PL EN


Preferences help
enabled [disable] Abstract
Number of results
2020 | 149 | 23-35
Article title

Quaternionic operational infinity

Authors
Content
Title variants
Languages of publication
EN
Abstracts
EN
By analogy to the complex analytic representation of infinity that I have already proposed in my prior paper, it is demonstrated here by examples that a hypercomplex representation of infinity can also be conjectured and formulated in a formwise similar way. Unlike the two-dimensional (2D) complex infinity, which holds in 2D spaces equipped with an orthogonal homogeneous algebraic basis, the 4D quaternionic infinity requires an algebraic or quasigeometric basis that would be either heterogenous if established within a single space or, if it would be established upon a quasigeometric multispatial structure then at least two homogeneous bases are needed for the latter structure. In the latter case orthogonality and isometry are preserved. The multispatiality not only conforms to the experimentally confirmed reality of the quantum Cheshire Cat whose grin was located in a separate beam path from the beam path of the cat but also implies the feasibility of prospective mathematical reformulation of the basically complex (and thus essentially 2D) quantum mechanics in terms of certain hypercomplex 4D quasispatial structures under auspices of the multispatial reality paradigm.
Discipline
Year
Volume
149
Pages
23-35
Physical description
Contributors
author
  • Science/Mathematics Education Department, Southern University and A&M College, Baton Rouge, LA 70813, USA
References
  • [1] Czajko J. Quaternionic division by zero is implemented as multiplication by infinity in 4D hyperspace. World Scientific News 94(2) (2018) 190-216
  • [2] Mandic D.P., Jahanchahi C. & Took C.C. A quaternion gradient operator and its applications. IEEE Signal Processing Letters 18/1 (2011) 47-50
  • [3] Xu D. et al. Quaternion algebras: The GHR calculus. arXiv:1409.8168v1 [math.GM]
  • [4] Xu D. et al. Enabling quaternion derivatives. The generalized HR calculus. The Royal Society Publishing (2015) Doi.org/10.1098/rsos150255
  • [5] Morris D. The physics of empty space. Revised edition. Port Mulgrave: Abane & Right, 2015, p. 57.
  • [6] Czajko J. Finegrained 3D differential operators hint at the inevitability of their dual reciprocal portrayals. World Scientific News 132 (2019) 98-120
  • [7] Czajko J. New product differentiation rule for paired scalar reciprocal functions. World Scientific News 144 (2020) 358-371
  • [8] Deavours C.A. The quaternion calculus. Am. Math. Month. 80(9) (1973) 995-1008
  • [9] Eberly D. Quaternion algebra and calculus. Posted on Geometrictools.com (2010)
  • [10] Macfarlane A. Vector analysis and quaternions. New York: Wiley, 1906, p. 16.
  • [11] Hestenes D. Space-Time Algebra. New York: Gordon & Breach Publishers, 1966, p. 2.
  • [12] Imaeda K. Quaternionic formulation of classical electromagnetic fields and theory of functions of a biquaternion variable. [pp. 495-500 in: Chisholm J.S.R. & Common A.K. (Eds.) Clifford Algebra and their Application to Mathematical Physics. Dordrecht: D. Reidel Publishing, 1986].
  • [13] Morris D. Quaternions. Port Mulgrave, UK: Abane & Right, 2015, pp. 28ff, 63ff.
  • [14] Latimer C.G. Quaternion algebras. Duke Math. J. 15 (1948) 357
  • [15] Rédei L. Algebra. I. Oxford, Pergamon Press, 1967, p. 298
  • [16] Mal’tsev A.I. Algebraic systems. Moscow: Nauka Publishing, 1970 [in Russian], p. 124.
  • [17] Peirce J.M. On certain complete systems of quaternionic expressions, and on the removal of metric limitations from the calculus of quaternions. Trans. AMS 5 (1904) 411
  • [18] Ebbinghaus H.-D. et al. Numbers. New York: Springer-Verlag, 1991, p.197.
