Aharonov-Bohm的圈量子引力
Loop Quantum Gravity a la Aharonov-Bohm
摘要：
圈量子引力的状态空间可以分解为与自旋网络图的同构等价类相关的正交子空间。在本文中，我调查的可能性，获得这种状态空间的量子化的拓扑场论与多个自由度。起始点是一个3-流形与网络的缺陷线。这个流形上的局部平坦联络可以在非可收缩回路周围具有非平凡的完整性。这实际上是Aharonov-Bohm效应的数学起源。我用标准场论方法证明了这个理论。示出的功能积分定义的标量积减少到一个有限维积分的模空间。由Faddeev-Popov行列式给出的一个非平凡的测度。我认为所得到的标积与圈量子引力中所用的标积是一致的。我提供了一个明确的推导的情况下，一个单一的缺陷线，对应于一个单一的循环量子引力。此外，我讨论了自旋泡沫模型的背景下使用的自旋网络的关系。
英文摘要：
The state space of Loop Quantum Gravity admits a decomposition into orthogonal subspaces associated to diffeomorphism equivalence classes of spin-network graphs. In this paper I investigate the possibility of obtaining this state space from the quantization of a topological field theory with many degrees of freedom. The starting point is a 3-manifold with a network of defect-lines. A locally-flat connection on this manifold can have non-trivial holonomy around non-contractible loops. This is in fact the mathematical origin of the Aharonov-Bohm effect. I quantize this theory using standard field theoretical methods. The functional integral defining the scalar product is shown to reduce to a finite dimensional integral over moduli space. A non-trivial measure given by the Faddeev-Popov determinant is derived. I argue that the scalar product obtained coincides with the one used in Loop Quantum Gravity. I provide an explicit derivation in the case of a single defect-line, corresponding to a single loop in Loop Quantum Gravity. Moreover, I discuss the relation with spin-networks as used in the context of spin foam models.
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