modules. Cambridge University Press. London Mathematical Society, Student Texts 33, 1995. 3 | |
| dc.relation.references | P. A. M. Dirac. Fundamental Equations of Quantum Mechanics. Proceedings A Royal Society, 109(752):642–653, 1925. 6 | |
| dc.relation.references | P. A. M. Dirac. The Principles of Quantum Mechanics. Clarendon Press, 1958. 48 | |
| dc.relation.references | R. Díaz and E. Pariguan. On the q-meromorphic Weyl algebra. São Paulo Journal of Mathematical Sciences, 3(2):283–298, 2009. 3 | |
| dc.relation.references | W. Fajardo, C. Gallego, O. Lezama, A. Reyes, H. Suárez, and H. Venegas. Skew PBW Extensions: Ring and Module-Theoretic Properties, Matrix and Gröbner Methods, and Applications, volume 28. Algebra and Applications. Springer, Cham, 2020 | |
| dc.relation.references | W. Fajardo, O. Lezama, C. Payares, A. Reyes, and C. Rodríguez. Introduction to Algebraic Analysis on Ore Extensions. In A. Martsinkovsky, editor, Functor Categories, Model Theory, Algebraic Analysis and Constructive Methods FCMTCCT2 2022, Almería, Spain, July 11-15, Invited and Selected Contributions, volume 450 of Springer Proceedings in Mathematics & Statistics, pages 45–116. Springer, Cham, 2024 | |
| dc.relation.references | J. Gaddis. Two-parameter analogs of the Heisenberg Enveloping Algebra. Communications in Algebra, 44(11):4637–4653., 2016. 11 | |
| dc.relation.references | D. Galetti. A realization of the q-Deformed Harmonic Oscillator: Rogers-Szego and Stieltjes - Wigert Polynomials. Brazilian Journal of Physics, 33(1), 2003 | |
| dc.relation.references | K. R. Goodearl and R. B. Warfield Jr. An Introduction to Noncommutative Noetherian Rings, volume 61 of London Mathematical Society Student Texts. Cambridge University Press, 2004 | |
| dc.relation.references | K. R. Goodearl and E.S. Letzter. Prime Ideals in Skew and q-Skew Polynomial Rings, volume 109 of Memoirs of the American Mathematical Society 521. Providence, Rhode Island: American Mathematical Society, Second edition, 1994. 3 | |
| dc.relation.references | K. R. Goodearl and R. B. Warfield. An Introduction to Noncommutative Noetherian Rings. Cambridge University Press, 2004. | |
| dc.relation.references | A. Giaquinto and J. J. Zhang. Quantum Weyl Algebras. Journal of Algebra, 176(3):861–881, 1995. | |
| dc.relation.references | B. Hall. Lie Groups, Lie Algebras, and Representations. Springer, 2010 | |
| dc.relation.references | Werner Heisenberg. Uber quantentheoretische umdeutung kinematischer und mechanischer beziehungen. Zeitschrift für Physik, (33):873–879, 1925 | |
| dc.relation.references | I. Volovich I. Y. Aref’eva. Quantum groups particles and non-archimedean geometry. Physics Letters B, 268(2):179–187, 1991 | |
| dc.relation.references | C. Itzykson and J. B. Zuber. Quantum Field Theory. McGraw Hill Interational Book Company, 1980 | |
| dc.relation.references | F. H. Jackson. On q-functions and a certain difference operator. Messenger of Mathematics, 38:57, 1909 | |
| dc.relation.references | J. C. Jaramillo. An approach to derivatives for non-monogenic functions. Revista Integración. Temas de matemáticas, 42(2):11–21, 2024. | |
| dc.relation.references | J. C. Jaramillo. q-differential Operators and Derivations on the Quadratic Relativistic Invariant Algebras. Revista Momento, 69:116–135, 2024 | |
| dc.relation.references | J. C. Jaramillo. q-relativistic wave equation of the form i∂ q · ψq + mψ0 = Eψ. Revista Integración. Temas de matemáticas, 42(2):31 – 43, 2024 | |
| dc.relation.references | J. C. Jaramillo. q-Deformed Heisenberg Picture Equation. arXiv:2504.09829, pages 1 – 8, 2025 | |
| dc.relation.references | J. C. Jaramillo. q-Heisenberg Algebra in ⊗2 -Tensor Space. arXiv:2504.10741, pages 1 – 9, 2025 | |
| dc.relation.references | V. A. Jategaonkar. A Multiplicative Analogue of the Weyl Algebra. Communications in Algebra, 12(14):1669–1688, 1984. | |
| dc.relation.references | A. Jannussis, G. Brodimas, and D. Sourlas. Remarks on the q-quantization. Lettere al Nuovo Cimento, 30:123–127, 1981 | |
| dc.relation.references | J. C. Jaramillo and P. Nechev. On the Deformed Algebras Applications to Quantum Physics. arXiv:2504.04606, pages 1 – 10, 2025 | |
| dc.relation.references | D. A. Jordan. Finite-dimensional simple modules over certain iterated skew polynomial rings. Journal of Pure and Applied Algebra, 98(1):45–55, 1995 | |
| dc.relation.references | D. A. Jordan. Down-Up Algebras and Ambiskew Polynomial Rings. Journal of Algebra, 228(1):311–346, 2000 | |
