Relativistic mean field theory: methods and applications

We develop a method for performing one-loop calculations in finite systems that is based on using the WKB approximation for the high energy states. This approximation allows us to absorb all the counterterms analytically and thereby avoids the need for extreme numerical precision that was required by previous methods. In addition, the local approximation makes this method well suited for self-consistent calculations. We then discuss the application of relativistic mean field methods to the atomic nucleus. Self-consistent, one loop calculations in the Walecka model are performed and the role of the vacuum in this model is analyzed. This model predicts that vacuum polarization effects are responsible for up to five percent of the local nucleon density. Within this framework the possible role of strangeness degrees of freedom is studied. We find that strangeness polarization can increase the kaon-nucleus scattering cross section by ten percent. By introducing a cutoff into the model, the dependence of the model on short-distance physics, where its validity is doubtful, is calculated. The model is very sensitive to cutoffs around one GeV.

Branes and supersymmetric quantum field theories

Since the discovery of D-branes as non-perturbative, dynamic objects in string theory, various configurations of branes in type IIA/B string theory and M-theory have been considered to study their low-energy dynamics described by supersymmetric quantum field theories.

One example of such a construction is based on the description of Seiberg-Witten curves of four-dimensional N = 2 supersymmetric gauge theories as branes in type IIA string theory and M-theory. This enables us to study the gauge theories in strongly-coupled regimes. Spectral networks are another tool for utilizing branes to study non-perturbative regimes of two- and four-dimensional supersymmetric theories. Using spectral networks of a Seiberg-Witten theory we can find its BPS spectrum, which is protected from quantum corrections by supersymmetry, and also the BPS spectrum of a related two-dimensional N = (2,2) theory whose (twisted) superpotential is determined by the Seiberg-Witten curve. When we don’t know the perturbative description of such a theory, its spectrum obtained via spectral networks is a useful piece of information. In this thesis we illustrate these ideas with examples of the use of Seiberg-Witten curves and spectral networks to understand various two- and four-dimensional supersymmetric theories.

First, we examine how the geometry of a Seiberg-Witten curve serves as a useful tool for identifying various limits of the parameters of the Seiberg-Witten theory, including Argyres-Seiberg duality and Argyres-Douglas fixed points. Next, we consider the low-energy limit of a two-dimensional N = (2, 2) supersymmetric theory from an M-theory brane configuration whose (twisted) superpotential is determined by the geometry of the branes. We show that, when the two-dimensional theory flows to its infra-red fixed point, particular cases realize Kazama-Suzuki coset models. We also study the BPS spectrum of an Argyres-Douglas type superconformal field theory on the Coulomb branch by using its spectral networks. We provide strong evidence of the equivalence of superconformal field theories from different string-theoretic constructions by comparing their BPS spectra.

Geometric integration applied to moving mesh methods and degenerate Lagrangians

Moving mesh methods (also called r-adaptive methods) are space-adaptive strategies used for the numerical simulation of time-dependent partial differential equations. These methods keep the total number of mesh points fixed during the simulation, but redistribute them over time to follow the areas where a higher mesh point density is required. There are a very limited number of moving mesh methods designed for solving field-theoretic partial differential equations, and the numerical analysis of the resulting schemes is challenging. In this thesis we present two ways to construct r-adaptive variational and multisymplectic integrators for (1+1)-dimensional Lagrangian field theories. The first method uses a variational discretization of the physical equations and the mesh equations are then coupled in a way typical of the existing r-adaptive schemes. The second method treats the mesh points as pseudo-particles and incorporates their dynamics directly into the variational principle. A user-specified adaptation strategy is then enforced through Lagrange multipliers as a constraint on the dynamics of both the physical field and the mesh points. We discuss the advantages and limitations of our methods. The proposed methods are readily applicable to (weakly) non-degenerate field theories---numerical results for the Sine-Gordon equation are presented.

