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.

This thesis explores the theoretical representation of localised electrons in magnetic systems, using Majorana fermions. A motivation is provided for the Majorana fermion representation, which is then developed and applied as a mean-field theory and in the path-integral formalism to the Ising model in transversal-field (TFIM) in one, two and three dimensions, on an orthonormal lattice. In one dimension the development of domain walls precludes long-range order in discrete systems; this is as free energy savings due to entropy outweigh the energetic cost of a domain wall. An argument due to Peierls exists in 2D which allows the formation of domains of ordered spins amidst a disordered background, however, which may be extended to 3D. The forms of the couplings to the bosons used in the Random Phase Analysis (RPA) are considered and an explanation for the non-existence of the phases calculated in this thesis is discussed, in terms of spare degrees of freedom in the Majorana representation. This thesis contains the first known application of Majorana fermions at the RPA level.

La statistique des valeurs extrêmes est une question majeure dans divers contextes scientifiques. Cependant, bien que la description de la statistique d'un extremum global soit certainement une caractéristique importante, celle-ci ne se concentre que sur une seule variable parmi un grand nombre de variables aléatoires. Une question naturelle qui se pose alors est la suivante: ces valeurs extrêmes sont-elles isolées, loin des autres variables ou bien au contraire existe-t-il un grand nombre d'autres variables proches de ces valeurs extrêmes ? Ces questions ont suscité l'étude de la densité d'état de ces événements quasi-extrêmes. Il existe pour cette quantité peu de résultats pour des variables fortement corrélées, qui est pourtant le cas rencontré dans de nombreux modèles fondamentaux. Deux pistes de modèles physiques de variables fortement corrélées pouvant être étudiés analytiquement se démarquent alors: les positions d’une marche aléatoire et les valeurs propres de matrice aléatoire. Cette thèse est ainsi consacrée à l’étude de statistique d’extrêmes pour ces deux modèles de variables fortement corrélées. Dans une première partie, j’étudie le cas où la collection de variables aléatoires est la position au cours du temps d’un mouvement brownien, qui peut être contraint à être périodique, positif... Ce mouvement brownien est vu comme la limite d’un marcheur aléatoire classique après un grand nombre de pas. Il est alors possible d’interprèter ce problème comme celui d’une particule quantique dans un potentiel ce qui permet d’utiliser des méthodes puissantes issues de la mécanique quantique comme l’utilisation de propagateurs et de l’intégrale de chemin. Ces outils permettent de calculer la densité moyenne à partir du maximum pour les différents mouvements browniens contraints et même la distribution complète de cette quantité pour certains cas. Il est également possible de généraliser cette démarche à l’étude de plusieurs marches aléatoires indépendantes ou avec interaction. Cette démarche permet également d’effectuer une étude temporelle, ainsi que de généraliser à l’étude d’autres fonctionnelle du maximum. Dans la seconde partie, j’étudie le cas où la collection de variables aléatoires est composée des valeurs propres d’une matrice aléatoire. Ce travail se concentre sur l’études des matrices des ensembles gaussiens (GOE, GUE et GSE) ainsi qu’à l’étude des matrices de Wishart. L’étude du voisinage de la valeur propre maximale pour ces deux modèles est faite en utilisant une méthode fondée sur les propriétés des polynômes orthogonaux. Dans le cas des matrices gaussiennes unitaires GUE, j’ai obtenu une formule analytique pour la distribution à partir du maximum ainsi qu’une nouvelle expression de la statistique du gap entre les deux plus grandes valeurs propres en termes d’une fonction transcendante de Painlevé. Ces résultats, et plus particulièrement leurs généralisations aux cas GOE, sont alors appliqués à un modèle de verre de spin sphérique en champs moyen. Dans le cas des matrices de Wishart, l’analyse des polynômes orthogonaux dans le régime de double échelle m’a permis de retrouver les différentes statistiques de la valeur propre minimale et également de prouver une conjecture sur la première correction de taille finie pour des grandes matrices de la distribution de la valeur propre minimale dans la limite dite de «hard edge». / Extreme value statistics plays a keyrole in various scientific contexts. Although the description of the statistics of a global extremum is certainly an important feature, it focuses on the fluctuations of a single variable among many others. A natural question that arises is then the following: is this extreme value lonely at the top or, on the contrary, are there many other variables close to it ? A natural and useful quantity to characterize the crowding is the density of states near extremes. For this quantity, there exist very few exact results for strongly correlated variables, which is however the case encountered in many situations. Two physical models of strongly correlated variables have attracted much attention because they can be studied analytically : the positions of a random walker and the eigenvalues of a random matrix. This thesis is devoted to the study of the statistics near the maximum of these two ensembles of strongly correlated variables. In the first part, I study the case where the collection of random variables is the position of a Brownian motion, which may be constrained to be periodic or positive. This Brownian motion is seen as the limit of a classical random walker after a large number of steps. It is then possible to interpret this problem as a quantum particle in a potential which allows us to use powerful methods from quantum mechanics as propagators and path integral. These tools are used to calculate the average density from the maximum for different constrained Brownian motions and the complete distribution of this observable in certain cases. It is also possible to generalize this approach to the study of several random walks, independent or with interaction, as well as to the study of other functional of the maximum. In the second part, I study the case of the eigenvalues of random matrices, belonging to both Gaussian and Wishart ensembles. The study near the maximal eigenvalues for both models is performed using a method based on semi-classical orthogonal polynomials. In the case of Gaussian unitary matrices, I have obtained an analytical formula for the density near the maximum as well as a new expression for the distribution of the gap between the two largest eigenvalues. These results, and in particular their generalizations to different Gaussian ensembles, are then applied to the relaxational dynamics of a mean-field spin glass model. Finally, for the case of Wishart matrices I proposed a new derivation of the distribution of the smallest eigenvalue using orthogonal polynomials. In addition, I proved a conjecture on the first finite size correction of this distribution in the «hard edge» limit.

