Dynamical realizations of the Lifshitz group

TL;DR

This paper explores the group-theoretic construction of dynamical systems with Lifshitz symmetry, generalizing conformal mechanics and the Ermakov-Milne-Pinney equation, and introduces invariant structures and metrics with applications to gravity.

Contribution

It develops a unified group-theoretic framework for Lifshitz-invariant dynamical systems, including new generalizations and invariant metrics with additional symmetries.

Findings

Constructed a generalized 1D conformal mechanics with arbitrary dynamical exponent z.

Proposed a generalized Ermakov-Milne-Pinney equation linked to Einstein equations.

Built Lorentzian metrics with Lifshitz symmetry and extra Galilei invariance.

Abstract

Dynamical realizations of the Lifshitz group are studied within the group-theoretic framework. A generalization of the 1d conformal mechanics is constructed, which involves an arbitrary dynamical exponent z. A similar generalization of the Ermakov-Milne-Pinney equation is proposed. Invariant derivative and field combinations are introduced, which enable one to construct a plethora of dynamical systems enjoying the Lifshitz symmetry. A metric of the Lorentzian signature in (d+2)-dimensional spacetime and the energy-momentum tensor are constructed, which lead to the generalized Ermakov-Milne-Pinney equation upon imposing the Einstein equations. The method of nonlinear realizations is used for building Lorentzian metrics with the Lifshitz isometry group. In particular, a (2d+2)-dimensional metric is constructed, which enjoys an extra invariance under the Galilei boosts.