Loop Quantum Gravity

This article presents an "in-a-nutshell" yet self-contained introductory review on loop quantum gravity (LQG) -- a background-independent, nonperturbative approach to a consistent quantum theory of gravity. Instead of rigorous and systematic derivations, it aims to provide a general picture of LQG, placing emphasis on the fundamental ideas and their significance. The canonical formulation of LQG, as the central topic of the article, is presented in a logically orderly fashion with moderate details, while the spin foam theory, black hole thermodynamics, and loop quantum cosmology are covered briefly. Current directions and open issues are also summarized.Comment: Invited review article. A large part based on arXiv:gr-qc/0404018 by A. Ashtekar and J. Lewandowski and "Quantum Gravity" by C. Rovelli. To appear in Int. J. Mod. Phys. D and in "One Hundred Years of General Relativity: Cosmology and Gravity," edited by Wei-Tou Ni (World Scientific, Singapore, 2015

Effective Dynamics for the Cosmological Bounces in Bianchi Type I Loop Quantum Cosmology

The detailed formulation for loop quantum cosmology (LQC) in the Bianchi I model with a massless scalar field was recently constructed. In this paper, its effective dynamics with the LQC discreteness corrections is studied and the equations of motion are analytically solved, showing that the big bang singularity is replaced by the big bounces, which take place up to three times, once in each diagonal direction, whenever each of the area scale factors approaches its critical value in the Planck regime measured by the reference of the scalar field momentum.Comment: 7 pages, 4 figures; typos corrected, one reference adde

Mathematical Modeling of Competition in Sponsored Search Market

Sponsored search mechanisms have drawn much attention from both academic community and industry in recent years since the seminal papers of [13] and [14]. However, most of the existing literature concentrates on the mechanism design and analysis within the scope of only one search engine in the market. In this paper we propose a mathematical framework for modeling the interaction of publishers, advertisers and end users in a competitive market. We first consider the monopoly market model and provide optimal solutions for both ex ante and ex post cases, which represents the long-term and short-term revenues of search engines respectively. We then analyze the strategic behaviors of end users and advertisers under duopoly and prove the existence of equilibrium for both search engines to co-exist from ex-post perspective. To show the more general ex ante results, we carry out extensive simulations under different parameter settings. Our analysis and observation in this work can provide useful insight in regulating the sponsored search market and protecting the interests of advertisers and end users.Comment: A short version would appear at 2010 Workshop on the Economics of Networks, Systems, and Computation (NetEcon '10

Loop quantum cosmology with higher order holonomy corrections

With a well-motivated extension of higher order holonomy corrections, the quantum theory of loop quantum cosmology (LQC) for the $k=0$ Friedmann-Robertson-Walker model (with a free massless scalar) is rigorously formulated. The analytical investigation reveals that, regardless of the order of holonomy corrections and for any arbitrary states, the matter density remains finite, bounded from above by an upper bound, which equals the critical density obtained at the level of heuristic effective dynamics. Particularly, with all orders of corrections included, the dynamical evolution is shown to follow the bouncing scenario in which two Wheeler-DeWitt (WDW) solutions (expanding and contracting) are bridged together through the quantum bounce. These observations provide further evidence that the quantum bounce is essentially a consequence of the intrinsic discreteness of LQC and LQC is fundamentally different from the WDW theory. Meanwhile, the possibility is also explored that the higher order holonomy corrections can be interpreted as a result of admitting generic SU(2) representations for the Hamiltonian constraint operators.Comment: 32 pages, 2 figures; typos corrected; version to appear in PR

An elementary proof and detailed investigation of the bulk-boundary correspondence in the generic two-band model of Chern insulators

With the inclusion of arbitrary long-range hopping and (pseudo)spin-orbit coupling amplitudes, we formulate a generic model that can describe any two-dimensional two-band bulk insulators, thus providing a simple framework to investigate arbitrary adiabatic deformations upon the systems of any arbitrary Chern numbers. Without appealing to advanced techniques beyond the standard methods of solving linear difference equations and applying Cauchy's integral formula, we obtain a mathematically elementary yet rigorous proof of the bulk-boundary correspondence on a strip, which is robust against any adiabatic deformations upon the bulk Hamiltonian and any uniform edge perturbation along the edges. The elementary approach not only is more transparent about the underlying physics but also reveals various intriguing nontopological features of Chern insulators that have remained unnoticed or unclear so far. Particularly, if a certain condition is satisfied (as in most renowned models), the loci of edge bands in the energy spectrum and their (pseudo)spin polarizations can be largely inferred from the bulk Hamiltonian alone without invoking any numerical computation for the energy spectrum of a strip.Comment: 55 pages, 18 figures; follow-up to arXiv:1705.06913, some material thereof repeated; v3: various minor improvements made, version to appear in IJMP

