Spinor Algebra

Abstract

In this chapter a unified treatment of the algebraic properties of the spinors in three-dimensional spaces is given. In Section 5.1 it is shown that every tensor index can be replaced by a pair of spinor indices that take two values only and, using this correspondence, in Section 5.2 all the orthogonal transformations are expressed in terms of 2 × 2 matrices with unit determinant. In Section 5.3 the conditions satisfied by the spinor equivalent of a real tensor are obtained and it is shown that spinors can be classified according to the repetitions of their principal spinors. In Section 5.4 it is shown that, under certain conditions, a one-index spinor defines a basis for the original three-dimensional space.