| dc.relation.references | B. Zumino J. Wess. Covariant Differential Calculus on the quantum hyperplane. Nuclear Physics B - Proceedings Supplements, 18:302–312, 1991 | |
| dc.relation.references | D. A. Jordan and I. Wells. Invariants for automorphisms of certain iterated skew polynomial rings. Proceedings of the Edinburgh Mathematical Society, 39(3):461–472, 1996 | |
| dc.relation.references | N. P. Kisby. Números de Stirling. Revista de Eduación Matemática, 13(2), 1998. | |
| dc.relation.references | M. V. Kuryshkin. Opérateurs quantiques généralisés de création et d’annihilation. Annales de la Fondation Louis de Broglie, 5:111–125, 1980 | |
| dc.relation.references | A. Lavagno and G. Gervino. Quantum Mechanics in q-deformed calculus. Journal of Physics Conference series, 174, 2009. | |
| dc.relation.references | H. Li. Noncommutative Gröbner Bases and Filtered-Graded Transfer. Springer Verlag, 2000 | |
| dc.relation.references | S. Lopes. Noncommutative Algebra and Representation Theory: Symmetry, Structure & Invariants. Communications in Mathematics, 32(3):63–117, 2024 | |
| dc.relation.references | A. Lorek, A. Ruffing, and J. Wess. A q-Deformation of the Harmonic Oscillator. Z Phys C - Particles and Fields, (74):369–377, 1997 | |
| dc.relation.references | A. Lavagno, A. Scarfone, and N. Swamy. Classical and quantum q-deformed physical systems. European Physical Journal C, (47):253–261, 2006 | |
| dc.relation.references | Y. I. Manin. Quantum Groups and Noncommutative Geometry. With a Contribution by Theo Raedschelders and Michel Van den Bergh. Centre de Recherches Mathématiques Short Courses. Springer Cham, Second edition, 2018 | |
| dc.relation.references | A. Marinho, F. Brito, and C. Chesman. Thermal properties of a solid through q-deformed algebra. Physica A, (391):3424– 3434, 2012 | |
| dc.relation.references | J. C. McConnell and J. J. Pettit. Crossed Products and Multiplicative Analogues of Weyl Algebras. Journal of the London Mathematical Society, s2–38(1):47–55, 1988 | |
| dc.relation.references | J. McConnell and J. Robson. Noncommutative Noetherian Rings, volume 30 of Graduate Studies in Mathematics. American Mathematical Society, Second edition, 2001 | |
| dc.relation.references | T. Mansour and M. Schork. Commutation Relations, Normal Ordering, and Stirling Numbers. CRC Press Francis and Taylor Group, 2016. | |
| dc.relation.references | Y. Miao and Z. Xu. Investigation of non-hermitian Hamiltonians in the Heisenberg picture. Physics Letters A, (380):1805–1810., 2016 | |
| dc.relation.references | Zettili N. Quantum Mechanics: concepts and applications. John Wiley and Sons, Ltd, 2009. | |
| dc.relation.references | E. Noether and W. Schmeidler. Moduln in nichtkommutativen bereichen, insbesondere aus differential - und differenzenausdrücken. Math. Zeitschrift, 8:1–35, 1920 | |
| dc.relation.references | M. Ngu, N. Vinh, N. Lan, L. Thanh, and N. Viet. First, Second Quantization and Q-deformed Harmonic Oscillator. Journal of Physics and Conferences Series, 2016 | |
| dc.relation.references | L. Ouyang, J. Liu, and Y. Xiang. Σ-associated prime over extension rings. Journal of Mathemathics Reserarch Applied, 35(5):505–520, 2015 | |
| dc.relation.references | O. Ore. Linear Equations in Non-commutative Fields. Annals of Mathematics, 32(3):463–477, 1931 | |
| dc.relation.references | O. Ore. Theory of non-commutative polynomials. Annals of Mathematics Second Series, 34(3):480–508, 1933. | |
| dc.relation.references | Nobel Prize Outreach. Werner heisenberg – biographical. Nobel Prize Website, 1932. | |
| dc.relation.references | Nobel Prize Outreach. Paul dirac – biographical. Nobel Prize Website, 1933. | |
| dc.relation.references | R. Porter. Quantum Field Theory and Introduction to Chemical Physicist. World Scientific, 2021. | |
| dc.relation.references | M. Pimentel, T. Suzuki, and L. Tomazelli. QED3 radiative corrections in the Heisenberg picture. International Journal of Theoretical Physics, (33):2199 – 2206, 1999 | |
| dc.relation.references | Facts from Functional Analysis. Pure and applied mathematics, 74(25):3–63, 1977 | |
| dc.relation.references | G. Ramesh. Banach Algebra. 2013 | |
| dc.relation.references | A. Reyes. Jacobson conjecture and skew PBW extensions. Revista Integración Temas de Matemáticas, 32(2):139–152, 2014. | |
| dc.relation.references | A. Reyes. Uniform dimension over skew PBW extensions. Rev. Colombiana Mat., 48(1):79–96, 2014. | |
