Hamiltonian dynamics of purely affine fields (EINSTEIN-SCHRÖDINGER Theory)

Abstract

The Lagrangian of the general-relativistic affine field theory of the non-symmetric connection field Γikl is Schrödinger scalar density \documentclass{article}\pagestyle{empty}\begin{document}$ H = \frac{2}{\lambda }\sqrt { - \det [R} _{ik} ] $\end{document}, and the field variables (canonical coordinates) are Einstein's affine tensors Umnl = Γlmn - δnlΓrmn. The field equations are the Einstein-Schrödinger equations The minors give by definiton gmn = λ−1 Rmn, and λ becomes the cosmological constant. The Hamiltonian density is the V00-component of the Einstein energy-momentum complex and the tensor-density components are the canonically conjugated momentum densities of the field coordinates Ulmn. The canonical equations are , and we have no constraints. The affine field theory is invariant with respect to all transformations which preserve the Levi-Civita parallelism (Einstein's unified T-A group), and the field equations possess transposition invariance: With Ũlmn = Ulnm we get R̃mn = Rnm, g̃mn = gnm, and Ñmnl = Unml. The symmetry conditions Γimn = Γinm reduce the space to the general-relativistic Einstein spaces with Rik = Rki. The equation Rik = λgik yields Γikl = {ikl}, and the pathes of test particles define geodesic world lines of the Einstein spaces.