In the exploration of chemical systems through quantum mechanics, accurate treatment of the electron wavefunction, and the related electron density, is fundamental to extracting information concerning properties of a system. This work examines challenges in achieving accurate chemical information through manipulation of the one-electron basis set.

<p>Bridging the accuracy of ab initio (AI) QM/MM with the efficiency of semi-empirical<br> (SE) QM/MM methods has long been a goal in computational chemistry. This dissertation<br> presents four ∆-Machine learning schemes aimed at achieving this objective. Firstly, the in-<br> corporation of negative force observations into the Gaussian process regression (GPR) model,<br> resulting in GPR with derivative observations, demonstrates the remarkable capability to<br> attain high-quality potential energy surfaces, accurate Cartesian force descriptions, and reli-<br> able free energy profiles using a training set of just 80 points. Secondly, the adaptation of the<br> sparse streaming GPR algorithm showcases the potential of memory retention from previous<br> phasespace, enabling energy-only models to converge using simple descriptors while faith-<br> fully reproducing high-quality potential energy surfaces and accurate free energy profiles.<br> Thirdly, the utilization of GPR with atomic environmental vectors as input features proves<br> effective in enhancing both potential energy surface and free energy description. Further-<br> more, incorporating derivative information on solute atoms further improves the accuracy<br> of force predictions on molecular mechanical (MM) atoms, addressing discrepancies arising<br> from QM/MM interaction energies between the target and base levels of theory. Finally, a<br> comprehensive comparison of three distinct GPR schemes, namely GAP, GPR with an aver-<br> age kernel, and GPR with a system-specific sum kernel, is conducted to evaluate the impact<br> of permutational invariance and atomistic learning on the model’s quality. Additionally, this<br> dissertation introduces the adaptation of the GAP method to be compatible with the sparse<br> variational Gaussian processes scheme and the streaming sparse GPR scheme, enhancing<br> their efficiency and applicability. Through these four ∆-Machine learning schemes, this dis-<br> sertation makes significant contributions to the field of computational chemistry, advancing<br> the quest for accurate potential energy surfaces, reliable force descriptions, and informative<br> free energy profiles in QM/MM simulations.<br> </p>

No / We show that the multidimensional Zeno effect combined with non-holonomic control allows one to efficiently protect quantum systems from decoherence by a method similar to classical random coding. The method is applicable to arbitrary error-inducing Hamiltonians and general quantum systems. The quantum encoding approaches the Hamming upper bound for large dimension increases. Applicability of the method is demonstrated with a seven-qubit toy computer.

The study of magnons is an essential part of combining quantum information science and spintronics, allowing for the investigation of quantum properties such as entanglement in solid-state devices. Magnons are commonly described using the theory of T. Holstein and H. Primakoff, associating the spin operators with bosonic creation and annihilation operators. The quantum mechanical properties inherent to this description are the commutation relations. These relations must be conserved under transformation of the basis. This requires the application of pseudo-unitary transformations U (p, q) when studying the magnon eigenspectrum for example. Depending on the system at hand, the groups U (1, 1) and U (2, 2) are of particular interest and will be the focus of this work. We present a general formalism that leads to a representation of pseudo-unitary matrices via their self-adjoint elements. We apply this representation in examples involving magnons in antiferromagnets to find an explicit picture in connection to material properties. Finally, we explore numerical methods for determining magnon energies and compare them to the analytical counterpart.

