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Elementary Classical and Quantum Lifshitz Systems

Jarah Fluxman

TL;DR

This work provides a complete classification of elementary classical and quantum Lifshitz systems in 2 and 3 dimensions by examining the Lifshitz algebras (and their one-dimensional central extensions), their coadjoint orbits, and their unitary irreducible representations via the Mackey framework. Classical Lifshitz systems correspond to symplectic $G$-manifolds given by coadjoint orbits, while quantum Lifshitz systems correspond to projective representations that lift to true representations of central extensions. A central result is that, for building-block groups, coadjoint orbits and UIRs are in exact correspondence (with notable exceptions like complementary series); the indecomposable groups exhibit richer structures with explicit orbit-UIR classifications. The paper assembles a comprehensive dictionary linking geometric (coadjoint) data to quantum (UIR) data, enabling systematic construction of Lifshitz systems and their dispersions from symmetry principles. This framework has potential applications in holographic and condensed matter contexts where Lifshitz scaling and anisotropic dynamics arise.

Abstract

We classify the elementary classical and quantum Lifshitz systems. Lifshitz systems are systems where space and time scale anisotropically. That is, there is a constant $z$ such that under scaling by a factor of $λ$, \begin{equation*} \boldsymbol{x}\rightarrow λ\boldsymbol{x} \text{ and } t\rightarrow λ^{z}t \end{equation*} There are seven Lie groups, called the Lifshitz groups, which characterise all Lifshitz symmetries. Elementary classical Lifshitz systems are the symplectic manifolds with a transitive Lifshitz action, which turn out to be coadjoint orbits of the Lifshitz groups and their one-dimensional central extensions up to covering. Elementary quantum Lifshitz systems are the projective unitary irreducible representations of the Lifshitz groups.

Elementary Classical and Quantum Lifshitz Systems

TL;DR

This work provides a complete classification of elementary classical and quantum Lifshitz systems in 2 and 3 dimensions by examining the Lifshitz algebras (and their one-dimensional central extensions), their coadjoint orbits, and their unitary irreducible representations via the Mackey framework. Classical Lifshitz systems correspond to symplectic -manifolds given by coadjoint orbits, while quantum Lifshitz systems correspond to projective representations that lift to true representations of central extensions. A central result is that, for building-block groups, coadjoint orbits and UIRs are in exact correspondence (with notable exceptions like complementary series); the indecomposable groups exhibit richer structures with explicit orbit-UIR classifications. The paper assembles a comprehensive dictionary linking geometric (coadjoint) data to quantum (UIR) data, enabling systematic construction of Lifshitz systems and their dispersions from symmetry principles. This framework has potential applications in holographic and condensed matter contexts where Lifshitz scaling and anisotropic dynamics arise.

Abstract

We classify the elementary classical and quantum Lifshitz systems. Lifshitz systems are systems where space and time scale anisotropically. That is, there is a constant such that under scaling by a factor of , \begin{equation*} \boldsymbol{x}\rightarrow λ\boldsymbol{x} \text{ and } t\rightarrow λ^{z}t \end{equation*} There are seven Lie groups, called the Lifshitz groups, which characterise all Lifshitz symmetries. Elementary classical Lifshitz systems are the symplectic manifolds with a transitive Lifshitz action, which turn out to be coadjoint orbits of the Lifshitz groups and their one-dimensional central extensions up to covering. Elementary quantum Lifshitz systems are the projective unitary irreducible representations of the Lifshitz groups.

Paper Structure

This paper contains 42 sections, 1 theorem, 141 equations, 1 figure, 8 tables.

Table of Contents

  1. Introduction
  2. The Lifshitz Algebras and their Central Extensions
  3. Coadjoint Orbits and UIRs as Classical and Quantum Lifshitz Systems
  4. Coadjoint Orbits as Symplectic $G-$manifolds
  5. UIRs as quantum $G-$systems
  6. The Method of Mackey
  7. The "Building-Block" Groups
  8. The Two-dimensional Non-Abelian Group $\mathop{\mathrm{Aff}}\nolimits(1)$
  9. Coadjoint Orbits
  10. UIRs
  11. The Euclidean Groups $\mathop{\mathrm{Euc}}\nolimits(d)$
  12. Coadjoint Orbits
  13. UIRs
  14. The groups $\mathop{\mathrm{Spin}}\nolimits(d+1)$
  15. The groups $\mathop{\mathrm{Spin}}\nolimits(d,1)$
  16. ...and 27 more sections

Key Result

Theorem 3.1

Let $(M,\omega)$ be a simply-connected symplectic manifold admitting a transitive action of a connected Lie group $G$ via symplectomorphisms. Then $(M,\omega)$ is the universal cover of a coadjoint orbit of $G$ or one of its one-dimensional central extensions

Figures (1)

  • Figure 1: The unitary dual of the Heisenberg Group

Theorems & Definitions (1)

  • Theorem 3.1