Even though no direct experimental evidence of any new heavy particles is in sight 2 (see, e.g., [KS+16]), important conceptual problems still remain in high energy physics. One example is our lack of detailed understanding of the interplay between **quantum** mechanics **and** black hole physics. Following a recent proposal by Dvali **and** Gomez [DG13b; DG14], we have pursued the idea that black hole physics may, after all, be a manifestation of collec- tive **quantum** eﬀects in high energy physics. In this work, we present some of the results obtained for simplified model systems **and** the conclusion we draw for black holes. From this point, we were intrigued to further explore the relevance of **quantum** **collective** eﬀects. During our eﬀorts to unravel the phenomena using techniques of integrability, we discovered a surprising equivalence between our model system (Lieb-Liniger) **and** an otherwise seem- ingly unrelated **theory** in two dimensions (Yang-Mills). Finally, we turned to particle collisions, in which many particles are produced. These may only be accessible at future experiments, but until then, we need to dramatically improve upon our capabilities to calculate such processes, where **quantum** col- lective eﬀects are dominant. In this domain, the present work contains some formal developments that aim to further our understanding of the required mathematical tools.

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The second main **theory** that is nowadays considered the standard model for **gravity** is General Relativity (GR). GR is a classical **theory**. Its quanti- zation encounters a series of problems that, at present time, are still being investigated. One of the main problems is given by the fact that if one tries to quantize the **theory** starting from a classical Lagrangian as one would do for a gauge **theory**, one obtains an innite series of interaction terms which make in turn the **theory** non renormalizable **and** dicult to treat perturba- tively. In addition to the proliferation of terms, we have to face also a more complicated expression for the Feynman rules, which further encumbers cal- culations. How to quantize the **theory** of relativity in order to look for a unied **theory** including **gravity** **and** the other fundamental interactions, is one of the most thrilling open questions **and** challenges of modern, **and** most likely future, physics **and** constituted one of the main motivations for my application **and** devotion to this eld of research. In section 3.1 we will sum- marize the problem **and** present an intriguing idea oered by Zvi Bern **and** others [16].

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xed point in the bare se
tor for any
hoi
e of the measure parameter M **and** any dimension d , we will investigate the ow of the bare
ouplings in more detail, in parti
ular near 2 dimensions, **and** we try to simplify the map by
hoosing a suitable value of M . This way we will demonstrate that M
an always be xed su
h that the bare
osmologi
al
onstant vanishes. As we will show, this implies in d = 2 + ε that at rst order the bare Newton
onstant equals the ee
tive one.

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In the case of the κ-deformed space we concentrate on the noncommutative SU (N ) theories. Using the enveloping algebra approach **and** the Seiberg-Witten map [32], [41], the noncommutative gauge **theory** is constructed perturbatively order by order in the deforma- tion parameter. In this way we obtain an effective **theory** which provides corrections to the commutative **theory** up to first order in the deformation parameter. These corrections are given in terms of the commutative fields, so the **field** content of the **theory** is not changed. However, new interactions arise **and** the deformation parameter enters as a coupling con- stant. This approach has been used to construct the noncommutative gauge **theory** on the θ-deformed space [41], [42], as well as the generalisation of the Standard Model [43], [44]. Using these results some new **effects** which do not appear in the commutative Standard Model were calculated in [45], [46]. Also, it was shown that the theories obtained in this perturbative way are anomaly free [47], [48], [49]. It is interesting to note that cutting the **theory** at some order in the deformation parameter one avoids the UV/IR mixing. It only appears in the ”summed-up” theories, that is theories to all orders in the deformation pa- rameter. Also, the ”summed-up” models allow generalisation of the U (N ) gauge theories only, with some exceptions [50], [51].