In an attempt to extend our approach to degenerate field theories, in the last part of this thesis we construct higher-order variational integrators for a class of degenerate systems described by Lagrangians that are linear in velocities. We analyze the geometry underlying such systems and develop the appropriate theory for variational integration. Our main observation is that the evolution takes place on the primary constraint and the 'Hamiltonian' equations of motion can be formulated as an index 1 differential-algebraic system. We then proceed to construct variational Runge-Kutta methods and analyze their properties. The general properties of Runge-Kutta methods depend on the 'velocity' part of the Lagrangian. If the 'velocity' part is also linear in the position coordinate, then we show that non-partitioned variational Runge-Kutta methods are equivalent to integration of the corresponding first-order Euler-Lagrange equations, which have the form of a Poisson system with a constant structure matrix, and the classical properties of the Runge-Kutta method are retained. If the 'velocity' part is nonlinear in the position coordinate, we observe a reduction of the order of convergence, which is typical of numerical integration of DAEs. We also apply our methods to several models and present the results of our numerical experiments.

Topics in quantum gravity

We study some aspects of conformal field theory, wormhole physics and two-dimensional random surfaces. Inspite of being rather different, these topics serve as examples of the issues that are involved, both at high and low energy scales, in formulating a quantum theory of gravity. In conformal field theory we show that fusion and braiding properties can be used to determine the operator product coefficients of the non-diagonal Wess-Zumino-Witten models. In wormhole physics we show how Coleman's proposed probability distribution would result in wormholes determining the value of ^θQCD. We attempt such a calculation and find the most probable value of ^θQCD to be π. This hints at a potential conflict with nature. In random surfaces we explore the behaviour of conformal field theories coupled to gravity and calculate some partition functions and correlation functions. Our results throw some light on the transition that is believed to occur when the central charge of the matter theory gets larger than one.

Collective behavior of asperities as a model for friction and adhesion

Understanding friction and adhesion in static and sliding contact of surfaces is important in numerous physical phenomena and technological applications. Most surfaces are rough at the microscale, and thus the real area of contact is only a fraction of the nominal area. The macroscopic frictional and adhesive response is determined by the collective behavior of the population of evolving and interacting microscopic contacts. This collective behavior can be very different from the behavior of individual contacts. It is thus important to understand how the macroscopic response emerges from the microscopic one. In this thesis, we develop a theoretical and computational framework to study the collective behavior. Our philosophy is to assume a simple behavior of a single asperity and study the collective response of an ensemble. Our work bridges the existing well-developed studies of single asperities with phenomenological laws that describe macroscopic rate-and-state behavior of frictional interfaces. We find that many aspects of the macroscopic behavior are robust with respect to the microscopic response. This explains why qualitatively similar frictional features are seen for a diverse range of materials. We first show that the collective response of an ensemble of one-dimensional independent viscoelastic elements interacting through a mean field reproduces many qualitative features of static and sliding friction evolution. The resulting macroscopic behavior is different from the microscopic one: for example, even if each contact is velocity-strengthening, the macroscopic behavior can be velocity-weakening. The framework is then extended to incorporate three-dimensional rough surfaces, long- range elastic interactions between contacts, and time-dependent material behaviors such as viscoelasticity and viscoplasticity. Interestingly, the mean field behavior dominates and the elastic interactions, though important from a quantitative perspective, do not change the qualitative macroscopic response. Finally, we examine the effect of adhesion on the frictional response as well as develop a force threshold model for adhesion and mode I interfacial cracks.

Numerical Studies of Topological Phases

The topological phases of matter have been a major part of condensed matter physics research since the discovery of the quantum Hall effect in the 1980s. Recently, much of this research has focused on the study of systems of free fermions, such as the integer quantum Hall effect, quantum spin Hall effect, and topological insulator. Though these free fermion systems can play host to a variety of interesting phenomena, the physics of interacting topological phases is even richer. Unfortunately, there is a shortage of theoretical tools that can be used to approach interacting problems. In this thesis I will discuss progress in using two different numerical techniques to study topological phases.

