The Extended Born’s Reciprocal Relativity Theory : Division, Jordan, N-ary Algebras, and Higher Order Finsler spaces

We extend the construction of Born’s Reciprocal Relativity theory in ordinary phase spaces to an extended phase space based on Quaternions. The invariance symmetry group is the (pseudo) unitary quaternionic group U(N+, N−,H) which is isomorphic to the unitary symplectic group USp(2N+, 2N−,C). It is explicitly shown that the quaternionic group U(N+, N−,H) leaves invariant both the quadratic norm (corresponding to the generalized Born-Green interval in the extended phase space) and the tri-symplectic 2-form. The study of Octonionic, Jordan and ternary algebraic structures associated with generalized spacetimes (and their phase spaces) described by Gunaydin and collaborators is reviewed. A brief discussion on n-plectic manifolds whose Lie n-algebra involves multi-brackets and n-ary algebraic structures follows. We conclude with an analysis on the role of higher-order Finsler geometry in the construction of extended relativity theories with an upper and lower bound to the higher order accelerations (associated with the higher order tangent and cotangent spaces).