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Institute
This thesis deals with several aspects of non-perturbative calculations in low-dimensional quantum field theories. It is split into two main parts:
The first part focuses on method development and testing. Using exactly integrable QFTs in zero spacetime dimensions as toy models, the need for non-perturbative methods in QFT is demonstrated. In particular, we focus on the functional renormalization group (FRG) as a non-perturbative exact method and present a novel fluid-dynamic reformulation of certain FRG flow equations. This framework and the application of numerical schemes from the field of computational fluid dynamics (CFD) to the FRG is tested and benchmarked against exact results for correlation functions. We also draw several conclusions for the qualitative understanding and interpretation of renormalization group (RG) flows from this fluid-dynamic reformulation and discuss the generalization of our findings to realistic higher-dimensional QFTs.
The topics discussed in the second part are also manifold. In general, the second part of this thesis deals with the Gross-Neveu (GN) model, which is a prototype of a relativistic QFT. Even though being a model in two spacetime dimensions, it shares many features of realistic models and theories for high-energy particle physics, but also emerges as a limiting case from systems in solid state physics. Especially, it is interesting to study the model at non-vanishing temperatures and densities, thus, its thermodynamic properties and phase structure.
First, we use this model to test and apply our findings of the first part of this thesis in a realistic environment. We analyze how the fluid-dynamic aspects of the FRG realize themselves in the RG flow of a full-fledged QFT and how we profit from this numeric framework in actual calculations. Thereby, however, we also aim at answering a long-standing question: Is there still symmetry breaking and condensation at non-zero temperatures in the GN model, if one relaxes the commonly used approximation of an infinite number of fermion species and works with a finite number of fermions? In short: Is matter (in the GN model) in a single spatial dimension at non-zero temperature always gas-like?
In general, we also use the GN model to learn about the correct description of QFTs at non-zero temperatures and densities. This is of utmost relevance for model calculations in low-energy quan- tum chromodynamics (QCD) or other QFTs in medium and we draw several conclusions for the requirements for stable calculations at non-zero chemical potential.
Binary neutron star mergers represent unique observational phenomena because all four fundamental interactions play an important role at various stages of their evolution by leaving imprints in astronomical observables. This makes their accurate numerical modeling a challenging multiphysics problem that promises to increase our understanding of the high-energy astrophysics at play, thereby providing constraints for the underlying fundamental theories such as the gravitational interaction or the strong interaction of dense matter. For example, the first and so far only multi-messenger observation of the binary neutron star merger GW170817 resulted in numerous bounds on the parameters of isolated non-rotating neutron stars, e.g., their maximum mass or their distribution in radii, which can be directly used to constrain the equation of state of cold nuclear matter. While many of these results stem from the observation of the inspiral gravitational-wave signal, the postmerger phase of binary neutron star mergers encodes even more details about the extreme physics of hot and dense neutron star matter. In this Thesis we focus on the exploration of dissipative and shearing effects in binary neutron star mergers in order to identify novel approaches to constrain hot and dense neutron star matter.
The first effect is the well-motivated dissipation of energy due to the bulk viscosity which arises from violations of weak chemical equilibrium. We start by exploring the impact of bulk viscosity on black-hole accretion. This simplified problem gives us the opportunity to develop a test case for future codes taking into account the effects of dissipation in a fully general-relativistic setup and build intuition in the physics of relativistic dissipation. Next, we move on to isolated neutron stars and binary neutron star mergers by developing a robust implementation of bulk-viscous dissipation for numerical relativity simulations. We test our implementation by calculating the damping of eigenmodes of isolated neutron stars and the violent migration scenario. Finally, we present the first results on the impact of bulk viscosity on binary neutron star mergers. We identify a number of ways how bulk viscosity impacts the postmerger phase, out of which the suppression of gravitational-wave emission and dynamical mass ejection are the most notable ones.
In the last part of this Thesis we investigate how the shearing dynamics at the beginning of the merger affects the amplification of different initial magnetic-field topologies. We explore the hypothesis that magnetic fields which are located only in a small region near the stellar surface prior to merger lead to a weaker magnetic-field amplification. We show first evidence which confirms this hypothesis and discuss possible implications for constraining the physics of superconduction in cold neutron stars.
This thesis provides a detailed derivation of dissipative spin hydrodynamics from quantum field theory for systems composed of spin-0, spin-1/2, or spin-1 particles.
