Extremely Correlated Fermi Liquids in the limit of infinite dimensions

We study the infinite spatial dimensionality limit of the recently developed Extremely Correlated Fermi Liquid (ECFL) theory for the t-J model. We directly analyze the Schwinger equations of motion for the Gutzwiller projected electron Green's function. From simplifications arising in this limit, we are able to make several exact statements about the theory. The ECFL Green's function is shown to have a momentum independent Dyson (Mori) self energy. For practical calculations we introduce a partial projection parameter \lambda, and obtain the complete set of ECFL integral equations to second order in \lambda. In a related publication, these equations are compared in detail with the dynamical mean field theory for the large U Hubbard model. Paralleling the well known mapping for the Hubbard model, we find that the infinite dimensional t-J model can be mapped to the infinite-U Anderson impurity model with a self-consistently determined set of parameters. This mapping extends individually to the auxiliary Green's function and the caparison factor of the ECFL theory. Additionally, the optical conductivity is shown to be obtainable from the Green's function with negligibly small vertex corrections. These results are shown to hold to each order in \lambda.