Several microscopic theories point out that Bose-Einstein condensation (BEC), i.e., a macroscopic occupation of the lowest energy single particle state in many-boson systems, may appear also in quantum fluids and solids and that it is at the origin of the phenomenon of superfluidity. Nevertheless, the connection between BEC and superfluidity is still matter of debate, since the experimental evidences indicating a non zero condensate fraction in superfluid helium are only indirect. In the theoretical study of BEC in quantum fluids and solids, perturbative approaches are useless because of the strong correlations between the atoms, arising both from the interatomic potential and from the quantum nature of the system. Microscopic Quantum Monte Carlo simulations provide a reliable description of these systems. In particular, the Path Integral Monte Carlo (PIMC) method is very suitable for this purpose. This method is able to provide exact results for the properties of the quantum system, both at zero and finite temperature, only with the definition of the Hamiltonian and of the symmetry properties of the system, giving an easy picture for superfluidity and BEC in many-boson systems. In this thesis, we apply PIMC methods to the study of several quantum fluids and solids. We describe in detail all the features of PIMC, from the sampling methods to the estimators of the physical properties. We present also the most recent techniques, such as the high-order approximations for the thermal density matrix and the worm algorithm, used in PIMC to provide reliable simulations. We study the liquid phase of condensed 4He, providing unbiased estimations of the one-body density matrix g1(r). We analyze the model for g1(r) used to fit the experimental data, highlighting its merits and its faults. In particular we see that, even if it presents some difficulties in the description of the overall behavior of g1(r), it can provide an accurate estimation of the kinetic energy K and of the condensate fraction n0 of the system. Furthermore, we show that our results for n0 as a function of the pressure are in a good agreement with the most recent experimental results. The study of the solid phase of 4He is the most significant part of this thesis. The recent observation of non classical rotational inertia (NCRI) effects in solid helium has generated big interest in the study of an eventual supersolid phase, characterized at the same time by crystalline order and superfluidity. Nevertheless, until now it has been impossible to give a theoretical model able to describe all the experimental evidences. In this work, we perform PIMC simulations of 4He at high densities, according to different microscopic configurations of the atoms. In commensurate crystals we see that BEC does not appear, our model being able to reproduce the momentum distribution obtained form neutron scattering experiments. In a crystal with vacancies, we have been able to see a transition to a superfluid phase at temperatures in agreement with experimental results if the vacancy concentration is low enough. In amorphous solids, superfluid effects are enhanced but appear at temperatures higher than the experimental estimation for the transition temperature. Finally, we study also metastable disordered configurations in molecular para-hydrogen at low temperature. The aim of this study is to investigate if a Bose liquid other than helium can display superfluidity. Choosing accurately a ¿quantum liquid¿ initial configuration and the dimensions of the simulation box, we have been able to frustrate the formation of the crystal and to calculate the temperature dependence of the superfluid density, showing a transition to a superfluid phase at temperatures close to 1 K.

