Quantizing the Eisenhart Lift

Abstract

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 Schrödinger 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.