论文标题
通过凸起的凸面变形代数，u（1）$^3 $量子重力的精确定量
Exact quantisation of U(1)$^3$ quantum gravity via exponentiation of the hypersurface deformation algebroid
论文作者
论文摘要
3+1 Euclidian标志性一般相对论的U（1）$^3 $模型是一种相互作用的，共同的场理论，具有两个物理极化，具有Lorentzian一般相对论的许多特征。特别是，它显示出具有非平凡的高压变形代数的非平凡实现，即相位空间依赖性结构函数而不是结构常数。 在本文中，我们表明该模型允许{\ IT确切的定量}。定量基于这样的观察结果，即对于该模型和规范换向关系的选定表示，密度统一超出表面代数{\可以在非脱位状态上启用。这些状态代表非分类量子公制，从经典的角度来看，是高度表面代数可表示的相关状态。 代数的表示是准确的，没有歧义性及其无异常。量子约束可以使用{\ it groupoid平均}进行精确求解，而解决方案则可以接受希尔伯特空间结构，该结构与最近发现的减少相位空间公式的定量相吻合。使用最近发现的该模型的协变量动作，我们启动了一个路径积分或自旋泡沫公式，由于量规组的Abelian特征，它比Lorentzian Signature一般相对论要简单得多，并为一般旋转泡沫模型提供了理想的测试地面。
The U(1)$^3$ model for 3+1 Euclidian signature general relativity is an interacting, generally covariant field theory with two physical polarisations that shares many features of Lorentzian general relativity. In particular, it displays a non-trivial realisation of the hypersurface deformation algebroid with non-trivial, i.e. phase space dependent structure functions rather than structure constants. In this paper we show that the model admits {\it an exact quantisation}. The quantisation rests on the observation that for this model and in the chosen representation of the canonical commutation relations the density unity hypersurface algebra {\it can be exponentiated on non-degenerate states}. These are states that represent a non-degenerate quantum metric and from a classical perspective are the relevant states on which the hypersurface algebra is representable. The representation of the algebra is exact, with no ambiguities involved and anomaly free. The quantum constraints can be exactly solved using {\it groupoid averaging} and the solutions admit a Hilbert space structure that agrees with the quantisation of a recently found reduced phase space formulation. Using the also recently found covariant action for that model, we start a path integral or spin foam formulation which, due to the Abelian character of the gauge group, is much simpler than for Lorentzian signature general relativity and provides an ideal testing ground for general spin foam models.