The Wigner function formalism is introduced for quantum fields in the respective representations of the Poincaré group, and the conserved currents, i.e., the energy-momentum tensor and the total angular momentum tensor, in various so-called pseudogauges are derived. An expansion around the semiclassical limit in powers of the Planck constant is performed.
Subsequently, kinetic equations are obtained for binary elastic scattering, using both the de Groot-van Leeuwen-van Weert and Kadanoff-Baym method, with the latter retaining the effect of quantum statistics. The resulting collision term features both local and nonlocal contributions, with the latter providing a relaxation mechanism for the spin degrees of freedom of the quasiparticles. The local-equilibrium distribution function is derived from the requirement that the local part of the collision term vanishes.
From quantum kinetic theory, dissipative spin hydrodynamics is then constructed via the method of moments, extended to particles with spin. The system of moment equations is closed via the Inverse-Reynolds Dominance (IReD) approach, resulting in a set of equations of motion describing the evolution of both ideal and dissipative degrees of freedom. The application to polarization phenomena relevant to heavy-ion collisions is discussed.
We discuss aspects of the phase structure of a three-dimensional effective lattice theory of Polyakov loops derived from QCD by strong coupling and hopping parameter expansions. The theory is valid for the thermodynamics of heavy quarks where it shows all qualitative features of nuclear physics emerging from QCD. In particular, the SU(3) pure gauge effective theory also exhibits a first-order thermal deconfinement transition due to spontaneous breaking of its global Z₃ center symmetry. The presence of heavy dynamical quarks breaks this symmetry explicitly and consequently, the transition weakens with decreasing quark mass until it disappears at a critical endpoint. At non-zero baryon density, the effective theory can be evaluated either analytically by the so-called high-temperature expansion which does not suffer from the sign problem, or numerically by standard Monte-Carlo methods due to its mild sign problem. The first part of this work devotes to a systematic derivation of the effective theory up to the 6th order in the hopping parameter κ. This method combined with the SU(3) link update algorithm provides a way to simulate the O(κ⁶) effective theory. The second part involves a study of the deconfinement transition of the pure gauge effective theory, with and without static quarks, at all chemical potentials with help of the high-temperature expansion. Our estimate of the deconfinement transition and its critical endpoint as a function of quark mass and all chemical potentials agrees well with recent Monte-Carlo simulations. In the third part, we investigate the N ſ ∈ {1,2} effective theory with zero chemical potential up to O(κ⁴). We determine the location of the critical hopping parameter at which the first-order deconfinement phase transition terminates and changes to a crossover. Our results for the critical endpoint of the O(κ²) effective theory are in excellent agreement with the determinations from simulations of four-dimensional QCD with a hopping expanded determinant by the WHOT-QCD collaboration. For the O(κ⁴) effective theory, our estimate suggests that the critical quark mass increases as the order of κ-contributions increases. We also compare with full lattice QCD with N ſ = 2 degenerate standard Wilson fermions and thus obtain a measure for the validity of both the strong coupling and the hopping expansion in this regime.
The aim of this thesis is to provide a complete and consistent derivation of second-order dissipative relativistic spin hydrodynamics from quantum field theory. We will proceed in two main steps. The first one is the formulation of spin kinetic theory from quantum field theory using the Wigner-function formalism and performing an expansion in powers of the Planck constant. The essential ingredient here is the nonlocal collision term. We will find that the nonlocality of the collision term arises at first order in the Planck constant and is responsible for the spin alignment with vorticity, as it allows for conversion between spin and orbital angular momentum.
In the second step, this kinetic theory is used as the starting point to derive hydrodynamics including spin degrees of freedom. The so-called canonical form of the conserved currents follows from Noether’s theorem.
Applying an HW pseudo-gauge transformation, we obtain a spin tensor and energy-momentum tensor with obvious physical interpretation. Promoting all components of the HW tensors to be dynamical, we derive
second-order dissipative spin hydrodynamics. The additional equations of motion for the dissipative currents are obtained from kinetic theory generalizing the method of moments to include spin degrees of freedom.
We study the polarization of relativistic fluids using the relativistic density operator at global and local equilibrium. In global equilibrium, a new technique to compute exact expectation values is introduced, which is used to obtain the exact polarization vector for fields of any spin. The same result has been extended to the case of massless fields. Furthermore, it is demonstrated that at local equilibrium not only the thermal vorticity but also the thermal shear contribute to the polarization vector. It is shown that assuming an isothermal local equilibrium, the new term can solve the polarization sign puzzle in heavy ion collisions.
