  The classical Eisenhart lift is a method by which the dynamics of a classical
system subject to a potential can be recreated by means of a free system
evolving in a higher-dimensional curved manifold, known as the lifted manifold.
We extend the formulation of the Eisenhart lift to quantum systems, and show
that the lifted manifold recreates not only the classical effects of the
potential, but also its quantum mechanical effects. In particular, we find that
the solutions of the Schrodinger equations of the lifted system reduce to those
of the original system after projecting out the new degrees of freedom. In this
context, we identify a conserved quantum number, which corresponds to the
lifted momentum of the classical system. We further apply the Eisenhart lift to
Quantum Field Theory (QFT). We show that a lifted field space manifold is able
to recreate both the classical and quantum effects of a scalar field potential.
We find that, in the case of QFT, the analogue of the lifted momentum is a
quantum charge that is conserved not only in time, but also in space. The
different possible values for this charge label an ensemble of Fock spaces that
are all disjoint from one another. The relevance of these extended Fock spaces
to the cosmological constant and gauge hierarchy problems is considered.
