The Eisenhart Lift for Field Theories

TL;DR

This paper extends the Eisenhart lift formalism to field theories, allowing the dynamics of systems with potentials to be represented as free systems in curved space, with potential applications in fundamental physics.

Contribution

The paper introduces a fully dynamical 'fictitious' field in the Eisenhart lift for field theories and generalizes the approach to higher-dimensional theories using a mixed vielbein.

Findings

Eisenhart lift formalism applied to simple harmonic motion.

Extension of the Eisenhart lift to homogeneous field theories with a new dynamical field.

Potential applications in gauge hierarchy and inflation initial conditions.

Abstract

We present the Eisenhart-lift formalism in which the dynamics of a system that evolves under the influence of a conservative force is equivalent to that of a free system embedded in a curved manifold with one additional generalised coordinate. As an illustrative example in Classical Mechanics, we apply this formalism to simple harmonic motion. We extend the Eisenhart lift to homogeneous field theories by adding one new field. Unlike an auxiliary field, this field is fully dynamical and is therefore termed fictitious. We show that the Noether symmetries of a theory with a potential are solutions of the Killing equations in the lifted field space. We generalise this approach to field theories in four and higher spacetime dimensions by virtue of a mixed vielbein that links the field space and spacetime. Possible applications of the extended Eisenhart-lift formalism including the gauge hierarchy problem and the initial conditions problem in inflation are briefly discussed.