Feynman path integrals introduced heuristically in the 1940s are a powerful tool used in many areas of physics, but also an intriguing mathematical challenge. In this work we used techniques of infinite dimensional integration (i.e. the infinite dimensional oscillatory integrals) in two different, but strictly connected, directions. On the one hand we construct a functional integral representation for solutions of a general high-order heat-type equations exploiting a recent generalization of infinite dimensional Fresnel integrals; in this framework we prove a a Girsanov-type formula, which is related, in the case of Schrödinger equation, to the Feynman path integral representation for the solution in presence of a magnetic field; eventually a new phase space path integral solution for higher-order heat-type equations is also presented. On the other hand for the three dimensional Schrödinger equation with magnetic field we provide a rigorous mathematical Feynman path integral formula still in the context of infinite dimensional oscillatory integrals; moreover, the requirement of independence of the integral on the approximation procedure forces the introduction of a counterterm, which has to be added to the classical action functional (this is done by the example of a linear vector potential). Thanks to that, it is possible to give a natural explanation for the appearance of the Stratonovich integral in the path integral formula for both the Schrödinger and the heat equation with magnetic field.

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.

When bulk helium-4 is cooled below T = 2.18 K, it undergoes a phase transition to a superfluid, characterized by a complex wave function with a macroscopic phase and exhibits inviscid, quantized flow. The macroscopic phase coherence can be probed in a container filled with helium-4, by reducing one or more of its dimensions until they are smaller than the coherence length, the spatial distance over which order propagates. As this dimensional reduction occurs, enhanced thermal and quantum fluctuations push the transition to the superfluid state to lower temperatures. However, this trend can be countered via the proximity effect, where a bulk 3-dimensional (3d) superfluid is coupled to a low (2d) dimensional superfluid via a weak link producing superfluid correlations in the film at temperatures above the Kosterlitz-Thouless temperature. Recent experiments probing the coupling between 3d and 2d superfluid helium-4 have uncovered an anomalously large proximity effect, leading to an enhanced superfluid density that cannot be explained using the correlation length alone. In this work, we have determined the origin of this enhanced proximity effect via large scale quantum Monte Carlo simulations of helium-4 in a topologically non-trivial geometry that incorporates the important aspects of the experiments. We find that due to the bosonic symmetry of helium-4, identical particle permutations lead to correlations between contiguous spatial regions at a length scale greater than the coherence length. We show that quantum exchange plays a large role in explaining the anomalous experimental results while simultaneously showing how classical arguments fall short of this task.

An analytical solution for the coupled telegrapher’s equations in terms of the voltage and current on a homogeneous lossy transmission line and multiconductor transmission line is presented. The resulting telegrapher’s equation solution is obtained in the form of an exact time domain propagator operating on the line voltage and current. It is shown that the analytical equations lead to a stable numerical method that can be used in the analysis of both homogeneous and inhomogeneous transmission lines. A numerical dispersion relation is derived proving that this method has no numerical dispersion down to the two points per wavelength Nyquist limit. Examples are presented showing that exceptionally accurate results are obtained for lossy single and multiconductor transmission lines. The method is extended to represent the general solution to Maxwell’s differential equations in vector matrix form. It is shown that, given the electromagnetic field and boundary conditions at a given instant in time, the free space time domain propagator and corresponding dyadic Green’s functions in 1-, 2-, and 3-dimensions can be used to calculate the field at all subsequent times.

In the first part of this dissertation, recent efforts to understand quantum mechanical effects in biological systems are discussed. Especially, long-lived quantum coherences observed during the electronic energy transfer process in the Fenna-Matthews-Olson complex at physiological condition are studied extensively using theories of open quantum systems. In addition to the usual master equation based approaches, the effect of the protein structure is investigated in atomistic detail through the combined application of quantum chemistry and molecular dynamics simulations. To evaluate the thermalized reduced density matrix, a path-integral Monte Carlo method with a novel importance sampling approach is developed for excitons coupled to an arbitrary phonon bath at a finite temperature. In the second part of the thesis, simulations of molecular systems and applications to vibrational spectra are discussed. First, the quantum dynamics of a molecule is simulated by combining semiclassical initial value representation and density funcitonal theory with analytic derivatives. A computationally-tractable approximation to the sum-of-states formalism of Raman spectra is subsequently discussed.