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However, in recent years, there have been a number of attempts to establish modified spacetime backgrounds [1–7]. Most of them were motivated from approaches to **quantum** **gravity** [8–18], but many were also motivated from other particular physical **and** mathematical models [19–39]. Besides the need for testing grounds for **quantum** **gravity** results, the interest in spacetime back- grounds beyond Lorentzian geometry stems partly from the diversity of observations made over the last few years suggesting that there is something wrong with our understanding of the mat- ter content of the universe or gravitational dynamics or both. There are observations of the gravitational lensing of galaxies [40], high redshift supernovae [41, 42] **and** the cosmic microwave background [43, 44] that suggest that an overwhelming 83% of the matter **and** 95% of the total mass-energy in the universe is of unknown type. Even worse, one has to assume that this un- known 83% of the total matter in the universe is not interacting with the electromagnetic **field** to match the observations, which earned it the name “dark matter” [45]. This can be interpreted as a hint of a new particle physics [46] or a new gravitational physics [47]. Considering modified backgrounds, one would of course expect the new particle physics **and** gravitational physics to arise at the same time: On the one hand, the restriction to Lorentzian spacetimes also severely restricts the type of matter fields that can be considered [48], **and** this restriction is strongly used in particle physics [49]. On the other hand, general relativity is fundamentally based on the metric concept [50].

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Clearly, following [21, 121] such degeneracy of the horizon as given in (i) is not given exactly. Nevertheless, within RN approximation for A < 0 **and** B > 0, only the case (iii) is possible indeed ( ˜ Q 2 < 0), leading to a quasi-Schwarzschild behavior for low-**field** regimes. Nevertheless, the analogy to RN solutions is an interesting subject which reminds that for a massive object whose charge is not neutralized by further **effects**, the Schwarzschild radius itself loses its meaning of dominant property of the system. Here, the generalized charge is an intrinsic quality which affects the Schwarzschild radius itself, **and** the weakening of the latter appears indeed as consequence of the correction terms which already weaken gravitational fields for weak- **field** regimes (cf. figures). Taking this fact into account, it may be possible to establish measurably relevant distinctions of this induced-**gravity** model to usual dynamics even at long-scale regimes such as those of galactic bulges as well as relevant indications for intermediate regimes towards strong gravitational fields. It may be established that in all orders, the evolution of gravitational potentials (i.e. the metric components) strongly depends on the possible relations between A **and** B. Such relations are helpful to understand how new physical correction terms act within low gravitational regimes in order to finally break the gravitational collapse onto a Grey Star.

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From **quantum** electrodynamics it is well known that the anomalous magnetic moment of the electron is a relativistic effect. In the spirit of Wigner particles are irreducible repre- sentations of the covering group of the Poincar´e group **and** the electron is a representation of a relativistic particle with intrinsic SU (2) symmetry. For electrons in the low energy regime there appear for this reason correction terms such as spin orbit coupling or zitterbewegung. It is well known that such **effects** have influence of the behaviour of electrons in two dimensional electron systems. For instance the Rashba spin orbit coupling is discussed in the context of spin Hall **effects** [Win03]. Having realized that the spin of the electron is a relativistic effect we may wonder how the spin of a composite Fermion should be understood. This is the origin of the motivation of this thesis. We will discuss the possibility to derive composite Fermions with spin from relativistic **quantum** electrodynamics. The main problem is that **quantum** electrodynamics is defined in four rather than in three dimensions. Furthermore we have to explain how to attach flux quanta in this regime. We will provide an answer to both how to attach flux quanta **and** how to project the **quantum** fields to three dimensions. This leads to a relativistic composite Fermion **theory** with spin in three dimensions, however relativistic then means covariance under a subgroup of the Poincar´e group. Then we push the analysis forward to quasi relativistic composite Fermions in a lowest Landau level or lowest composite Fermion Landau level **and** quasi relativistic now means that the covariance is violated at the scale of the magnetic length. This quasi covariant model is a realization of a **quantum** **field** **theory** on a noncommutative plane **and** therefore highly nonlocal which should lead to a distortion of the dispersion relation.