An elementary rigorous proof of bulk-boundary correspondence in the generalized Su-Schrieffer-Heeger model

We generalize the Su-Schrieffer-Heeger (SSH) model with the inclusion of arbitrary long-range hopping amplitudes, providing a simple framework to investigate arbitrary adiabatic deformations that preserve the chiral symmetry upon the bulk energy bands with any arbitrary winding numbers. Using only elementary techniques of solving linear difference equations and applying Cauchy's integral formula, we obtain a mathematically rigorous and physically transparent proof of the bulk-boundary correspondence for the generalized SSH model. The multiplicity of robust zero-energy edge modes is shown to be identical to the winding number. On the other hand, nonzero-energy edge modes, if any, are shown to be unstable under adiabatic deformations and not related to the topological invariant. Furthermore, under deformations of small spatial disorder, the zero-energy edge modes remain robust.Comment: 15 pages, 3 figures; v5: various improvements made, version to appear in Physics Letters

Exact Foldy-Wouthuysen transformation of the Dirac-Pauli Hamiltonian in the weak-field limit by the method of direct perturbation theory

We apply the method of direct perturbation theory for the Foldy-Wouthuysen (FW) transformation upon the Dirac-Pauli Hamiltonian subject to external electromagnetic fields. The exact FW transformations exist and agree with those obtained by Eriksen's method for two special cases. In the weak-field limit of static and homogeneous electromagnetic fields, by mathematical induction on the orders of $1/c$ in the power series, we rigorously prove the long-held speculation: the FW transformed Dirac-Pauli Hamiltonian is in full agreement with the classical counterpart, which is the sum of the orbital Hamiltonian for the Lorentz force equation and the spin Hamiltonian for the Thomas-Bargmann-Michel-Telegdi equation.Comment: 37 pages; follow-up to arXiv:1311.3432; contains some of the same review materials in arXiv:1310.8513. v4: revised with various improvements; version appears in PR

Correspondence between classical and Dirac-Pauli spinors in view of the Foldy-Wouthuysen transformation

The classical dynamics for a charged spin particle is governed by the Lorentz force equation for orbital motion and by the Thomas-Bargmann-Michel-Telegdi (T-BMT) equation for spin precession. In static and homogeneous electromagnetic fields, it has been shown that the Foldy-Wouthuysen (FW) transform of the Dirac-Pauli Hamiltonian, which describes the relativistic quantum theory for a spin-1/2 particle, is consistent with the classical Hamiltonian (with both the orbital and spin parts) up to the order of $1/m^{14}$ ($m$ is the particle's mass) in the low-energy/weak-field limit. In this paper, we extend this correspondence to the case of inhomogeneous fields. Regardless of the field gradient (e.g., Stern-Gerlach) force, the T-BMT equation is unaltered and thus the classical Hamiltonian remains the same, but subtleties arise and need to be clarified. For the relativistic quantum theory, we apply Eriksen's method to obtain the exact FW transformations for the two special cases, which in conjunction strongly suggest that, in the weak-field limit, the FW transformed Dirac-Pauli Hamiltonian (except for the Darwin term) is in agreement with the classical Hamiltonian in a manner that classical variables correspond to quantum operators via a specific Weyl ordering. Meanwhile, the Darwin term is shown to have no classical correspondence.Comment: 14 pages, 1 table; one subsection added; version to appear in PR

Exact correspondence between classical and Dirac-Pauli spinors in the weak-field limit of static and homogeneous electromagnetic fields

It has long been speculated that the Dirac or, more generally, the Dirac-Pauli spinor in the Foldy-Wouthuysen (FW) representation should behave like a classical relativistic spinor in the low-energy limit when the probability of particle-antiparticle pair creation and annihilation is negligible. In the weak-field limit of static and homogeneous electromagnetic fields, by applying the method of direct perturbation theory inductively on the orders of $1/c$ in the power series, we rigorously prove that it is indeed the case: the FW transformation of the Dirac-Pauli Hamiltonian is in full agreement with the classical counterpart, which is the sum of the orbital Hamiltonian for the Lorentz force equation and the spin Hamiltonian for the Thomas-Bargmann-Michel-Telegdi equation.Comment: 6 pages; brief summary of the main result of arXiv:1405.4495; v3: minor improvements made; the main reference update

Exploring Network Economics

In this paper, we explore what \emph{network economics} is all about, focusing on the interesting topics brought about by the Internet. Our intent is make this a brief survey, useful as an outline for a course on this topic, with an extended list of references. We try to make it as intuitive and readable as possible. We also deliberately try to be critical at times, and hope our interpretation of the topic will lead to interests for further discussions by those doing research in the same field.Comment: It is a position paper, about what we might teach in a Network Economics course, and the type of research we found useful. Therefore, it is not an extensive survey pape