It is well established that stock market volatility has a memory of the past, moreover it is found that volatility correlations are long ranged. As a consequence, volatility cannot be characterized by a single correlation time in general. Recent empirical work suggests that the volatility correlation functions of various assets actually decay as a power law. Moreover it is well established that the distribution functions for the returns do not obey a Gaussian distribution, but follow more the type of distributions that incorporate what are commonly known as fat–tailed distributions. As a result, if one is to model the evolution of the stock price, stock market or any financial derivative, then standard Brownian motion models are inaccurate. One must take into account the results obtained from empirical studies and work with models that include realistic features observed on the market. In this thesis we show that it is possible to derive the path integral for a non-Gaussian option pricing model that can capture fat–tails. However we find that the path integral technique can only be used on a very small set of problems, as a number of situations of interest are shown to be intractable. / http://proxy.library.adelaide.edu.au/login?url= http://library.adelaide.edu.au/cgi-bin/Pwebrecon.cgi?BBID=1378473 / Thesis (Ph.D.) -- University of Adelaide, School of Electrical and Electronic Engineering, 2010

It is well established that stock market volatility has a memory of the past, moreover it is found that volatility correlations are long ranged. As a consequence, volatility cannot be characterized by a single correlation time in general. Recent empirical work suggests that the volatility correlation functions of various assets actually decay as a power law. Moreover it is well established that the distribution functions for the returns do not obey a Gaussian distribution, but follow more the type of distributions that incorporate what are commonly known as fat–tailed distributions. As a result, if one is to model the evolution of the stock price, stock market or any financial derivative, then standard Brownian motion models are inaccurate. One must take into account the results obtained from empirical studies and work with models that include realistic features observed on the market. In this thesis we show that it is possible to derive the path integral for a non-Gaussian option pricing model that can capture fat–tails. However we find that the path integral technique can only be used on a very small set of problems, as a number of situations of interest are shown to be intractable. / http://proxy.library.adelaide.edu.au/login?url= http://library.adelaide.edu.au/cgi-bin/Pwebrecon.cgi?BBID=1378473 / Thesis (Ph.D.) -- University of Adelaide, School of Electrical and Electronic Engineering, 2010

