  An equation proposed by Levy, Perdew and Sahni in 1984 [PRA 30, 2745 (1984)]
is an orbital--free formulation of density functional theory. However, this
equation describes a bosonic system. Here, we analyze on a very fundamental
level, how this equation could be extended to yield a formulation for a general
fermionic distribution of charge and spin. This analysis starts at the level of
single electrons and with the question, how spin actually comes into a charge
distribution in a non-relativistic model. To this end we present a space-time
model of extended electrons, which is formulated in terms of geometric algebra.
Wave properties of the electron are referred to mass density oscillations. We
provide a comprehensive and non-statistical interpretation of wavefunctions,
referring them to mass density components and internal field components. It is
shown that these wavefunctions comply with the Schr\"odinger equation, for the
free electron as well as for the electron in electrostatic and vector
potentials. Spin-properties of the electron are referred to intrinsic field
components and it is established that a measurement of spin in an external
field yields exactly two possible results. The model agrees with the results of
standard theory concerning the hydrogen atom. Finally, we analyze many-electron
systems and derive a set of coupled equations suitable to characterize the
system without any reference to single electron states. The model is expected
to have the greatest impact in condensed matter theory, where it allows to
describe an $N$-electron system by a many-electron wavefunction $\Psi$ of four,
instead of 3N variables. The many-body aspect of a system is in this case
encoded in a bivector potential.
