The kinematical phase space of the classical gravitational field
is flat (affine) and unbounded. Consequently, field variables may
tend to infinity, leading to the appearance of singularities, which
plague Einstein's theory of gravity. In loop quantum gravity (LQG),
the holonomies partially solve the problem by compacting the
connection variables. However, the conjugated fluxes remain
unbounded. The slides summarize some of our attempts toward
formulating LQG in a fully compact phase space manner, which
implies the finite-dimensional Hilbert space of the theory. The
compactification procedure is discussed first on the example
of a minisuperspace cosmological model (loop quantum cosmology)
with a single degree of freedom. The slides contain an overview of our recent results on applying the compact phase space approach to solving the Hamiltonian constraint using variational quantum methods.
The kinematical phase space of classical gravitational field is flat (affine) and unbounded. Because of this, field variables may tend to infinity leading to appearance of singularities, which plague Einstein's theory of gravity. At the slides an idea of generalizing theory of gravity by compactification of the phase space is discussed. The procedure of compactification of the phase space of de Sitter cosmological model with two dimensional phase space is introduced. The model is analyzed at both classical and quantum level.
Density Driven Groundwater Flow with Uncertain Porosity and PermeabilityAlexander Litvinenko
In this work, we solved the density driven groundwater flow problem with uncertain porosity and permeability. An accurate solution of this time-dependent and non-linear problem is impossible because of the presence of natural uncertainties in the reservoir such as porosity and permeability.
Therefore, we estimated the mean value and the variance of the solution, as well as the propagation of uncertainties from the random input parameters to the solution.
We started by defining the Elder-like problem. Then we described the multi-variate polynomial approximation (\gPC) approach and used it to estimate the required statistics of the mass fraction.
Utilizing the \gPC method allowed us
to reduce the computational cost compared to the classical quasi Monte Carlo method.
\gPC assumes that the output function $\sol(t,\bx,\thetab)$ is square-integrable and smooth w.r.t uncertain input variables $\btheta$.
Many factors, such as non-linearity, multiple solutions, multiple stationary states, time dependence and complicated solvers, make the investigation of the convergence of the \gPC method a non-trivial task.
We used an easy-to-implement, but only sub-optimal \gPC technique to quantify the uncertainty. For example, it is known that by increasing the degree of global polynomials (Hermite, Langange and similar), Runge's phenomenon appears. Here, probably local polynomials, splines or their mixtures would be better. Additionally, we used an easy-to-parallelise quadrature rule, which was also only suboptimal. For instance, adaptive choice of sparse grid (or collocation) points \cite{ConradMarzouk13,nobile-sg-mc-2015,Sudret_sparsePCE,CONSTANTINE12,crestaux2009polynomial} would be better, but we were limited by the usage of parallel methods. Adaptive quadrature rules are not (so well) parallelisable. In conclusion, we can report that: a) we developed a highly parallel method to quantify uncertainty in the Elder-like problem; b) with the \gPC of degree 4 we can achieve similar results as with the \QMC method.
In the numerical section we considered two different aquifers - a solid parallelepiped and a solid elliptic cylinder. One of our goals was to see how the domain geometry influences the formation, the number and the shape of fingers.
Since the considered problem is nonlinear,
a high variance in the porosity may result in totally different solutions; for instance, the number of fingers, their intensity and shape, the propagation time, and the velocity may vary considerably.
The number of cells in the presented experiments varied from $241{,}152$ to $15{,}433{,}728$ for the cylindrical domain and from $524{,}288$ to $4{,}194{,}304$ for the parallelepiped. The maximal number of parallel processing units was $600\times 32$, where $600$ is the number of parallel nodes and $32$ is the number of computing cores on each node. The total computing time varied from 2 hours for the coarse mesh to 24 hours for the finest mesh.
Efficient Simulations for Contamination of Groundwater Aquifers under Uncerta...Alexander Litvinenko
1. Solved time-dependent density driven flow problem with uncertain porosity and permeability in 2D and 3D
2. Computed propagation of uncertainties in porosity into the mass fraction.
3. Computed the mean, variance, exceedance probabilities, quantiles, risks.
4. Such QoIs as the number of fingers, their size, shape, propagation time can be unstable
5. For moderate perturbations, our gPCE surrogate results are similar to qMC results.
6. Used highly scalable solver on up to 800 computing nodes,
This poster was created in LaTeX on a Dell Inspiron laptop with a Linux Fedora Core 4 operating system. The background image and the animation snapshots are dxf meshes of elastic waveform solutions, rendered on a Windows machine using 3D Studio Max.
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
Evidence for baryon acoustic oscillations from galaxy–ellipticity correlationsSérgio Sacani
The baryon acoustic oscillation feature in the clustering of galaxies
or quasars provides a ‘standard ruler’ for distance measurements in
cosmology. In this work, we report a 2–3σ signal of the baryon acoustic
oscillation dip feature in the galaxy density–ellipticity cross-correlation
functions using the spectroscopic sample of the Baryon Oscillation
Spectroscopic Survey CMASS, combined with the deep Dark Energy
Spectroscopic Instrument Legacy Imaging Surveys for precise galaxy shape
measurements. We measure the galaxy–ellipticity correlation functions and
model them using the linear alignment model. We constrain the distance
DV/rd to redshift 0.57 to a precision of 3–5%, depending on the details of
modelling. The galaxy–ellipticity measurement reduces the uncertainty
of distance measurement by ~10% on top of that derived from the galaxy–
galaxy correlation. More importantly, for future large and deep galaxy
surveys, the independent galaxy–ellipticity measurements can help sort out
the systematics in the baryon acoustic oscillation studies.
This document summarizes research into understanding the molecular mechanisms that allow the motor protein myosin 7a to regulate its activity. Researchers constructed an all-atom model of myosin 7a and used coarse-grained simulations to investigate how flexible the protein needs to be for its head and tail domains to interact in the regulated compact structure. The simulations suggest the neck region, particularly the SAH domain, requires extremely high flexibility. Experiments are ongoing to determine which light chain proteins bind to myosin 7a under different conditions.
The kinematical phase space of classical gravitational field is flat (affine) and unbounded. Because of this, field variables may tend to infinity leading to appearance of singularities, which plague Einstein's theory of gravity. At the slides an idea of generalizing theory of gravity by compactification of the phase space is discussed. The procedure of compactification of the phase space of de Sitter cosmological model with two dimensional phase space is introduced. The model is analyzed at both classical and quantum level.
Density Driven Groundwater Flow with Uncertain Porosity and PermeabilityAlexander Litvinenko
In this work, we solved the density driven groundwater flow problem with uncertain porosity and permeability. An accurate solution of this time-dependent and non-linear problem is impossible because of the presence of natural uncertainties in the reservoir such as porosity and permeability.
Therefore, we estimated the mean value and the variance of the solution, as well as the propagation of uncertainties from the random input parameters to the solution.
We started by defining the Elder-like problem. Then we described the multi-variate polynomial approximation (\gPC) approach and used it to estimate the required statistics of the mass fraction.
Utilizing the \gPC method allowed us
to reduce the computational cost compared to the classical quasi Monte Carlo method.
\gPC assumes that the output function $\sol(t,\bx,\thetab)$ is square-integrable and smooth w.r.t uncertain input variables $\btheta$.
Many factors, such as non-linearity, multiple solutions, multiple stationary states, time dependence and complicated solvers, make the investigation of the convergence of the \gPC method a non-trivial task.
We used an easy-to-implement, but only sub-optimal \gPC technique to quantify the uncertainty. For example, it is known that by increasing the degree of global polynomials (Hermite, Langange and similar), Runge's phenomenon appears. Here, probably local polynomials, splines or their mixtures would be better. Additionally, we used an easy-to-parallelise quadrature rule, which was also only suboptimal. For instance, adaptive choice of sparse grid (or collocation) points \cite{ConradMarzouk13,nobile-sg-mc-2015,Sudret_sparsePCE,CONSTANTINE12,crestaux2009polynomial} would be better, but we were limited by the usage of parallel methods. Adaptive quadrature rules are not (so well) parallelisable. In conclusion, we can report that: a) we developed a highly parallel method to quantify uncertainty in the Elder-like problem; b) with the \gPC of degree 4 we can achieve similar results as with the \QMC method.
In the numerical section we considered two different aquifers - a solid parallelepiped and a solid elliptic cylinder. One of our goals was to see how the domain geometry influences the formation, the number and the shape of fingers.
Since the considered problem is nonlinear,
a high variance in the porosity may result in totally different solutions; for instance, the number of fingers, their intensity and shape, the propagation time, and the velocity may vary considerably.
The number of cells in the presented experiments varied from $241{,}152$ to $15{,}433{,}728$ for the cylindrical domain and from $524{,}288$ to $4{,}194{,}304$ for the parallelepiped. The maximal number of parallel processing units was $600\times 32$, where $600$ is the number of parallel nodes and $32$ is the number of computing cores on each node. The total computing time varied from 2 hours for the coarse mesh to 24 hours for the finest mesh.
