arXiv 1209.53582012-09gr-qchep-th

嘉当引力、物质场和规范原理

Cartan gravity, matter fields, and the gauge principle

H. F. Westman, T. G. Zlosnik

备注：: Updated to match content of published version
DOI：: 10.1016/j.aop.2013.03.012

摘要：

引力通常被认为是自然界中的四大力场之一。然而，在标准公式中，它的数学结构与粒子物理学中控制电磁、弱和强相互作用的杨-米尔斯场相当不同。本文探讨了这种不和谐，特别侧重于如何从引力的Cartan几何公式的角度重力耦合到物质。在那里，引力场由一对变量表示：1）一个“接触矢量”$V^A$，它在几何上被形象化为时空流形和在其上“滚动”的模型时空之间的接触点，以及2）一个规范连接$A_μ^{\phμAB}$，取SO（2，3）或SO（1，4）的李代数中的值，这在数学上决定了模型时空在滚动时旋转了多少。通过坚持两个原则，{ \em规范原则}和{ \em多项式简单性}，我们展示了如何可以重新制定的标准物质场的行动方式是和谐的嘉当的几何结构。这就产生了一个一阶偏微分方程的所有物质场的公式。我们详细展示了如何标准的二阶配方可以恢复。此外，能量-动量和自旋-密度三种形式自然地结合成一个单一的物体，这里表示为自旋-能量-动量三种形式。最后，我们强调一个特殊的数学结构，我们的一阶制定杨米尔斯领域。这就提出了一种方法，可以利用Cartan几何的自然推广，将具有引力的U（1）规范场统一为SO（1，5）值规范场，在Cartan几何中，较大的对称群自发地分解为$SO(1,3)\times U(1)$。这个统一理论与物质场的耦合以及对非阿贝尔规范场的可能扩展都是悬而未决的问题。

英文摘要：

Gravity is commonly thought of as one of the four force fields in nature. However, in standard formulations its mathematical structure is rather different from the Yang-Mills fields of particle physics that govern the electromagnetic, weak, and strong interactions. This paper explores this dissonance with particular focus on how gravity couples to matter from the perspective of the Cartan-geometric formulation of gravity. There the gravitational field is represented by a pair of variables: 1) a `contact vector' $V^A$ which is geometrically visualized as the contact point between the spacetime manifold and a model spacetime being `rolled' on top of it, and 2) a gauge connection $A_μ^{\phμAB}$, taken to be valued in the Lie algebra of SO(2,3) or SO(1,4), which mathematically determines how much the model spacetime is rotated when rolled. By insisting on two principles, the {\em gauge principle} and {\em polynomial simplicity}, we show how one can reformulate the standard matter field actions in a way that is harmonious with Cartan's geometric construction. This yields a formulation of all matter fields in terms of first order partial differential equations. We show in detail how the standard second order formulation can be recovered. Furthermore, the energy-momentum and spin-density three-forms are naturally combined into a single object here denoted the spin-energy-momentum three-form. Finally, we highlight a peculiarity in the mathematical structure of our first-order formulation of Yang-Mills fields. This suggests a way to unify a U(1) gauge field with gravity into a SO(1,5)-valued gauge field using a natural generalization of Cartan geometry in which the larger symmetry group is spontaneously broken down to $SO(1,3)\times U(1)$. The coupling of this unified theory to matter fields and possible extensions to non-Abelian gauge fields are left as open questions.

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摘要：

英文摘要：

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