2,176 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
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
Loop quantization of spherically symmetric midisuperspaces and loop quantum geometry of the maximally extended Schwarzschild spacetime
We elaborate on the Ashtekar's formalism for spherically symmetric
midisuperspaces and, for loop quantization, propound a new quantization scheme
which yields a graph-preserving Hamiltonian constraint operator and by which
one can impose the fundamental discreteness of loop quantum gravity a la the
strategy of "improved" dynamics in loop quantum cosmology (LQC). Remarkable
consequences are inferred at the heuristic level of effective dynamics: first,
consistency of the constraint algebra regarding the Hamiltonian and
diffeomorphism constraints fixes the improved quantization scheme to be of the
form reminiscent of the improved scheme in LQC which preserves scaling
invariance; second, consistency regarding two Hamiltonian constraints further
demands the inclusion of higher order holonomy corrections and fixes a ratio
factor of 2 for the improved scheme. It is suggested that the classical
singularity is resolved and replaced by a quantum bounce which bridges a
classical solution to another classical phase. However, the constraints violate
briefly during the bouncing period, indicating that one cannot make sense of
symmetry reduction by separating the degrees of freedom of the full theory into
spherical and non-spherical ones in the vicinity of the quantum bounce,
although the heuristic effective dynamics can still give a reliable
semiclassical description of large-scale physics. Particularly, for the
Schwarzschild solution in accordance with the Kruskal coordinates, revealing
insights lead us to conjecture the complete quantum extension of the
Schwarzschild spacetime: the black hole is evaporated via the Hawking radiation
and meanwhile the quantum spacetime is largely extended from the classical one
via the quantum bounce, suggesting that the information paradox might be
resolved.Comment: 46 pages, 3 figures, 2 tables; revised with one more reference adde
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
Larmor precession of the octet baryons in view of the general QCD parametrization
We apply general QCD parametrization to describe an octet baryon at rest in a
static and uniform magnetic field, in order to demonstrate a dynamical
implementation of the hadron model. The derived evolution has the asymptote of
Larmor precession, governed by an effective magnetic dipole moment coupling.
The parameters appear in the effective magnetic dipole moment has been studied
in the original works in kinematic settings. Here we show that the parameters
have actual dynamical meaning, and are indeed measured by the process.Comment: 16 page
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
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