We give an explicit algebraic description of finite Lorentz transformations of vectors in 10-dimensional Minkowski space by means of a parameterization in terms of the octonions. The possible utility of these results for superstring theory is mentioned. Along the way we describe automorphisms of the two highest dimensional normed division algebras, namely the quaternions and the octonions, in terms of conjugation maps. We use similar techniques to define SO(3) and SO(7) via conjugation, SO(4) via symmetric multiplication, and SO(8) via both symmetric multiplication and one-sided multiplication. The non-commutativity and non-associativity of these division algebras plays a crucial role in our constructions.

We argue that all Einstein-Maxwell or Einstein-Proca solutions to general relativity may be used to construct a large class of solutions (involving torsion and non-metricity) to theories of non-Riemannian gravitation that have been recently discussed in the literature. -2 - IntroductionNon-Riemannian geometries feature in a number of theoretical descriptions of the interactions between fields and gravitation. Since the early pioneering work by Weyl, Cartan, Schroedinger and others such geometries have often provided a succinct and elegant guide towards the search for unification of the forces of nature [1]. In recent times interactions with supergravity have been encoded into torsion fields induced by spinors and dilatonic interactions from low energy effective string theories have been encoded into connections that are not metric-compatible [2], [3], [4], [5]. However theories in which the non-Riemannian geometrical fields are dynamical in the absence of matter are more elusive to interpret. It has been suggested that they may play an important role in certain astrophysical contexts [6]. Part of the difficulty in interpreting such fields is that there is little experimental guidance available for the construction of a viable theory that can compete effectively with general relativity in domains that are currently accessible to observation. In such circumstances one must be guided by the classical solutions admitted by theoretical models that admit dynamical non-Riemannian structures [7], [8], [9], [10], [11]. A number of recent papers have pursued this approach and have found static spherically symmetric solutions to particular models [12], [13]. In [6] it was pointed out that in a particularly simple model all EinsteinMaxwell solutions to general relativity could be used to generate dynamic non-Riemannian geometries and a tentative interpretation was offered for the matter couplings in such a model. Particular solutions have also been found to more complex models, provided the coupling constants in the action are correlated [12], [13], [14], [15]. It is the purpose of this note to point out that, if such correlations are maintained then solutions may be generated from all Einstein-Maxwell solutions of general relativity. Furthermore the correlations may be discarded and solutions generated from all Einstein-Proca solutions of general relativity with or without the inclusion of a cosmological term in the action.

The theory of representations of Clifford algebras is extended to employ the division algebra of the octonions or Cayley numbers. In particular, questions that arise from the non-associativity and non-commutativity of this division algebra are answered. Octonionic representations for Clifford algebras lead to a notion of octonionic spinors and are used to give octonionic representations of the respective orthogonal groups. Finally, the triality automorphisms are shown to exhibit a manifest $\perm_3 \times SO(8)$ structure in this framework.Comment: 33 page

The theory of vectors and spinors in 9+1 dimensional spacetime is introduced in a completely octonionic formalism based on an octonionic representation of the Clifford algebra Cl(9, 1). The general solution of the classical equations of motion of the CBS superparticle is given to all orders of the Grassmann hierarchy. A spinor and a vector are combined into a 3 × 3 Grassmann, octonionic, Jordan matrix in order to construct a superspace variable to describe the superparticle. The combined Lorentz and supersymmetry transformations of the fermionic and bosonic variables are expressed in terms of Jordan products.

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We examine some of the subtleties inherent in formulating a theory of spinors on a manifold with a smooth degenerate metric. We concentrate on the case where the metric is singular on a hypersurface that partitions the manifold into Lorentzian and Euclidean domains. We introduce the notion of a complex spinor fibration to make precise the meaning of continuity of a spinor field and give an expression for the components of a local spinor connection that is valid in the absence of a frame of local orthonormal vectors. These considerations enable one to construct a Dirac equation for the discussion of the behavior of spinors in the vicinity of the metric degeneracy.We conclude that the theory contains more freedom than the spacetime Dirac theory and we discuss some of the implications of this for the continuity of conserved currents. PACS:04.20.Gz Spacetime topology, causal structure, spinor structure 11.90.+t Other topics in general field and particle theory 03.65.Pm Relativistic wave equations 03.50.Kk Other special classical field theories Typeset using REVT E X 1

We demonstrate that the left (and right) invariant Maurer-Cartan forms for any semi-simple Lie group enable one to construct solutions of the Yang-Mills equations on the group manifold equipped with the natural Cartan-Killing metric. For the unitary unimodular groups the Yang-Mills action integral is finite for such solutions. This is explicitly exhibited for the case of SU (3).