  We derive a rigorous, quantum mechanical map of fermionic creation and
annihilation operators to continuous Cartesian variables that exactly
reproduces the matrix structure of the many-fermion problem. We show how our
scheme can be used to map a general many-fermion Hamiltonian and then consider
two specific models that encode the fundamental physics of many fermionic
systems, the Anderson impurity and Hubbard models. We use these models to
demonstrate how efficient mappings of these Hamiltonians can be constructed
using a judicious choice of index ordering of the fermions. This development
provides an alternative exact route to calculate the static and dynamical
properties of fermionic systems and sets the stage to exploit the
quantum-classical and semiclassical hierarchies to systematically derive
methods offering a range of accuracies, thus enabling the study of problems
where the fermionic degrees of freedom are coupled to complex anharmonic
nuclear motion and spins which lie beyond the reach of most currently available
methods.
