Abstract
Clifford algebras and quantum mechanics are closely associated to each other. We are familiar with some popular terms, such as algebra, light polarization, and quantum spin. Dirac spinors, Majorana spinors, and Weyl spinors are discussed in particular as subspaces of Clifford algebras with some remarkable algebraic features. Furthermore, we are interested in demonstrating how the quantum spin state and classical polarization of light waves can be derived from one another along with the Bloch sphere and Poincare sphere representations.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Veblen, O.: Spinors. Science 80(2080), 415–419 (1934)
Cartan, É.: Les groupes projectifs qui ne laissent invariante aucune multiplicité plane. Bull. Soc. Math. France 41, 53–96 (1913)
De Sabbata, V., Datta, B.K.: Geometric Algebra and Applications to Physics. CRC Press (2006)
Ficek, Z.: Quantum Physics for Beginners. CRC Press (2017)
Held, A., Newman, E.T., Posadas, R.: The Lorentz group and the sphere. J. Math. Phys. 11(11), 3145–3154 (1970)
van der Waerden, B.L.: Group Theory and Quantum Mechanics, vol. 214. Springer Science & Business Media (2012)
Penrose, R., Rindler, W.: Spinors and Space-time: Volume 1, Two-spinor Calculus and Relativistic Fields, vol. 1. Cambridge University Press (1984)
Penrose, R., Rindler, W.: Spinors and Space-time. Volume 2: Spinor and Twistor Methods in Space-time Geometry (1986)
Klein, F., Sommerfeld, A.: The Theory of the Top. Volume IV: Technical Applications of the Theory of the Top, vol. 4. Springer Science & Business Media (2014)
Eaton, G.R., Eaton, S.S.: Foundations of Modern EPR. World Scientific (1998)
Sebens, C.T.: How electrons spin. Stud. Hist. Philos. Sci. B Stud. Hist. Philos. Mod. Phys. 68, 40–50 (2019)
Uhlenbeck, G.E.: Personal reminiscences. Phys. Today 29(6), 43–48 (1976)
Howard Haber Stephen Reucroft and Chris Quigg Michael Dine. What exactly is the Higgs Boson? Scientific American.
Greene, B.: How the Higgs Boson was Found. Smithsonian Magazine (2013)
Abe, K., Abgrall, N., Aihara, H., Ajima, Y., Albert, J. B., Allan, D., et al.: The T2K experiment. Nucl. Instrum. Methods Phys. Res. Sect. A Acceler. Spectrom. Detect. Assoc. Equip. 659(1), 106–135 (2011)
Kaneyuki, K., T2K Collaboration: T2K experiment. Nucl. Phys. B Proc. Suppl. 145, 178–181 (2005)
Valish, L.: Why is the Universe Made Up Almost Exclusively of Matter? neutrinos may hold the key. University of Rochester News Center (2020)
Sacchi, R., H1 Collaboration, & ZEUS Collaboration: Leading baryons at HERA. Nucl. Phys. B (Proc. Suppl.) 191, 214–220 (2009)
Trimble, V.: Existence and nature of dark matter in the universe. Annu. Rev. Astron. Astrophys. 25(1), 425–472 (1987)
Penrose, R., Rindler, W.: Spinors and Space-time: Volume 1, Two-spinor Calculus and Relativistic Fields, vol. 1. Cambridge University Press (1984)
González-Miret Zaragoza, L.: Möbius Transformations (TrabajoFindeGradoInédito). Universidad de Sevilla, Sevilla (2019)
Yur’ev, D.V.E.: Complex projective geometry and quantum projective field theory. Theoret. Math. Phys. 101(3), 1387–1403 (1994)
Griffiths, D.J., Schroeter, D.F.: Introduction to Quantum Mechanics. Cambridge University Press (2018)
Newman, M.H.A.: Hermann Weyl, 1885–1955 (1957)
Gravel, P., Gauthier, C.: Classical applications of the Klein–Gordon equation. Am. J. Phys. 79(5), 447–453 (2011)
Kaup, D.J.: Klein-gordon geon. Phys. Rev. 172(5), 1331 (1968)
Shatah, J.: Normal forms and quadratic nonlinear Klein-Gordon equations. Commun. Pure Appl. Math. 38(5), 685–696 (1985)
Furey, C.: Unified theory of ideals. Phys. Rev. D 86(2), 025024 (2012)
Bongaarts, P., Bongaarts, P.: Quantum Field Theory and Particle Physics: An Introduction. Quantum Theory: A Mathematical Approach, pp. 247–264 (2015)
Carmeli, M., Malin, S.: Theory of Spinors: An Introduction. World Scientific Publishing Company (2000)
Cottingham, W.N., Greenwood, D.A.: An Introduction to the Standard Model of Particle Physics. Cambridge University Press (2023)
Abłamowicz, R.: Construction of spinors via Witt decomposition and primitive idempotents: A review. In: Clifford Algebras and Spinor Structures: A Special Volume Dedicated to the Memory of Albert Crumeyrolle (1919–1992), pp. 113–123 (1995)
Harari, H.: Quarks and leptons. Phys. Rep. 42(4), 235–309 (1978)
Shupe, M.A.: A composite model of leptons and quarks. Phys. Lett. B 86(1), 87–92 (1979)
Raitio, R.: A model of lepton and quark structure. Phys. Scr. 22(3), 197 (1980)
Schray, J., Tucker, R.W., Wang, C.H.T.: LUCY: a Clifford algebra approach to spinor calculus approach to spinor calculus, pp. 121–143 (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Manzoor, T., Hasan, S.N. (2024). Algebraic and Quantum Mechanical Approach to Spinors. In: Gayoso MartÃnez, V., Yilmaz, F., Queiruga-Dios, A., Rasteiro, D.M., MartÃn-Vaquero, J., MierluÅŸ-Mazilu, I. (eds) Mathematical Methods for Engineering Applications. ICMASE 2023. Springer Proceedings in Mathematics & Statistics, vol 439. Springer, Cham. https://doi.org/10.1007/978-3-031-49218-1_21
Download citation
DOI: https://doi.org/10.1007/978-3-031-49218-1_21
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-49217-4
Online ISBN: 978-3-031-49218-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)