Franca, Andre (2016): Quantum many-body effects in gravity and Bosonic theories. Dissertation, LMU München: Faculty of Physics
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### Abstract

Many-body quantum effects play a crucial role in many domains of physics, from condensed matter to black-hole evaporation. The fundamental interest and difficulty in studying this class of systems is the fact that their effective coupling constant become rescaled by the number of particles involved $g= \alpha N$, and thus we observe a breakdown of perturbation theory even for small values of the $\ttt$ coupling constant. We will study three very different systems which share the property that their behaviour is dominated by non-perturbative effects. \\ The strong CP problem - the problem of why the $\theta$ angle of QCD is so small - can be solved by the Peccei-Quinn mechanism, which promotes the $\theta$ angle to a physical particle, the axion. The essence of the PQ mechanism is that the coupling will generate a mass gap, and thus the expectation value of the axion will vanish at the vacuum. It has been suggested that topological effects in gravity can spoil the axion solution. By using the dual formulation of the Peccei-Quinn mechanism, we are able to show that even in the presence of such dangerous contributions from gravity, the presence of light neutrinos can stabilize the axion potential. This effect also puts an upper bound on the lightest neutrino mass.\\ We know that at high energies, gravitational scattering is dominated by black-hole formation. The typical size of black-holes is a growing function of the total center-of-mass energy involved in the scattering process. In the asymptotic future, these black-holes will decay into Hawking radiation, which has a typical wave-length of the size of the black-hole. Thus high energy gravitational scattering is dominated by low energy out states. It has been suggested that gravity is self-complete due to this effect, and that furthermore, there is a class of bosonic theories which can also be self-complete due to the formation of large classical field configurations: UV completion by Classicalization. \\ We explore the idea of Classicalization versus Wilsonian UV completion in derivatively coupled scalars. We seek to answer the following question: how does the theory decide which road to take at high energies? We find out that the information about the high energy behaviour of the theory is encoded in the sign of the quartic derivative coupling. There is one sign that allows for a consistent Wilsonian UV-completion, and another sign that contains continuous classical field configurations for localized sources. \\ In the third part of the thesis we explore non-perturbative properties of black holes. We consider the model proposed by Dvali and Gomez where black holes are described as Bose-Einstein condensates of $N$ gravitons. These gravitons are weakly interacting, however their collective coupling constant puts them exactly at the critical point of a quantum phase transition $\alpha N = 1$. We focus on a toy model which captures some of the features of information storage and processing of black holes. The carriers of information and entropy are the Bogoliubov modes, which we are able to map to pseuo-Goldstone bosons of a broken SU(2) symmetry. At the quantum phase transition this gap becomes $1/N$, which implies that the cost of information storage disappears in the $\Ninf$ limit. Furthermore, the storage capacity and lifetime of the modes increases with $N$, becoming infinite in the $\Ninf$ limit.\\ The attractive Bose gas which we considered is integrable in 1+1d. All the eigenstates of the system can be constructed using the Bethe ansatz, which transforms the Hamiltonian eigenvalue problem into a set of algebraic equations - the Bethe equations - for $N$ parameters which play the role of generalize momenta. While the ground state and excitation spectrum are known in the repulsive regime, in the attractive case the system becomes more complicated due to the appearance of bound states. In order to solve the Bethe equations, we restrict ourselves to the $\Ninf$ limit and transform the algebraic equations into a constrained integral equation. By solving this integral equation, we are able to study the phase transition from the point of view of the Bethe ansatz. We observe that the phase transition happens precisely when the constraint is saturated, and manifests itself as a change in the functional form of the density of momenta. Furthermore, we are able to show that the ground state of this system can be mapped to the saddle-point equation of 2-dimensional Yang--Mills on a sphere, with a gauge group U(N).