The Eisenhart Lift for Field Theories

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.

arXiv comment7 pages, 1 figure, to appear in PRD V2: Title Changed, relation to the Eisenhart lift clarified V3: Title Changed, references added

References in corpus (24)

• False Vacuum Inflation with Einstein Gravity
• Kaluza-Klein Gravity
• Adiabatic and entropy perturbations from inflation
• Super-Horizon Scale Dynamics of Multi-Scalar Inflation
• Inflationary paradigm in trouble after Planck2013
• Primordial non-gaussianities from multiple-field inflation
• Scalar perturbations during multiple-field slow-roll inflation
• Primordial perturbations and non-Gaussianities in DBI and general multi-field inflation
• Perturbations in generalized multi-field inflation
• Multifield Inflation after Planck: The Case for Nonminimal Couplings
• A covariant approach to general field space metric in multi-field inflation
• Non-gaussianity from the bispectrum in general multiple field inflation
• The Power Spectrum for a Multi-Component Inflaton to Second-Order Corrections in the Slow-Roll Expansion
• The inflationary bispectrum with curved field-space
• Density perturbations arising from multiple field slow-roll inflation
• Constraints for the existence of flat and stable non-supersymmetric vacua in supergravity
• Frame Covariant Nonminimal Multifield Inflation
• Cosmological aspects of the Eisenhart-Duval lift
• Eisenhart lifts and symmetries of time-dependent systems
• Multiple-field inflation and the CMB
• The Eisenhart lift: a didactical introduction of modern geometrical concepts from Hamiltonian dynamics
• Eisenhart's theorem and the causal simplicity of Eisenhart's spacetime
• The metric on field space, functional renormalization, and metric-torsion quantum gravity
• Noether symmetry approach in the cosmological alpha-attractors

Cited by in corpus (10)

• Frame Covariant Formalism for Fermionic Theories
• Beyond the Poles in Attractor Models of Inflation
• A Finite Measure for the Initial Conditions of Inflation
• Eisenhart--Duval lift for minisuperspace quantum cosmology
• Effective Field Theories as Lagrange Spaces
• Quantizing the Eisenhart Lift
• Dynamics in wormhole spacetimes: a Jacobi metric approach
• Initial Conditions of Inflation in a Bianchi I Universe
• More on Jacobi metric: Randers-Finsler metrics, frame dragging and geometrisation techniques
• Equivalence of a harmonic oscillator to a free particle and Eisenhart lift