L'objectif de ce travail est une construction perturbative rigoureuse de la théorie de la Yang-Mills SU(2) dans l'espace euclidien à quatre dimensions. La technique d'intégration fonctionnelle donne une basemathématique pour établir les équations de flot différentielles du groupe de renormalisation pour l'action efficace. Si l'introduction de régulateurs dans l'espace de moments permet de donner une définition mathématique des fonctions de Schwinger, la difficulté importante de l'approche est le fait que cesrégulateurs brisent l'invariance de jauge. Ainsi, le travail principal est alors de prouver à tous les ordres en perturbation l'existence de ces fonctions de correlation et la validité des identités de Slavnov-Taylor pour la théorie renormalisée. / The goal of this work is a rigorous perturbative construction of the SU(2) Yang-Mills theory in four dimensional Euclidean space. The functional integration technique gives a mathematical basis for establishing the differential Flow Equations of the renormalization group for the effective action. While the introduction of momentum space regulators permits to give a mathematical definition of the Schwinger functions, the important difficulty of the approach is the fact that these regulators break gauge invariance. Thus the main part of the work is to prove at all loop orders the existence of the vertex functions and the restoration of the Slavnov-Taylor identities in the renormalised theory.

An investigation into open problems related to black hole evaporation in the semi-classical framework, concerning the existence of quantum field theories on spacetimes modelling evaporating black holes as well as the existence of black hole solutions to the semi-classical Einstein equations. / Eine Untersuchung offener Probleme zur Verdampfung schwarzer Löcher im semi-klassischen Modell, bezüglich der Existenz von Quantenfeldtheorien auf Raumzeiten, die verdampfende schwarze Löcher beschreiben, sowie der Existenz von Lösungen der semi-klassischen Einstein Gleichungen, welche schwarze Löcher darstellen.

Negative energy densities are an abundant and necessary feature of quantum field theory (QFT) and may lead to surprising measurable effects. Some of these stand in contrast to classical physics, so that the accumulation of negative energy, also in quantum field theory, must be subject to some constraints. One class of such constraints is commonly referred to as quantum energy inequalities (QEI). These are lower bounds on the averaged stress-energy tensor which have been established quite generically in quantum field theory, however, mostly excluding models with self-interaction. A rich but mathematically tractable class of interacting models are those subject to integrability. In this thesis, we give an overview of the construction of integrable models via the inverse scattering approach, extending previous results on the char- acterization of local observables to models with more than one particle species and inner degrees of freedom. We apply these results to the stress-energy tensor, leading to a characterization of the stress-energy tensor at one-particle level. In models with simple interaction, where the S-matrix is independent of the particles’ momenta, this suffices to con- struct the full stress-energy tensor and provide a state-independent QEI. In models with generic interaction, we obtain QEIs at the one-particle level and find that they substantially constrain the choice of reasonable stress-energy tensors, in some cases fixing it uniquely.:Acknowledgements 4 Contents 5 1 Introduction 7 2 Constructive aspects of integrable quantum field theories 13 2.1 General notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Particle spectrum and one-particle space . . . . . . . . . . . . . . . . 15 2.3 The scattering function . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Full state space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 Asymptotic completeness; closing the circle . . . . . . . . . . . . . . . 29 2.6 Connection to algebraic quantum field theory . . . . . . . . . . . . . 33 3 Locality and the form factor series 37 3.1 Locality and the form factor series . . . . . . . . . . . . . . . . . . . 38 3.2 Local commutativity theorem for one- and two-particle form factors . 44 3.3 Transformation properties of the form factors . . . . . . . . . . . . . 58 3.3.1 Form factors of invariant operators and derivatives . . . . . . 62 4 Structure of form factors and the minimal solution 64 4.1 Classification of two-particle form factors . . . . . . . . . . . . . . . . 64 4.2 Existence of the minimal solutions and asymptotic growth . . . . . . 68 4.3 Computing a characteristic function . . . . . . . . . . . . . . . . . . . 74 5 The stress-energy tensor 77 5.1 The stress-energy tensor from first principles . . . . . . . . . . . . . . 77 5.2 The stress-energy tensor at one-particle level . . . . . . . . . . . . . . 83 5.3 Characterization at one-particle level . . . . . . . . . . . . . . . . . . 88 6 State-independent QEI for constant scattering functions 94 6.1 Candidate for the stress-energy tensor . . . . . . . . . . . . . . . . . 94 6.2 A generic estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.3 Derivation of the QEI . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.4 Discussion of the QEI . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.5 Supplementary computations . . . . . . . . . . . . . . . . . . . . . . 108 7 QEIs at one-particle level for generic scattering functions 110 7.1 Derivation of the QEI at one-particle level . . . . . . . . . . . . . . . 111 7.2 Extending the scope of the QEI result . . . . . . . . . . . . . . . . . 117 7.3 A general recipe to obtain QEIs at one-particle level . . . . . . . . . 119 8 Examples 123 8.1 Models with one scalar particle type without bound states . . . . . . 123 8.2 Generalized Bullough-Dodd model . . . . . . . . . . . . . . . . . . . 125 8.3 Federbush model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 8.4 O(n)-nonlinear sigma model . . . . . . . . . . . . . . . . . . . . . . . 130 9 Conclusion, discussion, and outlook 134 A Constructive aspects of integrable quantum field theory 137 A.1 Representation theory of the Poincaré group in 1+1d . . . . . . . . . 137 A.2 Discrete symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 A.3 S-function and ZF operators in a basis . . . . . . . . . . . . . . . . . 143 A.4 Improper rapidity eigenstates . . . . . . . . . . . . . . . . . . . . . . 145 A.5 Bound states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 A.6 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 B Literature survey: Form factor conventions 158 C Stress-energy tensor 159 C.1 Stress-energy tensors for the free scalar field . . . . . . . . . . . . . . 159 C.2 A weaker notion for the density property . . . . . . . . . . . . . . . . 163 C.3 Stress-energy tensor at one-particle level generating the boosts . . . . 164 Bibliography 166

