Quaternionic Quantum Mechanics: the Particles, Their q-Potentials and Mathematical Electron Model
DOI:
https://doi.org/10.7494/jcme.2026.10.1.16-29Keywords:
ideal elastic solid, electron, quaternionic potential, vectorial potential, proton, quark chainsAbstract
In this work we show the quaternionic quantum descriptions of physical processes from the Planck to macro scale. The results presented here are based on the concepts of the Cauchy continuum and the elementary cell at the Planck scale. The structurally symmetric quaternion relations and the postulate of the quaternion velocity have been important in the present development. The momentum of the expansion and compression u̇0(t, x) is the consequence of the scalar term σ0(t, x) in the quaternionic deformation potential. The quaternionic G0(m)(σ0 + φ̂ ), vectorial G0(m)φ̂ and scalar G0(m)σ0 propagators are used to generate the second order PDE systems for the proton, electron and neutron. A mathematical model of an electron is formulated. It is described by the hyperbolic-elliptic partial differential system of quaternion equations with the initial-boundary conditions. The boundary conditions are generated by the quaternion energy flux that is found with the use of the Gauss theorem, the Cauchy–Riemann derivative and other mathematical formulas. The rigorous assessment of the second order PDE systems allows the proposal of two second order PDE systems for the u and d quarks from the up and down groups. It was verified that both the proton and the neutron obey experimental findings and are formed by three quarks. The proton and neutron are formed by the d-u-u and d-d-u complexes, respectively. The u and d quarks do not comply with the Cauchy equation of motion. The inconsistencies of the quarks’ PDE with the quaternion forms of the Cauchy equation of motion account for their short lifetime and the observed Quarks Chains. That is, they explain the Wilczek phenomenological paradox: Quarks are Born Free, but everywhere they are in Chains.
Downloads
References
[1] von Neumann J. (1996). Mathematische Grundlagen der Quantenmechanik. Zweite Auflage. Berlin – Heidelberg – New York: Springer. DOI: https://doi.org/10.1007/978-3-642-61409-5.
[2] Birkhoff G. & von Neumann J. (1936). The logic of quantum mechanics. Annals of Mathematics, 37(4), 823–843. DOI: https://doi.org/10.48550/arXiv.2510.12502.
[3] Lanczos C. (1919). Die Funktionentheoretischen Beziehungen der Maxwellschen Æthergleichungen Ein Beitrag zur Relativitäts und Elektronentheorie, Verlagsbuchhandlung Josef Németh, Budapest. In: Davis W.R., Chu M.T., Dolan P., McConnell J.R., Norris L.K., Ortiz E., Plemmons R.J., Ridgeway D.,Scaife B.K.P., Stewart W.J., York J.W.Jr., Dogget W.O., Gellai B.M, Gsponer A.A. & Prioli C.A. (Eds.), Cornelius Lanczos Collected Published Papers with Commentaries, VI, Releigh, NC, USA: North Carolina State University, A1-A82.
[4] Lanczos C. (1932). Electricity as a natural property of Riemannian geometry. Physical Review, 39(4), 716–736. DOI: https://doi.org/10.1103/PhysRev.39.716.
[5] Lanczos C. (1933). Die Wellenmechanik als Hamiltonsche Dynamik des Funktionraumes. Eine neue Ableitung der Diracschen Gleichung. Zeitschrift für Physik, 81, 703–732, DOI: https://doi.org/10.1007/BF01342068.
[6] Fueter R. (1935). The Funktionentheorie der Differentialgleichungen Δu = 0 und ΔΔu = 0 mit vier reellen Variablen. Commentarii Mathematici Helvetici, 7, 307–330. URL: http://eudml.org/doc/138642.
[7] Adler S.L. (1996). Quaternionic quantum mechanics and noncommutative dynamics. DOI: https://doi.org/10.48550/arXiv.hep-th/9607008.
[8] Christianto V. (2006). A new wave quantum relativistic equation from quaternionic representation of Maxwell–Dirac equation as an alternative to Barut–Dirac equation. Electronic Journal of Theoretical Physics, 3 (12), 117–144. URL: https://vixra.org/pdf/1411.0300v1.pdf.
[9] Harari H. (1979). A schematic model of quarks and leptons. Physics Letters B, 86(1), 83–86. DOI: https://doi.org/10.1016/0370-2693(79)90626-9.
[10] Adler S.L. (1994). Composite leptons and quarks constructed as triply occupied quasiparticles in quaternionic quantum mechanics. Physics Letters B, 332(3–4), 358–365. DOI:https://doi.org/10.1016/0370-2693(94)91265-3.
[11] Horwitz L.P. & Biedenharn L.C. (1984). Quaternion quantum mechanics: Second quantization and gauge fields. Annals of Physics, 157(2), 432–488. DOI: https://doi.org/10.1016/0003-4916(84)90068-X.
[12] Adler S.L. (1995). Quaternionic quantum mechanics and quantum fields. Oxford: Oxford University Press. DOI: https://doi.org/10.1093/oso/9780195066432.001.0001.
[13] Arbab A. (2011). The quaternionic quantum mechanics Applied Physics Research, 3(2), 160-170. DOI: https://doi.org/10.5539/apr.v3n2p160.
[14] Arbab A. & Al Ajmi M. (2018). The quaternioning commutator bracket and its implications. Symmetry, 10(10), 513. DOI: https://doi.org/10.3390/sym10100513.
[15] Giardino S. (2025). Self-interacting quantum particles. International Journal of Geometric Methods in Modern Physics, 22(12), 2550088. DOI: https://doi.org/10.1142/S0219887825500884.
