Higher order gravity, gauge invariant variables, and quantum behavior in   Weyl-like geometries
  This paper presents the detailed, standard treatment of a simple, gauge invariant action for Weyl and Weyl-like Cartan geometries outlined in a previous paper. In addition to the familiar scalar curvature squared and Maxwell terms, the action chosen contains the logarithmic derivative of the scalar curvature combined with the intrinsic four vector (Weyl vector) in a gauge invariant fashion. This introduces higher order derivative terms directly into the action. No separate, "matter" fields are introduced. As the usual Weyl metric and four vector are varied, certain gauge invariant combinations of quantities arise naturally as the results are collected, provided the scalar curvature is nonzero. This paper demonstrates the general validity of these results for any gauge choice. Additionally, "matter" terms appear in the field equations. Furthermore, the resulting forms isolate the familiar mathematical structure of a coupled Einstein-Maxwell-Schroedinger (relativistic) system of classical fields, with the exception of additional, second derivative terms in the stress tensor for the Schroedinger field, and the algebraic independence of the conjugate wavefunction. This independence is found to be equivalent to the presence of a second, negative energy, Schroedinger field. A detailed comparison is made between this model, and the standard Einstein-Maxwell-Schroedinger field theory. The possible use of such continuum models as a basis for quantum phenomena, and some generalizations of the model are discussed. 
J.E. Rankin