This thesis explores systems that exhibit strong coupling between an optical cavity field and a many-particle system. To treat the drive and dissipative nature of the cavity on the same footing as the dynamics of the many-particle system, we use a non-equilibrium field theoretic approach. The first system considered is an ultracold bosonic gas trapped inside a cavity. The dispersive coupling between the cavity field and the atoms' motion leads to the formation of a polariton. We show how a modulation of the pump laser on the energy scale of the transverse cavity mode splitting can be used to create effective interactions between different cavity modes. This effective interaction results in the polariton acquiring a multimode nature, exemplified by avoided crossings in the cavity spectrum. As the laser power is increased, the polariton softens and at a critical power becomes unstable. This instability signals the transition into a superradiant state. If the multimode polariton contains a cavity mode with an effective negative detuning, then the transition does not happen through a mode softening but at a finite frequency. To investigate this, classical non-linear equations are constructed from the action and from these we derive the critical couplings and frequencies. It is shown how the superradiant transition happening at a finite frequency is a consequence of a competition between the negatively and the positively detuned cavity modes making up the polariton. The finite-frequency transition is found to be equivalent to a Hopf bifurcation and leads to the emergence of limit cycles. Our analysis shows that the system can exhibit both bistabilities and evolution constricted to a two-torus. We end the investigation by showing how interactions among the atoms combined with the emerging limit cycle open new phonon scattering channels. The second system considered in the thesis is inspired by the recent experiments on gated Transition-metal dichalcogenides (TMD) monolayers inside cavities. An exciton within the TMD can couple strongly to the cavity and, due to the electronic gating, also interact strongly with the conduction electrons. To treat the strong interactions of the excitons with both cavity and electrons, we solve the coupled equations for the correlation functions non-perturbatively within a ladder approximation. The strong interactions give rise to new quasiparticles known as polaron-polaritons. By driving the system through the cavity, we show how the competition between electron-induced momentum relaxation and cavity loss leads to the accumulation of polaritons at a small but finite momentum, which is accompanied by significant decrease of the polariton linewidth Due to the hybrid nature of the polaron-polariton, we show that this behavior can by qualitatively modified by changing the cavity detuning.

