Nach oben pdf Non-perturbative effects in field theory and gravity

Non-perturbative effects in field
theory and gravity

Non-perturbative effects in field theory and gravity

This work is split into three main parts, the first parts discusses non-perturbative aspects of the dynamics of Yang-Mills theories, the second part contains a discussion of aspects of black hole dynamics in the context of the graviton condensate picture proposed by Dvali and Gomez while the last part of the thesis consists of reprints of the peer reviewed publications of the author. The first part starts with a review of the dynamics of Yang-Mills theories and their behaviour in the so called ’t Hooft limit of a large number of colors. We also review the relation of the ’t Hooft expansion to the genus expansion known from string theory. From there it proceeds with a short review of non local operators and topological field theories. Following this there is a review of how to localize massless gauge fields on topological defects as well as a short description of the mechanism for localizing topologically massive gauge fields on domain walls invented by the author. The corre- sponding more detailed explanation can be found in the third part of this thesis. The next section deals with a topological field theory description of the low energy dynamics of ordinary Yang-Mills theories as well as discussing the behaviour of domain walls appearing in these theories extending previous results by Seiberg and collaborators. Here we will also be able to shed light on the appearance of topological degrees of freedom on the world volume of these walls. The subsequent two sections are devoted to an extension of these results to supersymmetric gauge theories. We first review the prop- erties of supersymmetric Yang-Mills theories and the existing computations of the tension of domain walls in supersymmetric Yang-Mills theories. Then we follow up with an extension of our work in the previous chapter to show that there are topological degrees of freedom living on these domain walls as well. We conclude this part with a summary of the results achieved in this thesis with respect to domain walls in Yang-Mills theory and furthermore show how several puzzling properties of Yang-Mills theories can be seen to have natural analogs in critical string theories. We also point out a striking analogy to fractional quantum hall systems. Parts of the second part will be basis for an upcoming publication together with Markus Dierigl.
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Geometric symmetries and topological terms in F-theory and field theory

Geometric symmetries and topological terms in F-theory and field theory

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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Non-linear massive gravity

Non-linear massive gravity

The main purpose of this paper is to investigate the existence of a smooth limit of our model to Einstein gravity, when the mass of the graviton vanishes. It was noticed long ago by van Dam, Veltman and Zakharov [ 4 , 5 ] that in linearized massive gravity the extra scalar mode of the graviton did not disappear and remained coupled to matter even in the limit of a vanishing graviton mass. In turn, this spoils predictions of General Relativity either for the perihelion precession or deflection of starlight. This effect is known as the van Dam-Veltman-Zakharov (vDVZ) discontinuity and was first thought to be a no-go the- orem for massive theories of gravity [ 4 , 5 ]. However, it was pointed out by Vainshtein that the discontinuity could be an artifact due to the breakdown of the perturbation theory of massive gravity in the massless limit [ 6 ]. He has shown that in the case of gravitational field produced by a source of mass M 0 the nonlinear corrections become important at scales r < R V ≡ M 0 1/5 m −4/5 g (in Planck units) and conjectured that in the strong coupling regime General Relativity is restored. When the mass of the graviton m g vanishes the Vainshtein radius R V grows and becomes infinite, thus providing a continuous limit to General Rel- ativity in case the Vainshtein conjecture is correct. At distances r ≪ R V , around a static spherically symmetric massive source of mass M 0 the full non-linear strongly coupled mas- sive gravity has to be considered in order to recover the Einstein theory, which makes the proof of the Vainshtein conjecture non trivial. The question of continuous matching of the solutions below and above the Vainshtein radius have been extensively addressed in recent literature. The first model where such a transition was demonstrated is Dvali-Gabadadze- Porrati (DGP) model which imitates many features of massive gravity [ 13 , 20 ]. There was a claim that in the bigravity version of massive graviton the corresponding solutions do not match [ 7 ], but it was recently shown that this claim is not justified [ 8 – 10 ].
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Kappa deformed gauge theory and theta deformed gravity

Kappa deformed gauge theory and theta deformed gravity

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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Approaches to non-perturbative problems in hadron physics / Joseph P. Day

Approaches to non-perturbative problems in hadron physics / Joseph P. Day

There has been a deep historical connection between the pursuit of understanding the physics of strongly coupled systems and string theory. String theory was proposed as an explanation for the Regge trajectory pattern on bound states. This is the noted pattern that when the angular momentum of hadronic excitations J are plotted versus the mass or energy squared they form a pattern of lines. This pattern could be easily explained by representing the mesons for example as two quarks bound together by a string. With the emergence of QCD in the early 70’s this stringy explanation was abandoned however the ideas of string theory persisted into a theory of quantum gravity. In 1997 it seems these ideas finally came full circle. When Maldacena first conjectured the correspondence be- tween a geometrical (gravitational) theory in anti-de Sitter space (AdS) and a conformal
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Quantum many-body effects in gravity and Bosonic theories

