1,816 research outputs found
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
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 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 ( 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 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
- …