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arXiv 0907.2880hep-thgr-qcmath-phmath.MP
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2+1维伽利略量子引力的Chern-Simons方法

A Chern-Simons approach to Galilean quantum gravity in 2+1 dimensions

Georgios Papageorgiou, Bernd J. Schroers

备注：
38 pages, 1 figure, references updated
发表信息：
JHEP 0911:009,2009
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摘要：

本文定义并讨论了2+1维伽利略极限下的经典引力和量子引力。虽然在（2+1）维引力中，大质量物体之间不存在牛顿力，但伽利略极限并不是微不足道的。根据时空的拓扑结构，在大质量物体之间通常有许多拓扑自由度以及Aharonov-Bohm类型的拓扑相互作用。为了捕捉这些拓扑方面，我们考虑了一个双重的中心扩展的伽利略群的李代数具有不变的和非退化的内积。利用这个内积，我们将伽利略引力定义为双倍扩展的伽利略群的Chern-Simons理论。我们所考虑的伽利略群的特殊扩张是一个被广泛研究的群的经典二重群，即扩张的齐次伽利略群，它也通常被称为Nappi-Witten群。我们展示了双扩展Galilei群的Poisson-Lie结构，并使用Hamilton方法量子化了Chern-Simons理论。量子理论的许多方面都是由扩展的齐次伽利略群的量子偶决定的，简称伽利略偶。我们研究的表示理论的伽利略双，解释如何相关的辫子群表示占理论中的拓扑相互作用，并简要评论相关的非对易伽利略时空。

英文摘要：

We define and discuss classical and quantum gravity in 2+1 dimensions in the Galilean limit. Although there are no Newtonian forces between massive objects in (2+1)-dimensional gravity, the Galilean limit is not trivial. Depending on the topology of spacetime there are typically finitely many topological degrees of freedom as well as topological interactions of Aharonov-Bohm type between massive objects. In order to capture these topological aspects we consider a two-fold central extension of the Galilei group whose Lie algebra possesses an invariant and non-degenerate inner product. Using this inner product we define Galilean gravity as a Chern-Simons theory of the doubly-extended Galilei group. The particular extension of the Galilei group we consider is the classical double of a much studied group, the extended homogeneous Galilei group, which is also often called Nappi-Witten group. We exhibit the Poisson-Lie structure of the doubly extended Galilei group, and quantise the Chern-Simons theory using a Hamiltonian approach. Many aspects of the quantum theory are determined by the quantum double of the extended homogenous Galilei group, or Galilei double for short. We study the representation theory of the Galilei double, explain how associated braid group representations account for the topological interactions in the theory, and briefly comment on an associated non-commutative Galilean spacetime.

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