68,928 research outputs found
Conceptual Unification of Gravity and Quanta
We present a model unifying general relativity and quantum mechanics. The
model is based on the (noncommutative) algebra \mbox{{\cal A}} on the groupoid
\Gamma = E \times G where E is the total space of the frame bundle over
spacetime, and G the Lorentz group. The differential geometry, based on
derivations of \mbox{{\cal A}}, is constructed. The eigenvalue equation for the
Einstein operator plays the role of the generalized Einstein's equation. The
algebra \mbox{{\cal A}}, when suitably represented in a bundle of Hilbert
spaces, is a von Neumann algebra \mathcal{M} of random operators representing
the quantum sector of the model. The Tomita-Takesaki theorem allows us to
define the dynamics of random operators which depends on the state \phi . The
same state defines the noncommutative probability measure (in the sense of
Voiculescu's free probability theory). Moreover, the state \phi satisfies the
Kubo-Martin-Schwinger (KMS) condition, and can be interpreted as describing a
generalized equilibrium state. By suitably averaging elements of the algebra
\mbox{{\cal A}}, one recovers the standard geometry of spacetime. We show that
any act of measurement, performed at a given spacetime point, makes the model
to collapse to the standard quantum mechanics (on the group G). As an example
we compute the noncommutative version of the closed Friedman world model.
Generalized eigenvalues of the Einstein operator produce the correct components
of the energy-momentum tensor. Dynamics of random operators does not ``feel''
singularities.Comment: 28 LaTex pages. Substantially enlarged version. Improved definition
of generalized Einstein's field equation
Quantization of systems with temporally varying discretization I: Evolving Hilbert spaces
A temporally varying discretization often features in discrete gravitational
systems and appears in lattice field theory models subject to a coarse graining
or refining dynamics. To better understand such discretization changing
dynamics in the quantum theory, an according formalism for constrained
variational discrete systems is constructed. While the present manuscript
focuses on global evolution moves and, for simplicity, restricts to Euclidean
configuration spaces, a companion article discusses local evolution moves. In
order to link the covariant and canonical picture, the dynamics of the quantum
states is generated by propagators which satisfy the canonical constraints and
are constructed using the action and group averaging projectors. This projector
formalism offers a systematic method for tracing and regularizing divergences
in the resulting state sums. Non-trivial coarse graining evolution moves lead
to non-unitary, and thus irreversible, projections of physical Hilbert spaces
and Dirac observables such that these concepts become evolution move dependent
on temporally varying discretizations. The formalism is illustrated in a toy
model mimicking a `creation from nothing'. Subtleties arising when applying
such a formalism to quantum gravity models are discussed.Comment: 45 pages, 1 appendix, 6 figures (additional explanations, now matches
published version
A global picture of quantum de Sitter space
Perturbative gravity about a de Sitter background motivates a global picture
of quantum dynamics in `eternal de Sitter space,' the theory of states which
are asymptotically de Sitter to both future and past. Eternal de Sitter physics
is described by a finite dimensional Hilbert space in which each state is
precisely invariant under the full de Sitter group. This resolves a
previously-noted tension between de Sitter symmetry and finite entropy.
Observables, implications for Boltzmann brains, and Poincare recurrences are
briefly discussed.Comment: 17 pages, 1 figure. v2: minor changes, references added. v3: minor
changes to correspond to PRD versio
Geometry of the ergodic quotient reveals coherent structures in flows
Dynamical systems that exhibit diverse behaviors can rarely be completely
understood using a single approach. However, by identifying coherent structures
in their state spaces, i.e., regions of uniform and simpler behavior, we could
hope to study each of the structures separately and then form the understanding
of the system as a whole. The method we present in this paper uses trajectory
averages of scalar functions on the state space to: (a) identify invariant sets
in the state space, (b) form coherent structures by aggregating invariant sets
that are similar across multiple spatial scales. First, we construct the
ergodic quotient, the object obtained by mapping trajectories to the space of
trajectory averages of a function basis on the state space. Second, we endow
the ergodic quotient with a metric structure that successfully captures how
similar the invariant sets are in the state space. Finally, we parametrize the
ergodic quotient using intrinsic diffusion modes on it. By segmenting the
ergodic quotient based on the diffusion modes, we extract coherent features in
the state space of the dynamical system. The algorithm is validated by
analyzing the Arnold-Beltrami-Childress flow, which was the test-bed for
alternative approaches: the Ulam's approximation of the transfer operator and
the computation of Lagrangian Coherent Structures. Furthermore, we explain how
the method extends the Poincar\'e map analysis for periodic flows. As a
demonstration, we apply the method to a periodically-driven three-dimensional
Hill's vortex flow, discovering unknown coherent structures in its state space.
