Quantum gravity in 2+epsilon dimensions and renormalization group

本研究的目的是通过2+epsilon维度展开方法加深我们对量子引力的理解。引力相互作用的量子化是粒子物理学中的主要问题之一。这个问题的根本困难在于不可重整性。我们可以通过二维附近的2+的展开方法来避免这个问题。由于这个原因，2+epsilon维度的量子引力在理论上是有趣的，并且可能具有与四维量子引力在质量上相似的特征。我和我的合作者在一个形式主义中构造了一个可重整性的证明，该形式主义保留了明显的保体积微分同态。在这个证明中，我们使用了这样一个事实，即有效作用的摄动评价中出现的散度的结构受到BRS不变性的强烈约束。二维附近的量子异常打破了共形不变性。然而，它的结构已被很好地理解，辐散必须与异常一致。我和我的合作者在两个循环级别进一步执行了显式计算。我们可以计算出引力耦合常数在时空的紫外不动点附近的标度指数，该指数成为标度不变量。我们采用了一种明显的协变计算方法，我们能够在这样一个不动点附近估计一个大的量子效应。我们首先推导了量子引力中不动点处的非平凡标度指数，因为为此需要进行两环水平的计算。

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.