  • [19] Cohn P.M. Further algebra and applications. London: Springer, 2003, p.202.
  • [20] Ivanova T.A. Self-dual Yang-Mills connections and generalized Nahm equations. [pp.57-70 in: Fomenko A.T., Manturov O.V. & Trofimov V.V. Tensor and vector analysis. Geometry, mechanics and physics. Amsterdam: Gordon and Breach Science Publishers, 1998, see; pp. 61ff, 66ff].
  • [21] Okubo S. Introduction to octonion and other non-associative algebras in physics. Cambridge: Cambridge Univ. Press, 1995, p. 47ff.
  • [22] Czajko J. Radial and nonradial effects of radial fields in Frenet frame. Appl. Phys. Res. 3(1) (2011) 2-7
  • [23] Czajko J. Path-independence of work done theorem is invalid in center-bound force fields. Stud. Math. Sci. 7(2) (3013) 25-39
  • [24] Czajko J. On Conjugate Complex Time II: Equipotential Effect of Gravity Retrodicts Differential and Predicts Apparent Anomalous Rotation of the Sun. Chaos Solit. Fract. 11 (2000) 2001-2016
  • [25] Czajko J. On the Hafele-Keating Experiment. Ann. Phys. (Leipzig) 47 (1990) 517-518
  • [26] Czajko J. Experiments with Flying Atomic Clocks. Experim. Techn. Phys. 39 (1991) 145-147
  • [27] Czajko J. Galilei was wrong: Angular nonradial effects of radial gravity depend on density of matter. Int. Lett. Chem. Phys. Astron. 30 (2014) 89-105
  • [28] Czajko J. Equipotential energy exchange depends on density of matter. Stud. Math. Sci. 7(2) (3013) 40-54
  • [29] Czajko J. Complexification of operational infinity. World Scientific News 148 (2020) 60-71
  • [30] Czajko J. Algebraic division by zero implemented as quasigeometric multiplication by infinity in real and complex multispatial hyperspaces. World Scientific News 92(2) (2018) 171-197
  • [31] Czajko J. Unrestricted division by zero as multiplication by the – reciprocal to zero – infinity. World Scientific News 145 (2020) 180-197
  • [32] Kolata G. Topologists startled by new results. First a famous problem was solved, and now there is proof that there is more than one four-dimensional spacetime. Science 217 (1982) 432-433
  • [33] Czajko J. Operational constraints on dimension of space imply both spacetime and timespace. Int. Lett. Chem. Phys. Astron. 36 (2014) 220-235
  • [34] Czajko J. Operational restrictions on morphing of quasi-geometric 4D physical spaces. Int. Lett. Chem. Phys. Astron. 41 (2015) 45-72
  • [35] Serret J.-A. Cours d’algèbre supérieure 2. Paris, 1928, p. 512.
  • [36] Czajko J. Dual reciprocal scalar potentials paired via differential operators in Frenet frames make the operators to act simultaneously in each of two paired 3D reciprocal spaces. World Scientific News 137 (2019) 96-118
  • [37] Klein F. Vergleichende Betrachtungen über neuere geometrische Forschungen. Programm. Erlangen, 1872, pp. 7, 16.
  • [38] Klein F. Ausgewählte Kapitel der Zahlentheorie I. Göttingen, 1896, p. 51.
  • [39] Lautman A. Essai sur les notions de structure et d’existence en mathématiques. Paris: 1937, p. 22.
  • [40] Lobachevskii N.I. Pangeometry. [pp. 435-524 in: Lobachevskii N.I. Wholly Collected Works III. Moscow: 1951, in Russian, see p. 521].
  • [41] Kodomtsev S.B. Group of motions of the Lobachevskii plane. Math. Notes Acad. Sci. USSR 35 (1984) 59-61
  • [42] Norden A.P. Elementare Einführung in die Lobatschewskische Geometrie. Berlin: VEB Deutscher Verlag der Wissenschaften, 1958, p. 204.
