plies that the latter should be composite as well. Using this mapping we explicitly showed how semi-classical instanton results are easily obtained in terms of the coherent state de- scription of the corresponding soliton to leading order in 1/N . This was done in detail in several cases such as instantons in quantum mechanics, Yang-Mills theory or 3-dimensional electrodynamics. In addition, in order to make the mapping manifest we constructed the- ories which naturally embed instanton physics in d dimensions into theories in one more dimension describing evolving solitons. Using the insight that instantons should have a quantum description, we further argued that the concept of resurgence should follow as a consequence of the basic principles of quantum mechanics such as unitarity. As a next step, we were concerned with higher order corpuscular effects in the case of solitons in SUSY theories. In the example of a Wess-Zumino model in 1 + 1 dimensions we worked out in detail that these effects lead to a novel mechanism of SUSY breaking which can never be discovered in the semi-classical treatment. We argued that these correction can naturally be understood in terms a corpuscular renormalization of the classical profile induced by corpuscular loops. Alternatively, we explained that these effects can also be understood in the many-body language. Indeed, in Bogoliubov approximation, quantum corrections are encoded in the dynamics of small fluctuations (quasi-particle excitations) around the mean field data. Finally, we applied the coherent state picture to the physics of AdS space-time. To leading order in 1/N , we explained how geometric properties such as local flatness or stability of AdS with respect to decay into Minkowski space-time are mapped to the occu- pation of corpuscles in AdS. In addition, we saw that the central charge of the dual CFT can be understood as a collective effects of corpuscles constituting a portion of AdS with volume set by the curvature radius. Based on these results, we proceeded with a discussion of higher order correction. In particular, we investigated how corpuscular effects correct propagators and Wightman functions in an AdS space-time. On the one hand, it was shown that the corpuscular effects on the propagator can be resummed in a Dyson-type series. On the other hand, using the KMS condition as a tool, we demonstrated that there are corrections to thermality of the spectrum that an accelerated observer measures in AdS which can never be uncoverer in the semi-classical treatment.
In this thesis we gave an analytic derivation of the two-loop correction to bulk/boundary two-point functions for a conformally coupled λφ 4 theory in Euclidean AdS, as well as the one-loop correction for the four-point boundary-to-boundary correlation function, by directly computing the related integrals in position space. The final result can be reduced to a single integral expression which is not given by elementary functions. Nonetheless, it can either be evaluated numerically or, more importantly, be evaluated analytically in a short-distance expansion on the boundary. We have then shown that the theory describes a fully consistent one-parameter family of dual conformal field theories on the boundary of AdS whose OPE coefficients and dimensions are parametrized by the renormalized coupling λ R . The structure of the dual CFT turns out to be that of a deformed generalized free field of dimension ∆ = 1 and ∆ = 2. The OPE of the CFT contains an infinite number of further primary double-trace operators which have anomalous dimensions and anomalous OPE coefficients that we are able to compute from our boundary correlation functions. This is the AdS equivalent of determining the masses and branching ratios in flat space-time. In order for the interpretation of our result to work out correctly in terms of a dual CFT, our loop corrected boundary correlation functions have to pass some nontrivial consistency tests. For example, the first order anomalous dimension enters not just at tree-level, but also in the bulk four-point function at one loop multiplying log(v) 2 . Similarly, the conformal spin expansion [ 129 , 130 , 131 ] implies a certain asymptotic fall-off behavior of the anomalous dimensions for large spin. All of these conditions are fulfilled by our bulk correlators. In addition, our bulk calculation gives manifestly finite results for all anomalous dimensions in terms of the renormalized bulk coupling, something that is more difficult to achieve in an approach that reconstructs the bulk correlators from the boundary CFT (e.g., refs. [ 39 , 41 , 40 ]).
