四维欧氏引力的Smolins $G\rightarrow 0$极限中的约束代数
The constraint algebra in Smolins' $G\rightarrow 0$ limit of 4d Euclidean Gravity
摘要：
Smolin广义协变的$G_{\mathrm{Newton}}\rightarrow0$极限是研究圈量子引力中约束代数的一个有用的玩具模型。特别地，其哈密顿约束之间的交换子具有度量依赖的结构函数。虽然存在一个先前的LQG一样的非平凡的异常自由约束的构造模型，该工作遭受两个缺陷。首先，斯莫林关于量子动力学无法产生传播效应的评论适用。第二，构造只产生一个单一的哈密顿约束的作用连同其交换子的作用，通过相应的离散逼近的连续极限; 2个或更多个约束的乘积的连续极限不存在。在这里，我们将在量子动力学的变化，通过结构修改的离散近似的量子哈密顿约束的选择。这种新的结构是由LQG中负责传播的参数化场论量子化所激发的，并显著地改变了物理状态的空间。我们研究了在这些结构变化的背景下，该模型的脱壳约束代数，并表明，连续极限行动的多个产品的哈密顿约束是（a）支持在一个适当的域的状态（b）产生异常免费之间的哈密顿约束和（c）是同胚协变。我们的许多考虑似乎足够强大，可以应用于4d欧几里德引力的设置。
英文摘要：
Smolin's generally covariant $G_{\mathrm{Newton}}\rightarrow0$ limit of 4d Euclidean gravity is a useful toy model for the study of the constraint algebra in Loop Quantum Gravity. In particular, the commutator between its Hamiltonian constraints has a metric dependent structure function. While a prior LQG like construction of non-trivial anomaly free constraint commutators for the model exists, that work suffers from two defects. First, Smolin's remarks on the inability of the quantum dynamics to generate propagation effects apply. Second, the construction only yields the action of a single Hamiltonian constraint together with the action of its commutator through a continuum limit of corresponding discrete approximants; the continuum limit of a product of 2 or more constraints does not exist. Here, we incorporate changes in the quantum dynamics through structural modifications in the choice of discrete approximants to the quantum Hamiltonian constraint. The new structure is motivated by that responsible for propagation in an LQG like quantization of Paramaterized Field Theory and significantly alters the space of physical states. We study the off shell constraint algebra of the model in the context of these structural changes and show that the continuum limit action of multiple products of Hamiltonian constraints is (a) supported on an appropriate domain of states (b) yields anomaly free commutators between pairs of Hamiltonian constraints and (c) is diffeomorphism covariant. Many of our considerations seem robust enough to be applied to the setting of 4d Euclidean gravity.
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