Recently much research in topological phases has focused on phases made up of bosons. Unlike fermions, free bosons form a condensate and so interactions are vital if the bosons are to realize a topological phase. Since these phases are difficult to study, much of our understanding comes from exactly solvable models, such as Kitaev's toric code, as well as Levin-Wen and Walker-Wang models. We may want to study systems for which such exactly solvable models are not available. In this thesis I present a series of models which are not solvable exactly, but which can be studied in sign-free Monte Carlo simulations. The models work by binding charges to point topological defects. They can be used to realize bosonic interacting versions of the quantum Hall effect in 2D and topological insulator in 3D. Effective field theories of "integer" (non-fractionalized) versions of these phases were available in the literature, but our models also allow for the construction of fractional phases. We can measure a number of properties of the bulk and surface of these phases.

Few interacting topological phases have been realized experimentally, but there is one very important exception: the fractional quantum Hall effect (FQHE). Though the fractional quantum Hall effect we discovered over 30 years ago, it can still produce novel phenomena. Of much recent interest is the existence of non-Abelian anyons in FQHE systems. Though it is possible to construct wave functions that realize such particles, whether these wavefunctions are the ground state is a difficult quantitative question that must be answered numerically. In this thesis I describe progress using a density-matrix renormalization group algorithm to study a bilayer system thought to host non-Abelian anyons. We find phase diagrams in terms of experimentally relevant parameters, and also find evidence for a non-Abelian phase known as the "interlayer Pfaffian".

Dynamics of non-Abelian Aharonov-Bohm systems

This thesis examines several examples of systems in which non-Abelian magnetic flux and non-Abelian forms of the Aharonov-Bohm effect play a role. We consider the dynamical consequences in these systems of some of the exotic phenomena associated with non-Abelian flux, such as Cheshire charge holonomy interactions and non-Abelian braid statistics. First, we use a mean-field approximation to study a model of U(2) non-Abelian anyons near its free-fermion limit. Some self-consistent states are constructed which show a small SU(2)-breaking charge density that vanishes in the fermionic limit. This is contrasted with the bosonic limit where the SU(2) asymmetry of the ground state can be maximal. Second, a global analogue of Chesire charge is described, raising the possibility of observing Cheshire charge in condensedmatter systems. A potential realization in superfluid He-3 is discussed. Finally, we describe in some detail a method for numerically simulating the evolution of a network of non-Abelian (S₃) cosmic strings, keeping careful track of all magnetic fluxes and taking full account of their non-commutative nature. I present some preliminary results from this simulation, which is still in progress. The early results are suggestive of a qualitatively new, non-scaling behavior.

Efficient methods for empirical tests of behavioral economics theories in laboratory and field experiments

In the quest for a descriptive theory of decision-making, the rational actor model in economics imposes rather unrealistic expectations and abilities on human decision makers. The further we move from idealized scenarios, such as perfectly competitive markets, and ambitiously extend the reach of the theory to describe everyday decision making situations, the less sense these assumptions make. Behavioural economics has instead proposed models based on assumptions that are more psychologically realistic, with the aim of gaining more precision and descriptive power. Increased psychological realism, however, comes at the cost of a greater number of parameters and model complexity. Now there are a plethora of models, based on different assumptions, applicable in differing contextual settings, and selecting the right model to use tends to be an ad-hoc process. In this thesis, we develop optimal experimental design methods and evaluate different behavioral theories against evidence from lab and field experiments.

We look at evidence from controlled laboratory experiments. Subjects are presented with choices between monetary gambles or lotteries. Different decision-making theories evaluate the choices differently and would make distinct predictions about the subjects' choices. Theories whose predictions are inconsistent with the actual choices can be systematically eliminated. Behavioural theories can have multiple parameters requiring complex experimental designs with a very large number of possible choice tests. This imposes computational and economic constraints on using classical experimental design methods. We develop a methodology of adaptive tests: Bayesian Rapid Optimal Adaptive Designs (BROAD) that sequentially chooses the "most informative" test at each stage, and based on the response updates its posterior beliefs over the theories, which informs the next most informative test to run. BROAD utilizes the Equivalent Class Edge Cutting (EC²) criteria to select tests. We prove that the EC² criteria is adaptively submodular, which allows us to prove theoretical guarantees against the Bayes-optimal testing sequence even in the presence of noisy responses. In simulated ground-truth experiments, we find that the EC² criteria recovers the true hypotheses with significantly fewer tests than more widely used criteria such as Information Gain and Generalized Binary Search. We show, theoretically as well as experimentally, that surprisingly these popular criteria can perform poorly in the presence of noise, or subject errors. Furthermore, we use the adaptive submodular property of EC² to implement an accelerated greedy version of BROAD which leads to orders of magnitude speedup over other methods.

