arXiv 1605.093542016-05-30hep-th

量子引力渐近安全性中的泛函截断

Functional truncations in asymptotic safety for quantum gravity

Juergen Dietz

备注：: PhD Thesis

摘要：

有限维截断和单场近似在量子引力的渐近安全性研究中起着主导作用。本论文致力于探索无穷维或泛函截断的有效作用量的渐近安全性，以及在这种情况下，单场近似可能引起的影响。首先对单场f（R）截断的三个现有流动方程进行全面分析，确定它们的整体不动点解空间，并在适当的情况下确定相应的本征算子解空间。作为第二个结果，它表明，一个化身的单场f（R）近似实际上打破了意义上说，没有物理内容留下探索。为了澄清这种激烈的调查结果是否可以造成的近似在建立重整化群流，我们确定单场近似作为一个主要的候选人，并显示在更熟悉的背景下的标量场理论，它确实可以导致许多类型的非物理结果。作为一种方法，以避免这种非物理行为，我们强调了以前已知的分裂沃德恒等式的重要性，并通过完全恢复正确的物理图像在标量场理论中的实用性。以此证明分裂的沃德恒等式在引力场中也可以导出良好的泛函截断，我们导出了涨落场中有效作用量的双场截断的共形引力流方程，它超越了局域势近似.结果发现，分裂沃德恒等式导致一组简化的重整化群方程的共形因子，而不同的关键点，承担密切相似的流方程中得到的标量场理论。

英文摘要：

Finite dimensional truncations and the single field approximation have thus far played dominant roles in investigations of asymptotic safety for quantum gravity. This thesis is devoted to exploring asymptotic safety in infinite dimensional, or functional, truncations of the effective action as well as the effects that can be caused by the single field approximation in this context. It begins with a comprehensive analysis of the three existing flow equations of the single field f(R) truncation by determining their spaces of global fixed point solutions and, where applicable, of corresponding eigenoperator solutions. As a second result, it is then shown that one incarnation of the single field f(R) approximation actually breaks down in the sense that there is no physical content left to explore. In order to clarify whether such drastic findings can be caused by the approximations used in setting up the renormalisation group flow, we identify the single field approximation as a prime candidate and show in the more familiar context of scalar field theory that it can indeed lead to many types of non-physical results. As a way to avoid such non-physical behaviour we highlight the importance of the previously known split Ward identity and exemplify its usefulness by fully restoring the correct physical picture in scalar field theory. Taking this result as evidence that the split Ward identity may lead to well behaved functional truncations also in gravity, we derive the flow equations of conformal gravity in a bi-field truncation of the effective action that goes beyond the local potential approximation in the fluctuation field. It is found that the split Ward identity leads to a simplified set of renormalisation group equations for the conformal factor that, while differing at crucial points, bear close resemblance to flow equations obtained in scalar field theory.

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