Quantum many-body effects in gravity and Bosonic theories

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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Non-perturbative approach to calculation of correlation functions in 1D Fermi gases

Non-perturbative approach to calculation of correlation functions in 1D Fermi gases

Quantum low dimensional systems attract huge interest since they provide a test area for the investigation of the general behaviour of complicated nonlinear systems. Indeed, a lot of inter- esting phenomena in condensed matter, high-energy physics and quantum field theory emerge due to the presence of complicated non inear interactions. The study of these systems is often an extremely complicated task that is out of the reach of perturbative approaches. However, it is known that in many cases interesting nontrivial many body effects prevail even if the system is restricted to 1D. Moreover, the simplified but still nontrivial cases of 1D system allows one to concentrate on key properties of a system while avoiding bulky technical problems. Sometimes this allows to make crucial simplifications because the system will exhibit integrability, i.e. can be solved exactly. We also see that a lot of specific, interesting properties emerge in 1D systems, especially in integrable models, thus making 1D integrable models interesting in their own right. Among these interesting systems it is worth noting t-J [16, 73–79] and Hubbard [6, 7, 71, 72] models, which describe lattice gases of strongly-correlated electrons and are expected to exhibit the high-T superconductivity behaviour. While this phenomenon was intensively studied over the last 50 years, still most of the mechanisms of high-T superconductivity are unknown. One of the main open questions of the field is the mechanism of formation of electron pairs. The derivation of exact solutions of the t-J and Hubbard models can help to answer such a question and thus to understand the nature of the high temperature superconductivity. These are important models from the perspective of studying the entanglement entropy and Kondo model [9] which describes an anomalous low-T behaviour of conductivity of doped metals (Kondo problem).
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Symmetries in four-dimensional multi-spin-two field theory: relations to Chern-Simons gravity and further applications

Symmetries in four-dimensional multi-spin-two field theory: relations to Chern-Simons gravity and further applications

Whether the generalisation of general relativity to N non-interacting metrics is consistent was investigated in ref. [ 114 ], to find out indeed, that such a theory is inconsistent unless N = 1. Hinterbichler and Rosen first developed then a set of theories for N interacting vielbeine [ 91 ]. However, it was later realized that only those theories whose vielbeine interact pair-wisely through the bimetric potential are free of ghost instabilities [ 115 ]. Furthermore, loop interactions, i.e., when a vielbein interacts only with a next one and this latter in addition with a next one and so on until closing the circle, also lead to in- consistent theories (see refs. [ 91 , 116 – 118 ]). Those consistent theories were first formulated by Hinterbichler and Rosen using the language of differential forms for N vielbeine e a (p) , with p = 1, . . . , N . Their equivalent theories in tensors is established when a generalised symmetry condition (e a (p) ) [µ| (e b (q) ) |ν] η ab = 0 is
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Gutenberg Open Science: Field space parametrization in quantum gravity and the identification of a unitary conformal field theory at the heart of 2D Asymptotic Safety

Gutenberg Open Science: Field space parametrization in quantum gravity and the identification of a unitary conformal field theory at the heart of 2D Asymptotic Safety

larly interesting: The mass dimension of the running Newton onstant, [G k ] = 2 − d , vanishes in exa tly d = 2 spa etime dimensions, and a perturbative treatment be- omes feasible. This approa h involves omputing the β -fun tions (i.e. the ve tor eld whi h drives the RG ow) in d = 2 + ε > 2 dimensions and expanding them in terms of ε . A general onsideration [4℄ shows that the β -fun tion of the dimensionless Newton onstant, g k ≡ k

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The welfare effects of persuasion and taxation: Theory and evidence from the field

The welfare effects of persuasion and taxation: Theory and evidence from the field

Tables A9–A13 report results for the subsample for which data on vote shares for the environmental party is available. We assign each subject to the group of ”green consumers” andnon-green consumers” based on whether she comes from a region with above- or below-median support for the Green party. Non-green consumers have larger (structural) misperceptions of energy efficiency. They overvalue LEDs by around 1.25 euros per bulb and undervalue Non-LEDs by 2.62 euros per bulb. Green consumers undervalue LEDs by 0.43 and overvalue Non-LEDs by 1.03 euros per bulb. The less informative signal increases the overvaluation for LEDs for both groups to around 1.80 per bulb. It also decreases consumers’ undervaluation of Non-LEDs but leaves consumers with weaker environmental preferences with a larger bias of 2.32 euros per bulb (as opposed to 0.89 for green consumers). In terms of consumer surplus, households with lower green preferences benefit more from the fully informative signal and are hurt less by partial information disclosure than households with strong green preferences. The tax vector is similar to the scenario in which income is assumed to identify homogeneous subgroups but is characterized by both a slightly lower subsidy and a lower tax. Much of the tax burden is borne by green consumers who lose 5.94 euros per bulb per consumer, which is around 3.7 times more than non-green consumers. The reason for this asymmetry is again primarily attributable to the difference in own-price elasticities for Non-LED products.
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Giving in a Large Economy: Price vs. Non-Price Effects in a Field Experiment