| dc.relation.references | F. Razavinia and S. Lopes. Structure and isomorphisms of quantum generalized Heisenberg algebras. Journal of Algebra and Its Applications, 21(10), 2022 | |
| dc.relation.references | A. Rosenberg. Non-commutative Algebraic Geometry and Representations of Quantized Algebras. Springer Dordrecht, 1995 | |
| dc.relation.references | E. Schrodinger. An Undulatory Theory of the Mechanics of Atoms and Molecules. Physical Review., 28(6):1049–1070, 1926 | |
| dc.relation.references | A. Schirrmacher. Quantum Groups, Quantum Spacetime, and Dirac Equation, volume 315. Springer Link, 1993. | |
| dc.relation.references | K. Schmüdgen. Operator Representations of a q-Heisenberg algebra. arXiv:math/9805131 [math.QA], 1:1–17, 1998 | |
| dc.relation.references | L. Hellström S. D. Silvestrov. Commuting Elements in q-Deformed Heisenberg Algebra. World Scientific, 2000 | |
| dc.relation.references | W. M. Seiler. Involution. The Formal Theory of Differential Equations and its Applications in Computer Algebra, volume 24 of Algorithms and Computation in Mathematics (AACIM). Springer Berlin, Heidelberg, 2010. | |
| dc.relation.references | A. Sinha. Dispersion relations, knots polynomials, and the q- deformed harmonic oscillator. Physical Review D, 106(126019), 2022. | |
| dc.relation.references | S. P. Smith. Quantum Groups: An Introduction and Survey for Ring Theorists. In S. Montgometry and L. Small, editors, Noncommutative Rings, volume 24 of Mathematical Science Research Institute Publications, pages 131–178. SringerVerlag, New York, 1992 | |
| dc.relation.references | S. Stenberg. Lectures on Differential Geometry. Prentice Hall, 1972 | |
| dc.relation.references | D. Tristand and F. A. Brito. Some electronic properties of metals through q−deformed algebras. Physica A: Statistical Mechanics and its Applications, 407(1):276–286, 2014 | |
| dc.relation.references | J.A. Tuszyński, J.L. Rubin, J. Meyer, and M. Kibler. Statistical mechanics of a q−deformed boson gas. Physics Letters A, 175:173–177, 1993 | |
| dc.relation.references | I. O. Vakarchuk. On Dirac theory in the space with deformed Heisenberg algebra: exact solutions. Journal of Physics A: Mathematical and General, 38:7567–7576, 2005 | |
| dc.relation.references | V.Kac and P. Cheung. Quantum Calculus. Springer, 2002 | |
| dc.relation.references | V.Kac. Lectures 22- The Universal Enveloping Algebra. Scribe: Aaron Potechin, 2010 | |
| dc.relation.references | V. S. Vladimirov. Equations of Mathematical Physics, volume 315. Marcel Dekker, INC, New York, 1971 | |
| dc.relation.references | J. Wess. q-Deformed Heisenberg Algebra. In JHEP Proceedings, pages 1–4, 1999. I | |
| dc.relation.references | J. Wess. q-Deformed Heisenberg Algebra, volume 543. Springer, 2000. | |
| dc.relation.references | J. Wess and J. Schwenk. A q-deformed quantum mechanical toy model. Phys. Lett. B., 291(3):273–277, 1992. | |
| dc.relation.references | J Wess and B. Zumino. Covariant differential calculus on the quantum hyperplane. Nuclear Physics B Proceedings Supplements, 18(2):302–312, 1991 | |
| dc.rights.accessrights | info:eu-repo/semantics/openAccess | |
| dc.rights.license | Reconocimiento 4.0 Internacional | |
| dc.subject.ddc | 510 - Matemáticas | |
| dc.subject.lemb | Algebra | spa |
| dc.subject.lemb | Teoria cuántica | spa |
| dc.subject.lemb | Quantum theory | eng |
| dc.subject.lemb | álgebra de Heisenberg | spa |
| dc.subject.lemb | Espacios generalizados | spa |
| dc.subject.lemb | Spaces, generalized | eng |
| dc.subject.proposal | álgebra de Heisenberg, propiedades combinatoriales, mecánica cuántica | spa |
| dc.subject.proposal | Heisenberg algebra, combinatorial properties, quantum mechanics | eng |
| dc.subject.wikidata | Heisenberg algebra | eng |
| dc.title | Physical and Mathematical properties of some Deformations of Heisenberg Algebra | eng |
| dc.title.translated | Propiedades Físicas y Matemáticas de algunas deformaciones del álgebra de Heisenberg | |
| dc.type | Trabajo de grado - Maestría | |
| dc.type.coarversion | http://purl.org/coar/version/c_ab4af688f83e57aa | |
| dc.type.content | Text | |
| dc.type.driver | info:eu-repo/semantics/masterThesis | |
| dc.type.redcol | http://purl.org/redcol/resource_type/TM | |
| dc.type.version | info:eu-repo/semantics/acceptedVersion | |
| oaire.accessrights | http://purl.org/coar/access_right/c_abf2 |