Die Pfadintegral-Quantisierung der semi-klassischen Gravitation ist einer der vielversprechendsten Ansätze zur Vereinheitlichung von Quantenmechanik und allgemeiner Relativitätstheorie. In dieser Arbeit untersuchen wir die Konsequenzen der Anwendung dieses Pfadintegralansatzes auf die Kosmologie des sehr frühen Universums. Im ersten Teil konzentrieren wir uns auf den no-boundary proposal, der einen nicht-singulären Anfang des Universums konstruiert, indem er sich auf das gravitative Pfadintegral der allgemeinen Relativitätstheorie stützt. Wir beweisen, dass die no-boundary Lösung das Hinzufügen von Korrekturen höherer Ordnung zur Gravitationswirkung überlebt. Unsere Analyse deutet also darauf hin, dass semi-klassische Ergebnisse auch in der perturbative Störungstheorie der vollständigen Quantengravitation gültig sind. Anschließend beziehen wir ein Skalarfeld in den neuen no-boundary proposal ein, der im Lorentz-Formalismus als Summe über Geometrien mit festem Anfangsimpuls definiert ist. Unsere Ergebnisse sind der Schlüssel zur Bestätigung der Gültigkeit des neuen no-boundary proposals, denn Skalarfelder sind das einfachste Beispiel für Materiefelder, die in einer realistischen Theorie des frühen Universums enthalten sein müssen. Der zweite Teil der Arbeit befasst sich mit der Pfadintegralansatz für allgemeineren Modellen des frühen Universums. Zunächst testen wir die Gültigkeit des semi-klassischen Limits dieser Modelle mit dem Kriterium der endlichen Amplitude, das z.B. Theorien höherer Ordnung der Gravitation stark einschränkt und den no-boundary proposal sowie emergente Universen begünstigt. Schließlich wenden wir das Kriterium der komplexen Metrik von Kontsevich und Segal auf kosmologische Hintergründe an. Im Kontext der Quantenkosmologie angewandt, führt es zu einem neuen Verständnis des gravitativen Pfadintegrals im no-boundary proposal und schließt generische quantum bounces aus. / The path integral quantization of gravity is one of the most promising approaches to unify quantum mechanics and general relativity. This thesis pursues the consequences of the path integral approach applied to the cosmology of the very early universe, for which this unification is crucial. The first part focuses on the no-boundary proposal, which constructs a non-singular beginning of the universe by relying on the gravitational path integral of general relativity. We prove that the no-boundary solution survives the addition of higher-order corrections to the gravity action, usually found in high-energy completions of general relativity such as string theory. This indicates that semi-classical results may still hold at the perturbative level of full quantum gravity. We then include a scalar field in the new no-boundary proposal, defined in the Lorentzian formalism as a sum over geometries with fixed initial momentum flow. Our results are key to confirming the viability of the proposal, but also highlight the non-locality puzzle of the no-boundary proposal in the presence of matter fields, for which we offer new perspectives. The second part of the thesis deals with the path integral treatment of more general early universe models. First we test the validity of the semi-classical limit of these models with a finite amplitude criterion, which severely constrains e.g. higher-order theories of gravity and globally favors the no-boundary proposal and emergent-like universes. At last, we apply Kontsevich and Segal’s complex metric criterion to cosmological backgrounds. This criterion tests the path integral convergence of any quantum field theory on a given metric background. Applied in the context of quantum cosmology, it leads to a new understanding of the path integral in the no-boundary proposal, rules out generic quantum bounces, and stresses the limitation of minisuperspace for classical transitions in de Sitter spacetime.

No description available.