In particular, we will adapt the coherent state mechanism to promote classical solutionsto bound states of quantum degrees of freedom. As we would expect, we can apply the same procedure to the components of the classical AdS metric as to the soliton and instanton profile presented in chapters 3 and 4. Accordingly, the bound state quanta referred to as corpuscles have similar features in the soliton as well as in the AdS case. Namely, the corpuscles can only exist inside the bound state, but not as asymptotic S-matrix quanta. Accordingly, they are interaction eigenstates. However, there are some differences between the corpuscular AdS and the corpuscular soliton. First of all, the corpuscular dispersion can, in general, be very different in both cases because AdS is not diagonalized on the semi- classical level. Secondly, we observe a divergence of the occupation number of corpuscles of small wavelengths. On the one hand, this can be understood in terms of the infinite blue-shift occurring at the boundary of AdS. On the other hand, from the corpuscular point view this effect directly leads to the stability of AdS with respect to decay into Minkowski space-time. Furthermore, let us remark that in the linear limit of AdS we can develop an interpretation of the AdS constituents in terms of on-shell massive gravitons.
Finally, we will use the background-independent geometric quantization scheme that we will introduce for the holomorphic representation of the GBF for an example of field quantization on tensorial spacetimes. For that purpose, we do not need the full generality of the GBF; we are dealing with initial and final data on Cauchy hypersurfaces. However, the geometric quan- tization scheme suits perfectly for situations with generalized backgrounds. More specifically, we will use the geometric quantization scheme to quantize a generalization of the Klein-Gordon field on a non-metric tensorial spacetime with a dispersion relation of fourth order. That means in particular that the corresponding field equations will be of fourth order. We will find that additional solutions not corresponding to classical particles have to be included in order to obtain a microcausal theory when canonical commutation relations are imposed. We will obtain that Lorentzian spacetimes are the only tensorial spacetimes on which one can consistently establish a microcausal, unitary quantum scalar fieldtheory fulfilling canonical commutation relations (CCRs) such that only classical interpretable particles exist. Including the non-classical modes, however, leads to mathematical problems and conceptual problems concerning the interpretation of these modes. Comparing this result to results obtained for the imaginary mass Klein-Gordon field, we will argue that different inertial observers would see a different content of non-classical particles in the same state of the field.
The passage to the quantumtheory required a canonical transformation so as to be able to write the holonomies as a product of strictly and almost periodic functions. A Cauchy completion then led to a Hilbert space given by square integrable functions over both ¯ R B and U(1). However the drawback of the canonical transformation is a much more complicated expression for the components of the densitized triad containing both the momentum and the conﬁguration variables. Following the standard procedure of LQC we substituted these conﬁguration variables with the sine thereof and were able to solve the eigenvalue problem analytically. Surprisingly it turned out that the spectrum of the triad operators can be either discrete or continuous, depending on the initial value problem. On the other hand we were also able to ﬁnd almost periodic solutionsto the eigenvalue problem of the triad operators without performing the substitution just described, but once again the spectrum depends can be either continuous or discrete. The reason why both ways can lead to a continuous spectrum is the non-cubical form of the torus, for if we set the angles θ 1,2 = 0 in Eq. (2.26) the triads correspond to the ones obtained in isotropic models.
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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85 Evaluating the two common sources of bilateral FDI data, UNCTAD data is not used for several reasons; firstly, the time-frame only covers 2001-2012 which is perceived as being insufficient for general gravity panel studies, origin-destination reports differ too much for a large share of developing- and tiger states but also for industrialized countries, and a large number of no-observations is found for implausible country-pairs. 55 In opposition to that, OECD macro-data is compiled in a more uniform matter and available from 1985-2017, however the dataset is gathered with two different benchmark definitions (1985-2012 BMD3 and 2013- 2017 BMD4) and therefore the two datasets have to be merged. The difference for the BMD4 is the introduction of splitting FDI on the basis of Special Purpose Entities (SPE) and non-SPE FDIs, where an SPE is defined as an entity with little or no physical presence in the respective country and which serves primarily for holding assets and liabilities or raising capital for the multinational firm (OECD 2015). Discussing the SPE FDI split in general makes sense for FDI gravity research, especially in the field of tax (avoidance), however this has to be left open for future research as most countries do not report splits as recommended by the OECD but instead report total FDI equal to non-SPE, indicating that the BMD4 guideline has not yet been successfully implemented. This however simplifies merging both datasets; in addition, a trend-break variable is introduced to control for a potential bias. We convert negative flow values to zero and exclude missing values, as explained in Welfens and Baier (2019).