Efficient Simulations for Contamination of Groundwater Aquifers under Uncerta...Alexander Litvinenko
1. Solved time-dependent density driven flow problem with uncertain porosity and permeability in 2D and 3D
2. Computed propagation of uncertainties in porosity into the mass fraction.
3. Computed the mean, variance, exceedance probabilities, quantiles, risks.
4. Such QoIs as the number of fingers, their size, shape, propagation time can be unstable
5. For moderate perturbations, our gPCE surrogate results are similar to qMC results.
6. Used highly scalable solver on up to 800 computing nodes,
This poster was created in LaTeX on a Dell Inspiron laptop with a Linux Fedora Core 4 operating system. The background image and the animation snapshots are dxf meshes of elastic waveform solutions, rendered on a Windows machine using 3D Studio Max.
UCSD NANO 266 Quantum Mechanical Modelling of Materials and Nanostructures is a graduate class that provides students with a highly practical introduction to the application of first principles quantum mechanical simulations to model, understand and predict the properties of materials and nano-structures. The syllabus includes: a brief introduction to quantum mechanics and the Hartree-Fock and density functional theory (DFT) formulations; practical simulation considerations such as convergence, selection of the appropriate functional and parameters; interpretation of the results from simulations, including the limits of accuracy of each method. Several lab sessions provide students with hands-on experience in the conduct of simulations. A key aspect of the course is in the use of programming to facilitate calculations and analysis.
Evidence for baryon acoustic oscillations from galaxy–ellipticity correlationsSérgio Sacani
The baryon acoustic oscillation feature in the clustering of galaxies
or quasars provides a ‘standard ruler’ for distance measurements in
cosmology. In this work, we report a 2–3σ signal of the baryon acoustic
oscillation dip feature in the galaxy density–ellipticity cross-correlation
functions using the spectroscopic sample of the Baryon Oscillation
Spectroscopic Survey CMASS, combined with the deep Dark Energy
Spectroscopic Instrument Legacy Imaging Surveys for precise galaxy shape
measurements. We measure the galaxy–ellipticity correlation functions and
model them using the linear alignment model. We constrain the distance
DV/rd to redshift 0.57 to a precision of 3–5%, depending on the details of
modelling. The galaxy–ellipticity measurement reduces the uncertainty
of distance measurement by ~10% on top of that derived from the galaxy–
galaxy correlation. More importantly, for future large and deep galaxy
surveys, the independent galaxy–ellipticity measurements can help sort out
the systematics in the baryon acoustic oscillation studies.
This document summarizes research into understanding the molecular mechanisms that allow the motor protein myosin 7a to regulate its activity. Researchers constructed an all-atom model of myosin 7a and used coarse-grained simulations to investigate how flexible the protein needs to be for its head and tail domains to interact in the regulated compact structure. The simulations suggest the neck region, particularly the SAH domain, requires extremely high flexibility. Experiments are ongoing to determine which light chain proteins bind to myosin 7a under different conditions.
The document discusses modeling and simulation of active soft matter physics models using interactive simulations and knowledge sharing. It describes modeling biological systems using objects like cells and processes like chemical reactions, transport, growth and motion. It presents the Mechanica modeling language which describes the physical world in terms of objects and processes, and uses a simple syntax based on TypeScript to define computational models in a way that is close to physical reality. The language aims to facilitate model interchange and be easily learned based on physical concepts.
The document summarizes research on quantifying uncertainty in groundwater contamination modeling. It discusses using stochastic methods and surrogate models to estimate how uncertainties in geological parameters, like porosity, propagate and influence quantities of interest in groundwater flow and contaminant transport simulations. Numerical experiments were conducted in 2D and 3D domains using parallel computing to analyze mean concentrations, variances, and other statistics over time under different uncertainty scenarios.
The build up_of_the_c_d_halo_of_m87_evidence_for_accretion_in_the_last_gyrSérgio Sacani
Observações recentes obtidas com o Very Large Telescope do ESO mostraram que Messier 87, a galáxia elíptica gigante mais próximo de nós, engoliu uma galáxia inteira de tamanho médio no último bilhão de anos. Uma equipe de astrônomos conseguiu pela primeira vez seguir o movimento de 300 nebulosas planetárias brilhantes, encontrando evidências claras deste evento e encontrando também excesso de radiação emitida pelos restos da vítima completamente desfeita.
This document discusses research into the structure and organization of chromatin at the centromere in Saccharomyces cerevisiae. Key findings include:
- Cohesin is enriched approximately 3-fold in a 50kb region flanking the centromere.
- Deconvolution and model convolution techniques are used to visualize cohesin enrichment and resolve its barrel-shaped organization around the mitotic spindle.
- Heatmap analysis indicates centromere proximal chromatin occupies a similar volumetric space as predicted by cohesin visualization, with decreased localization at the spindle axis.
- Kinetochore components like Cse4 show a more anisotropic localization pattern compared to Ndc80, suggesting regulated anisotropy
ADAPTIVE SEGMENTATION OF CELLS AND PARTICLES IN FLUORESCENT MICROSCOPE IMAGEJournal For Research
The document presents an adaptive segmentation method for segmenting cells and particles in fluorescent microscope images. It involves applying a coherence-enhancing diffusion filter to reduce noise and enhance structures, followed by using the Chan-Vese model to detect cell boundaries. The method allows simultaneous tracking of multiple cells over time by integrating both fast level set and graph cut frameworks with a topological prior. It is demonstrated on 2D and 3D time-lapse images of stem cells and carcinoma cells.
TOMOGRAPHY OF HUMAN BODY USING EXACT SIMULTANEOUS ITERATIVE RECONSTRUCTION AL...cscpconf
In this paper an Exact Simultaneous Iterative Reconstruction Algorithm is developed and applied on a large semi human size normal biological model and a diseased model (liver region affected) to verify the efficiency of the algorithm. The algorithm is successfully reconstructed the normal model having 15%-20% perturbation i.e. change in permittivity during disease. In diseased case, reconstructed imaginary part of complex permittivity clearly detects the affected zone and it may help the medical diagnosis. Hence it may be a powerful tool for early detection of cancerous tumors as the interrogating wave is a noninvasive one at the ultra high frequency range. The resolution of this system is increased with the reduction of
wavelength by immersing the antenna system and the model in saline water region. The advantage of this algorithm is that the calculation of cofactor are done offline to save the computational time and cofactors are expressed as a function of distances irrespective of their positions
APPLICATION OF PARTICLE SWARM OPTIMIZATION TO MICROWAVE TAPERED MICROSTRIP LINEScseij
This document discusses using Particle Swarm Optimization (PSO) to design a tapered microstrip transmission line to match an arbitrary load to a 50Ω line. PSO was used to optimize the impedances of a three section tapered line to minimize reflections. Simulations found impedances that gave good matching at 5GHz. PSO converged to solutions in under 1000 iterations. This demonstrates PSO's effectiveness in solving multi-objective microwave engineering optimization problems.
Application of particle swarm optimization to microwave tapered microstrip linescseij
Application of metaheuristic algorithms has been of continued interest in the field of electrical engineering
because of their powerful features. In this work special design is done for a tapered transmission line used
for matching an arbitrary real load to a 50Ω line. The problem at hand is to match this arbitray load to 50
Ω line using three section tapered transmission line with impedances in decreasing order from the load. So
the problem becomes optimizing an equation with three unknowns with various conditions. The optimized
values are obtained using Particle Swarm Optimization. It can easily be shown that PSO is very strong in
solving this kind of multiobjective optimization problems.
This document summarizes a talk on solving density-driven groundwater flow problems with uncertain porosity and permeability coefficients. The major goal is to estimate pollution risks in subsurface flows. The presentation covers: (1) setting up the groundwater flow problem; (2) reviewing stochastic modeling methods; (3) modeling uncertainty in porosity and permeability; (4) numerical methods to solve deterministic problems; and (5) 2D and 3D numerical experiments. The experiments demonstrate computing statistics of contaminant concentration and its propagation under uncertain parameters.
The stelar mass_growth_of_brightest_cluster_galaxies_in_the_irac_shallow_clus...Sérgio Sacani
This document describes a study of the stellar mass growth of brightest cluster galaxies (BCGs) between z=1.5 and z=0.5 using data from the Spitzer IRAC Shallow Cluster Survey (ISCS). The researchers developed a method to select high-redshift clusters as progenitors of lower-redshift clusters to study the evolution of BCG stellar masses. They find that between z=1.5 and z=0.5, the BCGs grew in stellar mass by a factor of 2.3, matching predictions from a semi-analytic galaxy formation model. Below z=0.5, there are hints of differences between the observed BCG growth and the model predictions.
