• higher derivative gravity
• agravity
• conformal invariance
• fixed point
• UV-completion
• unitarity
• ghosts

This thesis deals with a tricky problem yet unsolved in Theoretical Physics, that is the search for a consistent quantum theory of gravity, which is the last fundamental force of nature which is still lacking a quantum description. Indeed, the other three fundamental forces, which are the electromagnetic, the weak and the strong nuclear forces, are well described at the quantum level by the Standard Model of particle physics, which enjoys the fundamental properties of locality, renormalizability and unitarity. The classical model of gravity is General Relativity, the theory formulated by Einstein and described by the Einstein-Hilbert action that has been the starting point of several past attempts to quantize gravity. Unfortunately, the Einstein-Hilbert action violates one of the fundamental requirement for a quantum field theory: renormalizability. To overcome this problem a straightforward solution has been proposed, that is the model of Higher Derivative Gravity. It is called in this way because its kinematic term contains four derivatives instead of the usual two. However, the price for gaining renormalizability is the loss of another fundamental requirement: unitarity. Furthemore, various analyses have shown that there could also be an obstacle to the UV completeness of the theory, because the coupling f_0 has a positive Beta Function and only an unstable fixed point in f_0 = 0, so the coupling f_0 could grow unbounded according to the renormalization group. This is the starting point of this thesis, because the purpose of this work is not to find a different outset in the search of a quantum theory of gravity, as other approaches do, but it is instead to insist on \eqref{hdg action} as a viable model and work to overcome its problems. Therefore, after an explanation of why Higher Derivative Gravity is needed, its problems are first described and then addressed, analyzing and comparing different proposals present in literature and recent papers, including also some original contributions. After the introduction of Higher Derivative Gravity, the thesis is divided into two main parts.

In the first main part it is addressed the issue of the behaviour of the coupling f_0, from different points of view. The first point of view foresees to start with the complete higher derivative action and to study the Beta Function of the coupling f_0. An original computation is done in this section, which has been inspired by the so-called ''Agravity Mechanism'' discussed by Salvio and Strumia. They suggest that the strong self-coupling of the conformal mode of the graviton, caused by the growth of the coupling f_0 that controls its interaction, is not a problem for the theory because, at high energy ,the conformal mode decouples from the rest of the theory. The way in which this might take place is discussed and it is suggested that in the UV limit the theory should become the so-called Weyl Gravity, whose action is invariant under the Local Conformal (or Weyl) Transformations. The computation done in the thesis confirms the result of on the positivity of the Beta Function of f_0, and the implications of its functional form are discussed in detail.

The second point of view on the issue regarding the coupling f_0 is a change of perspective on Weyl symmetry and it follows the work of Oda. It is shown that a ''Restricted Weyl Symmetry'', which consists in a Local Weyl Symmetry whose parameter obeys a differential constraint, can be obtained performing a partial gauge-fixing of the Local Weyl Symmetry of the Conformal action. In this process, starting from the Conformal action, at the end of the gauge-fixing procedure one obtains the action plus two extra scalar fields, whose nature is discussed. This idea might offer a solution to the f_0 problem changing the perspective. Indeed, in the first approach one considers the coupling f_0 as a physical coupling, so one must understand how to interpret its problematic behaviour at high energies, instead, in this second approach, the $R^2$ term arises from the partial gauge-fixing procedure, so the parameter accompanying it is not physical. This is of course just a starting point, and other steps have to be taken in this direction, but it might represent an interesting change of paradigm. At the end of this part of the thesis, a link is outlined between the proposal of Salvio-Strumia and Oda, because it is shown that both discuss the idea that Conformal Gravity is equivalent to Higher Derivative Gravity plus two scalar fields. A comparison between these two approaches is provided.

The last approach presented on this topic refers to the work of Percacci and Ohta. They find a set of Beta Functions for the higher derivative coupling constants, which depend on the spacetime dimension. The interesting characteristic of this method is that the spacetime dimension is not a regularization tool, because these Beta Functions are automatically finite in every dimension, and d can be treated as a continuous parameter. Thus, they analyze the behaviour of the fixed points of the theory with changing d. The interesting result of this work is that working in d = 4 + epsilon a fixed point is found, whose natural continuation in d=4 brings to a stable fixed point in omega = 0, where omega = -((d-1) f_2)/f_0. This fixed point corresponds to a Weyl-invariant four-derivative sector, and can be roughly interpreted as the indication of a stable fixed point in f_0 = +infinity, instead of the usual well-known unstable fixed point in f_0 = 0. This might represent a direct computational confirmation of the Agravity Mechanism. The reason why this result has not be obtained before, working directly in four dimension, is discussed. In this section is also shown a different version of the original computation of this thesis, that is a d dependant one, whose result, unfortunately, seems not to be applicable in the perturbative framework.

The second main part of the thesis focuses on the unitarity problem of Higher Derivative Gravity, again from two different points of view.

The first point of view follows the idea of Mannheim, who tries to show that the loss of unitarity of the theory is caused by the fact that the usual quantization procedure does not apply to a fourth-order derivative theory. Instead of exploiting the property of Hermiticity, which the theory does not enjoy, one has to exploit the PT symmetry of the theory, which is a more generic property that nevertheless ensures real eigenvalues of the Hamiltonian. The gain is that, defining the inner-product according to this new symmetry, the problem of the loss of unitarity is circumvented.

The second point of view on the issue of unitarity is based on the prescription proposed by Anselmi and Piva, that represents a new way of performing the integration domain deformation and the Wick rotation. With this prescription, the unitarity condition becomes valid in a subspace of the whole Fock space of the theory, while the rest of the Fock space is considered to contain ''fake'' degrees of freedom. As a conclusion of the second main part, a comparison between the two points of view is provided.

In conclusion, the aim of this thesis is to show the need for and the problems of the Higher Derivative Gravity model, to collect some of their possible solutions and to provide comparisons between some of these solutions. The hope is that all of these pieces can fit together, and can contribute to complete the puzzle in order to provide a consistent quantum theory of gravity.