abstract: One dimensional (1D) and quasi-one dimensional quantum wires have been a subject of both theoretical and experimental interest since 1990s and before. Phenomena such as the "0.7 structure" in the conductance leave many open questions. In this dissertation, I study the properties and the internal electron states of semiconductor quantum wires with the path integral Monte Carlo (PIMC) method. PIMC is a tool for simulating many-body quantum systems at finite temperature. Its ability to calculate thermodynamic properties and various correlation functions makes it an ideal tool in bridging experiments with theories. A general study of the features interpreted by the Luttinger liquid theory and observed in experiments is first presented, showing the need for new PIMC calculations in this field. I calculate the DC conductance at finite temperature for both noninteracting and interacting electrons. The quantized conductance is identified in PIMC simulations without making the same approximation in the Luttinger model. The low electron density regime is subject to strong interactions, since the kinetic energy decreases faster than the Coulomb interaction at low density. An electron state called the Wigner crystal has been proposed in this regime for quasi-1D wires. By using PIMC, I observe the zig-zag structure of the Wigner crystal. The quantum fluctuations suppress the long range correla- tions, making the order short-ranged. Spin correlations are calculated and used to evaluate the spin coupling strength in a zig-zag state. I also find that as the density increases, electrons undergo a structural phase transition to a dimer state, in which two electrons of opposite spins are coupled across the two rows of the zig-zag. A phase diagram is sketched for a range of densities and transverse confinements. The quantum point contact (QPC) is a typical realization of quantum wires. I study the QPC by explicitly simulating a system of electrons in and around a Timp potential (Timp, 1992). Localization of a single electron in the middle of the channel is observed at 5 K, as the split gate voltage increases. The DC conductance is calculated, which shows the effect of the Coulomb interaction. At 1 K and low electron density, a state similar to the Wigner crystal is found inside the channel. / Dissertation/Thesis / Ph.D. Physics 2012

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-04-26T17:27:53Z No. of bitstreams: 1 monicacristinamelquiadesduque.pdf: 697556 bytes, checksum: bf32f50f1172cd3d5154fa9a1bbb5219 (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-05-13T12:03:30Z (GMT) No. of bitstreams: 1 monicacristinamelquiadesduque.pdf: 697556 bytes, checksum: bf32f50f1172cd3d5154fa9a1bbb5219 (MD5) / Made available in DSpace on 2017-05-13T12:03:30Z (GMT). No. of bitstreams: 1 monicacristinamelquiadesduque.pdf: 697556 bytes, checksum: bf32f50f1172cd3d5154fa9a1bbb5219 (MD5) Previous issue date: 2013-02-20 / CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Apresenta-se aqui uma discussão sobre três métodos para o cálculo do propagador de Feynman para alguns modelos em mecânica quântica não relativística. O formalismo de Feynman é apenas um dos vários existentes para a abordagem de problemas na mecânica quântica. O primeiro método é um cálculo da integral de caminho, que é baseado em uma relação de recorrência para a produção de propagadores infinitesimais. Essa relação de recorrência não tem aparecido em discussões anteriores da integral de caminho do oscilador harmônico unidimensional, embora seja inspirada por uma relação similar em um sistema tridimensional. O segundo método foi desenvolvido por Schwinger em 1951 para tratar ações efetivas na eletrodinâmica quântica baseado na solução das equações de movimento do operador de Heisenberg. Com o uso adequado do operador ordenado e subordinadas as condições iniciais produz o propagador. Por fim, o terceiro método, que usa-se de técnicas algébricas baseadas na fatoração do operador evolução temporal usando a fórmula Baker-Campbell-Hausdorff. / Here we present a discussion of three methods to calculate the Feynman propagator for some models in non-relativistic quantum mechanics. The formalism of Feynman is just one of several available for addressing problems in quantum mechanics. The first method is a calculation of the integral path, which is based on a recurrence relation for the production of infinitesimal propagators. This recurrence relation has not appeared in previous discussions of the full path of the one-dimensional harmonic oscillator, although inspired by a similar relationship in a three-dimensional system. The second method was developed by Schwinger in 1951 for treating effective action in quantum electrodynamics based on the solution of the equations of motion of the Heisenberg operator. With the proper use of the operator ordained and subordinated the initial conditions produces the propagator. Finally, the third method, which uses the algebraic techniques are based on factorization of the time evolution operator using the formula Baker-Campbell-Hausdorff.

In this thesis we offer a broad introduction into loop quantum gravity against the backdrop of the quantum gravity research as a whole. We focus on both the canonical and covariant version of the theory. In the latter version we investigate the dynamics of some simple configurations in the simplified setting of Ponzano-Regge model. We ascertain that the naïve approach to define a consistent dynamics, where the path integral's partition function is computed as a sum of amplitudes corresponding to all boundary and bulk states, fails in this case, on account of an appearance of divergences. This opens up space for the utilization of some more sophisticated methods.

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.