Octonionic Methods in Field Theory.

Some applications of octonion algebra and octonionic analysis to group theory and higher dimensional field theories are presented. To this end an eight dimensional covariant treatment of the octonion algebra is needed. The existing formulations which are covariant only in seven dimensions are reviewed. In this work the eight dimensional formulation is developed through the introduction of fourth rank tensors f_{rm abcd} and {rm f}^'abcd in eight dimensions that generalize the octonionic structure constants. The seven octonion units e_ {alpha} are generalized to an 8 -vector e_{rm a} and two second rank tensors e_{rm ab} and {rm e}^ '_{rm ab}. Higher rank tensors associated with e_{alpha } are also introduced. Chirality and duality properties of the structure tensors f,f^ ' and the octonionic tensors e _{rm a}, e_ {rm ab}, etc. are discussed and various new identities relating these quantities are derived. New vector products for two, three and four octonions are introduced and their duality properties with respect to the eight -dimensional Levi-Civita tensor as well as their orthogonality properties are studied. The foregoing covariant treatment of octonions is first applied to a reformulation of the Lie algebra of the group 0(8) in which the triality property of octonions is manifest as the triality property of some generators associated with the following decomposition of 0(8)(UNFORMATTED TABLE OR EQUATION FOLLOWS)0(8) = {0(8) over{rm S0(7)_{x}} } times {{rm S0(7)_{x }}over{rm G_2 }} times {rm G_2 }(TABLE/EQUATION ENDS)where the label x stands for usual, (L) or right(R). The physical importance of 0(8) arises from its being the helicity group in 10-dimensional field theories as well as its giving rise to models in 2 dimensions with triality. As a second application self-dual Yang-Mills theories are generalized to a set of field equations in eight dimensions for a non-abelian version of Kalb-Ramond fields associated with high rank antisymmetrical tensors. Such equations have instanton type solutions. Such a class of solutions is found by using octonionic analytic functions that generalize Fueter's theory of quaternionic analytic functions.