On the spinor formalism for even n

Details On the spinor formalism for even n On the spinor formalism for even n

2011-10-21

arxiv.org/abs/1110.4737v2

Physics

Mathematical Physics

MiKTeX v2.7, 119 pages, 3 tables, 2 figures. Rus edition: On the spinor formalism for the base space of even dimension. VINITI

Scientific paper

Spinor formalism is the formalism induced by solutions of the Clifford equation (the connection operators). For the space-time manifold (n = 4), those operators that connect the tangent and spinor bundle are operators that are represented by the Dirac matrices in the special basis. Reduced connecting operators are represented by the Pauli matrices. In order to uniquely extended the Killing equation from the tangent bundle to the spin bundle over the space-time manifold, it is necessary to pass to the complexification of the manifold and the corresponding bundles, and then to pass to the real representation. Returning reverse motion, it is possible to receive already two copies of the spinor bundle. Whether their set (the pair-spinor) allows to construct the Lee operator analogues for the spinors (and the pair-spinors). Similar procedure is feasible for any even n. For n=6 the specified formalism is closely connected with the Bogolyubov-Valatin transformations. For n mod 8=0, being based on the Bott periodicity, the reduced connecting operators generate the structural constants of an hypercomplex algebra (without division for n> 8) with the alternative-elastic, flexible (Jordan), and "norm" identities. For n = 8 such the algebra is the octonion algebra. In addition, in the article the various options of continuing of the connection to the spinor bundle with even-dimensional base are considered and constructed the corresponding curvature spinors.

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