In dieser Arbeit werden die Grundlagen von zwei häufig auftretenden Merkmalen unserer Naturgesetze untersucht: Eichsymmetrien und Quantisierung. Durch die Betrachtung dieser Merkmale im mathematischen Rahmen von Homotopie-Algebren wollen wir neue Methoden zur Berechnung physikalischer Observablen beschreiben, insbesondere in der Kosmologie und der Quantenmechanik. Zunächst befassen wir uns mit dem Problem der Eichredundanzen, die es schwer machen zu erkennen, welche Größen eine physikalische Bedeutung haben. Im Jahr 1980 erreichte Bardeen dieses Ziel in der kosmologischen Störungstheorie zu erster Ordnung. Die Frage, ob dieses Verfahren auf die perturbative Expansion von Eichtheorien aller Ordnungen ausgedehnt werden kann, ist seitdem jedoch offen geblieben. Wir zeigen, dass die Umformulierung von Eichtheorien in eichinvariante Felder als ein Transfer von homotopie-algebraischer Strukturen verstanden werden kann. Unter Verwendung dieses mathematischen Rahmens erweitern wir dann die Gültigkeit der Bardeen-Variablen auf perturbative Eichtheorien zu allen Ordnungen. Nach der Einführung eines systematisches Verfahrens für die eichinvariante Störungstheorie betrachten wir die Berechnung von Observablen in der Doppelfeldtheorie um zeitabhängige Hintergründe. Indem wir die Doppelfeldtheorie um zeitabhängige Hintergründe quadratischer und kubischer Ordnung erweitern und die quadratische Wirkung in den eichinvarianten Variablen ausdrücken, schaffen wir eine Grundlage für zukünftige Berechnungen, insbesondere zur Untersuchung des Einflusses massiver Stringmoden in kosmologischen Hintergründen. Zum Schluss betrachten wir einen anderen Ansatz zur Berechnung von Erwartungswerten in der Quantenmechanik. Obwohl die Pfadintegralformulierung der Quantenmechanik für den Fortschritt der Quantentheorie von entscheidender Bedeutung war, fehlt ihr immer noch eine strenge mathematische Definition. Die Reduktion eines unendlich-dimensionalen Raums von klassisch erlaubten Trajektorien auf einen Erwartungswert, der lediglich eine Funktion der Anfangs- und Endrandbedingungen ist, hat jedoch eine homotopiealgebraische Interpretation. Mit Hilfe des Batalin-Vilkovisky-Formalismus, der eng mit Homotopie-Lie-Algebren verwandt ist, entwickeln wir einen homologischen Ansatz zur Berechnung von Quantenerwartungswerten. Als Beispiel betrachten wir den harmonischen Oszillator und zeigen, dass unsere Methode auch im Kontext der Quantenfeldtheorie in gekrümmter Raumzeit verwendet werden kann, indem wir den Unruh-Effekt berechnen. / This thesis examines the underpinnings of two frequently manifest features of our laws of nature: gauge symmetries and quantization. By studying these features through the mathematical framework of homotopy algebras, we aim to describe new methods towards the computation of physical observables, in particular for cosmology and quantum mechanics. First, we deal with the problem of gauge redundancies, which make it difficult to discern which quantities have physical meaning. In 1980, Bardeen introduced a procedure to achieve this goal in first order cosmological perturbation theory. However, the question whether this procedure can be extended to the perturbative expansion of gauge theories to all orders has remained open since then. We show that, in general, the reformulation of gauge theories in gauge invariant fields has the interpretation of transferring homotopy algebraic structure. Utilising this mathematical framework, we then generalize Bardeen’s procedure to perturbative expansions of gauge theories to all orders in perturbations. After establishing a systematic procedure for gauge invariant perturbation theory, we set up the stage for computing observables in double field theory around time-dependent backgrounds. Double field theory not only has T-duality as a manifest symmetry, which is expected to be important in string cosmology proposals, but is also (in its weakly constrained form) a description of massive string modes, and hence is a suitable arena to investigate the imprint of massive string modes in cosmological backgrounds. By expanding double field theory around time-dependent backgrounds to quadratic and cubic order and expressing the quadratic action in terms of gauge invariant variables, we provide a basis for future computations. Finally, we describe a different approach for computing expectation values in quantum mechanics. Though having been essential for the progress of quantum theory, the path integral formulation of quantum mechanics still lacks a rigorous mathematical definition. However, the act of reducing an infinite-dimensional space of classically allowed trajectories into an expectation value which is merely a function of the initial and final boundary conditions does have a homotopy algebraic interpretation. Through the Batalin-Vilkovisky formalism, which is closely related to homotopy Lie algebras, we build a homological approach for computing quantum expectation values. We demonstrate our method for the harmonic oscillator and we show that our method can also be used in the context of quantum field theory in curved spacetime by rederiving the Unruh effect.

Specific non-relativistic quantum mechanical one-dimensional systems, interacting via point interactions, are discussed within the resolvent algebra setting.

