Hamiltonian dynamics of purely affine fields (E INSTEIN ‐S CHRÖDINGER Theory)

The Lagrangian of the general‐relativistic affine field theory of the non‐symmetric connection field Γ i kl 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 U mn l = Γ l mn ‐ δ n l Γ r mn . The field equations are the Einstein‐Schrödinger equations\documentclass{article}\pagestyle{empty}\begin{document}$$ \frac{{\delta H}}{{\delta U^i _{mn} }} = N^{mn} _i = 0 $$\end{document}The minors\documentclass{article}\pagestyle{empty}\begin{document}$$ \frac{{\partial H}}{{\partial R_{mn} }} = G^{mn} = \sqrt { - gg^{mn} } $$\end{document} give by definiton g mn = λ −1 R mn , and λ becomes the cosmological constant. The Hamiltonian density is the V 0 0 ‐component of the Einstein energy‐momentum complex\documentclass{article}\pagestyle{empty}\begin{document}$$ V_i ^k = \frac{{\partial H}}{{\partial U^i _{mn,k} }}U^i _{mn,i} = - \frac{1}{{2\lambda ^2 }}HR^{mn} \partial _i^k U^i _{mn.i} $$\end{document} and the tensor‐density components\documentclass{article}\pagestyle{empty}\begin{document}$$ \gamma ^{mn} _i = - \frac{1}{{2\lambda ^2 }}HR^{mn} \partial _i^0 = - \sqrt { - gg^{mn} } \partial _i^0 = g^{mn} \partial _1^0 $$\end{document} are the canonically conjugated momentum densities of the field coordinates U l mn . The canonical equations are\documentclass{article}\pagestyle{empty}\begin{document}$$ ( - g)^{ - \frac{1}{2}} N^{mn} _{iV_0^0 } = 0 $$\end{document} , 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 Ũ l mn = U l nm we get R̃ mn = R nm , g̃ mn = g nm , and Ñ mn l = U nm l . The symmetry conditions Γ i mn = Γ i nm reduce the space to the general‐relativistic Einstein spaces with R ik = R ki . The equation R ik = λ g ik yields Γ i kl = { i kl }, and the pathes of test particles define geodesic world lines of the Einstein spaces.