A high-mass X-ray binary descended from an ultra-stripped supernovaSérgio Sacani
Ultra-stripped supernovae are diferent from other terminal explosions of massive
stars, as they show little or no ejecta from the actual supernova event1,2
. They are
thought to occur in massive binary systems after the exploding star has lost its surface
through interactions with its companion2
. Such supernovae produce little to no kick,
leading to the formation of a neutron star without loss of the binary companion,
which itself may also evolve into another neutron star2
. Here we show that a recently
discovered high-mass X-ray binary, CPD −29 2176 (CD −29 5159; SGR 0755-2933)3–6
, has
an evolutionary history that shows the neutron star component formed during an
ultra-stripped supernova. The binary has orbital elements that are similar both in
period and in eccentricity to 1 of 14 Be X-ray binaries that have known orbital periods
and eccentricities7
. The identifcation of the progenitors systems for ultra-stripped
supernovae is necessary as their evolution pathways lead to the formation of binary
neutron star systems. Binary neutron stars, such as the system that produced the
kilonova GW170817 that was observed with both electromagnetic and gravitational
energy8
, are known to produce a large quantity of heavy elements
Hybrid functionals admix a certain amount of Hartree-Fock exchange to density functionals in order to provide a more accurate treatment of electronic correlation effects. Popular hybrid functionals include PBE0, HSE, and B3LYP, which contain different proportions of Hartree-Fock exchange and have been empirically optimized. Hybrid functionals show improved descriptions of properties like band gaps, atomization energies, and reaction barriers compared to standard DFT functionals. However, their descriptions of metallic and strongly correlated systems can be less accurate compared to semi-local functionals.
The document summarizes simulations of liquid crystal molecular organizations at different resolutions, from atomistic to mesoscale. It discusses using the Gay-Berne potential to model nematic and columnar liquid crystal phases. It also describes using genetic algorithms to parameterize effective charges to mimic electrostatic interactions and reproduce ordering in columnar phases. Simulations are able to form nanoscale liquid crystal droplets and demonstrate how molecular shape and charge distributions influence self-assembly.
Serendipitous discovery of an extended xray jet without a radio counterpart i...Sérgio Sacani
A recent Chandra observation of the nearby galaxy cluster Abell 585 has led to the discovery of
an extended X-ray jet associated with the high-redshift background quasar B3 0727+409, a luminous
radio source at redshift z = 2:5. This is one of only few examples of high-redshift X-ray jets known
to date. It has a clear extension of about 1200, corresponding to a projected length of 100 kpc, with
a possible hot spot located 3500 from the quasar. The archival high resolution VLA maps surprisingly
reveal no extended jet emission, except for one knot about 1:400 from the quasar. The high X-ray to
radio luminosity ratio for this source appears consistent with the / (1 + z)4 amplication expected
from the inverse Compton radiative model. This serendipitous discovery may signal the existence
of an entire population of similar systems with bright X-ray and faint radio jets at high redshift, a
selection bias which must be accounted for when drawing any conclusions about the redshift evolution
of jet properties and indeed about the cosmological evolution of supermassive black holes and active
galactic nuclei in general.
This document discusses the relationship between two approaches to modeling the dynamics of open quantum systems - the Nakajima-Zwanzig time-nonlocal quantum master equation and the time-convolutionless time-local quantum master equation. It is shown that these two descriptions are connected by a simple fixed-point relation, where the time-local generator G is equal to the memory kernel K applied to G. This fixed-point relation allows insights into both the stationary and transient dynamics, providing a nonperturbative understanding of how memory effects are generated in the time-local description from the time-nonlocal memory kernel.
Heterogeneous relaxation dynamics in amorphous materials under cyclic loadingNikolai Priezjev
1) Molecular dynamics simulations were performed on a binary mixture under oscillatory shear strain with varying amplitudes.
2) At small strains, particle motion was nearly reversible over thousands of cycles. At large strains, intermittent bursts of particle displacements led to faster structural relaxation.
3) Mobile particles were found to cluster together, with cluster sizes increasing at higher strains. This is unlike prior results for steadily sheared systems.
4) The fraction of mobile particles facilitated by neighboring particles also increased with strain amplitude.
Finite Element Simulation of Microfluidic Biochip for High Throughput Hydrody...TELKOMNIKA JOURNAL
In this paper, a microfluidic device capable of trapping a single cell in a high throughput manner
and at high trapping efficiency is designed simply through a concept of hydrodynamic manipulation. The
microfluidic device is designed with a series of trap and bypass microchannel structures for trapping
individual cells without the need for microwell, robotic equipment, external electric force or surface
modification. In order to investigate the single cell trapping efficiency, a finite element model of the
proposed design has been developed using ABAQUS-FEA software. Based on the simulation, the
geometrical parameters and fluid velocity which affect the single cell trapping are extensively optimized.
After optimization of the trap and bypass microchannel structures via simulations, a single cell can be
trapped at a desired location efficiently.
Measurement-induced long-distance entanglement of superconducting qubits usin...Ondrej Cernotik
Although superconducting systems provide a promising platform for quantum computing, their networking poses a challenge as they cannot be interfaced to light---the medium used to send quantum signals through channels at room temperature. We show that mechanical oscillators can mediated such coupling and light can be used to measure the joint state of two distant qubits. The measurement provides information on the total spin of the two qubits such that entangled qubit states can be postselected. Entanglement generation is possible without ground-state cooling of the mechanical oscillators for systems with optomechanical cooperativity moderately larger than unity; in addition, our setup tolerates a substantial transmission loss. The approach is scalable to generation of multipartite entanglement and represents a crucial step towards quantum networks with superconducting circuits.
Constraints on the Universe as a Numerical Simulationsolodoe
This document discusses the possibility that our observed universe is a numerical simulation performed on a cubic space-time lattice. It first motivates this idea by extrapolating trends in computational resources needed for lattice QCD simulations into the future. It then investigates potentially observable consequences if the universe is an early simulation with unimproved Wilson fermion discretization. The most stringent bound on the inverse lattice spacing of the universe is derived from the high-energy cutoff of the cosmic ray spectrum. The numerical simulation scenario could be revealed in distributions of the highest energy cosmic rays exhibiting symmetry breaking reflecting the underlying lattice structure.
Describing and Interpreting an Immersive Learning Case with the Immersion Cub...Leonel Morgado
Current descriptions of immersive learning cases are often difficult or impossible to compare. This is due to a myriad of different options on what details to include, which aspects are relevant, and on the descriptive approaches employed. Also, these aspects often combine very specific details with more general guidelines or indicate intents and rationales without clarifying their implementation. In this paper we provide a method to describe immersive learning cases that is structured to enable comparisons, yet flexible enough to allow researchers and practitioners to decide which aspects to include. This method leverages a taxonomy that classifies educational aspects at three levels (uses, practices, and strategies) and then utilizes two frameworks, the Immersive Learning Brain and the Immersion Cube, to enable a structured description and interpretation of immersive learning cases. The method is then demonstrated on a published immersive learning case on training for wind turbine maintenance using virtual reality. Applying the method results in a structured artifact, the Immersive Learning Case Sheet, that tags the case with its proximal uses, practices, and strategies, and refines the free text case description to ensure that matching details are included. This contribution is thus a case description method in support of future comparative research of immersive learning cases. We then discuss how the resulting description and interpretation can be leveraged to change immersion learning cases, by enriching them (considering low-effort changes or additions) or innovating (exploring more challenging avenues of transformation). The method holds significant promise to support better-grounded research in immersive learning.
The document discusses modeling and simulation of active soft matter physics models using interactive simulations and knowledge sharing. It describes modeling biological systems using objects like cells and processes like chemical reactions, transport, growth and motion. It presents the Mechanica modeling language which describes the physical world in terms of objects and processes, and uses a simple syntax based on TypeScript to define computational models in a way that is close to physical reality. The language aims to facilitate model interchange and be easily learned based on physical concepts.
The document summarizes research on quantifying uncertainty in groundwater contamination modeling. It discusses using stochastic methods and surrogate models to estimate how uncertainties in geological parameters, like porosity, propagate and influence quantities of interest in groundwater flow and contaminant transport simulations. Numerical experiments were conducted in 2D and 3D domains using parallel computing to analyze mean concentrations, variances, and other statistics over time under different uncertainty scenarios.
The build up_of_the_c_d_halo_of_m87_evidence_for_accretion_in_the_last_gyrSérgio Sacani
Observações recentes obtidas com o Very Large Telescope do ESO mostraram que Messier 87, a galáxia elíptica gigante mais próximo de nós, engoliu uma galáxia inteira de tamanho médio no último bilhão de anos. Uma equipe de astrônomos conseguiu pela primeira vez seguir o movimento de 300 nebulosas planetárias brilhantes, encontrando evidências claras deste evento e encontrando também excesso de radiação emitida pelos restos da vítima completamente desfeita.
This document discusses research into the structure and organization of chromatin at the centromere in Saccharomyces cerevisiae. Key findings include:
- Cohesin is enriched approximately 3-fold in a 50kb region flanking the centromere.
- Deconvolution and model convolution techniques are used to visualize cohesin enrichment and resolve its barrel-shaped organization around the mitotic spindle.
- Heatmap analysis indicates centromere proximal chromatin occupies a similar volumetric space as predicted by cohesin visualization, with decreased localization at the spindle axis.
- Kinetochore components like Cse4 show a more anisotropic localization pattern compared to Ndc80, suggesting regulated anisotropy
ADAPTIVE SEGMENTATION OF CELLS AND PARTICLES IN FLUORESCENT MICROSCOPE IMAGEJournal For Research
The document presents an adaptive segmentation method for segmenting cells and particles in fluorescent microscope images. It involves applying a coherence-enhancing diffusion filter to reduce noise and enhance structures, followed by using the Chan-Vese model to detect cell boundaries. The method allows simultaneous tracking of multiple cells over time by integrating both fast level set and graph cut frameworks with a topological prior. It is demonstrated on 2D and 3D time-lapse images of stem cells and carcinoma cells.