Giving in a Large Economy: Price vs. Non-Price Effects in a Field Experiment

While the presence and direction of the price e¤ect are mostly clear, theory provides less guidance on its strength. There are a number of arguments, such as lack of substitutes and salience of price in deciding on public goods contribu- tions, that support the notion that the price elasticity for contributing to public goods should be low (Green 1992). Also, both in a pure and impure altruism model in the spirit of Andreoni (1988) and Andreoni (1990), respectively, the subjects’ strategic interdependence in providing the public good reduces the price elasticity of the Nash contributions as long as subjects believe that all subjects face the same change in price. Support for predicting low price elastic- ity comes from experimental studies that examine a limited number of discrete price variations and report low estimates at the extensive margin of contribut- ing: Smith, Kehoe and Cremer (1995) …nd that the decision whether to make a charitable contribution for a rural health care facility is insensitive to price. Likewise, examining contribution choices for an unmatched baseline and three match ratios, Karlan and List (2007) …nd that while the probability of donating 1 2 Standard deviation is 6.4. The smallest group consisted of 31, the largest of 66 sub jects. 1 3 The same authors also discuss that a broader class of models makes for more equivocal
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The Welfare Effects of Persuasion and Taxation: Theory and Evidence from the Field

The Welfare Effects of Persuasion and Taxation: Theory and Evidence from the Field

Tables A9–A13 report results for the subsample for which data on vote shares for the environmental party is available. We assign each subject to the group of ”green consumers” andnon-green consumers” based on whether she comes from a region with above- or below-median support for the Green party. Non-green consumers have larger (structural) misperceptions of energy efficiency. They overvalue LEDs by around 1.25 euros per bulb and undervalue Non-LEDs by 2.62 euros per bulb. Green consumers undervalue LEDs by 0.43 and overvalue Non-LEDs by 1.03 euros per bulb. The less informative signal increases the overvaluation for LEDs for both groups to around 1.80 per bulb. It also decreases consumers’ undervaluation of Non-LEDs but leaves consumers with weaker environmental preferences with a larger bias of 2.32 euros per bulb (as opposed to 0.89 for green consumers). In terms of consumer surplus, households with lower green preferences benefit more from the fully informative signal and are hurt less by partial information disclosure than households with strong green preferences. The tax vector is similar to the scenario in which income is assumed to identify homogeneous subgroups but is characterized by both a slightly lower subsidy and a lower tax. Much of the tax burden is borne by green consumers who lose 5.94 euros per bulb per consumer, which is around 3.7 times more than non-green consumers. The reason for this asymmetry is again primarily attributable to the difference in own-price elasticities for Non-LED products.
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E-gravity theory

E-gravity theory

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-theory to gravitational physics called “E-gravity” is made clear. Similarities between this quantum gravity theory and Topological Dipole Field Theory (Linker 2015) are also shown in this research paper. Topological Dipole Field Theory (TDFT) is a model that describes a modification of the dynamics of gauge bosons which implies distinct behavior of quantum fluctuations.
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Atmospheric and Oceanic Mass Variations and their role for gravity field determination

Atmospheric and Oceanic Mass Variations and their role for gravity field determination

including a mean atmosphere and ocean potential is introduced. Further potential fields generating pertur- bations on a satellite’s orbit are tides. Therefore, also direct tides, solid Earth tides, ocean tides, and pole tides are included in the force model. In addition - and which is the most important part for this work - short-term atmospheric and oceanic mass variations cause potential variations and act as disturbance forces on a satellite’s orbit. During GRACE gravity field analysis these variations are removed by the so-called atmosphere and ocean de-aliasing (AOD) product. This is done in order to avoid aliasing due to temporal undersampling, because GRACE is not able to adequately sample these short-term atmospheric and oceanic mass variations. Thus, these variations are modelled in 6-hourly intervals (see chapter 3) via geophysical models and are, after interpolation on the integration interval which is usually 5 seconds, ’removed’ during gravity field determination. This process is called de-aliasing. Having modelled them correctly and realis- tically, the resulting GRACE gravity field solutions should not contain any atmospheric and oceanic signal with periods shorter than the sampling period of the gravity field solution (which is usually one month). The mentioned empirical forces are introduced in the disturbance force models in order to treat the mismod- elled or umodelled forces acting on a satellite. The intent of these empirical non-gravitational parameters, introduced during gravity field determination, is to absorb the poorly modelled parts of the force fields and to treat the measuring effects. In principle there are several types of empirical parameters like dynamic and kinematic ones. E.g., these empirical models are often used to absorb the biased accelerometer mea- surements. A more detailed description of the role of the empirical forces within orbit and gravity field determination is given, e.g., in Kim (2000), J¨ aggi (2006) and Beutler et al. (2010b). The main aspect con- cerning these empirical forces, and being relevant for this work is the fact, that each empirical parameter introduced in the orbit or gravity field determination might disturb the solution for the other non-empirical parameters, e.g. gravity field coefficients, as these empirical forces may absorb in addition geophysical sig- nals, also model errors and unmodelled effects. This issue is revisited in chapter III.
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Non-perturbative gravity at different length scales

