Abstract
Representations of Z 2-graded algebras acting on their graded subspaces are described, and applied to Clifford algebras in particular. The derivation of Maxwell’s equations from the massless Dirac equation is characterized in terms of graded representations, and the consequences of graded representations for the Dirac equation are investigated.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. Ablamowicz, P. Lounesto, and J. Maks: “Conference Report, Second Workshop on Clifford Algebras and Their Applications in Mathematical Physics”, Found. Phys. 21, 735–748 (1991).
I.M. Benn and R.W. Tucker: An introduction to Spinors and Geometry with applications in physics, (Adam Hilger, Bristol, 1987).
P. Budinich and A. Trautman: The spinorial Chessboard, (Springer, Berlin, 1988).
E. Cartan: Leçons sur la théorie des spineurs, (Hermann, Paris, 1938),
E. Cartan: The theory of Spinors, (Hermann, Paris, 1966).
C. Chevalley: The Algebraic Theory of Spinors, (Columbia University Press, New York, 1954).
A. Crumeyrolle: Algèbres de Clifford et Spineurs, (Université Paul Sabatier, Toulouse, 1974).
A. Crumeyrolle: Orthogonal and Symplectic Clifford Algebras, (Kluwer, Dordrecht, 1990).
R. Delanghe, F. Sommen and V. Souček: Clifford Algebra and Spinor-Valued Functions, (Kluwer, Dordrecht, 1992).
A. Dimakis: “A New Representation of Clifford Algebras”, J. Phys. A22, 3171–3193 (1989).
C. Doran, A. Lasenby and S. Gull: “States and Operators in the Spacetime Algebra”, Found. Phys. 23, 1239–1264 (1993).
W.I. Fushchich, W.M. Shtelen and S.V. Spichak: “On the connection between solutions of Dirac and Maxwell equations, dual Poincaré invariance and superalgebras of invariance and solutions of nonlinear Dirac equations” , J. Phys. A, 24, 1683–1698 (1991).
W. Graf: “Differential forms as spinors”, Ann. Inst. Henri Poincaré 29, 85–109 (1978).
D. Hestenes: “Observables, operators, and complex numbers in the Dirac theory”, J. Math. Phys. 16, 556–572 (1975).
C. Itzykson and J.-B. Zuber: Quantum Field Theory, (McGraw-Hill, New York, 1980).
A.I. Kostrikin and I.R. Shafarevich (Eds.): Algebra I, (Springer, Berlin, 1990).
I.Yu. Krivskii and V.M. Simulik: “Dirac equation and spin 1 representations, a connection with symmetries of the Maxwell equations”, Theor. and Math. Phys. 90, 265–276 (1992).
K. Ljolje: “Some Remarks on Variational Formulations of Physical Fields”, Fortschr. Phys. 36, 9–32 (1988).
P. Lounesto and G.P. Wene: “Idempotent structure of Clifford algebras,” Acta Applicandea Mathematicae 9, 165–173 (1987).
P. Lounesto: “Clifford Algebras and Hestenes Spinors”, Found. Phys. 23, 1203–1237 (1993).
W. Pauli: “On the Conservation of the Lepton Charge”, Nuovo Cimento 6, 204–215 (1957).
I.R. Porteous: Topological Geometry, (Van Nostrand Reinhold, London, 1969; Cambridge University Press, 1981).
M.F. Ross: “Representation-free Calculations in Relativistic Quantum Mechanics”, in J.S.R. Chisholm and A.K. Common (eds.), Clifford Algebras and Their Applications in Mathematical Physics, 347–352 (D.Reidel, 1986).
H. Weyl: The Classical Groups, (Princeton University Press, 1939).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Morgan, P. (1995). The Massless Dirac Equation, Maxwell’s Equation, and the Application of Clifford Algebras. In: Ablamowicz, R., Lounesto, P. (eds) Clifford Algebras and Spinor Structures. Mathematics and Its Applications, vol 321. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8422-7_17
Download citation
DOI: https://doi.org/10.1007/978-94-015-8422-7_17
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-4525-6
Online ISBN: 978-94-015-8422-7
eBook Packages: Springer Book Archive