TOMOGRAPHY OF HUMAN BODY USING EXACT SIMULTANEOUS ITERATIVE RECONSTRUCTION AL...cscpconf
In this paper an Exact Simultaneous Iterative Reconstruction Algorithm is developed and applied on a large semi human size normal biological model and a diseased model (liver region affected) to verify the efficiency of the algorithm. The algorithm is successfully reconstructed the normal model having 15%-20% perturbation i.e. change in permittivity during disease. In diseased case, reconstructed imaginary part of complex permittivity clearly detects the affected zone and it may help the medical diagnosis. Hence it may be a powerful tool for early detection of cancerous tumors as the interrogating wave is a noninvasive one at the ultra high frequency range. The resolution of this system is increased with the reduction of
wavelength by immersing the antenna system and the model in saline water region. The advantage of this algorithm is that the calculation of cofactor are done offline to save the computational time and cofactors are expressed as a function of distances irrespective of their positions
APPLICATION OF PARTICLE SWARM OPTIMIZATION TO MICROWAVE TAPERED MICROSTRIP LINEScseij
This document discusses using Particle Swarm Optimization (PSO) to design a tapered microstrip transmission line to match an arbitrary load to a 50Ω line. PSO was used to optimize the impedances of a three section tapered line to minimize reflections. Simulations found impedances that gave good matching at 5GHz. PSO converged to solutions in under 1000 iterations. This demonstrates PSO's effectiveness in solving multi-objective microwave engineering optimization problems.
Application of particle swarm optimization to microwave tapered microstrip linescseij
Application of metaheuristic algorithms has been of continued interest in the field of electrical engineering
because of their powerful features. In this work special design is done for a tapered transmission line used
for matching an arbitrary real load to a 50Ω line. The problem at hand is to match this arbitray load to 50
Ω line using three section tapered transmission line with impedances in decreasing order from the load. So
the problem becomes optimizing an equation with three unknowns with various conditions. The optimized
values are obtained using Particle Swarm Optimization. It can easily be shown that PSO is very strong in
solving this kind of multiobjective optimization problems.
This document summarizes a talk on solving density-driven groundwater flow problems with uncertain porosity and permeability coefficients. The major goal is to estimate pollution risks in subsurface flows. The presentation covers: (1) setting up the groundwater flow problem; (2) reviewing stochastic modeling methods; (3) modeling uncertainty in porosity and permeability; (4) numerical methods to solve deterministic problems; and (5) 2D and 3D numerical experiments. The experiments demonstrate computing statistics of contaminant concentration and its propagation under uncertain parameters.
The stelar mass_growth_of_brightest_cluster_galaxies_in_the_irac_shallow_clus...Sérgio Sacani
This document describes a study of the stellar mass growth of brightest cluster galaxies (BCGs) between z=1.5 and z=0.5 using data from the Spitzer IRAC Shallow Cluster Survey (ISCS). The researchers developed a method to select high-redshift clusters as progenitors of lower-redshift clusters to study the evolution of BCG stellar masses. They find that between z=1.5 and z=0.5, the BCGs grew in stellar mass by a factor of 2.3, matching predictions from a semi-analytic galaxy formation model. Below z=0.5, there are hints of differences between the observed BCG growth and the model predictions.
A high-mass X-ray binary descended from an ultra-stripped supernovaSérgio Sacani
Ultra-stripped supernovae are diferent from other terminal explosions of massive
stars, as they show little or no ejecta from the actual supernova event1,2
. They are
thought to occur in massive binary systems after the exploding star has lost its surface
through interactions with its companion2
. Such supernovae produce little to no kick,
leading to the formation of a neutron star without loss of the binary companion,
which itself may also evolve into another neutron star2
. Here we show that a recently
discovered high-mass X-ray binary, CPD −29 2176 (CD −29 5159; SGR 0755-2933)3–6
, has
an evolutionary history that shows the neutron star component formed during an
ultra-stripped supernova. The binary has orbital elements that are similar both in
period and in eccentricity to 1 of 14 Be X-ray binaries that have known orbital periods
and eccentricities7
. The identifcation of the progenitors systems for ultra-stripped
supernovae is necessary as their evolution pathways lead to the formation of binary
neutron star systems. Binary neutron stars, such as the system that produced the
kilonova GW170817 that was observed with both electromagnetic and gravitational
energy8
, are known to produce a large quantity of heavy elements
Hybrid functionals admix a certain amount of Hartree-Fock exchange to density functionals in order to provide a more accurate treatment of electronic correlation effects. Popular hybrid functionals include PBE0, HSE, and B3LYP, which contain different proportions of Hartree-Fock exchange and have been empirically optimized. Hybrid functionals show improved descriptions of properties like band gaps, atomization energies, and reaction barriers compared to standard DFT functionals. However, their descriptions of metallic and strongly correlated systems can be less accurate compared to semi-local functionals.
The document summarizes simulations of liquid crystal molecular organizations at different resolutions, from atomistic to mesoscale. It discusses using the Gay-Berne potential to model nematic and columnar liquid crystal phases. It also describes using genetic algorithms to parameterize effective charges to mimic electrostatic interactions and reproduce ordering in columnar phases. Simulations are able to form nanoscale liquid crystal droplets and demonstrate how molecular shape and charge distributions influence self-assembly.
Serendipitous discovery of an extended xray jet without a radio counterpart i...Sérgio Sacani
A recent Chandra observation of the nearby galaxy cluster Abell 585 has led to the discovery of
an extended X-ray jet associated with the high-redshift background quasar B3 0727+409, a luminous
radio source at redshift z = 2:5. This is one of only few examples of high-redshift X-ray jets known
to date. It has a clear extension of about 1200, corresponding to a projected length of 100 kpc, with
a possible hot spot located 3500 from the quasar. The archival high resolution VLA maps surprisingly
reveal no extended jet emission, except for one knot about 1:400 from the quasar. The high X-ray to
radio luminosity ratio for this source appears consistent with the / (1 + z)4 amplication expected
from the inverse Compton radiative model. This serendipitous discovery may signal the existence
of an entire population of similar systems with bright X-ray and faint radio jets at high redshift, a
selection bias which must be accounted for when drawing any conclusions about the redshift evolution
of jet properties and indeed about the cosmological evolution of supermassive black holes and active
galactic nuclei in general.
This document discusses the relationship between two approaches to modeling the dynamics of open quantum systems - the Nakajima-Zwanzig time-nonlocal quantum master equation and the time-convolutionless time-local quantum master equation. It is shown that these two descriptions are connected by a simple fixed-point relation, where the time-local generator G is equal to the memory kernel K applied to G. This fixed-point relation allows insights into both the stationary and transient dynamics, providing a nonperturbative understanding of how memory effects are generated in the time-local description from the time-nonlocal memory kernel.
Heterogeneous relaxation dynamics in amorphous materials under cyclic loadingNikolai Priezjev
1) Molecular dynamics simulations were performed on a binary mixture under oscillatory shear strain with varying amplitudes.
2) At small strains, particle motion was nearly reversible over thousands of cycles. At large strains, intermittent bursts of particle displacements led to faster structural relaxation.
3) Mobile particles were found to cluster together, with cluster sizes increasing at higher strains. This is unlike prior results for steadily sheared systems.
4) The fraction of mobile particles facilitated by neighboring particles also increased with strain amplitude.
Finite Element Simulation of Microfluidic Biochip for High Throughput Hydrody...TELKOMNIKA JOURNAL
In this paper, a microfluidic device capable of trapping a single cell in a high throughput manner
and at high trapping efficiency is designed simply through a concept of hydrodynamic manipulation. The
microfluidic device is designed with a series of trap and bypass microchannel structures for trapping
individual cells without the need for microwell, robotic equipment, external electric force or surface
modification. In order to investigate the single cell trapping efficiency, a finite element model of the
proposed design has been developed using ABAQUS-FEA software. Based on the simulation, the
geometrical parameters and fluid velocity which affect the single cell trapping are extensively optimized.
After optimization of the trap and bypass microchannel structures via simulations, a single cell can be
trapped at a desired location efficiently.
Measurement-induced long-distance entanglement of superconducting qubits usin...Ondrej Cernotik
Although superconducting systems provide a promising platform for quantum computing, their networking poses a challenge as they cannot be interfaced to light---the medium used to send quantum signals through channels at room temperature. We show that mechanical oscillators can mediated such coupling and light can be used to measure the joint state of two distant qubits. The measurement provides information on the total spin of the two qubits such that entangled qubit states can be postselected. Entanglement generation is possible without ground-state cooling of the mechanical oscillators for systems with optomechanical cooperativity moderately larger than unity; in addition, our setup tolerates a substantial transmission loss. The approach is scalable to generation of multipartite entanglement and represents a crucial step towards quantum networks with superconducting circuits.
