Alternativity and reciprocity in the Cayley-Dickson algebra

We calculate the eigenvalue \rho of the multiplication mapping R on the Cayley-Dickson algebra A_n. If the element in A_n is composed of a pair of alternative elements in A_{n-1}, half the eigenvectors of R in A_n are still eigenvectors in the subspace which is isomorphic to A_{n-1}. The invariant under the reciprocal transformation A_n \times A_{n} \ni (x,y) -> (-y,x) plays a fundamental role in simplifying the functional form of \rho. If some physical field can be identified with the eigenspace of R, with an injective map from the field to a scalar quantity (such as a mass) m, then there is a one-to-one map \pi: m \mapsto \rho. As an example, the electro-weak gauge field can be regarded as the eigenspace of R, where \pi implies that the W-boson mass is less than the Z-boson mass, as in the standard model.Comment: To be published in J. Phys. A: Mathematical and Genera

A Non-Associative Deformation of Yang-Mills Gauge Theory

An ansatz is presented for a possible non-associative deformation of the standard Yang-Mills type gauge theories. An explicit algebraic structure for the deformed gauge symmetry is put forward and the resulting gauge theory developed. The non-associative deformation is constructed in such a way that an apparently associative Lie algebraic structure is retained modulo a closure problem for the generators. It is this failure to close which leads to new physics in the model as manifest in the gauge field kinetic term in the resulting Lagrangian. A possible connection between this model and quantum group gauge theories is also investigated.Comment: 18 pages, RevTeX, also uses aps.st

Magnetic Charge in a Nonassociative Field Theory

The violation of the Jacobi identity by the presence of magnetic charge is accomodated by using an explicitly nonassociative theory of octonionic fields. It is found that the dynamics of this theory is simplified if the Lagrangian contains only dyonic charges, but certain problems in the constrained quantisation remain. The extension of these concepts to string theory may however resolve these difficulties.Comment: 10 pages, REVTeX, no figure