zbMATH — the first resource for mathematics

Covariant, algebraic, and operator spinors. (English) Zbl 0699.53019
The authors review three basic approaches to the notion of a spinor which they term the covariant, algebraic, and operator definitions respectively. They then consider in what sense these are equivalent and indicate cases when such relationships can be established. Their results include the types of spinors employed by Pauli, Dirac, Majorana and Weyl. This leads to a new approach to the spinor structure of space-time and a representation of the Dirac and Maxwell equations in terms of Clifford and spin-Clifford bundles over the space-time.
Reviewer: J.D.Zund

53C27 Spin and Spin\({}^c\) geometry
83C60 Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism
Full Text: DOI
[1] Atiyah, M. F., Bott, R., and Shapiro, A. (1984). Clifford modules,Topology 3(Supp. 1), 3. · Zbl 0146.19001
[2] Benn, I. M., and Tucker, R. W. (1983a).Physics Letters,130B, 177.
[3] Benn, I. M., and Tucker, R. W. (1983b). Fermions without spinors,Communications in Mathematical Physics,89, 34. · Zbl 0527.58023
[4] Benn, I. M., and Tucker, R. W. (1983b). A local right-spin covariant Kähler equation,Physics Letters,130B, 177.
[5] Benn, I. M., and Tucker, R. W. (1985a). The differential approach to spinor and their symmetries,Nuovo Cimenta,88A, 273. · Zbl 0587.58002
[6] Benn, I. M., and Tucker, R. W. (1985b). The Dirac equation in exterior form,Communications in Mathematical Physics,98, 53. · Zbl 0587.58002
[7] Bichteler, K. (1963). Global existence of spin structures for gravitational fields,Journal of Mathematical Physics,9, 198. · Zbl 0162.29902
[8] Blaine Lawson, Jr., H., and Michelsohn, M. L. (1983).Spin Geometry, Universidad Federal do Ceará, Brazil.
[9] Blau, M. (1987). Connections on Clifford bundles and the Dirac operator,Letters in Mathematical Physics,13, 83. · Zbl 0644.58028
[10] Bleecker, D. (1981).Gauge Theory and Variational Principles, Addison-Wesley, Reading, Massachusetts. · Zbl 0481.58002
[11] Brauer, R., and Weyl, H. (1935). Spinors inn dimensions,American Journal of Mathematics,57, 425. · Zbl 0011.24401
[12] Budinich, P., and Trautman, A. (1986). Remarks on pure spinors,Letters in Mathematical Physics,11, 315. · Zbl 0602.15021
[13] Bugajska, B. (1979). Spinor structure of space-time,International Journal of Theoretical Physics,18, 77. · Zbl 0441.53049
[14] Caianello, E. R. (1988). ?Spineurs Simples?, ?Urfelder? and factorizations of Dirac equations and spinors,Physica Scripta,37, 197. · Zbl 1063.81598
[15] Cartan, E. (1966).Theory of Spinors, Dover, New York. · Zbl 0147.40101
[16] Chevalley, C. (1954).The Algebraic Theory of Spinors, Columbia University Press, New York. · Zbl 0057.25901
[17] Coquereaux, R. (1982). Modulo 8 periodicity of real Clifford algebras and particle physics,Physics Letters B,115, 189.
[18] Crumeyrolle, A. (1969). Structures spinorielles,Annales de l’Institut Henri Poincarè, A,XI(1), 19. · Zbl 0188.26102
[19] Crumeyrolle, A. (1971).Annales de l’Institut Henri Poincarè,14, 309.
[20] Dimakis, A. (1986). InClifford Algebras and their Applications to Mathematical Physics J. S. R. Chisholm and A. K. Common, eds.), D. Reidel, Dordrecht.
[21] Felzenswalb, B. (1979).Algebras de Dimensão Finita, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil.
[22] Figueiredo, V. L., de Oliveira, E. C., Rodrigues, Jr., W. A. (1990). Clifford Algebras and the hidden geometrical nature of spinors,Hadronic Journal, to appear. · Zbl 0774.53008
[23] Geroch, R. (1968). Spinor structure of space-times in general relativity I,Journal of Mathematical Physics,9, 813. · Zbl 0162.29902
[24] Graf, W. (1978). Differential forms as spinors,Annales de l’Institut Henri Poincarè,XXIX, 85.