Quantum mechanics stands unmatched in its experimental success. However, significant gaps remain in the quantum description of nature, notably the absence of a satisfactory integration of gravity and an unresolved measurement problem. This thesis investigates a potential approach to resolving these issues known as superdeterminism. A superdeterministic theory is one which violates an assumption called Statistical Independence. Although this approach has historically been readily dismissed as harmful to science, superdeterminism has recently gained traction as researchers address these criticisms and explore its implications more deeply. This thesis aims to provide a unified resource on superdeterminism, compiling progress, identifying current gaps, and suggesting future research directions. We establish criteria for evaluating superdeterministic theories and apply these to existing models. Our analysis focuses on two main approaches: the Donadi-Hossenfelder path integral approach and Palmer's Invariant Set Theory. The path integral approach appears promising, particularly if a suitable measure for the "quantumness" of states can be developed. Invariant Set Theory provides a unique, geometric framework but is still too early in its development to show clear potential. Our findings underscore the early stage of superdeterminism research and the need for further theoretical development and empirical testing. By providing a structured framework for future work, this thesis seeks to advance the understanding and application of superdeterminism in addressing the foundational issues in quantum mechanics.

The problem of how quantum mechanics gives rise to classicality has been debated for more than a century. A commonly proposed solution is decoherence, i.e. the gradual decay of superpositions in open quantum systems due to their inevitable interaction with their environment. However, the ability of decoherence to account for all aspects of the classical world is often questioned. A recently proposed model suggests that decoherence can occur even in isolated composite systems subject to gravitational time dilation, something which has sparked a debate. In this thesis we attempt to identify the precise role of decoherence in the quantum-to-classical transition (QTCT) and then use the result to analyze the validity of the newly proposed time dilation-induced decoherence mechanism. We find that the problem of the QTCT can be divided into two parts and that decoherence solves the first of these whereas the second is unsolvable without fundamental modifications to quantum theory. Moreover, we argue that the effect is fundamentally frame-dependent and we find a general formula for the rate of decoherence of macroscopic superpositions in the case where both the system and observer use Rindler coordinates. The result suggests that the frame-dependence may be utilized to increase the strength of the effect in experimental settings. Finally, the possibilities of experimental verification are discussed and we argue that recent advances in quantum measurement techniques in gravitational-wave observatories may enable tests of gravitational decoherence in the near future, finally providing an empirical glimpse into the resolution of one of the most critical debates in all of physics. / Huruvida kvantfysiken kan ge uppkomst till den klassiska fysiken på stora skalor är ett problem som diskuterats under mer än ett århundrade. En föreslagen lösning är dekoherens, alltså det gradvisa sönderfallet av superpositioner i öppna kvantsystem på grund av den oundvikliga interaktionen med deras omgivning. Dekoherensens förmåga att förklara alla delar av den klassiska världen ifrågasätts emellertid fortfarande. De senaste åren har en ny effekt uppmärksammats som tyder på att dekoherens även kan uppstå i isolerade kompositsystem under påverkan av gravitationell tidsdilatation, något som orsakat en debatt i litteraturen. I detta arbete försöker vi identifiera dekoherensens roll i övergången från det kvantmekaniska till det klassiska, och vi använder sedan resultatet för att analysera den ovannämnda gravitationella dekoherensmekanismen. Det allmänna problemet med övergången från kvantfysik till klassisk fysik delas upp i två delar, och vi visar att dekoherens löser den första delen; den andra delen visar sig vara olösbar utan fundamentala förändringar av kvantfysikens ramverk. Vidare visas den gravitationella dekoherenseffekten vara observatörsberoende och vi härleder en allmän formel för takten med vilken makroskopiska superpositioner sönderfaller i de fall då både systemet och observatören använder Rindlerkoordinater. Resultaten tyder på att observatörsberoendet eventuellt kan utnyttjas för att öka effektens styrka i experimentalla sammanhang. Slutligen diskuteras möjligheter att experimentellt verifiera effekten; vi argumenterar för att nya genombrott inom kvantmätteknik i gravitationsvågsobservatorium kan möjliggöra tester av gravitationell dekoherens inom en snar framtid, vilket skulle ge oss en första empirisk inblick i lösningen till en av fysikens mest kritiska debatter.