Electronic coarse graining is a technique improving the predictive power of molecular dynamics simulations by representing electrons via a quantum harmonic oscillator. This construction, known as a Quantum Drude Oscillator, provides all molecular long-range responses by uniting many-body dispersion, polarisation and cross interactions to all orders. To demonstrate the predictive power of electronic coarse graining and provide insights into the physics of water, a molecular model of water based on Quantum Drude Oscillators is developed. The model is parametrised to the properties of an isolated molecule and a single cut through the dimer energy surface. Such a parametrisation makes the condensed phase properties of the model a prediction rather than a fitting target. These properties are studied in four environments via two-temperature adiabatic path integral molecular dynamics: a proton ordered ice, the liquid{vapour interface, supercritical and supercooled water. In all these environments, the model predicts a condensed phase in excellent agreement with experiment, showing impressive transferability. It predicts correct densities and pressures in liquid water from 220 K to 647 K, and a correct temperature of maximum density. Furthermore, it predicts the surface tension, the liquid-vapour critical point, density of ice II, and radial distribution functions across all conditions studied. The model also provides insight into the relationship between the molecular structure of water and its condensed phase properties. An asymmetry between donor and acceptor hydrogen bonds is identified as the molecular scale mechanism responsible for the surface orientation of water molecules. The dipole moment is identified as a molecular scale signature of liquid-like and gas-like regions in supercritical water. Finally, a link between the coordination number and the anomalous thermal expansion of the second coordination shell is also presented.

Orientador: Alex Antonelli / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin / Made available in DSpace on 2018-08-20T22:54:46Z (GMT). No. of bitstreams: 1 Brito_BraulioGabrielAlencar_D.pdf: 3210920 bytes, checksum: 1140e372f96bf86f5f06d96119eeb8e7 (MD5) Previous issue date: 2012 / Resumo: Neste trabalho apresentaremos a extensão dos métodos adiabatic switching (AS), reversible scaling (RS) e integração dinâmica de Clausius-Clapeyron (d-CCI) para o formalismo de integral de tragetória. Desenvolvemos programas de Monte Carlo de integrais de trajetória (PIMC) para implementar esses métodos a fim de incluir efeitos quânticos nos cálculos das energias livres e na determinação das curvas de coexistência de fase de sistemas a baixa temperatura. Aplicamos as aproximações primitivas e Li-broughton para a ação para escrever as matrizes densidade de alta temperatura dos sistemas estudados. Calculamos a curva de fusão do neônio utilizando o método de integração dinâmica de Clausius-Clapeyron quantico (q-dCCI) e comparamos nossos resultados com resultados encontrados na literatura. Determinamos a curva de coexistência diamante-grafite utilizando o potencial AIREBO e os métodos AS, RS e q-dCCI. Estudamos os efeitos da pressão sobre algumas propriedades termodinâmicas do grafite e do grafeno e a diversas temperaturas aplicando método PIMC juntamente dos métodos AS e RS / Abstract: In this work we present the extension of the methods adiabatic switching (AS), reversible scaling (RS), dynamical Clausius-Clapeyron integration (d-CCI) within the path integral formalism. We developed Path Integral Monte Carlo computer codes to implement these methods in order to include quantum effects in the calculation of free energies and in the determination of the phase coexistence curves of systems at low temperature. We applied the primitive and Li-Broughton approximations to the action to write the high temperature density matrices of the systems we studied. We calculated the melting curve of the neon using the quantum dynamical Clausius-Clapeyron (q-dCCI) and compare our results with results found at the literature. We determined the diamond-graphite coexistence curve using the AIREBO inter-atomic potential and the AS, RS e q-dCCI methods. We studied the pressure effects on some thermodynamic properties of the graphite and graphene at several temperatures using the method PIMC together with the AS and RS methods / Doutorado / Física / Doutor em Ciências

The concept of a propagator is useful and is a well-known object in diffusion NMR experiments. Here, we investigate the related concept; the propagator for the magnetization or the Green’s function of the Torrey-Bloch equations. The magnetization propagator is constructed by defining functions such as the Hamiltonian and Lagrangian and using these to define a path integral. It is shown that the equations of motion derived from the Lagrangian produce complex-valued trajectories (classical paths) and it is conjectured that the end-points of these trajectories are real-valued. The complex nature of the trajectories also suggests that the spin degrees of freedom are also encoded into the trajectories and this idea is explored by explicitly modeling the spin or precessing magnetization by anticommuting Grassmann variables. A pseudoclassical Lagrangian is constructed by combining the diffusive (bosonic) Lagrangian with the Grassmann (fermionic) Lagrangian, and performing the path integral over the Grassmann variables recovers the original Lagrangian that was used in the construction of the propagator for the magnetization. The trajectories of the pseudoclassical model also provide some insight into the nature of the end-points.