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The present Chapter approaches the problem of volume constraints in Sec. V.2 from the different perspective. It was already stressed that the normals are considered to be as characteristic for discrete picture as bivectors are, in our view. In what follows, we revisit the **field**-theoretic content of the linear formulation in Sec. IV.2.4. One is able to achieve the closer contact with Cartan geometric degrees of freedom, by switching from normals directly to tetrad/co-frame, in the classical continuum formulation. Following the approach adopted in Spin Foams, we first provide the characterization in Sec. VII.1 for the (unconstrained) Poincar´ e BF-type **theory** on the extended configuration space 1 . The focus is on gauge symmetries: we first verify the invariances of lagrangian, **and** then show how they are preserved also on the canonical level in the Hamiltonian analysis of Sec. VII.1.2. The canonical generator of gauge transformations is explicitly constructed. In Sec. VII.2, the equivalent set of linear simplicity constraints, reformulated in terms of co-frames, is presented **and** read geometrically as describing 3-volume of a hypersurface.

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The study of the spectral function out of equilibrium is of great interest in current research regarding transport **and**/or kinetic processes. For example, in the early universe a proper description of leptogenesis requires the inclusion of off-shell **effects** which could change the standard picture of the evolution [GHKL09]. Having tools available which allow to study those dynamics from first principles we will consider in this chapter an analysis of the spectral function out of equilibrium in the presence of an instability. We use our generic model introduced in Sec. 3.1. The main features might be transferred to more phenomenological models. Another interesting prop- erty of the spectral function is the scaling behavior in the four dimensional Fourier space. The exponent of the scaling behavior is connected to the so called anomalous dimension [PS95]. Such scaling solutions play an important role in studying critical phenomena in **and** out of equilibrium. The emergence of critical phenomena out of equilibrium gained recently a lot attention [BRS08, BSS09b]. The anomalous dimension is typically chosen to be zero as it is expected in relativistic scalar **field** theories. Till now there exist no numerical determination of the exponent out of equilibrium in the 2PI framework. With the method presented in this chapter it is possible to attack this problem. However, we will left this out **and** refere to future studies.

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E-**gravity** is a **quantum** **field** **theory**, where basic integrals of motion like energy are not conserved in general. Only in the low-energy regime these integrals of motions can exist. Scattering processes predicted by E-**gravity** contain also self-interactions of particles due to **gravity**. This modifies the behavior of **quantum** fluctuations in a similar way as TDFT does. Gravitational modifications of **quantum** fluctuations apply on all elementary particles **and** gauge bosons **and** not only on gauge bosons.

flow pattern was found in [4], **and** higher derivative truncations were analyzed in [3, 5, 10]. Matter fields were added in refs. [2, 9], **and** in [12] the beta-functions of [1] **and** [3] were used for finding optimized RG flows. The most remarkable result of these investigations is that the beta-functions of [1] predict a non-Gaussian RG fixed point [8]. After detailed studies of the reliability of the pertinent truncations [3, 4, 5, 12] it is now believed that it corresponds to a fixed point in the exact **theory** **and** is not an approximation artifact. It was found to possess all the nec- essary properties to make **quantum** **gravity** nonperturbatively renormalizable along the lines of Weinberg’s “asymptotic safety” scenario [17, 18], thus overcoming the notorious problems related to its nonrenormalizability in perturbation **theory**. We shall refer to the **quantum** **field** **theory** of the metric tensor whose infinite cutoff limit is taken at the non-Gaussian fixed point as **Quantum** Einstein **Gravity** or “QEG”. This **theory** should not be thought of as a quantization of classical general relativity. Its bare action is dictated by the fixed point condition **and** is therefore expected to contain more invariants than the Einstein-Hilbert term only. Independent evidence pointing towards a fixed point in the full **theory** came from the symmetry reduction approach of Ref. [19] where the 2-Killing subsector of the gravitational path integral was quantized exactly.