In this work a nonlinear evolution of pure states of a finite dimensional quantum system is introduced, in particular a Riccati evolution equation.
It is shown how this class of dynamics is actually a Hamiltonian dynamics in the complex projective space.
In this projective space it is shown that there is a nonlinear superposition rule, consistent with its linear counterpart in the Hilbert space. As an example, the developed nonlinear formalism is applied to the semiclassical Jaynes–Cummings model.
Later, it is shown that there is an inherent nonlinear evolution in the dynamics of the so-called generalized coherent states.
To show this, the fact that in quantum mechanics it is possible to immerse a ''classical'' manifold into the Hilbert space is employed, such that one may parametrize the time-dependence of the wave function through the variation of parameters in the classical manifold.
The immersion allows to consider the so-called principle of analogy, i.e. using the procedures and structures available from the classical setting to employ them in the quantum setting.
Finally, it is introduced the contact Hamiltonian mechanics, an extension of symplectic Hamiltonian mechanics, and it is showed that it is a natural candidate for a geometric description of non-dissipative and dissipative systems.
We study the Wigner function for massive spin-1/2 fermions in electromagnetic fields. The Wigner function is analytically solved in five cases when electromagnetic fields are constants. For a general space-time dependent field configuration, we use the method of semi-classical expansion and solved the Wigner function at linear order in the Planck's constant. At the same order, we obtained a generalized Boltzmann equation for particle distribution, and a generalized BMT equation for spin polarization. Using the Wigner function, we calculated some physical quantities in a thermal equilibrium system.
Das Standardmodell der Elementarteilchenphysik beschreibt nach aktuellem Kenntnisstand die Entstehung, den Aufbau und das Verhalten der Materie in unserem Universum am erfolgreichsten. Dennoch gibt es einige Phänomene, die sich nicht in dessen Rahmen beschreiben lassen, wie z. B. die Existenz von dunkler Materie und Energie, nicht-verschwindende Neutrinomassen oder die Baryonenasymmetrie. Speziell im Hinblick auf die starke Wechselwirkung, welche im Standardmodell durch die Quantenchromodynamik (QCD) beschrieben wird, gibt es noch immer viele offene Fragen.
Eine Umgebung, in der man die QCD experimentell ergründen kann, bieten vor allem Schwerionenkollisionen, die insbesondere am Large Hadron Collider (LHC) oder am Relativistic Heavy Ion Collider (RHIC) durchgeführt werden.
In dieser Arbeit soll ein Beitrag von theoretischer Seite aus hinsichtlich eines besseren Verständnisses dieser Schwerionenkollisionen und der zugrundeliegenden QCD erbracht werden. Der Fokus liegt dabei auf dem Isotropisierungsprozess unmittelbar nach der Kollision der beiden Kerne.
Neben etlichen effektiven Theorien, die sehr gute Ergebnisse in den entsprechenden Grenzbereichen liefern, ist die Beschreibung der QCD im Rahmen der Gittereichtheorie (Gitter-QCD) die am meisten etablierte. Diese beinhaltet in den meisten Fällen einen Übergang zur euklidischen Raumzeit, da somit ein Auswerten der hochdimensionalen Pfadintegrale mithilfe von Monte-Carlo-Simulation basierend auf dem sogenannten Importance Sampling ermöglicht wird. Aufgrund der Komplexwertigkeit der euklidischen Zeitkomponente ist man jedoch an das Studieren von statischen Observablen gebunden. Da wir aber gerade an einer Zeitentwicklung des Systems interessiert sind, sehen wir von dem Übergang zur euklidischen Raumzeit ab, was den Namen “real-time” im Titel der Arbeit erklärt.
Wir folgen dem sogenannten Hamilton-Ansatz und leiten damit Feldgleichungen in Form von partiellen Differentialgleichungen her, die wir dann mit den Methoden der Gitter-QCD numerisch lösen. Dabei bedienen wir uns der effektive Theorie des Farb-Glas-Kondensats (CGC, aus dem Englischen: “Color Glass Condensate”), um geeignete Anfangsbedingungen zu erhalten. Genauer gesagt basieren unsere Gitter-Anfangsbedingungen auf dem McLerran-Venugopalan-Modell (MV-Modell), das eine klassische Approximation in niedrigster Ordnung darstellt und nur Beiträge rein gluonischer Felder berücksichtigt.