We calculate tree-level scattering amplitudes in φ^3 theory and Yang-Mills theory by means of the perturbiner expansion. This involves solving the Euler-Lagrange equations of motion perturbatively via a multi-particle ansatz, and using Berends-Giele recursion relations to extract the solution from simple on-shell data. The results are Berends-Giele currents which are then used to calculate the scattering amplitudes. The theoretical calculations are implemented into a Mathematica script which effectively handles recursive calculations and allows us to calculate amplitudes for an arbitrary number of particles.

This thesis considers the problem of calculating and observing the mixing of modes of positive and negative frequency in inhomogeneous, dispersive media. Scattering of vacuum modes of the electromagnetic field at a moving interface in the refractive index of a dielectric medium is discussed. Kinematics arguments are used to demonstrate that this interface may, in a regime of linear dispersion, act as the analogue of the event horizon of a black hole to modes of the field. Furthermore, a study of the dispersion of the dielectric shows that five distinct configurations of modes of the inhomogeneous medium at the interface exist as a function of frequency. Thus it is shown that the interface is simultaneously a black- and white-hole horizon-like and horizonless emitter. The role, and importance, of negative-frequency modes of the field in mode conversion at the horizon is established and yields a calculation of the spontaneous photonic flux at the interface. An algorithm to calculate the scattering of vacuum modes at the interface is introduced. Spectra of the photonic flux in the moving and laboratory frame, for all modes and all realisable increase in the refractive index at the interface are computed. As a result of the various mode configurations, the spectra are highly structured in intervals with black-hole, white-hole and no horizon. The spectra are dominated by a negative-frequency mode, which is the partner in any Hawking-type emission. An experiment in which an incoming positive-frequency wave is populated with photons is assembled to observe the transfer of energy to outgoing waves of positive and negative frequency at the horizon. The effect of mode conversion at the interface is clearly shown to be a feature of horizon physics. This is a classical version of the quantum experiment that aims at validating the mechanism of Hawking radiation.

No description available.

