Abstract
We discuss the eigenvalue problem for 3 ×3 octonionic Hermitian matrices which is relevant to theJordan formulation of quantum mechanics. In contrast tothe eigenvalue problems considered in our previous work, all eigenvalues are real and solve theusual characteristic equation. We give an elementaryconstruction of the corresponding eigenmatrices, and wefurther speculate on a possible application to particle physics.
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REFERENCES
T. Dray and C.A. Manogue, The octonionic eigenvalue problem, Adv. Appl. Clifford Algebras 8, 341-364 (1998).
T. Dray and C.A. Manogue, Finding octonionic eigenvectors using Mathematica, Comput. Phys. Comm. 115, 536-547 (1998).
T. Dray, J. Janesky, and C.A. Manogue, Octonionic Hermitian matrices with non-real eigenvalues, in preparation.
C.A. Manogue and T. Dray, Dimensional reduction, Mod. Phys. Lett. A 14, 93-97 (1999).
C.A. Manogue and T. Dray, Quaternionic spin, in Clifford Algebras and Mathematical Physics, R. Abłamowicz and B. Fauser, eds., to appear; hep-th/9910010.
O. V. Ogievetskii, Kharakteristichesko e Uravnenie dlya Matrits 3 × 3 nad Oktavami, Uspekhi Mat. Nauk 36, 197-198 (1981); translated in: O. V. Ogievetskii, The Characteristic Equation for 3 × 3 Matrices over Octaves, Russian Math. Surveys 36, 189-190 (1981).
Susumu Okubo, Eigenvalue Problem for Symmetric 3 × 3 Octonionic Matrix, University of Rochester preprint, 1999.
P. Jordan, Über die Multiplikation quantenmechanisch er Größen, Z. Phys. 80, 285-291 (1933).
P. Jordan, J. von Neumann, and E. Wigner, On an algebraic generalization of the quantum mechanical formalism, Ann. Math. 35, 29-64 (1934).
H. Freudenthal, Oktaven, Ausnahmegruppen, und Oktavengeometrie, Mathematisch Instituut der Rijksuniversiteit te Utrecht (1951); new revised edition, 1960; reprinted, Geom. Dedicata 19, 1-63 (1985).
H. Freudenthal, Zur Ebenen Oktavengeometrie, Proc. Kon. Ned. Akad. Wet. A 56, 195-200 (1953).
H. Freudenthal, Lie groups in the foundations of geometry, Adv. Math. 1, 145-190 (1964).
N. Jacobson, Structure and Representations of Jordan Algebras (American Mathematical Society, Providence, Rhode Island, 1968).
R.D. Schafer, An Introduction to Nonassociative Algebras (Academic Press, New York, 1966, reprint, Dover, Mineola, New York, 1995).
F. Reese Harvey, Spinors and Calibrations (Academic Press, Boston, 1990).
B. Rosenfeld, Geometry of Lie Groups (Kluwer, Dordrecht, 1997).
C. Chevalley and R.D. Schafer, The exceptional simple Lie algebras F 4 and E 6, Proc. Nat. Acad. Sci. USA 36, 137-141 (1950).
A. Adrian Albert, On a certain algebra of quantum mechanics, Ann. Math. 35, 65-73 (1934).
F. Gürsey and C.-H. Tze, On the Role of Division, Jordan, and Related Algebras in Particle Physics (World Scientific, Singapore, 1996).
S. Okubo, Introduction to Octonion and Other Non-Associative Algebras in Physics (Cambridge University Press, Cambridge, 1995).
J. Schray, The general classical solution of the superparticle, Class. Quant. Grav. 13, 271-38 (1996).
J. Schray, Octonions and supersymmetry, Ph.D. thesis, Department of Physics, Oregon State University (1994).
C.A. Manogue and J. Schray, Finite Lorentz transformations, automorphisms, and division algebras, J. Math. Phys. 34 3746-3767 (1993).
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Dray, T., Manogue, C.A. The Exceptional Jordan Eigenvalue Problem. International Journal of Theoretical Physics 38, 2901–2916 (1999). https://doi.org/10.1023/A:1026699830361
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DOI: https://doi.org/10.1023/A:1026699830361