Elementary Classical and Quantum Lifshitz Systems

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 $\lambda$, \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.