Fiber with Intrinsic Action on a 1 + 1 Dimensional Spacetime

Abstract

I construct an algebraic model for a typical fiber on a 1 + 1 dimensional spacetime. The vector space comprising the fiber is composed of elements x ⊗ x formed from the direct product of two copies of an element x in the D ₂ = C ₂ ⊗ C ₂ finite group algebra over the real numbers. The fiber contains subspaces whose elements can be associated with the tangent and momentum vectors of trajectories in the manifold. The fiber also contains a subspace whose elements are associated with the local flow of action of each trajectory. The condition of minimum action translates into a constraint on the original vector x in the direct product structure.