However, in terms of the Lagrangian, the limit θ − → ∞ has to be taken carefully since this leads to an undefined singular Lagrangian term. Thus in that limit the field a µ− has to vanish simultaneously. Otherwise the Chern Simons term is no more defined and meaningless. It corresponds to the theories for partially polarized Hall states described in [MR96]. The incor- poration of SU (2) Chern Simons fields [LF95] and [BF91] respectively differs from that ap- proach since an additional cubic term is needed in the nonabelian Chern Simons Lagrangian, which is missing in this approach. Furthermore, if we perform a Chern Simons transformation with local spin 1/2 fluxes (flux-attachment to electrons or composite Fermions) we can not systematically obtain the Θ-matrix approach since for the attachment for SU (2) fields i.e. spins we have to perform a local SU (2)-Chern Simons transformation which automatically generates a corresponding coupling to a Lie(SU (2)) valued gauge field. The corresponding Lagrangian is a nonabelian Chern Simons Lagrangian. In this sense the Θ-matrix approach is a pure phenomenological approach which can only be justified by heuristic arguments rather than fundamental gauge principles. The implementation of a SU (2) symmetry need some care what concerns a quantization in a perturbative treatment. In [MR96] the quantization is performed in terms of Witten’s approachto invariants [Wit89], however a perturbative treatment requires a different approach [AS91]. In the next chapter we introduce a proper quantization procedure for general Chern Simons theories with semi simple compact gauge group as symmetry group in the composite Fermion picture and discuss the SU (2) example.
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.
Moreover, it is possible to add further D branes to these supergravity back- grounds as shown in [3; 9–12]. These additional branes were considered as probe branes, i.e., they do not deform the background and break some symme- try by adding open string states which could also end on the additional brane. Recently, some investigation in non-probe branes  has been published, how- ever beyond the scope of basic probe brane properties we are interested in here. The D brane intersections are very useful to describe flavour since the open strings – which can end on the D brane stack as well as on the probe brane – become massive and show a CFT dual in the fundamental representation. Moreover, due to the duality property of AdS/CFT correspondence, a weak coupling theory on the AdS side describes a strong coupling one on the CFT side which then can be considered as QCD. In this way an indirect perturba- tional approachto QCD has been presented and induced intense investigation [9; 11; 14; 15].
This research paper shows how E-theory can be applied to a scalar boson which underlies gravitational effects in 4-dimensional spacetime. The low-energy limit, conservation laws and the perturbative calculation of scattering amplitudes are shown. With these considerations the general procedure of an application of E-theoryto gravitational physics called “E-gravity” is made clear. Similarities between this quantumgravitytheoryand Topological Dipole FieldTheory (Linker 2015) are also shown in this research paper. Topological Dipole FieldTheory (TDFT) is a model that describes a modification of the dynamics of gauge bosons which implies distinct behavior of quantum fluctuations.
Serious difficulties in combining quantum mechanics with general relativity arise from the conceptual differences of these two theories and from the lack of physical experiments hinting to some kind of quantumgravity effects. Such effects, however, one would expect from a theory unifying gravitational andquantum nature of matter. So, at this point, unification can only be done on a purely theoretical level, and here one basically has the choice between the following two strategies. First, one can try to construct a completely new theory which, in the appropriate physical limits, reproduces general relativity andquantum mechanics. Second, one can try to quantize general relativity directly, hoping to end up with a unified theory or, to be more realistic, to get some hints on how such a theory should look like. Following these philosophies, promising candidates are string theory, a pertubative approachto the construction of a superordinated theory, and the loop quantumgravityapproach we will follow in this thesis.
The best known description of fundamental interactions is given in terms of gauge theories. Electromagnetic, weak and strong interactions are obtained by localising internal symmetries while gravity can be understood as the gauge theory with the Poincar´e group as the gauge group. Therefore, it is of importance to generalise this concept to deformed spaces as well. In this chapter we construct a general nonabelian gauge theory on the κ-deformed space. However, the construction is done in a very general way so that it can be applied to other deformed spaces as well. In our approach we have to introduce enveloping algebra-valued quantities (noncommutative gauge parameter, noncommutative gauge field, . . . ) . This leads to (apparently) infinitely many degrees of freedom in the theory. The problem is solved in terms of the Seiberg-Witten map . This map allows to express noncommutative variables in terms of the corresponding commutative ones and this reduces the number of degrees of freedom to the commutative ones.