Constraints on the Universe as a Numerical Simulationsolodoe
This document discusses the possibility that our observed universe is a numerical simulation performed on a cubic space-time lattice. It first motivates this idea by extrapolating trends in computational resources needed for lattice QCD simulations into the future. It then investigates potentially observable consequences if the universe is an early simulation with unimproved Wilson fermion discretization. The most stringent bound on the inverse lattice spacing of the universe is derived from the high-energy cutoff of the cosmic ray spectrum. The numerical simulation scenario could be revealed in distributions of the highest energy cosmic rays exhibiting symmetry breaking reflecting the underlying lattice structure.
Similar to Towards the loop quantum gravity with compact phase space (20)
Describing and Interpreting an Immersive Learning Case with the Immersion Cub...Leonel Morgado
Current descriptions of immersive learning cases are often difficult or impossible to compare. This is due to a myriad of different options on what details to include, which aspects are relevant, and on the descriptive approaches employed. Also, these aspects often combine very specific details with more general guidelines or indicate intents and rationales without clarifying their implementation. In this paper we provide a method to describe immersive learning cases that is structured to enable comparisons, yet flexible enough to allow researchers and practitioners to decide which aspects to include. This method leverages a taxonomy that classifies educational aspects at three levels (uses, practices, and strategies) and then utilizes two frameworks, the Immersive Learning Brain and the Immersion Cube, to enable a structured description and interpretation of immersive learning cases. The method is then demonstrated on a published immersive learning case on training for wind turbine maintenance using virtual reality. Applying the method results in a structured artifact, the Immersive Learning Case Sheet, that tags the case with its proximal uses, practices, and strategies, and refines the free text case description to ensure that matching details are included. This contribution is thus a case description method in support of future comparative research of immersive learning cases. We then discuss how the resulting description and interpretation can be leveraged to change immersion learning cases, by enriching them (considering low-effort changes or additions) or innovating (exploring more challenging avenues of transformation). The method holds significant promise to support better-grounded research in immersive learning.
Signatures of wave erosion in Titan’s coastsSérgio Sacani
The shorelines of Titan’s hydrocarbon seas trace flooded erosional landforms such as river valleys; however, it isunclear whether coastal erosion has subsequently altered these shorelines. Spacecraft observations and theo-retical models suggest that wind may cause waves to form on Titan’s seas, potentially driving coastal erosion,but the observational evidence of waves is indirect, and the processes affecting shoreline evolution on Titanremain unknown. No widely accepted framework exists for using shoreline morphology to quantitatively dis-cern coastal erosion mechanisms, even on Earth, where the dominant mechanisms are known. We combinelandscape evolution models with measurements of shoreline shape on Earth to characterize how differentcoastal erosion mechanisms affect shoreline morphology. Applying this framework to Titan, we find that theshorelines of Titan’s seas are most consistent with flooded landscapes that subsequently have been eroded bywaves, rather than a uniform erosional process or no coastal erosion, particularly if wave growth saturates atfetch lengths of tens of kilometers.
The cost of acquiring information by natural selectionCarl Bergstrom
This is a short talk that I gave at the Banff International Research Station workshop on Modeling and Theory in Population Biology. The idea is to try to understand how the burden of natural selection relates to the amount of information that selection puts into the genome.
It's based on the first part of this research paper:
The cost of information acquisition by natural selection
Ryan Seamus McGee, Olivia Kosterlitz, Artem Kaznatcheev, Benjamin Kerr, Carl T. Bergstrom
bioRxiv 2022.07.02.498577; doi: https://doi.org/10.1101/2022.07.02.498577
JAMES WEBB STUDY THE MASSIVE BLACK HOLE SEEDSSérgio Sacani
The pathway(s) to seeding the massive black holes (MBHs) that exist at the heart of galaxies in the present and distant Universe remains an unsolved problem. Here we categorise, describe and quantitatively discuss the formation pathways of both light and heavy seeds. We emphasise that the most recent computational models suggest that rather than a bimodal-like mass spectrum between light and heavy seeds with light at one end and heavy at the other that instead a continuum exists. Light seeds being more ubiquitous and the heavier seeds becoming less and less abundant due the rarer environmental conditions required for their formation. We therefore examine the different mechanisms that give rise to different seed mass spectrums. We show how and why the mechanisms that produce the heaviest seeds are also among the rarest events in the Universe and are hence extremely unlikely to be the seeds for the vast majority of the MBH population. We quantify, within the limits of the current large uncertainties in the seeding processes, the expected number densities of the seed mass spectrum. We argue that light seeds must be at least 103 to 105 times more numerous than heavy seeds to explain the MBH population as a whole. Based on our current understanding of the seed population this makes heavy seeds (Mseed > 103 M⊙) a significantly more likely pathway given that heavy seeds have an abundance pattern than is close to and likely in excess of 10−4 compared to light seeds. Finally, we examine the current state-of-the-art in numerical calculations and recent observations and plot a path forward for near-future advances in both domains.
TOPIC OF DISCUSSION: CENTRIFUGATION SLIDESHARE.pptxshubhijain836
Centrifugation is a powerful technique used in laboratories to separate components of a heterogeneous mixture based on their density. This process utilizes centrifugal force to rapidly spin samples, causing denser particles to migrate outward more quickly than lighter ones. As a result, distinct layers form within the sample tube, allowing for easy isolation and purification of target substances.
Sexuality - Issues, Attitude and Behaviour - Applied Social Psychology - Psyc...PsychoTech Services
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!
(June 12, 2024) Webinar: Development of PET theranostics targeting the molecu...Scintica Instrumentation
Targeting Hsp90 and its pathogen Orthologs with Tethered Inhibitors as a Diagnostic and Therapeutic Strategy for cancer and infectious diseases with Dr. Timothy Haystead.
ESA/ACT Science Coffee: Diego Blas - Gravitational wave detection with orbita...Advanced-Concepts-Team
Presentation in the Science Coffee of the Advanced Concepts Team of the European Space Agency on the 07.06.2024.
Speaker: Diego Blas (IFAE/ICREA)
Title: Gravitational wave detection with orbital motion of Moon and artificial
Abstract:
In this talk I will describe some recent ideas to find gravitational waves from supermassive black holes or of primordial origin by studying their secular effect on the orbital motion of the Moon or satellites that are laser ranged.
Discovery of An Apparent Red, High-Velocity Type Ia Supernova at 𝐳 = 2.9 wi...Sérgio Sacani
We present the JWST discovery of SN 2023adsy, a transient object located in a host galaxy JADES-GS
+
53.13485
−
27.82088
with a host spectroscopic redshift of
2.903
±
0.007
. The transient was identified in deep James Webb Space Telescope (JWST)/NIRCam imaging from the JWST Advanced Deep Extragalactic Survey (JADES) program. Photometric and spectroscopic followup with NIRCam and NIRSpec, respectively, confirm the redshift and yield UV-NIR light-curve, NIR color, and spectroscopic information all consistent with a Type Ia classification. Despite its classification as a likely SN Ia, SN 2023adsy is both fairly red (
�
(
�
−
�
)
∼
0.9
) despite a host galaxy with low-extinction and has a high Ca II velocity (
19
,
000
±
2
,
000
km/s) compared to the general population of SNe Ia. While these characteristics are consistent with some Ca-rich SNe Ia, particularly SN 2016hnk, SN 2023adsy is intrinsically brighter than the low-
�
Ca-rich population. Although such an object is too red for any low-
�
cosmological sample, we apply a fiducial standardization approach to SN 2023adsy and find that the SN 2023adsy luminosity distance measurement is in excellent agreement (
≲
1
�
) with
Λ
CDM. Therefore unlike low-
�
Ca-rich SNe Ia, SN 2023adsy is standardizable and gives no indication that SN Ia standardized luminosities change significantly with redshift. A larger sample of distant SNe Ia is required to determine if SN Ia population characteristics at high-
�
truly diverge from their low-
�
counterparts, and to confirm that standardized luminosities nevertheless remain constant with redshift.
Towards the loop quantum gravity with compact phase space
1. Towards the loop quantum gravity
with compact phase space
Jakub Mielczarek
Jagiellonian University
Featuring:
• G. Czelusta, J. M. Quantum variational solving of the Wheeler-DeWitt equation,
arXiv: 2111.03038 (2021)
• D. Artigas, J. M. and C. Rovelli, A minisuperspace model of compact phase
space gravity, Phys. Rev. D 100, 043533 (2019)
2. Compact phase spaces arise in the semiclassical description of
quantum systems with finite dimensional Hilbert spaces.