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The outline of this thesis is as follows. In chapter 2, we study the question of **quantum** breaking, i.e. of how long a given system can be approximated as clas- sical. We use simple scaling arguments **and** the analysis of a prototypical self- interacting scalar **field** to draw conclusions about **quantum** breaking in generic systems. Subsequently, we investigate two concrete examples. For hypothetical cosmic axions of QCD, we show that the classical approximation of today’s axion **field** is extremely accurate. Next, we study **quantum** breaking in de Sitter. First, we construct a concrete model for the corpuscular picture of de Sitter reviewed in 1.3.2, in which the spacetime is resolved as excited multi-graviton state on top of Minkowski vacuum. We show that our model is able to reproduce all known classical **and** semiclassical properties of de Sitter. Moreover, it allows us to explic- itly compute the **quantum** break-time, after which the description in terms of a classical metric ceases to be valid. Our result is in full agreement with Eq. (1.26). Additionally, we study implications of **quantum** breaking for the dark energy in today’s Universe **and** for inflationary scenarios. Whereas the discussion up to this point is independent of the question if **quantum** breaking is a sign of a fundamental inconsistency of de Sitter, we finally discuss some of the important consequences that arise if a consistent **theory** must not allow for de Sitter **quantum** breaking. In particular, this makes the existence of a QCD axion mandatory **and** excludes the self-reproduction regime in inflation as well as any extension of the Standard Model with a spontaneously-broken discrete symmetry.

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Since there is no one widely-accepted **theory**, many phenomenological models of **quantum** **gravity** have been proposed. One feature of these models is the existence of a minimum length scale of the order of Planck length [ 6 ]. Detecting the existence of such a minimal length scale is one of the main goals of the **field**, but has so far eluded experimental verification. Direct detection of the Planck length, 1.6 × 10 −35 m, is infeasible with current **and** foreseeable technology because the **effects** of **quantum** **gravity** are expected to become directly relevant only at energies of the order of Planck energy which is Ep = 1.2 × 10 19 GeV. This is 15 orders of magnitude larger than the energy scales achievable in the Large Hadron Collider today. Hence, it seems unlikely that these energy scales will be achieved in the near future **and** we must resort to indirect methods. So in order to experimentally probe **quantum**

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After their creation, neutron stars cool down by emitting neutrinos. In the later stage of their life, because of the high densities, or high Fermi momenta, of the nucleons, neutron-star matter can be assumed as cold T = 0 matter **and** thermal **effects** are then only a small correction. This simplifies modeling the EoS of neutron stars **and** makes them an ideal **and** exciting testing ground for predictions of strongly interacting matter. These predictions have to be tested against astronomical observations of neutron stars to discriminate between different models for the EoS. The neutron-star mass-radius relation has been studied in many works, e.g., using the liquid drop model [11], using relativistic mean-**field** models [12, 13, 14, 15] or with phenomenolog- ical potential models [16]. All of these calculations are based on different models for nuclear interactions that bind neutrons **and** protons together **and** lead to a large span in the mass- radius prediction of neutron stars, ranging for a typical 1.4M neutron star from 9 − 15 km, see Refs. [5, 17, 18] **and** references therein. In this Thesis, we will present a way of improving the theoretical predictions for the equation of state of neutron matter, making better predictions of the neutron-star mass-radius relation possible.

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Being a non-pertubative **and** background independent approach, loop **quantum** **gravity** (LQG) [6, 34] seems to be appropriate for understanding **quantum** **gravity** **effects** near classical singulari- ties where curvature is by no means small. Indeed, within this context it was possible to derive the Bekenstein-Hawking area law for a large class of black holes. [1,14] Now, being a canonical quanti- zation of **gravity**, LQG is based on a splitting of space-time into time **and** space, entailing that the four-dimensional covariance of general relativity is no longer manifest. [32, 34] In particular, the 4-dimensional diffeomorphism constraint splits up into a spatial **and** the Hamiltonian constraint, the latter one defining the dynamics of LQG. Unfortunately, the quantization of the Hamiltonian of the full **theory** turns out to be difficult **and** is, at this point, not completely understood. [33] Here, symmetry reduced versions of LQG like loop **quantum** cosmology [10] can help to better understand this quantization for the full **theory**. In addition to that, mathematical developments like the reduction concept to be developed in this work carry over to a bigger class of gauge **field** theories, so that LQG is not an isolated **field** of research but an approach to **quantum** **gravity** whose developments enhance other areas of theoretical physics **and** even mathematics.

