The Asymptotic Safety hypothesis states that the high-energy completion of gravity is provided by an interacting renormalization group fixed point. This implies non-trivial quantum corrections to the scaling dimensions of operators and correlation functions which are characteristic for the corresponding universality class. We use the composite operator formalism for the effective average action to derive an analytic expression for the scaling dimension of an infinite family of geometric operators $\int d^dx \sqrt{g} R^n$. We demonstrate that the anomalous dimensions interpolate continuously between their fixed point value and zero when evaluated along renormalization group trajectories approximating classical general relativity at low energy. Thus classical geometry emerges when quantum fluctuations are integrated out. We also compare our results to the stability properties of the interacting renormalization group fixed point projected to $f(R)$-gravity, showing that the composite operator formalism in the single-operator approximation cannot be used to reliably determine the number of relevant parameters of the theory.
Nonrelativistic scale anomaly, and composite operators with complex scaling dimensions
它表明，一个非相对论性的量子尺度异常表现在复合运营商的外观与复杂的标度尺寸。特别地，我们研究了具有平方反比势的非相对论性量子力学，并考虑了复合s波算符O= π π。我们解析地计算了这个算子的标度维数，并确定了传播子<0| T O O^{ \dagger }| 0>。算符O代表一个无限的束缚态塔，具有几何能谱。简要讨论了具有较高角动量的算符。
Rudjer Boskovic Institute（鲁德·博什科维奇研究所）；Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern（伯尔尼大学理论物理研究所爱因斯坦基础物理中心）；Southern Denmark University（南丹麦大学）；Università di Napoli Federico II（那不勒斯费德里科二世大学）