<latexit sha1_base64="vYDMe4ccJRre9xMa8Jwkjr6rBaI=">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</latexit>
<latexit sha1_base64="vYDMe4ccJRre9xMa8Jwkjr6rBaI=">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</latexit>
<latexit sha1_base64="vYDMe4ccJRre9xMa8Jwkjr6rBaI=">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</latexit>
<latexit sha1_base64="WPdPUMlSpvGUjKpqLEsfjgmdehA=">AAACyXicjVHLTsMwEJyGVymvAkcuERUSpyrhAkcEB5C4FIk+pLZCTuqW0LxwHARUnPgBrvBjiD+Av2BtXAmoEDhKMp7dGXt3vTQMMuk4rwVranpmdq44X1pYXFpeKa+uNbIkFz6v+0mYiJbHMh4GMa/LQIa8lQrOIi/kTW94qOLNay6yIInP5G3KuxEbxEE/8JkkqtE5YlHEzssVp+roZU8C14AKzKol5Rd00EMCHzkicMSQhEMwZPS04cJBSlwXI+IEoUDHOe5RIm1OWZwyGLFD+g5o1zZsTHvlmWm1T6eE9ApS2tgiTUJ5grA6zdbxXDsr9jfvkfZUd7ulv2e8ImIlLoj9SzfO/K9O1SLRx56uIaCaUs2o6nzjkuuuqJvbX6qS5JASp3CP4oKwr5XjPttak+naVW+Zjr/pTMWqvW9yc7yrW9KA3Z/jnASNnarrVN3Tncr+gRl1ERvYxDbNcxf7OEYNdfK+xCOe8GydWFfWjXX3mWoVjGYd35b18AFljZF0</latexit>
Compact phase space Finite Hilbert space
We are focused on the compact case rather on the phase space with boundaries.
V =
Z
!
<latexit sha1_base64="xTL/p8YT5xoSa0tUZfT8cwlJUaE=">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</latexit>
<latexit sha1_base64="xTL/p8YT5xoSa0tUZfT8cwlJUaE=">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</latexit>
<latexit sha1_base64="xTL/p8YT5xoSa0tUZfT8cwlJUaE=">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</latexit>
<latexit sha1_base64="VVaE4Zhqb2T2gdbIGK9eZ+MwARI=">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</latexit>
dim H ⇠
V
(2⇡})m
<latexit sha1_base64="00hKEIMIixq/X7jSgJtaZCbNJE4=">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</latexit>
<latexit sha1_base64="00hKEIMIixq/X7jSgJtaZCbNJE4=">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</latexit>
<latexit sha1_base64="00hKEIMIixq/X7jSgJtaZCbNJE4=">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</latexit>
<latexit sha1_base64="19319sJZKVy34wFFtEUMh88exUc=">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</latexit>
dim = 2m
<latexit sha1_base64="0C/rV5vg21JJjCKQVvfevV0kEZQ=">AAAC1XicjVHLSsNAFD3GV62vqEs3wSK4KkkRdCMUXeiygn1AKzJJp3VoXiSTQindiVt/wK3+kvgH+hfeGVNQi+iEJGfOvefM3Hvd2BeptO3XOWN+YXFpubBSXF1b39g0t7YbaZQlHq97kR8lLZel3Bchr0shfd6KE84C1+dNd3Cm4s0hT1IRhVdyFPPrgPVD0RMek0TdmGanKwKrc86CgFknViW4MUt22dbLmgVODkrIVy0yX9BBFxE8ZAjAEUIS9sGQ0tOGAxsxcdcYE5cQEjrOMUGRtBllccpgxA7o26ddO2dD2ivPVKs9OsWnNyGlhX3SRJSXEFanWTqeaWfF/uY91p7qbiP6u7lXQKzELbF/6aaZ/9WpWiR6ONY1CKop1oyqzstdMt0VdXPrS1WSHGLiFO5SPCHsaeW0z5bWpLp21Vum4286U7Fq7+W5Gd7VLWnAzs9xzoJGpezYZefysFQ9zUddwC72cEDzPEIVF6ihTt5DPOIJz0bTmBh3xv1nqjGXa3bwbRkPH0m7lN0=</latexit>
<latexit sha1_base64="0C/rV5vg21JJjCKQVvfevV0kEZQ=">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</latexit>
<latexit sha1_base64="0C/rV5vg21JJjCKQVvfevV0kEZQ=">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</latexit>
<latexit sha1_base64="oVAdWF4w/M7B3DtPDa8VFSPciJE=">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</latexit>
dim H < 1 , V < 1
<latexit sha1_base64="qiC27e7ZtqB1flBoM7YJk1jVmt8=">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</latexit>
<latexit sha1_base64="qiC27e7ZtqB1flBoM7YJk1jVmt8=">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</latexit>
<latexit sha1_base64="qiC27e7ZtqB1flBoM7YJk1jVmt8=">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</latexit>
<latexit sha1_base64="G30nvnJkyKBlKcCymNnrqTxJgV4=">AAACtXicjVLLSgMxFD0dX7VWrWs3g0VwVTJudCnowmUF+4BaZCZNa+y8TDJCKf6AWz9O/AP9C2/iCGoRzTAzJ+fec5KbmyiPpTaMvVS8peWV1bXqem2jXtvc2m7UuzorFBcdnsWZ6kehFrFMRcdIE4t+rkSYRLHoRdNTG+/dC6Vlll6aWS6GSThJ5Vjy0BDVvm40WYu54S+CoARNlCNrPOMKI2TgKJBAIIUhHCOEpmeAAAw5cUPMiVOEpIsLPKBG2oKyBGWExE7pO6HZoGRTmltP7dScVonpVaT0sU+ajPIUYbua7+KFc7bsb95z52n3NqN/VHolxBrcEPuX7jPzvzpbi8EYx64GSTXljrHV8dKlcKdid+5/qcqQQ06cxSOKK8LcKT/P2Xca7Wq3Zxu6+KvLtKyd8zK3wJvdJfU3+NnNRdA9bAWsFVwwVLGLPRxQG49wgnO00SHLER7x5J15t97dxz3wKuWF2MG34el34YWM3A==</latexit>
<latexit sha1_base64="PGKPZOIal0gMIRBH6acDc0z1IFo=">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</latexit>
<latexit sha1_base64="PGKPZOIal0gMIRBH6acDc0z1IFo=">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</latexit>
<latexit sha1_base64="G6/HARSyPMDh05QunC2DoE2EHXU=">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</latexit>
<latexit sha1_base64="qiC27e7ZtqB1flBoM7YJk1jVmt8=">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</latexit>
<latexit sha1_base64="qiC27e7ZtqB1flBoM7YJk1jVmt8=">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</latexit>
<latexit sha1_base64="qiC27e7ZtqB1flBoM7YJk1jVmt8=">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</latexit>
<latexit sha1_base64="qiC27e7ZtqB1flBoM7YJk1jVmt8=">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</latexit>
<latexit sha1_base64="qiC27e7ZtqB1flBoM7YJk1jVmt8=">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</latexit>
<latexit sha1_base64="OpiY0HaFH0NYbuZEVDOlyAY4cro=">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</latexit>
3. Nonlinear Field Space Theory (NFST)
J. M. and T. Trześniewski, The Nonlinear Field Space Theory, Phys. Lett. B 759, 424 (2016)
J. M., Spin-Field Correspondence, Universe 3, 29 (2017)
J. M. and T. Trześniewski, Nonlinear Field Space Cosmology, Phys. Rev. D 96, 043522 (2017)
J. Bilski, S. Brahma, A. Marcianò and J. M., Klein–Gordon field from the XXZ
Heisenberg model, Int. J. Mod. Phys. D 28, 1950020 (2018)
D. Artigas, J. M. and C. Rovelli, A minisuperspace model of compact phase space gravity,
Phys. Rev. D 100, 043533 (2019)
Grant Sonata Bis DEC-2017/26/E/ST2/00763 of the
National Science Centre, Compact phase space field theories,
2018-2022
D. Artigas, J. Bilski, S. Crowe, J. M. and T. Trześniewski,
General-relativistic spin system, Phys. Rev. D 102, 125029 (2020)
D. Artigas, S. Crowe and J. M., Quantum fluctuations of the compact phase
space cosmology, Class. Quantum Grav. 38, 085007 (2021)
- compact phase space generalization of field theories
4. Why compact phase space for gravity?
Locally finite dimensional Hilbert space of quantum gravitational
field is supported at the theoretical ground.
Spikes (IR divergences)
Singularities (UV divergences)
Spikes in Ponzano-Regge amplitudes
are cured by q-deformations, which
introduce finite dimension of a Hilbert
space -> Turaev-Viro amplitude.
Compact phase spaces introduce
bounds on the the phase space
variables, which regularize curvature,
energy density, etc.