Even though the paradigm of cosmic inflation fits very nicely with the observed CMB power spectrum , we have so far no idea about the “nature” of the mechanism behind inflation. In the standard slow-roll inflation approach, in- flation is driven by a scalar condensate with negative pressure [24–27]. This is realized by a scalar field whose potential energy dominates over its kinetic energy. This scalar field, the inflaton, effectively acts as a “clock” telling us when inflation ends. Its quantum fluctuations are stretched out to macroscopic scales and can directly be related to the temperature fluctuations of the CMB. But which particle is the inflaton, i. e. what are its quantum numbers and interactions? To get a handle on this problem, it is inevitable to consider infla- tion in a particle physics framework. It is particularly appealing to embed the inflaton into the matter sector. Then its interactions are not only constrained by cosmology but also by particle physics and astroparticle physics.
Although it was now possible to calculate observables like the Lamb shift or the g-factor of the electron perturbatively to a high precision, the situ- ation was not really satisfying. On the one hand, there was evidence that the meson coupling constants 1 are of order unity, while on the other hand even in quantum electrodynamics there are domains in momentum space where the effective coupling can grow to infinity, making non-perturbative considerations necessary . An important step in this direction was made by Stückelberg and Petermann [15, 16], who found that finite renormalization transformations of scattering-matrix elements form a Lie group and thus obey differential equations. Seemingly unaware of their work, Gell-Mann and Low  independently considered scale transformations in QED and showed that the electromagnetic coupling changes with the energy scale according to a differential equation, which in modern parlance is known as a beta function. These ideas were further developed by Bogolyubov and Shirkov ; the title of their paper “Charge Renormalization Group in QuantumFieldTheory” marks the first appearance of the term renormalization group (RG).
that oscillations and longitudinal movement are suppressed in the large separation limit. ∗ Integration of the Polyakov action along the string can then be performed, yielding effectively a centre-of-mass movement weighted by a factor from averaging over the geometry between the two D7s. To obtain a field equation, na¨ıve quantisation is performed, which results in a modified Klein–Gordon equation. (In a Minkowski space, this procedure yields the conventional, unmodified Klein–Gordon equations.) After the AdS case, the discussion will be moved on to the dilaton deformed background by Gubser introduced in Chapter 3 and a similar background by Constable–Myers. Both exhibit chiral symmetry breaking. While these are known to be far from perfect QCD gravity duals, experience shows that even simple holographic models can reproduce measured mass values with an accuracy of 10–20%. Assuming the two respective quark flavours associated to the D7-branes being up and bottom, the mass of the rho (u¯ u) and upsilon (b¯ b) meson can be used to fix all scales in the theoryand yield a numerical prediction for the B meson mass, which indeed is less than 20% from the experimental value.
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.
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
Next, one can include soft quanta in the final state. The number of soft quanta can be arbitrarily large but the total energy contained in them must be small. Then computation shows that the sum over all such final states that include soft quanta gives an infinite contribution. However, when combined with the vanishing contribution due to loop corrections, a finite total rate is obtained [57–59]. Thus, all divergent contributions cancel self-consistently. Moreover, it turns out that the tree level result, which includes neither loops nor soft emission, is a good approximation and infrared physics only gives a small correction to the total rate. This result could sound like the end of the story, but it is not. Shortly af- ter Weinberg’s computation , a different approachto infrared divergences was suggested, in which no soft emission was considered. Instead, charged asymp- totic states were modified by adding to each of them, i.e. to both final and initial states, a carefully chosen coherent state of soft photons [60–65]. The physical justification for this modification of asymptotic states is that in a theory with long-range interactions, approximate eigenstates of the asymptotic Hamiltonian can only be formed if the above-mentioned dressing by soft photons is included. Even though the procedure is very different, the combination of all dressing factors approximately results in the same contribution as the one from soft emission and therefore yields a finite total rate.