Abstract
The spacetime algebra (STA) is the natural, representation-free language for Dirac's theory of the electron. Conventional Pauli, Dirac, Weyl, and Majorana spinors are replaced by spacetime multivectors, and the quantum σ- and γ-matrices are replaced by two-sided multivector operations. The STA is defined over the reals, and the role of the scalar unit imaginary of quantum mechanics is played by a fixed spacetime bivector. The extension to multiparticle systems involves a separate copy of the STA for each particle, and it is shown that the standard unit imaginary induces correlations between these particle spaces. In the STA, spinors and operators can be manipulated without introducing any matrix representation or coordinate system. Furthermore, the formalism provides simple expressions for the spinor bilinear covariants which dispense with the need for the Fierz identities. A reduction to2+1 dimensions is given, and applications beyond the Dirac theory are discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. D. Bjorken and S. D. Drell,Relativistic Quantum Mechanics, Vol. 1 (McGraw-Hill, New York, 1964).
C. Itzykson and J-B. Zuber,Quantum Field Theory (McGraw-Hill, New York, 1980).
D. Hestenes,Space-Time Algebra (Gordon & Breach, New York, 1966).
D. Hestenes, “Vectors, spinors, and complex numbers in classical and quantum physics,”Am. J. Phys. 39, 1013 (1971).
D. Hestenes, “Observables, operators, and complex numbers in the Dirac theory,”J. Math. Phys. 16(3), 556 (1975).
D. Hestenes, “Clifford algebra and the interpretation of quantum mechanics,” in J. S. R. Chisholm and A. K. Common, eds.,Clifford Algebras and Their Applications in Mathematical Physics (Reidel, Dordrecht, 1986), p. 321.
S. F. Gull, A. N. Lasenby, and C. J. L. Doran, “Electron paths, tunnelling and diffraction in the spacetime algebra,”Found. Phys. 23(10), (1993).
R. Penrose and W. Rindler,Spinors and Space-time, Vol. I:Two-spinor Calculus and Relativistic Fields (Cambridge University Press, Cambridge, 1984).
S. F. Gull, A. N. Lasenby, and C. J. L. Doran, “Imaginary numbers are not real—the geometric algebra of spacetime,”Found. Phys. 23(9), 1175 (1993).
A. N. Lasenby, C. J. L. Doran, and S. F. Gull, “2-spinors, twistors and supersymmetry in the spacetime algebra,” in Z. Oziewicz, A. Borowiec, and B. Jancewicz, eds.,Spinors, Twistors and Clifford Algebras (Kluwer Academic, Dordrecht, 1993), p. 233.
D. Hestenes and R. Gurtler, “Consistency in the formulation of the Dirac, Pauli and Schrödinger theories,”J. Math. Phys. 16(3), 573 (1975).
D. Hestenes and G. Sobczyk,Clifford Algebra to Geometric Calculus (Reidel, Dordrecht, 1984).
R. Penrose and W. Rindler,Spinors and Space-Time, Vol. II:Spinor and Twistor Methods in Space-Time Geometry (Cambridge University Press, Cambridge, 1986).
I. W. Benn and R. W. Tucker,An Introduction to Spinors and Geometry (Adam Hilger, London, 1988).
V. L. Figueiredo, E. C. de Oliveira, and W. A. Rodrigues, Jr., “Covariant, algebraic, and operator spinors,”Int. J. Theor. Phys. 29(4), 371 (1990).
S. F. Gull, “Charged particles at potential steps,” in A. Weingartshofer and D. Hestenes, eds.,The Electron (Kluwer Academic, Dordrecht, 1991), p. 37.
A. N. Lasenby, C. J. L. Doran, and S. F. Gull, “A multivector derivative approach to Lagrangian field theory,”Found. Phys. 23(10), (1993).
D. Hestenes, “Real Dirac theory,” in preparation, 1993.
J. P. Crawford, “On the algebra of Dirac bispinor identities: factorization and inversion theorems,”J. Math. Phys. 26(7), 1439 (1985).
J. P. Crawford, “Dirac equation for bispinor densities,” in J. S. R. Chisholm and A. K. Common, eds.,Clifford Algebras and Their Applications in Mathematical Physics (Reidel, Dordrecht, 1986), p. 353.
R. P. Feynman,Quantum Electrodynamics (Addison-Wesley, Reading, Massachusetts, 1961).
H. A. Bethe and E. E. Salpeter,Quantum Mechanics of One- and Two-Electron Atoms (Springer, New York, 1957).
R. P. Martinez-Romero and A. L. Salas-Brito, “Conformal invariance in a Dirac oscillator,”J. Math. Phys. 33(5), 1831 (1992).
W. I. Fushchich and R. Z. Zhdanov, “Symmetry and exact solutions of nonlinear spinor equations,”Phys. Rep. 172(4), 125 (1989).
C. Daviau and G. Lochak, “Sur un modèle d'équation spinorielle non linéaire,”Ann. Fond. L. de Broglie 16(1), 43 (1991).
A. O. Barut and N. Zanghi, “Classical models of the Dirac electron,”Phys. Rev. Lett. 52(23), 2009 (1984).
P. West,An Introduction to Supersymmetry and Supergravity (World Scientific, Singapore, 1986).
F. Reifler, “A vector wave equation for neutrinos,”J. Math. Phys. 25(4), 1088 (1984).
E. Marx, ‘Spinor equations in relativistic quantum mechanics,”J. Math. Phys. 33(6), 2290 (1992).
M. S. Plyuschay, “Lagrangian formulation for the massless (super)particles in (super)twistor approach,”Phys. Lett. B 240, 133 (1990).
M. S. Plyuschay, “Spin from isospin: The model of a superparticle in a non-Grassmannian approach,”Phys. Lett. B 280, 232 (1992).
S. Deser and R. Jackiw, “Statistics without spin. MasslessD = 3 systems,”Phys. Lett. B 263, 431 (1991).
N. Salingaros, “On the classification of Clifford algebras and their relation to spinors inn-dimensions,”J. Math. Phys. 23(1), 1 (1982).
Z. Hasiewicz, P. Siemion, and F. Defever, “A Bosonic model for particles with arbitrary spin,”Int. J. Mod. Phys. A 7(17), 3979 (1992).
G. W. Gibbons, “Kummer's configuration, causal structures and the projective geometry of Majorana spinors,” in Z. Oziewicz, A. Borowiec, and B. Jacewicz, eds.,Spinors, Twistors and Clifford Algebras (Kluwer Academic, Dordrecht, 1993).
Author information
Authors and Affiliations
Additional information
Supported by a SERC studentship.
Rights and permissions
About this article
Cite this article
Doran, C., Lasenby, A. & Gull, S. States and operators in the spacetime algebra. Found Phys 23, 1239–1264 (1993). https://doi.org/10.1007/BF01883678
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01883678