Thermalization, the evolution of an interacting many-body system towards a thermal Gibbs ensemble after initialization in an arbitrary non-equilibrium state, is currently a phenomenon of great interest, both in theory and experiment. As the time evolution of a quantum system is unitary, the proposed mechanism of thermalization in quantum many-body systems corresponds to the so-called eigenstate thermalization hypothesis (ETH) and the typicality of eigenstates. Although this formally solves the contradiction of thermalizing but unitary dynamics in a closed quantum many-body system, it does neither make any statement on the dynamical process of thermalization itself nor in which way the coupling of the system to an environment can hinder or modify the relaxation dynamics. In this thesis, we address both the question whether or not a quantum system driven away from equilibrium is able to relax to a thermal state, which fulfills detailed balance, and if one can identify universal behavior in the non-equilibrium relaxation dynamics. As a first realization of driven quantum systems out of equilibrium, we investigate a system of Ising spins, interacting with the quantized radiation field in an optical cavity. For multiple cavity modes, this system forms a highly entangled and frustrated state with infinite correlation times, known as a quantum spin glass. In the presence of drive and dissipation, introduced by coupling the intra-cavity radiation field to the photon vacuum outside the cavity via lossy mirrors, the quantum glass state is modified in a universal manner. For frequencies below the photon loss rate, the dissipation takes over and the system shows the universal behavior of a dissipative spin glass, with a characteristic spectral density $\\mathcal{A}(\\omega)\\sim\\sqrt{\\omega}$. On the other hand, for frequencies above the loss rate, the system retains the universal behavior of a zero temperature, quantum spin glass. Remarkably, at the glass transition, the two subsystems of spins and photons thermalize to a joint effective temperature, even in the presence of photon loss. This thermalization is a consequence of the strong spin-photon interactions, which favor detailed balance in the system and detain photons from escaping the cavity. In the thermalized system, the features of the spin glass are mirrored onto the photon degrees of freedom, leading to an emergent photon glass phase. Exploiting the inherent photon loss of the cavity, we make predictions of possible measurements on the escaping photons, which contain detailed information of the state inside the cavity and allow for a precise, non-destructive measurement of the glass state. As a further set of non-equilibrium systems, we consider one-dimensional quantum fluids driven out of equilibrium, whose universal low energy theory is formed by the so-called Luttinger Liquid description, which, due to its large degree of universality, is of intense theoretical and experimental interest. A set of recent experiments in research groups in Vienna, Innsbruck and Munich have probed the non-equilibrium time-evolution of one-dimensional quantum fluids for different experimental realizations and are pushing into a time regime, where thermalization is expected. From a theoretical point of view, one-dimensional quantum fluids are particular interesting, as Luttinger Liquids are integrable and therefore, due to an infinite number of constants of motion, do not thermalize. The leading order correction to the quadratic theory is irrelevant in the sense of the renormalization group and does therefore not modify static correlation functions, however, it breaks integrability and will therefore, even if irrelevant, induce a completely different non-equilibrium dynamics as the quadratic Luttinger theory alone. In this thesis, we derive for the first time a kinetic equation for interacting Luttinger Liquids, which describes the time evolution of the excitation densities for arbitrary initial states. The resonant character of the interaction makes a straightforward derivation of the kinetic equation, using Fermi\'s golden rule, impossible and we have to develop non-perturbative techniques in the Keldysh framework. We derive a closed expression for the time evolution of the excitation densities in terms of self-energies and vertex corrections. Close to equilibrium, the kinetic equation describes the exponential decay of excitations, with a decay rate $\\sigma^R=\\mbox\\Sigma^R$, determined by the self-energy at equilibrium. However, for long times $\\tau$, it also reveals the presence of dynamical slow modes, which are the consequence of exactly energy conserving dynamics and lead to an algebraic decay $\\sim\\tau^$ with $\\eta_D=0.58$. The presence of these dynamical slow modes is not contained in the equilibrium Matsubara formalism, while they emerge naturally in the non-equilibrium formalism developed in this thesis. In order to initialize a one-dimensional quantum fluid out of equilibrium, we consider an interaction quench in a model of interacting, dispersive fermions in Chap.~\\ref. In this scenario, the fermionic interaction is suddenly changed at time $t=0$, such that for $t>0$ the system is not in an eigenstate and therefore undergoes a non-trivial time evolution. For the quadratic theory, the stationary state in the limit $t\\rightarrow\\infty$ is a non-thermal, or prethermal, state, described by a generalized Gibbs ensemble (GGE). The GGE takes into account for the conservation of all integrals of motion, formed by the eigenmodes of the Hamiltonian. On the other hand, in the presence of non-linearities, the final state for $t\\rightarrow\\infty$ is a thermal state with a finite temperature $T>0$. . The spatio-temporal, dynamical thermalization process can be decomposed into three regimes: A prequench regime on the largest distances, which is determined by the initial state, a prethermal plateau for intermediate distances, which is determined by the metastable fixed point of the quadratic theory and a thermal region on the shortest distances. The latter spreads sub-ballistically $\\sim t^$ in space with $0<\\alpha<1$ depending on the quench. Until complete thermalization (i.e. for times $t<\\infty$), the thermal region contains more energy than the prethermal and prequench region, which is expressed in a larger temperature $T_{t}>T_$, decreasing towards its final value $T_$. As the system has achieved local detailed balance in the thermalized region, energy transport to the non-thermal region can only be performed by the macroscopic dynamical slow modes and the decay of the temperature $T_{t}-T_\\sim t^$ again witnesses the presence of these slow modes. The very slow spreading of thermalization is consistent with recent experiments performed in Vienna, which observe a metastable, prethermal state after a quench and only observe the onset of thermalization on much larger time scales. As an immediate indication of thermalization, we determine the time evolution of the fermionic momentum distribution after a quench from non-interacting to interacting fermions. For this quench scenario, the step in the Fermi distribution at the Fermi momentum $k\\sub$ decays to zero algebraically in the absence of a non-linearity but as a stretched exponential (the exponent being proportional to the non-linearity) in the presence of a finite non-linearity. This can serve as a proof for the presence or absence of the non-linearity even on time-scales for which thermalization can not yet be observed. Finally, we consider a bosonic quantum fluid, which is driven away from equilibrium by permanent heating. The origin of the heating is atomic spontaneous emission of laser photons, which are used to create a coherent lattice potential in optical lattice experiments. This process preserves the system\'s $U(1)$-invariance, i.e. conserves the global particle number, and the corresponding long-wavelength description is a heated, interacting Luttinger Liquid, for which phonon modes are continuously populated with a momentum dependent rate $\\partial_tn_q\\sim\\gamma |q|$. In the dynamics, we identify a quasi-thermal regime for large momenta, featuring an increasing time-dependent effective temperature. In this regime, due to fast phonon-phonon scattering, detailed balance has been achieved and is expressed by a time-local, increasing temperature. The thermal region emerges locally and spreads in space sub-ballistically according to $x_t\\sim t^{4/5}$. For larger distances, the system is described by an non-equilibrium phonon distribution $n_q\\sim |q|$, which leads to a new, non-equilibrium behavior of large distance observables. For instance, the phonon decay rate scales universally as $\\gamma_q\\sim |q|^{5/3}$, with a new non-equilibrium exponent $\\eta=5/3$, which differs from equilibrium. This new, universal behavior is guaranteed by the $U(1)$ invariant dynamics of the system and is insensitive to further subleading perturbations. The non-equilibrium long-distance behavior can be determined experimentally by measuring the static and dynamic structure factor, both of which clearly indicate the exponents for phonon decay, $\\eta=5/3$ and for the spreading of thermalization $\\eta_T=4/5$. Remarkably, even in the presence of this strong external drive, the interactions and their aim to achieve detailed balance are strong enough to establish a locally emerging and spatially spreading thermal region. The physical setups in this thesis do not only reveal interesting and new dynamical features in the out-of-equilibrium time evolution of interacting systems, but they also strongly underline the high degree of universality of thermalization for the classes of models studied here. May it be a system of coupled spins and photons, where the photons are pulled away from a thermal state by Markovian photon decay caused by a leaky cavity, a one-dimensional fermionic quantum fluid, which has been initialized in an out-of-equilibrium state by a quantum quench or a one-dimensional bosonic quantum fluid, which is driven away from equilibrium by continuous, external heating, all of these systems at the end establish a local thermal equilibrium, which spreads in space and leads to global thermalization for $t\\rightarrow\\infty$. This underpins the importance of thermalizing collisions and endorses the standard approach of equilibrium statistical mechanics, describing a physical system in its steady state by a thermal Gibbs ensemble.