In the end, we discuss differences between the ergodic quotient and
alternatives, propose a generalization to analysis of (quasi-)periodic
structures, and lay out future research directions.Comment: Submitted to Elsevier Physica D: Nonlinear Phenomen
Mathematical Structure of Loop Quantum Cosmology: Homogeneous Models
The mathematical structure of homogeneous loop quantum cosmology is analyzed,
starting with and taking into account the general classification of homogeneous
connections not restricted to be Abelian. As a first consequence, it is seen
that the usual approach of quantizing Abelian models using spaces of functions
on the Bohr compactification of the real line does not capture all properties
of homogeneous connections. A new, more general quantization is introduced
which applies to non-Abelian models and, in the Abelian case, can be mapped by
an isometric, but not unitary, algebra morphism onto common representations
making use of the Bohr compactification. Physically, the Bohr compactification
of spaces of Abelian connections leads to a degeneracy of edge lengths and
representations of holonomies. Lifting this degeneracy, the new quantization
gives rise to several dynamical properties, including lattice refinement seen
as a direct consequence of state-dependent regularizations of the Hamiltonian
constraint of loop quantum gravity. The representation of basic operators -
holonomies and fluxes - can be derived from the full theory specialized to
lattices. With the new methods of this article, loop quantum cosmology comes
closer to the full theory and is in a better position to produce reliable
predictions when all quantum effects of the theory are taken into account
On the accuracy of retinal protonated Schiff base models
We investigate the molecular geometries of the ground state and the minimal
energy conical intersections (MECIs) between the ground and first excited
states of the models for the retinal protonated Schiff base in the gas phase
using the extended multistate complete active space second-order perturbation
theory (XMS-CASPT2). The biggest model in this work is the rhodopsin
chromophore truncated between the {\epsilon} and {\delta} carbon atoms, which
consists of 54 atoms and 12-orbital {\pi} conjugation. The results are compared
with those obtained by the state-averaged complete active space self-consistent
field (SA-CASSCF). The XMS-CASPT2 results suggest that the minimum energy
conical intersection associated with the so-called 13-14 isomerization is
thermally inaccessible, which is in contrast to the SA-CASSCF results. The
differences between the geometries of the conical intersections computed by
SA-CASSCF and XMS-CASPT2 are ascribed to the fact that the charge transfer
states are more stabilized by dynamical electron correlation than the
diradicaloid states. The impact of the various choices of active spaces, basis
sets, and state averaging schemes is also examined.Comment: Contribution to the special issue in honor of the 80th birthday of
Professor Michael Bae
On the resolution of the big bang singularity in isotropic Loop Quantum Cosmology
In contrast to previous work in the field, we construct the Loop Quantum
Cosmology (LQC) of the flat isotropic model with a massless scalar field in the
absence of higher order curvature corrections to the gravitational part of the
Hamiltonian constraint. The matter part of the constraint contains the inverse
triad operator which can be quantized with or without the use of a Thiemann-
like procedure. With the latter choice, we show that the LQC quantization is
identical to that of the standard Wheeler DeWitt theory (WDW) wherein there is
no singularity resolution. We argue that the former choice leads to singularity
resolution in the sense of a well defined, regular (backward) evolution through
and beyond the epoch where the size of the universe vanishes.
Our work along with that of the seminal work of Ashtekar, Pawlowski and Singh
(APS) clarifies the role, in singularity resolution, of the three `exotic'
structures in this LQC model, namely: curvature corrections, inverse triad
definitions and the `polymer' nature of the kinematic representation. We also
critically examine certain technical assumptions made by APS in their analysis
of WDW semiclassical states and point out some problems stemming from the
infrared behaviour of their wave functionsComment: 26 pages, no figure
Hilbert spaces built on a similarity and on dynamical renormalization
We develop a Hilbert space framework for a number of general multi-scale
problems from dynamics. The aim is to identify a spectral theory for a class of
systems based on iterations of a non-invertible endomorphism.
We are motivated by the more familiar approach to wavelet theory which starts
with the two-to-one endomorphism in the one-torus \bt, a
wavelet filter, and an associated transfer operator. This leads to a scaling
function and a corresponding closed subspace in the Hilbert space
L^2(\br). Using the dyadic scaling on the line \br, one has a nested family
of closed subspaces , n \in \bz, with trivial intersection, and with
dense union in L^2(\br). More generally, we achieve the same outcome, but in
different Hilbert spaces, for a class of non-linear problems. In fact, we see
that the geometry of scales of subspaces in Hilbert space is ubiquitous in the
analysis of multiscale problems, e.g., martingales, complex iteration dynamical
systems, graph-iterated function systems of affine type, and subshifts in
symbolic dynamics. We develop a general framework for these examples which
starts with a fixed endomorphism (i.e., generalizing ) in a
compact metric space . It is assumed that is onto, and
finite-to-one.Comment: v3, minor addition
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