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In the bosonic transport through a **quantum** critical system, the **quantum** Ising chain in a transverse magnetic **field**, a reduced subspace is found, where the dynam- ics can be described by a rate equation. The non-equilibrium steady-state of the rate equation can be obtained in closed form. This allows to calculate steady-state expectation values for the energy **and** magnetization. Both observables show in the limit of vanishing temperatures **and** in the thermodynamic limit the expected QPT. The bosonic steady-state current is obtained analytically **and** shows signatures of the underlying QPT for finite temperatures, where system observables are totally oblivious of it. This interesting behavior can also be found in a paradigmatic model for an avoided crossing, which in the thermodynamic limit shows a first-order QPT. It remains an interesting task for the future to check whether similar **effects** also arise with a different coupling geometry **and** in other **quantum**-critical systems. The evaluation of time-dependent correlation functions, e.g. by the EOM method, should reveal similarly signatures of the **quantum**-criticality at finite temperatures.

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As we discussed, the **theory** undergoes a large particle number phase transition [80]. This phase transition interpolates between a ho- mogeneous phase in the weak coupling limit to a phase dominated by a solitonic bound state in the strong coupling limit, known as a bright soliton. The dynamics of the phase transition has been extensively studied, both using the mean-**field** analysis [80] **and** also by a trunca- tion **and** numerical diagonalization of the Hamiltonian [96, 94, 76, 75]. Another interesting feature of this model is that it is exactly inte- grable [84]. As we’ll see, this implies that the Schr ¨odinger equation of the system can be mapped to a set of algebraic equations - the Bethe equations - which fully determine the complete spectrum of the the- ory. Despite the fact that the system can be in principle solved using this technique, in practice the equations are transcendental **and** cannot be analytically solved without any approximations. The only regime where it is possible to obtain exact solutions is in the c → ∞ limit, where we are in the deep solitonic regime. In this regime, it is possible to explicitly construct exact solutions of the Bethe equations, due to the string hypothesis [87], which we’ll revisit later.

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that time **and** still do: String **theory** is about replacing point particles by extended one- dimensional objects called strings, which can be either open or closed. These fields can vibrate, **and** different vibrational modes correspond to different particles, like the differ- ent vibrational modes of a violin generate different tones. Importantly, in the spectrum of vibrational modes there is always an excitation, which describes the fluctuation of a background spacetime metric. This was considered as a hint that string **theory** could be a candidate for a consistent **theory** of **quantum** **gravity**. Indeed, it is astonishing how string **theory** deals with the bad non-renormalizable infinities in **quantum** **field** **theory** associated to gravitational interactions. The extended nature of the string delocalizes interaction vertices, **and** the problematic ultraviolet regime is mapped by a so-called duality to the infrared regime which can be described easily. More precisely, this dual- ity states that the physics of long strings at high energies is the same as the physics of short strings at low energies. Via a precise ’dictionary’ these regimes can be mapped to each other. All these nice properties have already appeared in the early version of bosonic string **theory**. However, the latter suffers from a couple of important drawbacks which makes it impossible to consider it as a **theory** of the world around us. First, it cannot account for spacetime fermions, which are the fundamental building blocks of our world. Second, in the spectrum of the **theory** one finds tachyons, i.e. modes of imaginary mass. These signal an instability of the **theory**. While the tachyon in the sector of open strings is quite well understood (we are sitting at the maximum of a potential, **and** rolling down corresponds to so-called D-brane condensation), the impli- cations of the tachyon in the closed string sector are not clear but might most certainly render spacetime itself unstable. Both issues, the presence of tachyons **and** the absence of spacetime fermions, soon got resolved by moving from bosonic string **theory** to su- perstring **theory**. By introducing a fermionic partner string for the bosonic string the **theory** acquires a new symmetry, namely two-dimensional supersymmetry. The latter is powerful enough to allow for stable solutions, **and** at the same time also leads to spacetime fermions, while keeping the nice properties in the ultraviolet regime. Indeed, it was found that there even exist five different superstring theories, which all require for consistency a total number of exactly ten spacetime dimensions. They are called type I, type IIA, type IIB, SO(32) heterotic **and** E 8 × E 8 heterotic string **theory**.