No description available.

This thesis describes the neutron scattering measurements of magnetic excitations in spin-chains and ladders. The first part discusses an experimental investigation of the copper oxide family Sr₁₄Cu₂₄O₄₁ composed of edge-sharing chains and spin-ladders. The study of La₄Sr₁₀Cu₂₄O₄₁ comprises a slightly hole-doped chain and an undoped ladder structure where the chain can be modeled by a ferromagnetic nearest and an antiferromagnetic next-nearest neighbor coupling. The hole effects are apparent in gaps in the dispersion relation and can be described by a charge-density wave agreeing with the commensuration of the dispersion. Investigating the undoped ladder establishes the exchange constants including a cyclic exchange manifested by the two-magnon continuum and the suppression of the S = 1 bound mode. An orbital consideration provides an explanation for the exchanges including the different sizes of rung and leg coupling. The excitation spectrum of the doped ladder in Ca₂.₅Sr₁₁.₅Cu₂₄O₄₁ can be described by a direct comparison with the undoped ladder and the differences consisting of a higher energy mode and subgap scattering can be successfully modeled by the charge spectrum of the ladder calculated from the free electron model. The second part of the thesis investigates the alternating chain material Cu(NO₃)₂ · 2.5D2O and establishes the gapped one-magnon dispersion, the two-magnon continuum and for the first time the S =1 bound mode. Applying magnetic field drives the system through two critical field transitions, condensation of magnons into the ground state and saturation. The modes beyond saturation can be modeled by spin wave theory and the excitations at the first critical field follow Luttinger Liquid behavior. Additionally investigated are the temperature effects with the excitations being of a different nature but containing the signature of a strong correlated system. For an outlook the measurements including temperature and field are provided with further theoretical descriptions necessary.