Model-independent arguments: N. Bao, S. M. Carroll and A. Singh, The Hilbert
Space of Quantum Gravity Is Locally Finite-Dimensional,
Int. J. Mod. Phys. D 26 (2017) no.12, 1743013
Finite horizon entropy/holography
SBH =
1
4
A
l2
Pl
<latexit sha1_base64="hlSDegc/77QgWSJ7UTIAb6biyE4=">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</latexit>
<latexit sha1_base64="hlSDegc/77QgWSJ7UTIAb6biyE4=">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</latexit>
<latexit sha1_base64="hlSDegc/77QgWSJ7UTIAb6biyE4=">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</latexit>
<latexit sha1_base64="i7BJqtcG+TdUTkbTddlmrjfIhV4=">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</latexit>
SdS ⇠
1
l2
Pl⇤
<latexit sha1_base64="o8zLmax7SoV9k7RXd9Tl93kvyFA=">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</latexit>
<latexit sha1_base64="o8zLmax7SoV9k7RXd9Tl93kvyFA=">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</latexit>
<latexit sha1_base64="o8zLmax7SoV9k7RXd9Tl93kvyFA=">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</latexit>
<latexit sha1_base64="y+20wniumFcasvp6MEg/CagtrNE=">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</latexit>
S ⇠ ln dim H
<latexit sha1_base64="vQh4At5xWgezBnL4qEoPPDNlvgE=">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</latexit>
<latexit sha1_base64="vQh4At5xWgezBnL4qEoPPDNlvgE=">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</latexit>
<latexit sha1_base64="vQh4At5xWgezBnL4qEoPPDNlvgE=">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</latexit>
<latexit sha1_base64="cS4FNpjreVofrzM45JSgdm9N6mo=">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</latexit>
5. Loop Quantum Gravity
C. Rovelli and F. Vidotto, Compact phase space, cosmological constant, and discrete time,
Phys. Rev. D 91 (2015) no.8, 084037
⇣
e
R
A
, E
⌘
2 SU(2) ⇥ su(2)
<latexit sha1_base64="CdkIUU6wRZtB7sjCMykF2u3+RLU=">AAADAnicjVHLahRBFD3pqInxNerSTeEgjCBDdxB0E0gUwWVEJwmkY6ju1MwU0y+qqoUwzC5/kp07cesPuNWF+Af6F54qO6AG0Wq6+9S595yqe2/WFNq6OP66FC1fuHhpZfXy2pWr167f6N28tWPr1uRqlNdFbfYyaVWhKzVy2hVqrzFKllmhdrPZUx/ffaOM1XX1yh036qCUk0qPdS4dqcPeRlqosRsI9Xqe6sqJrcUD8Sw1ejJ19wUZ8XI0WCdyulRWpKV007GRs7ltSS8Oe/14GIclzoOkA310a7vufUGKI9TI0aKEQgVHXEDC8tlHghgNuQPMyRkiHeIKC6xR2zJLMUOSnfE74W6/YyvuvacN6pynFHwNlQL3qKmZZ4j9aSLE2+Ds2b95z4Onv9sx/1nnVZJ1mJL9l+4s8391vhaHMR6HGjRragLjq8s7lzZ0xd9c/FKVo0NDzuMjxg1xHpRnfRZBY0PtvrcyxL+FTM/6fd7ltvjub8kBJ3+O8zzYWR8m8TB58bC/+aQb9Sru4C4GnOcjbOI5tjGi9yk+4hM+RyfR2+hd9P5narTUaW7jtxV9+AFW4aY9</latexit>
<latexit sha1_base64="CdkIUU6wRZtB7sjCMykF2u3+RLU=">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</latexit>
<latexit sha1_base64="CdkIUU6wRZtB7sjCMykF2u3+RLU=">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</latexit>
<latexit sha1_base64="IPYPn5ZESeSSS5EWKQUGOC7lfpg=">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</latexit>
The phase space of LQG per link is partially compact:
The compact phase space LQG proposal:
⇣
e
R
A
, e
R
E
⌘
2 SU(2) ⇥ SU(2)
<latexit sha1_base64="6953ZNA4lMASasbAVideoDL3Vp8=">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</latexit>
<latexit sha1_base64="6953ZNA4lMASasbAVideoDL3Vp8=">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</latexit>
<latexit sha1_base64="6953ZNA4lMASasbAVideoDL3Vp8=">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</latexit>
<latexit sha1_base64="+fdNu4hgoRepBJkMxKA2uZcpx1w=">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</latexit>
It has been justified that the compacatification can be associated with the
q-deformations of the SU(2) group, leading to the upper limit on j.
In the 2+1 case the deformations introduce the IR scale, which is the
cosmological constant: q = ei⇤l2
Pl
<latexit sha1_base64="2nyuevNyoTUk1OqwYqHtMYBkUQg=">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</latexit>
<latexit sha1_base64="2nyuevNyoTUk1OqwYqHtMYBkUQg=">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</latexit>
<latexit sha1_base64="2nyuevNyoTUk1OqwYqHtMYBkUQg=">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</latexit>
<latexit sha1_base64="xU3POQp8IkwtwL1Z3/qJLos/6Q8=">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</latexit>
M. Artymowski and J.M, Quantum Hubble horizon, Eur. Phys. J. C 79 (2019) no.7, 632
The deformations lead to possible solution to the initial homogeneity problem:
(holonomies and fluxes)
6. Loop Quantum Cosmology
In the homogeneous and isotropic case, the holonomy-flux algebra reduces
to the „cylindrical algebra”:
[E, U] = U
<latexit sha1_base64="UnOmbfCvKbf+ke14qKoH2lMnUYU=">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</latexit>
<latexit sha1_base64="UnOmbfCvKbf+ke14qKoH2lMnUYU=">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</latexit>
<latexit sha1_base64="UnOmbfCvKbf+ke14qKoH2lMnUYU=">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</latexit>
<latexit sha1_base64="pNRfvwiVPWNQtqw00nzjRV7WgJQ=">AAACzXicjVHLSsNAFD2Nr1pfVZdugkVwISXpRjdCUQR3VjBtsRZJptM6NC+SiVCqbv0Bt/pb4h/oX3hnjKAW0QlJzpx7z5m593qxL1JpWS8FY2p6ZnauOF9aWFxaXimvrjXTKEsYd1jkR0nbc1Pui5A7Ukift+OEu4Hn85Y3PFTx1jVPUhGFZ3IU827gDkLRF8yVRJ13jnacrrlvOuZluWJVLb3MSWDnoIJ8NaLyMy7QQwSGDAE4QkjCPlyk9HRgw0JMXBdj4hJCQsc5blEibUZZnDJcYof0HdCuk7Mh7ZVnqtWMTvHpTUhpYos0EeUlhNVppo5n2lmxv3mPtae624j+Xu4VECtxRexfus/M/+pULRJ97OkaBNUUa0ZVx3KXTHdF3dz8UpUkh5g4hXsUTwgzrfzss6k1qa5d9dbV8VedqVi1Z3luhjd1Sxqw/XOck6BZq9pW1T6tVeoH+aiL2MAmtmmeu6jjGA045B3iAY94Mk6MzLgx7j5SjUKuWce3Zdy/A3gIkc0=</latexit>
[E, U†
] = U†
<latexit sha1_base64="6u236C5RAXB9QFezpyRlJWIM+r0=">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</latexit>
<latexit sha1_base64="6u236C5RAXB9QFezpyRlJWIM+r0=">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</latexit>
<latexit sha1_base64="6u236C5RAXB9QFezpyRlJWIM+r0=">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</latexit>
<latexit sha1_base64="e09Ffe0xrCrEK1hrx/3U1hKrT94=">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</latexit>
[U, U†
] = 0
<latexit sha1_base64="RR0bSU76aDhdcrEksVyqZAiZ0b4=">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</latexit>
<latexit sha1_base64="RR0bSU76aDhdcrEksVyqZAiZ0b4=">AAAC2HicjVHLSsNAFD2Nr1pf1S7dBIvgQkoigm6EohuXFUxbbGtJ0mkNzYvJRCil4E7c+gNu9YvEP9C/8M4YQS2iE5KcOfeeM3PvdWLfS4RhvOS0mdm5+YX8YmFpeWV1rbi+UU+ilLvMciM/4k3HTpjvhcwSnvBZM+bMDhyfNZzhiYw3rhlPvCg8F6OYdQJ7EHp9z7UFUd1iqWXtWpfjds8eDBifdPQj3egWy0bFUEufBmYGyshWLSo+o40eIrhIEYAhhCDsw0ZCTwsmDMTEdTAmjhPyVJxhggJpU8pilGETO6TvgHatjA1pLz0TpXbpFJ9eTkod26SJKI8TlqfpKp4qZ8n+5j1WnvJuI/o7mVdArMAVsX/pPjP/q5O1CPRxqGrwqKZYMbI6N3NJVVfkzfUvVQlyiImTuEdxTthVys8+60qTqNplb20Vf1WZkpV7N8tN8SZvSQM2f45zGtT3KqZRMc/2y9XjbNR5bGILOzTPA1Rxihos8h7hAY940i60G+1Wu/tI1XKZpoRvS7t/B9X2lkE=</latexit>
<latexit sha1_base64="RR0bSU76aDhdcrEksVyqZAiZ0b4=">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</latexit>
<latexit sha1_base64="cnHTk9HrnF7nVajidzm+Fvfga8w=">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</latexit>
