Abstract
We show that complex representations of Clifford algebra can always be reduced either to a real or to a quaternionic algebra depending on signature of complex space thus showing that spinors are unavoidably either real Majorana spinors or quaternionic spinors and complex spinors disappear. We use this result to support (1, 3) signature for Minkowski space.
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References
Adler, S.L.: Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press, Oxford (1995)
Benn, I.M., Tucker, R.W.: An Introduction to Spinors and Geometry with Applications in Physics. Adam Hilger, Bristol (1987)
Budinich, M.: On spinors transformations. J. Math. Phys. 57(7), 071703 (2016). arXiv:1603.02181 [math-ph]
Budinich, M.: On clifford algebras and binary integers. Adv. Appl. Clifford Algebras 27(2), 1007–1017 (2017). arXiv:1605.07062 [math-ph]
Budinich, P., Trautman, A.M.: The Spinorial Chessboard. Trieste Notes in Physics. Springer, Berlin (1988)
Cartan, É.: Les groupes projectifs qui ne laissent invariante aucune multiplicité plane. Bulletin de la Société Mathématique de France 41, 53–96 (1913)
Cartan, É.: The Theory of Spinors. Hermann, Paris (1966) (first edition: 1938 in French)
Chevalley, C.C.: Algebraic Theory of Spinors. Columbia University Press, New York (1954)
Conrad, K.: Complexification (2018). http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/complexification.pdf. Accessed 21 Dec 2018
Dabrowski, L.: Group Actions on Spinors. Lecture Notes. Bibliopolis, Naples (1988)
De Leo, S.: A one-component dirac equation. Int. J. Mod. Phys. A 11(21), 3973–3985 (1996)
De Leo, S., Rodrigues Jr., W.A.: Quaternionic electron theory: Dirac’s equation. Int. J. Theor. Phys. 37(5), 1511–1529 (1998)
Giardino, S.: Quaternionic quantum mechanics in real Hilbert space (2018). arXiv:1803.11523 [quant-ph]
Granville, A.: Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers. In: Canadian Mathematical Society Conference Proceedings, vol. 20, pp. 253–276. American Mathematical Society, Providence (1997)
Kugo, T., Townsend, P.: Supersymmetry and the division algebras. Nucl. Phys. B 221(2), 357–380 (1983)
Lawson, H.B., Michelsohn, M.-L.: Spin Geometry. Princeton Mathematical Series, vol. 38. Princeton University Press, Princeton (1990)
Okubo, S.: Real representations of finite Clifford algebras. I. Classification. J. Math. Phys. 32(7), 1657–1668 (1991)
Porteous, I.R.: Topological Geometry, II edn. Cambridge University Press, Cambridge (1981)
Porteous, I.R.: Clifford Algebras and the Classical Groups. Cambridge Studies in Advanced Mathematics, vol. 50. Cambridge University Press, Cambridge (1995)
Trautman, A.M.: On complex structures in physics. In: Harvey, A. (ed.) On Einstein’s Path: Essays in Honor of Engelbert Schucking, Chapter 34, pp. 487–495. Springer, New York (1999)
Varlamov, V.V.: Discrete symmetries and clifford algebras. Int. J. Theor. Phys. 40(4), 769–805 (2001)
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Budinich, M. On Complex Representations of Clifford Algebra. Adv. Appl. Clifford Algebras 29, 18 (2019). https://doi.org/10.1007/s00006-018-0930-3
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DOI: https://doi.org/10.1007/s00006-018-0930-3