Geometric Formulation of the Many-Electron Theory

We present a systematic analysis of the time-dependent problem with an accurate formulation based on geometric algebra. This provides a systematic definition of the configuration space, of the external potentials, of the one electron operators for a many electron system, and of the electron-electron interaction terms. We arrive both to a formal equation for the total energy and to the equation for the time-evolution of the wave function. From this using the new geometric notation and the indistinguishability and equivalence of the electrons and the fact that we are interested either in the ground state or in states near the ground state, we formulate a variational problem from which a set of tractable equations, which self-consistently define the many electron wave function and density, is obtained. The equation of motion for Ψ(t) is given.