Quantum gravity in 2+epsilon dimensions and renormalization group

本研究的目的是通过2+epsilon维展开法加深对量子引力的理解。引力相互作用的量子化是粒子物理学的基本问题之一。这个问题的根本困难在于不可重整化。在二维附近用2+epsilon展开法可以避免这一问题。由于这个原因，2+epsilon维的量子引力在理论上是很有趣的，它可能具有与四维Qunatum引力定性相似的特征。我用形式主义构造了可重正性的一个证明，它保持了显式体积，并保持了与我的合作者的微分同胚。在这个证明中，我们利用了这样一个事实，即出现有效作用的微扰估计的发散的结构强烈地受到BRS不变性的限制。二维附近的量子反常打破了共形不变性。然而，它的结构已经被很好地理解了，并且发散必须与反常一致。我和我的合作者进一步在两个环水平上进行了显式计算。我们能够在时空的紫外不动点附近计算引力耦合常数的标度指数，它变成了标度不变量。我们采用了明显的协变计算方法，并且我们能够估计在这样一个固定点附近的大量子效应。我们首次导出了量子引力中不动点的非平凡标度指数，这是因为两圈能级计算是必要的。

The purpose of this research is to deepen our understanding of quantum gravity by the 2+epsilon demensional expansion method. The quantization of the gravitational interaction is the one of the principle problem in particle physics. The fundamental difficulty of this problem resides in the nonrenormalizability. We can avoid this problem by the 2+epsilon dimensional expansion method near two dimensions. The quantum gravity in 2+epsilon dimension is theoretically interesting due to this reason and may possess qualitatively similar feature with four dimensional qunatum gravity. I have constructed a proof of the renormalizabilty in a formalism which preserves the manifest volume preserving diffeomorphism with my collaborators. In this proof we have used the fact that the structure of the divergences which appear the the perturbative evaluation of the effective action is strongly constrained by the BRS invariance. The conformal invariance is broken by the quantum anomaly near two dimension. However its structure is well understood and the divergences have to be consistent with the anomaly.I have further performed an explicit calculation at the two loop level with my collaborator. We are able to calculate the scaling exponent of the gravitational coupling constant near the untraviolet fixed point of the space time which becomes the scale invariant.We have adopted a manifestly covariant calculation procedure and we are able to estimate a large quntum effect near such a fixed point. We are the first to derive the nontrivial scaling exponent at the fixed point in quantum gravity since thw two loop level calculation is necesarily for this purpose.