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 quantumcollective 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.
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 gravityand 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 .
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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 , , 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 , , as well as the generalisation of the Standard Model , . Using these results some new effects which do not appear in the commutative Standard Model were calculated in , . Also, it was shown that the theories obtained in this perturbative way are anomaly free , , . 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 , .
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 quantumgravity [8–18], but many were also motivated from other particular physical and mathematical models [19–39]. Besides the need for testing grounds for quantumgravity 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 , 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” . This can be interpreted as a hint of a new particle physics  or a new gravitational physics . 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 , and this restriction is strongly used in particle physics . On the other hand, general relativity is fundamentally based on the metric concept .
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.
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 quantumfieldtheory on a noncommutative plane and therefore highly nonlocal which should lead to a distortion of the dispersion relation.
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.
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.
E-gravity is a quantumfieldtheory, 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 , and higher derivative truncations were analyzed in [3, 5, 10]. Matter fields were added in refs. [2, 9], and in  the beta-functions of  and  were used for finding optimized RG flows. The most remarkable result of these investigations is that the beta-functions of  predict a non-Gaussian RG fixed point . 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 theoryand is not an approximation artifact. It was found to possess all the nec- essary properties to make quantumgravity 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 quantumfieldtheory 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.  where the 2-Killing subsector of the gravitational path integral was quantized exactly.
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.
Since there is no one widely-accepted theory, many phenomenological models of quantumgravity 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 quantumgravity 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
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 , using relativistic mean-field models [12, 13, 14, 15] or with phenomenolog- ical potential models . 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.
Being a non-pertubative and background independent approach, loop quantumgravity (LQG) [6, 34] seems to be appropriate for understanding quantumgravityeffects 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.  Here, symmetry reduced versions of LQG like loop quantum cosmology  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 quantumgravity whose developments enhance other areas of theoretical physics and even mathematics.
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.
As we discussed, the theory undergoes a large particle number phase transition . 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  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 . 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 , which we’ll revisit later.
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 quantumgravity. Indeed, it is astonishing how string theory deals with the bad non-renormalizable infinities in quantumfieldtheory 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.
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  it was straightforward to show that in our description, classical geometry indeed emerges in the limit N → ∞.
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.