We use BROAD to perform two experiments. First, we compare the main classes of theories for decision-making under risk, namely: expected value, prospect theory, constant relative risk aversion (CRRA) and moments models. Subjects are given an initial endowment, and sequentially presented choices between two lotteries, with the possibility of losses. The lotteries are selected using BROAD, and 57 subjects from Caltech and UCLA are incentivized by randomly realizing one of the lotteries chosen. Aggregate posterior probabilities over the theories show limited evidence in favour of CRRA and moments' models. Classifying the subjects into types showed that most subjects are described by prospect theory, followed by expected value. Adaptive experimental design raises the possibility that subjects could engage in strategic manipulation, i.e. subjects could mask their true preferences and choose differently in order to obtain more favourable tests in later rounds thereby increasing their payoffs. We pay close attention to this problem; strategic manipulation is ruled out since it is infeasible in practice, and also since we do not find any signatures of it in our data.

In the second experiment, we compare the main theories of time preference: exponential discounting, hyperbolic discounting, "present bias" models: quasi-hyperbolic (α, β) discounting and fixed cost discounting, and generalized-hyperbolic discounting. 40 subjects from UCLA were given choices between 2 options: a smaller but more immediate payoff versus a larger but later payoff. We found very limited evidence for present bias models and hyperbolic discounting, and most subjects were classified as generalized hyperbolic discounting types, followed by exponential discounting.

In these models the passage of time is linear. We instead consider a psychological model where the perception of time is subjective. We prove that when the biological (subjective) time is positively dependent, it gives rise to hyperbolic discounting and temporal choice inconsistency.

We also test the predictions of behavioral theories in the "wild". We pay attention to prospect theory, which emerged as the dominant theory in our lab experiments of risky choice. Loss aversion and reference dependence predicts that consumers will behave in a uniquely distinct way than the standard rational model predicts. Specifically, loss aversion predicts that when an item is being offered at a discount, the demand for it will be greater than that explained by its price elasticity. Even more importantly, when the item is no longer discounted, demand for its close substitute would increase excessively. We tested this prediction using a discrete choice model with loss-averse utility function on data from a large eCommerce retailer. Not only did we identify loss aversion, but we also found that the effect decreased with consumers' experience. We outline the policy implications that consumer loss aversion entails, and strategies for competitive pricing.

In future work, BROAD can be widely applicable for testing different behavioural models, e.g. in social preference and game theory, and in different contextual settings. Additional measurements beyond choice data, including biological measurements such as skin conductance, can be used to more rapidly eliminate hypothesis and speed up model comparison. Discrete choice models also provide a framework for testing behavioural models with field data, and encourage combined lab-field experiments.

Three-dimensional superconformal field theory, Chern-Simons theory, and their correspondence

In this thesis, we discuss 3d-3d correspondence between Chern-Simons theory and three-dimensional N = 2 superconformal field theory. In the 3d-3d correspondence proposed by Dimofte-Gaiotto-Gukov information of abelian flat connection in Chern-Simons theory was not captured. However, considering M-theory configuration giving the 3d-3d correspondence and also other several developments, the abelian flat connection should be taken into account in 3d-3d correspondence. With help of the homological knot invariants, we construct 3d N = 2 theories on knot complement in 3-sphere for several simple knots. Previous theories obtained by Dimofte-Gaiotto-Gukov can be obtained by Higgsing of the full theories. We also discuss the importance of all flat connections in the 3d-3d correspondence by considering boundary conditions in 3d N = 2 theories and 3-manifold.