In this thesis we examine methodologies for determining free energy differences (FEDs) of phases via Monte Carlo simulation. We identify and address three generic issues that arise in FED calculations; the choice of representation, the choice of estimator, and the choice of sampling strategy. In addition we discuss how the classical framework may be extended to take into account quantum effects. Key words: Phase Mapping, Phase Switch, Lattice Switch, Simulated Tempering, Multi-stage, Weighted Histogram Analysis Method, Fast Growth, Jarzynski method, Umbrella, Multicanonical, Path Integral Monte Carlo, Path Sampling, Multihamiltonian, fluctuation theorem.

Dans cette thèse, nous présentons quelques analyses théoriques récentes ainsi que des observations expérimentales de l’effet tunnel quantique macroscopique et des tran- sitions de phase classique-quantique dans le taux d’échappement des systèmes de spins élevés. Nous considérons les systèmes de spin biaxial et ferromagnétiques. Grâce à l’approche de l’intégral de chemin utilisant les états cohérents de spin exprimés dans le système de coordonnées, nous calculons l’interférence des phases quantiques et leur distribution énergétique. Nous présentons une exposition claire de l’effet tunnel dans les systèmes antiferromagnétiques en présence d’un couplage d’échange dimère et d’une anisotropie le long de l’axe de magnétisation aisé. Nous obtenons l’énergie et la fonc- tion d’onde de l’état fondamentale ainsi que le premier état excité pour les systèmes de spins entiers et demi-entiers impairs. Nos résultats sont confirmés par un calcul utilisant la théorie des perturbations à grand ordre et avec la méthode de l’intégral de chemin qui est indépendant du système de coordonnées. Nous présentons aussi une explica- tion claire de la méthode du potentiel effectif, qui nous laisse faire une application d’un système de spin quantique vers un problème de mécanique quantique d’une particule. Nous utilisons cette méthode pour analyser nos modèles, mais avec la contrainte d’un champ magnétique externe ajouté. La méthode nous permet de considérer les transitions classiques-quantique dans le taux d’échappement dans ces systèmes. Nous obtenons le diagramme de phases ainsi que les températures critiques du passage entre les deux régimes. Nous étendons notre analyse à une chaine de spins d’Heisenberg antiferro- magnétique avec une anisotropie le long d’un axe pour N sites, prenant des conditions frontière périodiques. Pour N paire, nous montrons que l’état fondamental est non- dégénéré et donné par la superposition des deux états de Néel. Pour N impair, l’état de Néel contient un soliton, et, car la position du soliton est indéterminée, l’état fondamen- tal est N fois dégénéré. Dans la limite perturbative pour l’interaction d’Heisenberg, les fluctuations quantiques lèvent la dégénérescence et les N états se réorganisent dans une bande. Nous montrons qu’à l’ordre 2s, où s est la valeur de chaque spin dans la théorie des perturbations dégénérées, la bande est formée. L’état fondamental est dégénéré pour s entier, mais deux fois dégénéré pour s un demi-entier impair, comme prévu par le théorème de Kramer / This thesis presents recent theoretical analyses together with experimental observa- tions on macroscopic quantum tunneling and quantum-classical phase transitions of the escape rate in large spin systems. We consider biaxial ferromagnetic spin systems. Using the coordinate dependent spin coherent state path integral, we obtain the quantum phase interference and the energy splitting of these systems. We also present a lucid exposition of tunneling in antiferromagnetic exchange-coupled dimer, with easy-axis anisotropy. Indeed, we obtain the ground state, the first excited state, and the energy splitting, for both integer and half-odd integer spins. These results are then corroborated using per- turbation theory and the coordinate independent spin coherent state path integral. We further present a lucid explication of the effective potential method, which enables one to map a spin Hamiltonian onto a particle Hamiltonian; we employ this method to our models, however, in the presence of an applied magnetic field. This method enables us to investigate quantum-classical phase transitions of the escape rate of these systems. We obtain the phase boundaries, as well as the crossover temperatures of these phase transi- tions. Furthermore, we extend our analysis to one-dimensional anisotropic Heisenberg antiferromagnet, with N periodic sites. For even N, we show that the ground state is non-degenerate and given by the coherent superposition of the two Neél states. For odd N, however, the Neél state contains a soliton; as the soliton can be placed anywhere along the ring, the ground state is, indeed, N-fold degenerate. In the perturbative limit (weak exchange interaction), quantum fluctuation stemming from the interaction term lifts this degeneracy and reorganizes the states into a band. We show that this occurs at order 2s in (degenerate) perturbation theory. The ground state is non-degenerate for inte- ger spin, but degenerate for half-odd integer spin, in accordance with Kramers’ theorem