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In order to make our ideas transparent, we applied these constructions to concrete physical systems. In the case of the auxiliary current description, we were mostly inter- ested in black hole physics. Understanding the black hole **quantum** state in terms of a multi-graviton state on flat space-time, we were able to compute observables connected to the black hole interior such as the density of gravitons in momentum space or their energy density at the parton level **and** in the large mass limit. Consistency of our results then implied that the total mass of the black hole must scale as the number of fields composing the auxiliary current. We found that the distribution of gravitons in the black hole is dominated by quanta of large wavelength which resonates with the ideas put forward in the Black Hole **Quantum** N Portrait. Based on these results, we explained that the distri- bution function of gravitons is directly accessible in S-matrix processes. As a consequence, an outside observer can in principle reconstruct the internal structure of the black hole. Thus, no loss of information as well as the existence of **quantum** hair are expected in our approach. Finally, we showed how the Schwarzschild metric emerges from our approach. The basic insight was to replace a classical black hole source characterized by the mass by the corresponding microscopic **quantum** source. This **quantum** source was identified with the energy density of gravitons inside the black hole. On the one hand, the mass arises as a **collective** effect of N gravitons. On the other hand, for any finite N , we argued that there are corrections to the classical notion of mass which can be absorbed in a wave function renormalization. Using this result combined with the finding of [67] it was straightforward to show that in our description, classical geometry indeed emerges in the limit N → ∞.

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cw laser. It was found that this system is closely connected to two standard setups of **quantum** optics, namely cooperative resonance fluorescence **and** absorptive optical bistability. The rate equation **theory** predicts bistable behavior which is in contradiction to the full **quantum** solution that always exhibits a unique steady state. This contradiction is relieved when considering the large system size or thermodynamic limit, but this observation already hints at the importance of **quantum** correlations in this setup. Since the strength of the permutation symmetric method is the ability to compute the full non-approximate density matrix this serves as an ideal testing ground for the capabilities of this method. Furthermore this discussion also serves as an ideal testing ground for the solvers PETSc **and** SLEPc solvers available in PsiQuaSP: The population dynamics in Fig. 7.8 b) for N = 9 were calculated in parallel on four cpus, using the permu- tation symmetric method **and** an adaptive time step Runge-Kutta **and** the whole calculation took roughly 2 months. This corresponds to a single point in the steady state plots. For N = 5 at κ ∼ g the explicit time integration still takes roughly two weeks on four cpus. Using the SLEPc Krylov-Schur algorithm **and** the shift **and** invert spectral transformation as explained in Appendix A.6 the entire curve in the steady state plot can be computed over night on four cpus. The investigation of the population of the subradiant states revealed that the non-equilibrium phase transition known from cooperative resonance fluorescence experiences a stark qualita- tive change when leaving the bad cavity limit. This results in a new type of phase transition altogether. In this new phase transition the system undergoes a change from a steady state superradiant into a steady state subradiant state. Counterintuitively, the crucial parameter for this process is the cavity quality factor **and** not the individual spontaneous decay rate. This has not been reported before, in fact the results in this chapter provide the first study investigating the steady state population dynamics of subradiant states.

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