[16] Giardino S. (2023). Generalized imaginary units in quantum mechanics. URL: https://arXiv:2311.14162.
[17] Danielewski M. (2007). The Planck–Kleinert Crystal. Zeitschrift für Naturforschung A, 62(10–11), 564–568. DOI: https://doi.org/10.1515/zna-2007-10-1102.
[18] Danielewski M. & Sapa L. (2020). Foundations of the quaternion quantum mechanics. Entropy, 22(12), 1424, 14–24. DOI: https://doi.org/10.3390/e22121424.
[19] Danielewski M. & Sapa L. (2017). Nonlinear Klein–Gordon equation in Cauchy–Navier elastic solid. Вісник Черкаського університету. Серія: Фізико-математичні науки, 1, 22–29.
[20] Danielewski M. (2022). An ontological basis for the diffusion theory. Journal of Phase Equilibria and Diffusion, 43, 883–893. DOI: https://doi.org/10.1007/s11669-022-01006-y.
[21] Danielewski M., Sapa L. & Roth Ch. (2023). Quaternion quantum mechanics II: Resolving the problems of gravity and imaginary numbers. Symmetry, 15(9), 1672. DOI: https://doi.org/10.3390/sym15091672.
[22] Başkal S., Kim Y.S. & Noz M.E. (2015) Physics of the Lorentz Group. San Rafael, CA, USA: Morgan & Claypool Publishers. DOI: https://doi.org/10.1088/978-1-6817-4254-0.
[23] Gürlebeck K. & Sprössig W. (1997). Quaternionic and Clifford Calculus for Physicists and Engineers; Mathematical Methods in Practice. New York: Wiley.
[24] Altmann S.L. (1986). Rotations, Quaternions, and Double Groups. New York: Clarendon Press.
[25] Graves R.P. (1989). Life of Sir William Rowan Hamilton. Dublin: Hodges, Figgis & Co.
[26] Gürlebeck K. & Sprössig W. (1989) Quaternionic Analysis and Elliptic Boundary Value Problems. Berlin: Akademie-Verlag. DOI: https://doi.org/10.1002/zamm.19920720117.
[27] Cauchy A.L. (1823). Récherches sur L’équilibre et le Mouvement Intérieur des Corps Solides ou Fluides, Élastiques ou non Élastiques. Bulletin de la Société Philomathique de Paris, 9, 300–304.
[28] Poisson D. (1829). Mémoire sur L’équilibre et le Mouvement des Corps Élastiques. Mémoires de l’Académie des Sciences de Paris, 8, 357–570.
[29] Neumann F. (1885). Vorlesungen über die Theorie der Elasticität der Festen Körper und des Lichtäthers. Leipzig: B.G. Teubner.
[30] Duhem P. (1899). Sur l’intégrale des équations des petits mouvements d’un solide isotrope. Mémoires de la Société des Sciences de Bordeaux, Ser. V, t. III, 317– 329.
[31] Love A.E.H. (1944). Mathematical Theory of Elasticity, 4th Edition. New York: Dover Publications Inc.
[32] Marder M.P. (2000). Condensed Matter Physics. New York: John Wiley & Sons.
[33] Landau L.D. & Lifshitz E.M. (1986). Theory of Elasticity, 3rd Edition. Amsterdam: Butterworth-Heinemann Elsevier Ltd.
[34] National Institute of Standards and Technology (2018). URL: https://www.nist.gov/physics.
[35] Feynman R. (1963). The Feynman Lectures in Physics, Vol. 1. Boston: Addison-Wesley Publishing Co.
[36] Glötzl E. & Richters E.O. (2023). Helmholtz decomposition and potential functions for n-dimensional analytic vector fields. Journal of Mathematical Analysis and Applications, 525(2), 127–138. DOI: https://doi.org/10.1016/j.jmaa.2023.127138.
[37] Bodurov T. (1988). Generalized Ehrenfest theorem for nonlinear Schrödinger equations. International Journal of Theoretical Physics, 37, 1299–1306. DOI: https://doi.org/10.1023/A:1026632006040.
[38] Zeidler E. (1990). Nonlinear Functional Analysis and Its Applications II/A: Linear Monotone Operators. New York: Springer. DOI: https://doi.org/10.1007/978-1-4612-0985-0.
[39] Dirac P.A.M. (1952). Is there an aether? Nature, 168(4282), 906–907.
[40] Reid C. (1969). Hilbert. Berlin: Springer-Verlag.
[41] Derbes D. (1996). Feynman’s derivation of the Schrödinger equation. American Journal of Physics, 64(7), 881–884. URL: https://web.physics.utah.edu/~starykh/phys7640/Lectures/FeynmansDerivation.pdf.
[42] Wilczek F. (2005). Nobel lecture: Asymptotic freedom: from paradox to paradigm. Reviews of Modern Physics, 77(3), 857–870. DOI: https://doi.org/10.1103/RevModPhys.77.857.
[43] Kleinert H. (1989). Gauge Fields in Condensed Matter, Vol. II, Stresses and Defects. Singapore: World Scientific, Singapore. URL: https://hagenkleinert.de/documents/gf/HagenKleinert_GaugeFieldsInCondensedMatter_Part3.pdf.
Downloads
Published
Issue
Section
License
Copyright (c) 2026 Bogusław Bożek, Marek Danielewski, Lucjan Sapa
This work is licensed under a Creative Commons Attribution 4.0 International License.
