Conformal Symmetry in Field Theory and in Quantum Gravity

Lesław Rachwał

doi:10.3390/universe4110125

Abstract

Conformal symmetry always played an important role in field theory (both quantum and classical) and in gravity. We present construction of quantum conformal gravity and discuss its features regarding scattering amplitudes and quantum effective action. First, the long and complicated story of UV-divergences is recalled. With the development of UV-finite higher derivative (or non-local) gravitational theory, all problems with infinities and spacetime singularities might be completely solved. Moreover, the non-local quantum conformal theory reveals itself to be ghost-free, so the unitarity of the theory should be safe. After the construction of UV-finite theory, we focused on making it manifestly conformally invariant using the dilaton trick. We also argue that in this class of theories conformal anomaly can be taken to vanish by fine-tuning the couplings. As applications of this theory, the constraints of the conformal symmetry on the form of the effective action and on the scattering amplitudes are shown. We also remark about the preservation of the unitarity bound for scattering. Finally, the old model of conformal supergravity by Fradkin and Tseytlin is briefly presented.

Highlights

From the beginning of research on theories enjoying invariance under local spacetime-dependent transformations, conformal symmetry played a pivotal role—first introduced by Weyl related changes of meters to measure distances
What are killers? In the construction of a higher derivative theory so far we have considered only operators, which were quadratic in field strengths
Here we propose a gravitational theory without matter, which still may be one-loop conformal, but it must be defined in a dimension higher than four

Summary

Introduction

From the beginning of research on theories enjoying invariance under local spacetime-dependent transformations, conformal symmetry played a pivotal role—first introduced by Weyl related changes of meters to measure distances (and due to relativity changes of periods of clocks to measure time intervals). The absence of UV-divergences or vanishing of the beta functions (or in the language of the RG community the condition of being at a fixed point of RG) are all signals that we are dealing with conformally invariant theories on the quantum level These all are different sides of the same coin that can be investigated from the technical point of view by the analysis of invariance of the measure, of the quantum renormalization procedure and of the classical action of a theory. It is known that in d = 4 spacetime dimensions this theory on the classical level is not scale-invariant, because of the presence of a dimensionful coupling constant κ4 related to the Newton’s constant, the situation can be improved a bit by using a trick with a dilaton field Such dilatonic Einsteinian gravitation is scale-invariant classically, but the UV-divergences are inevitable on the quantum level and they destroy the conformal symmetry. In tight relation to previous topics, we discuss some virtues of the Fradkin–Tseytlin conformal gravity models in four dimensions

Long Story of Infinities

Conformal Quantum Gravity

Scattering Amplitudes in Scale-Invariant Theories at the Tree-Level

Conformal Supergravity

Conclusions