Local Temporal Symmetry

Abstract

Abstract A theory is reparameterization invariant when its equations of motion are invariant under the temporal relabelling symmetry given by locally rescaling a monotonic and smooth time function. This understanding of reparameterization invariance is more general than that provided by study of the action since the properties of the equations of motion generated by a variational principle depend only on the local terms, and not the boundary terms, of that variation principle. The treatment presented here outlines the connections between reparameterization invariance, various conditions on the Lagrangian and Hamiltonian functions, and the relevant generalized Bianchi identities. In the next crucial stage of the analysis a novel first-order geometric analysis of reparameterization invariance is provided in terms of the study of tangential transformations. This provides a basis for the unambiguous differentiation between the objects responsible for generating evolution and those that generate degeneracy along tangential directions.