Clifford algebras and the hidden geometrical nature of spinors
V. L. Figueiredo, E. C. de Oliveira, W. A. Rodrigues Jr.
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Agradecimentos: W. A. R. and E. C. O. are grateful to Professor G. Vigna Suria and Professor M. Toller for discussions and the hospitality at the Diparti- mento di Matematica, Univ. di Trento, where this work has been completed. The authors are also grateful to FAPESP, CNPq and CNR for financial...
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Agradecimentos: W. A. R. and E. C. O. are grateful to Professor G. Vigna Suria and Professor M. Toller for discussions and the hospitality at the Diparti- mento di Matematica, Univ. di Trento, where this work has been completed. The authors are also grateful to FAPESP, CNPq and CNR for financial support
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Abstract: Many different definitions and representations of spinors are given in the literature, but there are no single reference explaining how they are related, which may explain why considerable confusion on the subject persists. Here and in following papers (II and III) we deal with three...
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Abstract: Many different definitions and representations of spinors are given in the literature, but there are no single reference explaining how they are related, which may explain why considerable confusion on the subject persists. Here and in following papers (II and III) we deal with three different definitions for spinors, (i) the covariant definition (E. Cartan) based on group theory representation, (ii) the ideal definition, based on real. Clifford algebra (p,q) methods, and (iii) the operator definition, where spinors are interpreted as particular elements an appropriated Clifford algebras (not necessarily elements of lateral ideals). By introducing the concept of spinorial metric on the space of algebraic spinors (i.e., elements of lateral ideals in appropriated Clifford algebras) we prove that for p + q <5 that there exists an equivalence from the group theoretical point of view between covariant and algebraic spinors. Tho this end we study the Clifford and the twisted Clifford groups, a subject that is also necessary, e.g., for the construction of the Clifford bundle for Lorentzian manifolds with (p,q) (3,1). We give explicit construction of the representative of Pauli spinors in R3,0 and Dirac spinors, Majorana spinors, dotted and undotted two component spinors in R1,3, R3,1 and R4,1. The problem of the transformations laws of algebraic spinors is also treated in details anda satisfactory mathematical solution is presented. Our approach clears among others the geometrical meaning of spinors and shows that the usual claim that spinors are objects more fundamental than tensors is non-sequitur. Also our techniques permit, e.g., the construction of sets of Majorana or Dirac matrices in very time saving way. In paper II we show how to obtain an almost elementary proof of Geroch's theorem without using the sophisticated techniques of algebraic topology. In paper III we study the problem of algebraic spinor fields, the Spinors and Clifford bundles and Dirac equations
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FUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO - FAPESP
CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO - CNPQ
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Clifford algebras and the hidden geometrical nature of spinors
V. L. Figueiredo, E. C. de Oliveira, W. A. Rodrigues Jr.
Clifford algebras and the hidden geometrical nature of spinors
V. L. Figueiredo, E. C. de Oliveira, W. A. Rodrigues Jr.
Fontes
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Relatório técnico n. 27, p. 1-34, dez. 1988 |