Die klassische Näherung sowie das Vernachlässigen der fermionischen Felder wird insbesondere mit den hohen Besetzungszahlen der Feldmoden begründet. Einerseits dominieren Infrarot-Effekte, welche klassischer Natur sind, und andererseits ist dadurch der Einfluss der Fermionen, die dem Pauli-Prinzip gehorchen, unterdrückt. Gerade bei letzterer Aussage fehlt es jedoch an numerischen Belegen. Wir erweitern daher die klassische MV-Beschreibung durch stochastische Gitter-Fermionen, um diesem Punkt nachzugehen. Da sich Fermionen nicht klassisch beschreiben lassen, spricht man hierbei oft von einem semi-klassischen Ansatz.
Eines der Hauptziele dieser Arbeit liegt darin, den Isotropisierungsprozess, der bislang noch viele Fragen aufwirft, aber unter anderem Voraussetzung für das Anwenden von hydrodynamischen Modellen ist, zu studieren. Wir legen dabei einen besonderen Fokus auf die systematische Untersuchung der verschiedenen Parameter, die durch die CGC-Anfangsbedingungen in unsere Beschreibung einfließen, und deren Auswirkungen auf etwa die Gesamtenergiedichte des Systems oder die zugehörigen Isotropisierungszeiten. Währenddessen überprüfen wir zudem den Einfluss von unphysikalischen Gitter-Artefakten und präsentieren eine eichinvariante Methode zur Analyse der Güte unserer klassischen Näherung. Die Zeitentwicklung des Systems betrachten wir dabei sowohl in einer statischen Box als auch in einem expandierenden Medium, wobei Letzteres durch sogenannte comoving Koordinaten beschrieben wird. Zudem liefern wir einen Vergleich von der realistischen SU(3)-Eichgruppe und der rechentechnisch ökonomischeren SU(2)-Eichgruppe.
Mit unseren numerischen Ergebnissen zeigen wir, dass das System hochempfindlich auf die verschiedenen Modellparameter reagiert, was das Treffen quantitativer Aussagen in dieser Formulierung deutlich erschwert, insbesondere da einige dieser Parameter rein technischer Natur sind und somit keine zugehörigen physikalisch motivierten Größen, die den Definitionsbereich einschränken könnten, vorhanden sind. Es ist jedoch möglich, die Anzahl der freien Parameter zu reduzieren, indem man ihren Einfluss auf die Gesamtenergie des Systems analysiert und sich diesen zunutze macht. Dadurch gelingt es uns mithilfe von Konturdiagrammen einige Abhängigkeiten zu definieren und somit die Unbestimmtheit des Systems einzuschränken. Des Weiteren finden wir dynamisch generierte Filamentierungen in der Ortsdarstellung der Energiedichte, die ein starkes Indiz für die Präsenz von sogenannten chromo-Weibel-Instabilitäten sind. Unsere Studie des fermionischen Einflusses auf den Isotropisierungsprozess des CGC-Systems weist auf, dass dieser bei kleiner Kopplung vernachlässigbar ist. Bei hinreichend großen Werten für die Kopplungskonstante sehen wir allerdings einen starken Effekt hinsichtlich der Isotropisierungszeiten, was ein bemerkenswertes Resultat ist.
Light scalar mesons can be understood as dynamically generated resonances. They arise as 'companion poles' in the propagators of quark-antiquark seed states when accounting for hadronic loop contributions to the self-energies of the latter. Such a mechanism may explain the overpopulation in the scalar sector - there exist more resonances with total spin J=0 than can be described within a quark model.
Along this line, we study an effective Lagrangian approach where the isovector state a_{0}(1450) couples via both non-derivative and derivative interactions to pseudoscalar mesons. It is demonstrated that the propagator has two poles: a companion pole corresponding to a_{0}(980) and a pole of the seed state a_{0}(1450). The positions of these poles are in quantitative agreement with experimental data. Besides that, we investigate similar models for the isodoublet state K_{0}^{*}(1430) by performing a fit to pion-kaon phase shift data in the I=1/2, J=0 channel. We show that, in order to fit the data accurately, a companion pole for the K_{0}^{*}(800), that is, the light kappa resonance, is required. A large-N_{c} study confirms that both resonances below 1 GeV are predominantly four-quark states, while the heavy states are quarkonia.