Galilean quantum gravity with cosmological constant and the extended q-Heisenberg algebra
G Papageorgiou, B J Schroers
TL;DR
This work extends Galilean gravity in 2+1 dimensions to include a cosmological constant by framing it as a Chern-Simons theory for the doubly-extended Newton-Hooke group. It establishes the extended Newton-Hooke algebra as the classical double of the extended Heisenberg algebra and shows that quantisation is governed by the quantum double of the extended $q$-Heisenberg algebra, linking the deformation parameter to physical constants $( ext{Planck}, G, Lambda)$. The paper also delineates the corresponding noncommutative spacetime structure and outlines how combinatorial quantisation proceeds via the associated quantum group, setting the stage for a detailed representation-theoretic and geometric analysis in future work.
Abstract
We define a theory of Galilean gravity in 2+1 dimensions with cosmological constant as a Chern-Simons gauge theory of the doubly-extended Newton-Hooke group, extending our previous study of classical and quantum gravity in 2+1 dimensions in the Galilean limit. We exhibit an r-matrix which is compatible with our Chern-Simons action (in a sense to be defined) and show that the associated bi-algebra structure of the Newton-Hooke Lie algebra is that of the classical double of the extended Heisenberg algebra. We deduce that, in the quantisation of the theory according to the combinatorial quantisation programme, much of the quantum theory is determined by the quantum double of the extended q-deformed Heisenberg algebra.