Magnetic field and pressure effects in a saturated gas laser amplifier

Theoretical and experimental studies of a gas laser amplifier are presented, assuming the amplifier is operating with a saturating optical frequency signal. The analysis is primarily concerned with the effects of the gas pressure and the presence of an axial magnetic field on the characteristics of the amplifying medium. Semiclassical radiation theory is used, along with a density matrix description of the atomic medium which relates the motion of single atoms to the macroscopic observables. A two-level description of the atom, using phenomenological source rates and decay rates, forms the basis of our analysis of the gas laser medium. Pressure effects are taken into account to a large extent through suitable choices of decay rate parameters.

Two methods for calculating the induced polarization of the atomic medium are used. The first method utilizes a perturbation expansion which is valid for signal intensities which barely reach saturation strength, and it is quite general in applicability. The second method is valid for arbitrarily strong signals, but it yields tractable solutions only for zero magnetic field or for axial magnetic fields large enough such that the Zeeman splitting is much larger than the power broadened homogeneous linewidth of the laser transition. The effects of pressure broadening of the homogeneous spectral linewidth are included in both the weak-signal and strong-signal theories; however the effects of Zeeman sublevel-mixing collisions are taken into account only in the weak-signal theory.

The behavior of a He-Ne gas laser amplifier in the presence of an axial magnetic field has been studied experimentally by measuring gain and Faraday rotation of linearly polarized resonant laser signals for various values of input signal intensity, and by measuring nonlinearity - induced anisotropy for elliptically polarized resonant laser signals of various input intensities. Two high-gain transitions in the 3.39-μ region were used for study: a J = 1 to J = 2 (3s₂ → 3p₄) transition and a J = 1 to J = 1 (3s₂ → 3p₂) transition. The input signals were tuned to the centers of their respective resonant gain lines.

The experimental results agree quite well with corresponding theoretical expressions which have been developed to include the nonlinear effects of saturation strength signals. The experimental results clearly show saturation of Faraday rotation, and for the J = 1 t o J = 1 transition a Faraday rotation reversal and a traveling wave gain dip are seen for small values of axial magnetic field. The nonlinearity induced anisotropy shows a marked dependence on the gas pressure in the amplifier tube for the J = 1 to J = 2 transition; this dependence agrees with the predictions of the general perturbational or weak signal theory when allowances are made for the effects of Zeeman sublevel-mixing collisions. The results provide a method for measuring the upper (neon 3s₂) level quadrupole moment decay rate, the dipole moment decay rates for the 3s₂ → 3p₄ and 3s₂ → 3p₂ transitions, and the effects of various types of collision processes on these decay rates.

A theory of strong and weak scintillations with applications to astrophysics

The propagation of waves in an extended, irregular medium is studied under the "quasi-optics" and the "Markov random process" approximations. Under these assumptions, a Fokker-Planck equation satisfied by the characteristic functional of the random wave field is derived. A complete set of the moment equations with different transverse coordinates and different wavenumbers is then obtained from the characteristic functional. The derivation does not require Gaussian statistics of the random medium and the result can be applied to the time-dependent problem. We then solve the moment equations for the phase correlation function, angular broadening, temporal pulse smearing, intensity correlation function, and the probability distribution of the random waves. The necessary and sufficient conditions for strong scintillation are also given.

We also consider the problem of diffraction of waves by a random, phase-changing screen. The intensity correlation function is solved in the whole Fresnel diffraction region and the temporal pulse broadening function is derived rigorously from the wave equation.

The method of smooth perturbations is applied to interplanetary scintillations. We formulate and calculate the effects of the solar-wind velocity fluctuations on the observed intensity power spectrum and on the ratio of the observed "pattern" velocity and the true velocity of the solar wind in the three-dimensional spherical model. The r.m.s. solar-wind velocity fluctuations are found to be ~200 km/sec in the region about 20 solar radii from the Sun.

We then interpret the observed interstellar scintillation data using the theories derived under the Markov approximation, which are also valid for the strong scintillation. We find that the Kolmogorov power-law spectrum with an outer scale of 10 to 100 pc fits the scintillation data and that the ambient averaged electron density in the interstellar medium is about 0.025 cm^-3. It is also found that there exists a region of strong electron density fluctuation with thickness ~10 pc and mean electron density ~7 cm^-3 between the PSR 0833-45 pulsar and the earth.