论文标题
$ u（1）^3 $模型的欧几里得量子重力的协变点
Covariant Origin of the $U(1)^3$ model for Euclidean Quantum Gravity
论文作者
论文摘要
最近由于Varadarajan等人，最近在一系列论文中强调了U（1）$^3 $模型作为量子重力的测试实验室的实用性。从SU（2）到U（1）$^3 $的简化可以通过分别从高斯，空间差异和汉密尔顿约束中删除所有非亚伯式的术语来简单地在哈密顿公式中执行。但是，人们可能会问从哪种拉格朗日制定该理论下降。对于SU（2）理论，众所周知，人们可以选择Palatini动作，Holst Action或（反）自我行动（Euclidian Signature）作为起点，所有起点都导致了同等的Hamiltonian配方。在本文中，我们直接将这个问题直接针对U（1）$^3 $理论分析。令人惊讶的是，事实证明，帕拉蒂尼（Palatini）或霍尔斯特（Holst）表述的阿贝尔（Abelian）类似物是一种一致但拓扑理论，而没有传播自由度。另一方面，（反）自我配方的扭曲的Abelian类似物确实导致了所需的汉密尔顿配方。我们派生的一个新方面是，我们使用1个。半密度的价值四分法，简化了分析，2。没有简单的约束（通常接受一个通常被手动忽略的不希望的解决方案）和3。而不会从一开始就施加时间表。作为副产品，我们表明，非亚伯理论也承认了一种扭曲的（反）自我表述。最后，我们还通过扩展了由于山羊角，戴尔（Dell），雅各布森（Jacobson）和佩尔丹（Peldan）而扩大了先前的工作，从而获得了欧几里得GR的纯连接公式，这可能是路径积分研究（Euclidian）GR作为Yang-Mills理论的有趣起点，该起点是具有非级别的lagranmial lagranmial lagrangian。
The utility of the U(1)$^3$ model as a test laboratory for quantum gravity has recently been emphasized in a recent series of papers due to Varadarajan et al. The simplification from SU(2) to U(1)$^3$ can be performed simply by hand within the Hamiltonian formulation by dropping all non-Abelian terms from the Gauss, spatial diffeomorphism and Hamiltonian constraints respectively. However, one may ask from which Lagrangian formulation this theory descends. For the SU(2) theory it is known that one can choose the Palatini action, Holst action or (anti-)selfdual action (Euclidian signature) as starting point all leading to equivalent Hamiltonian formulations. In this paper we systematically analyse this question directly for the U(1)$^3$ theory. Surprisingly, it turns out that the Abelian analog of the Palatini or Holst formulation is a consistent but topological theory without propagating degrees of freedom. On the other hand, a twisted Abelian analog of the (anti-)selfdual formulation does lead to the desired Hamiltonian formulation. A new aspect of our derivation is that we work with 1. half-density valued tetrads which simplifies the analysis, 2. without the simplicity constraint (which admits one undesired solution that is usually neglected by hand) and 3. without imposing the time gauge from the beginning. As a byproduct we show that also the non-Abelian theory admits a twisted (anti-)selfdual formulation. Finally we also derive a pure connection formulation of Euclidian GR including a cosmological constant by extending previous work due to Capovilla, Dell, Jacobson and Peldan which may be an interesting starting point for path integral investigations and displays (Euclidian) GR as a Yang-Mills theory with non-polynomial Lagrangian.
