Generalized Minkowski metric and composition algebras

Abstract

New variables («new» as regards the space-time variables) are obtained starting from the requirement that their allowable values should leave the value of the Minkowski metric unchanged. To this purpose, the space-time variables are embedded into real and then complex nonassociative algebras (with an identity element) endowed with nondegenerate quadratic forms generalizing the Minkowski metric. The algebras are proved to fulfil the above-mentioned requirement if and only if they are composition algebras. Using the Hurwitz theorem, it is shown that in the real case there is only one allowable algebra, which is isomorphic to the eight-dimensional split real Cayley algebra, while, in the complex case, there are two allowable algebras: one is isomorphic to the four-dimensional complex quaternion algebra, the other is isomorphic to the eight-dimensional complex Cayley algebra. In the case of these three algebras, new variables and their allowable values are obtained. The essential role played in this context by the (multiplicative) noncommutativity and nonassociativity is pointed out in some final remarks.