Non-perturbative gravity at different length scales

In this thesis, we investigate different aspects of gravity as an effective field theory. Building on the arguments of self-completeness of Einstein gravity, we argue that any sensible theory, which does not propagate negative-norm states and reduces to General Relativity in the low energy limit is self-complete. Due to black hole formation in high energy scattering experiments, distances smaller than the Planck scale are shielded from any accessibility. Degrees of freedom with masses larger than the Planck mass are mapped to large classical black holes which are described by the already existing infrared theory. Since high energy (UV) modifications of gravity which are ghost-free can only produce stronger gravitational interactions than Einstein gravity, the black hole shielding is even more efficient in such theories. In this light, we argue that conventional attempts of a Wilsonian UV completion are severely constrained. Furthermore, we investigate the quantum picture for black holes which emerges in the low energy description put forward by Dvali and Gomez in which black holes are described as Bose-Einstein condensates of many weakly coupled gravitons. Specifically, we investigate a non-relativistic toy model which mimics certain aspects of the graviton condensate picture. This toy model describes the collapse of a condensate of attractive bosons which emits particles due to incoherent scattering. We show that it is possible that the evolution of the condensate follows the critical point which is accompanied by the appearance of a light mode.
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Collective quantum effects in field theory and gravity

Collective quantum effects in field theory and gravity

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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Quantum corpuscular approach to solutions in gravity and field theory

Quantum corpuscular approach to solutions in gravity and field theory

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.
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A Combined Entropy/Phase-Field Approach to Gravity

A Combined Entropy/Phase-Field Approach to Gravity

Theories of phase transitions have their origin in the early models of van der Waals (1893), Korteweg (1901), Ginzburg-Landau (1950), Cahn-Hilliard (1958), Allen-Cahn (1960), Halperin, Hohenberg and Ma (1977). While the early models did not include any spatial resolution, especially the Cahn-Hilliard equation which, for the first time, addressed demixing phenomena using a spatially-resolved approach. The order parameter entering into the equations was the—conserved—concentration of alloy elements. The Allen-Cahn equation then included the option of non-conserved order parameters for the first time. It seems essential to highlight that phase transitions are best described by non-conserved order parameters such as, for example, the fraction liquid of a system will change from 1 to 0 in a solidification process and is thus not a conserved quantity.
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Symmetries in classical field theory

Symmetries in classical field theory

An important feature of classical field theory lies in its relation to symmetries. On the one hand, implementing a symmetry into a given theory is rather simple, as we shall see in the following section. On the other hand, if a continuous symmetry is present, then there a exist corresponding conserved quantity (or set of conserved quantities), as we shall now see.

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Transversality Results and Computations in Symplectic Field Theory

Transversality Results and Computations in Symplectic Field Theory

Using the characterization of trivial curves as curves with trivial ω-energy we can prove that we indeed have a global obstruction bundle over the compactification of every moduli space of trivial curves. While in Gromov-Witten theory the count of elements in the moduli space, more general, the cobordism class of the moduli space, is independent of the chosen abstract perturbation of the Cauchy-Riemann operator, this no longer holds for the moduli spaces in symplectic field theory. This follows from the fact that the moduli spaces in symplectic field theory typically have codimension one boundary strata, while in Gromov-Witten theory the regular moduli spaces form pseudo-cycles in the sense that the boundary strata have codimension at least two, i.e., from the homological point of view have no boundary. So while in Gromov-Witten theory the moduli spaces can be studied separately, the interplay between the different moduli spaces is the reason why the algebraic invariants of symplectic field theory are defined as differential algebras, which can be shown to be independent of extra choices like the cylindrical almost complex structure and the compact perturbation. In our case this problem is expressed by the fact that we have to study sections in vector bundles over moduli spaces with codimension one boundary, so that the count of zeroes in general depends on the choice of sections in the boundary, i.e., the chosen perturbations of the Cauchy-Riemann operator used to define the regular moduli spaces in the boundary. However we outline below that in our case we indeed have a well-defined count of zeroes so that, as in the Gromov-Witten case, we can (iteratively) define Euler numbers for our Fredholm problems.
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