  • [43] Lobatschewsky N. Géométrie imaginaire. J. Reine angew. Math. 17 (1837) 295
  • [44] Lobachevskii N.I. On origins of geometry. [pp. 185-261 in: Lobachevskii N.I. Complete collected works I. Moscow, 1946, in Russian, see p.209].
  • [45] Rosenfeld B.A. A history of non-Euclidean geometry. Evolution of the concept of a geometric space. New York: Springer, 1988. p. 237.
  • [46] Licis N.A. Philosophical and scientific meaning of ideas of N.I. Lobachevskii. Riga, 1976 [in Russian], p. 372.
  • [47] Greenberg M.J. Euclidean and non-Euclidean geometries. Development and history. New York: Freeman, 1993, p. 147, 293.
  • [48] Kaluza Th. Zur Relativitätstheorie. Phys. Z. XI (1910) 977-978
  • [49] Barankin E.W. Heat flow and non-Euclidean geometry. Am. Math. Mon. 49 (1942) 4, see p. 12.
  • [50] Fock V. The theory of space time and gravitation. London: Pergamon Press, 1959, p. 39
  • [51] Rozendorn E.R. Surfaces of negative curvature. [pp. 87-178 in: Burago Yu.D. & Zalgaller V.A. (Eds.) Geometry III: Theory of surfaces. Berlin: Springer, 1992, see p. 97].
  • [52] Cartan E. Leçons sur la géométrie projective complexe. Paris: Gauthier-Villars, 1950, p. 187.
  • [53] Yaglom I.M. Complex numbers in geometry. New York: Academic Press, 1968, p. 13f.
  • [54] Shirokov P.A. A sketch of the fundamentals of Lobachevskian geometry. Groningen: Noordhoff, 1964.
  • [55] Graves L.M. A finite Bolyai-Lobachevsky plane. Am. Math. Month. 69 (1962) 130, see p. 131
  • [56] Chernikov N.A. Introduction of Lobachevskii geometry into the theory of gravitation. Sov. J. Part. Nucl. 23(5) (1992) 507-521
  • [57] Nut Yu.Yu. Lobachevsky geometry in analytical presentation. Moscow: AN SSSR Publising, 1961 [in Russian], p. 43.
  • [58] Nuut J. Eine nichteuklidische Deutung der relativistischen Welt. Tartu, 1935. Series: Acta et Commentationes Univ. Tartuensis (Dorpatensis), see p. 26
  • [59] Suppes P. et al. Foundations of measurement II. San Diego, CA: Academic Press, 1989, p. 111.
  • [60] Room T.G. & Kirkpatrick P.B. Miniquaternion geometry. Cambridge: At The Univ. Press, 1971, p.1.
  • [61] Klein F. Vorlesungen über nicht-Euklidische Geometrie. Berlin: Springer, 1928, p. 239.
  • [62] Willmore T.J. An introduction to differential geometry. Mineola, NY: Dover, 2012, pp. 133, 138.
  • [63] Beck H. Eine Cremonasche Geometrie. J. reine angew. Math. 175 (1936) 129
  • [64] Hestenes D. Spinor particle mechanics. Proceedings of the Fourth Int. Conference on Clifford Algebras and their applications to mathematical physics, Aachen, Germany: 1996. Also available online on modelingnts.la.asu.edu/pdf/spinorPM.pdf
  • [65] Denkmayr T. et al. Observation of a quantum Cheshire Cat in a matter-wave interferometer experiment. Nature Communications | 5:4492 | DOI: 10.1038/ncomms5492 (2014). Also available on arXiv:1312.3775v1 [quant-ph] 13 Dec 2013.
Document Type
article
Publication order reference
Identifiers
YADDA identifier
bwmeta1.element.psjd-784c1e8e-c9be-473f-9ef4-9f3cc653b888
JavaScript is turned off in your web browser. Turn it on to take full advantage of this site, then refresh the page.