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arXiv 0807.4405gr-qcmath-phmath.MP
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Riem（M）中的规范理论

Gauge Theory in Riem(M)

Henrique Gomes

备注：
Version virtually identical to the one published in J. Math. Phys. Contains corrections to a crucial proof, and new figures
发表信息：
J.Math.Phys. 52: 082501, 2011
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摘要：

在拉格朗日形式的广义相对论的几何动力学背景下，研究的对象是给定三维流形$M$上的{ \it}黎曼度量（及其时间导数）。本文的目的是研究$M$上所有度规的空间Riem（M）所具有的规范性质，特别是当它们与几何动力学约束有关时.例如，哈密尔顿约束不产生群，因此很难在规范设置中观察它在Riem（M）中的作用。然而，鉴于最近的结果代表GR作为一个对偶理论，不变的叶理下保持3-共形同态和3D共形变换，但不是根据refoliations，我们有理由在考虑规范结构只涉及到组$\mathcal{D}$的$M$的共形同态，和$\mathcal{C}$，共形的共形同态$M$。对于这些无穷维对称群，Riem（M）具有自然的主纤维丛（PFB）结构，这使得引力场可以用规范理论的全部处理。我们讨论了其中的一些结构，并构造了超度规诱导规范连接的显式公式。为了应用形式主义，我们计算一个特定的连接轴承强烈的相似性自然引起的德威特超度量，显示它具有理想的relationalist属性的一般属性。最后，我们发现，共形的共形同态组解决了固有的病理$\DD$组，也使它更接近Horava引力和对偶共形理论称为形状动力学。

英文摘要：

In the geometrodynamical setting of general relativity in Lagrangian form, the objects of study are the {\it Riemannian} metrics (and their time derivatives) over a given 3-manifold $M$. It is our aim in this paper to study the gauge properties that the space Riem(M) of all metrics over $M$ possesses, specially as they relate to the constraints of geometrodynamics. For instance, the Hamiltonian constraint does not generate a group, and it is thus hard to view its action in Riem(M) in a gauge setting. However, in view of the recent results representing GR as a dual theory, invariant under foliation preserving 3--diffeomorphisms and 3D conformal transformations, but not under refoliations, we are justified in considering the gauge structure pertaining only to the groups $\mathcal{D}$ of diffeomorphisms of $M$, and $\mathcal{C}$, of conformal diffeomorphisms on $M$. For these infinite-dimensional symmetry groups, Riem(M) has a natural principal fiber bundle (PFB) structure, which renders the gravitational field amenable to the full range of gauge-theoretic treatment. We discuss some of these structures and construct explicit formulae for supermetric-induced gauge connections. To apply the formalism, we compute general properties for a specific connection bearing strong resemblance to the one naturally induced by the deWitt supermetric, showing it has desirable relationalist properties. Finally, we find that the group of conformal diffeomorphisms solves the pathologies inherent in the $\DD$ group and also brings it closer to Horava gravity and the dual conformal theory called Shape Dynamics.

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