[25] Hestenes, D. (1967). Real spinor fields,Journal of Mathematical Physics,8, 798.
[26] Hestenes, D. (1971a). Vectors, spinors, and complex numbers in classical and quantum physics,American Journal of Physics,59, 1013.
[27] Hestenes, D. (1971b). Local observables in quantum theory,American Journal of Physics,39, 1028.
[28] Hestenes, D. (1975). Observables, operators, and complex numbers in the Dirac theory,Journal of Mathematical Physics,16, 556.
[29] Hestenes, D. (1986). InClifford Algebras and Their Applications to Mathematical Physics, J. S. R. Chisholm and A. K. Common, eds., D. Reidel, Dordrecht.
[30] Hestenes, D., and Sobezyk, G. (1984).Clifford Algebra to Geometrical Calculus, D. Reidel, Dordrecht.
[31] Landau, L. D., and Lifschitz, E. M. (1971).Relativistic Quantum Theory, Addison-Wesley, Reading, Massachusetts.
[32] Lichnerowicz, A. (1964). Champs Spinoriels et Propagateurs en Relativité Générale,Bulletin Société Mathématique France 92, 11.
[33] Lounesto, P. (1981). Scalar products of spinors and an extension of Brauer-Wall groups,Foundations of Physics,11, 721.
[34] Lounesto, P. (1986). InClifford Algebras and Their Applications in Mathematical Physics, J. S. R. Chisholm and A. K. Common, eds., D. Reidel, Dordrecht. · Zbl 0596.15028
[35] Micali, A. (1986). InClifford Algebras and Their Applications in Mathematical Physics, J. S. R. Chisholm and A. K. Common, eds., D. Reidel, Dordrecht.
[36] Miller, W., Jr. (1972).Symmetry Groups and Their Applications, Academic Press, New York.
[37] Milnor, J. W. (1963). Spin structures on manifolds,L’Enseignement Mathematique,9, 198. · Zbl 0116.40403
[38] Penrose, R., and Rindler, W. (1984).Spinors and Space-Time, Vols. I and II, Cambridge University Press, Cambridge. · Zbl 0538.53024
[39] Porteous, I. R. (1981).Topological Geometry, 2nd ed., Cambridge University Press, Cambridge. · Zbl 0446.15001
[40] Riesz, M. (1958).Clifford Numbers and Spinors, Lecture Notes No. 38, Institute for Fluid Mechanics and Applied Mathematics, University of Maryland. · Zbl 0103.38403
[41] Rodrigues, Jr., W. A., and de Oliveira, E. C. (1990). Dirac and Maxwell equations in the Clifford and spin Clifford bundles,International Journal of Theoretical Physics, this issue. · Zbl 0699.53020
[42] Rodrigues, Jr., W. A., and Figueiredo, V. L. (1989). InProceedings VIII Convegno Nazionale di Relatività Generale e Fisica della Gravitazione, M. Toller, M. Cerdonio, M. Francaviglia, and R. Cianci, eds., World Scientific, Singapore.
[43] Rodrigues, Jr., W. A., and Figueiredo, V. L. (1990). Real spin-Clifford bundle and the spinor structure of space-time,International Journal of Theoretical Physics, this issue. · Zbl 0699.53021
[44] Salingaros, N. A., and Wene, G. P. (1985). The Clifford algebra of differential forms,Acta Applicadae Mathematica,4, 271. · Zbl 0572.15011
[45] Santaló, L. A. (1976).Geometria Espinorial, Consejo Nacional de Inv. Cient. y Tecnica, Int. Argentino de Matematica, Buenos Aires, Argentina. · Zbl 0339.53001
[46] Srivastrava, P. P. (1974). On spinor representation of the Lorentz group,Revista Brasileira de Fisica,4, 507.
[47] Van der Waerden, B. L. (1932).Group Theory and Quantum Mechanics, Springer, Berlin. · Zbl 0433.22013
[48] Weyl, H. (1929). Elektron and Gravitation 1.2,Physik,56, 330. · JFM 55.0513.04
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.