Finsler geometry motivates a generalization of the Riemannian structure of spacetime to include dependence of the spacetime metric and associated invariant tensor fields on the four-velocity coordinates as well as the spacetime coordinates of the observer. It is then useful to consider the tangent bundle of spacetime with spacetime in the base manifold and four-velocity space in the fiber. A physical basis for the differential geometric structure of the spacetime tangent bundle is provided by the universal upper limit on proper acceleration relative to the vacuum. It is then natural to consider a quantum field having a vanishing eigenvalue when acted on by the Laplace-Beltrami operator of the spacetime tangent bundle. On this basis a quantum field theory can be constructed having a built-in intrinsic regularization at the Planck scale, and finite vacuum energy density.

Invited plenary lecture at the Fourth World Congress of Nonlinear Analysts in the session: Applications of Finsler Differential Geometry (in Engineering, Physics and Biology), July 5, 2004, Orlando, Florida