where (U, E) 2 U(1) ⇥ R
<latexit sha1_base64="IzpB4MSQP23IBr4Ui2W7WqIFiNE=">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</latexit>
<latexit sha1_base64="IzpB4MSQP23IBr4Ui2W7WqIFiNE=">AAAC5XicjVHLSsNAFD2Nr1pfVZduBotQQUoigi5FEVxWMbVgpSTptA7mRTIRpHTrzp249Qfc6q+If6B/4Z0xBbWITkhy5tx7zsy91419kUrTfC0YY+MTk1PF6dLM7Nz8QnlxqZFGWeJx24v8KGm6Tsp9EXJbCunzZpxwJ3B9fupe7qv46RVPUhGFJ/I65ueB0wtFV3iOJKpdZlV742CdtUTI7KpFQIqAp6wVOPLCdfvHg3a5YtZMvdgosHJQQb7qUfkFLXQQwUOGABwhJGEfDlJ6zmDBREzcOfrEJYSEjnMMUCJtRlmcMhxiL+nbo91Zzoa0V56pVnt0ik9vQkqGNdJElJcQVqcxHc+0s2J/8+5rT3W3a/q7uVdArMQFsX/phpn/1alaJLrY0TUIqinWjKrOy10y3RV1c/alKkkOMXEKdyieEPa0cthnpjWprl311tHxN52pWLX38twM7+qWNGDr5zhHQWOzZpk162irsruXj7qIFayiSvPcxi4OUYdN3jd4xBOejZ5xa9wZ95+pRiHXLOPbMh4+AD0Xmp8=</latexit>
<latexit sha1_base64="IzpB4MSQP23IBr4Ui2W7WqIFiNE=">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</latexit>
<latexit sha1_base64="sbXSnnPGHmF4tIja13kFciurF8g=">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</latexit>
Employing the fact that , one can equivalently write:
What we consider in what follows is the SU(2) generalization of the
above algebra, so that
[Sx, Sy] = iSz
<latexit sha1_base64="TfaTtnhgjobRuQ332kj0qtN8SB4=">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</latexit>
<latexit sha1_base64="TfaTtnhgjobRuQ332kj0qtN8SB4=">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</latexit>
<latexit sha1_base64="TfaTtnhgjobRuQ332kj0qtN8SB4=">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</latexit>
<latexit sha1_base64="PqwCuxVi4bc0TST3FEaDyljTv/0=">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</latexit>
[ Sz, Sy] = iSx
<latexit sha1_base64="tlmEsutil6w/TX/xEEwEAVbz1/M=">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</latexit>
<latexit sha1_base64="tlmEsutil6w/TX/xEEwEAVbz1/M=">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</latexit>
<latexit sha1_base64="tlmEsutil6w/TX/xEEwEAVbz1/M=">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</latexit>
<latexit sha1_base64="MFRVY5XfOc9ROQOKHua2kMEDA9E=">AAAC1nicjVHLSsNAFD2Nr1pfrS7dBIvgQkvSjW6EohuXldoH1BKSdFoH0yQkE7UW3Ylbf8CtfpL4B/oX3hlTUIvohCRnzr3nzNx7ndDjsTCM14w2NT0zO5edzy0sLi2v5AurjThIIpfV3cALopZjx8zjPqsLLjzWCiNmDxyPNZ3zQxlvXrAo5oF/IoYh6wzsvs973LUFUVa+0N6pWdfbNWvY2de5XrOurHzRKBlq6ZPATEER6aoG+RecoosALhIMwOBDEPZgI6anDRMGQuI6GBEXEeIqznCDHGkTymKUYRN7Tt8+7dop69NeesZK7dIpHr0RKXVskiagvIiwPE1X8UQ5S/Y375HylHcb0t9JvQbECpwR+5dunPlfnaxFoIc9VQOnmkLFyOrc1CVRXZE3179UJcghJE7iLsUjwq5SjvusK02sape9tVX8TWVKVu7dNDfBu7wlDdj8Oc5J0CiXTKNkHpeLlYN01FmsYwNbNM9dVHCEKurkfYlHPOFZa2m32p12/5mqZVLNGr4t7eEDZi+VTA==</latexit>
[ Sz, Sx] = iSy
<latexit sha1_base64="xcQnj3IKJcMU7jmF5FhQjM2t0yE=">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</latexit>
<latexit sha1_base64="xcQnj3IKJcMU7jmF5FhQjM2t0yE=">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</latexit>
<latexit sha1_base64="xcQnj3IKJcMU7jmF5FhQjM2t0yE=">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</latexit>
<latexit sha1_base64="TD7yzAuB2WBWvh7scwExLF4v7wc=">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</latexit>
The associated phase space is .
S2
<latexit sha1_base64="bINhW9wAWDrMgyeV7FavIc94V5M=">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</latexit>
<latexit sha1_base64="bINhW9wAWDrMgyeV7FavIc94V5M=">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</latexit>
<latexit sha1_base64="bINhW9wAWDrMgyeV7FavIc94V5M=">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</latexit>
<latexit sha1_base64="FQdV073B++8mJBySUjDjnu6Lhjs=">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</latexit>
U|'i = ei'
|'i
<latexit sha1_base64="feG20PHt+/O2ArMBR3S++OcJEYg=">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</latexit>
<latexit sha1_base64="feG20PHt+/O2ArMBR3S++OcJEYg=">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</latexit>
<latexit sha1_base64="feG20PHt+/O2ArMBR3S++OcJEYg=">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</latexit>
<latexit sha1_base64="nUzdepdA7p2pYh5nTBPWGaR0Ha8=">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</latexit>
[E, cos '] = i sin '
<latexit sha1_base64="ArEWWdXLtcY0nKzD5di8Ro+QadM=">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</latexit>
<latexit sha1_base64="ArEWWdXLtcY0nKzD5di8Ro+QadM=">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</latexit>
<latexit sha1_base64="ArEWWdXLtcY0nKzD5di8Ro+QadM=">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</latexit>
<latexit sha1_base64="6c7IDnr1TFztEgyO2wMw0p+gCrI=">AAAC63icjVHLLgRBFD3T3u/B0qZiQiyYdNuwkQiRWJKYIZmZSHcpVPQrVdUSEX9gZye2fsCW/xB/wF+4VZp4RKhOd586955Tde+N8lhq4/tPFa+ru6e3r39gcGh4ZHSsOj7R1FmhuGjwLM7UXhRqEctUNIw0sdjLlQiTKBa70cm6je+eCqVllu6Ys1x0kvAolYeSh4ao/epsa2OetXmmWfs0VPmxZB22whaYZG0t0w9yv1rz675b7CcISlBDubay6iPaOEAGjgIJBFIYwjFCaHpaCOAjJ66Dc+IUIeniAhcYJG1BWYIyQmJP6HtEu1bJprS3ntqpOZ0S06tIyTBDmozyFGF7GnPxwjlb9jfvc+dp73ZG/6j0Sog1OCb2L9175n91thaDQyy7GiTVlDvGVsdLl8J1xd6cfarKkENOnMUHFFeEuVO+95k5jXa1296GLv7sMi1r97zMLfBib0kDDr6P8ydoLtYDvx5sL9ZW18pR92MK05ijeS5hFZvYQoO8L3GHezx4iXflXXs3b6lepdRM4svybl8BKvSc1Q==</latexit>
[E, sin '] = i cos '
<latexit sha1_base64="QCiR9XdiI3qUegFXNexZeViGRAg=">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</latexit>
<latexit sha1_base64="QCiR9XdiI3qUegFXNexZeViGRAg=">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</latexit>
<latexit sha1_base64="QCiR9XdiI3qUegFXNexZeViGRAg=">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</latexit>
<latexit sha1_base64="zpIY28mD9p47A/6hmprfqY4oz6w=">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</latexit>
[cos ', sin '] = 0
<latexit sha1_base64="KiKNCC/QhNBJrj7vdJTsQ4tyfEk=">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</latexit>
<latexit sha1_base64="KiKNCC/QhNBJrj7vdJTsQ4tyfEk=">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</latexit>
<latexit sha1_base64="KiKNCC/QhNBJrj7vdJTsQ4tyfEk=">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</latexit>
<latexit sha1_base64="0+RerXOD/NAHY7TH+dpPpZup/L4=">AAAC5nicjVHLSsNAFD3G97vq0s1oEVxISbrRjSC6calgVWhLScbRDqZJmJkUirh2507c+gNu9VPEP9C/8M6YglpEJyQ599x7zsydG2Wx1Mb3X4e84ZHRsfGJyanpmdm5+dLC4rFOc8VFjadxqk6jUItYJqJmpInFaaZE2IlicRJd7tn8SVcoLdPkyPQy0eyEF4k8lzw0RLVKK/UGTzVrdEOVteUGa2iZ9CPWZNvMb5XKfsV3iw2CoABlFOsgLb2ggTOk4MjRgUACQzhGCE1PHQF8ZMQ1cUWcIiRdXuAaU6TNqUpQRUjsJX0vKKoXbEKx9dROzWmXmF5FSoY10qRUpwjb3ZjL587Zsr95XzlPe7Ye/aPCq0OsQZvYv3T9yv/qbC8G59hyPUjqKXOM7Y4XLrm7FXty9qUrQw4ZcRafUV4R5k7Zv2fmNNr1bu82dPk3V2lZG/OiNse7PSUNOPg5zkFwXK0EfiU4rJZ3dotRT2AZq1ineW5iB/s4QI28b/CIJzx7be/Wu/PuP0u9oUKzhG/Le/gA1A6bmA==</latexit>
7. Spherical phase space
2-sphere is a phase space of spin - SU(2)
variable. The 2-sphere is not a cotangent
bundle, but is equipped with symplectic
form and is a well defined symplectic
manifold.
Change of variables
R1R2 = S
<latexit sha1_base64="bIZhG1QqI09pAWDU9o2gQaghXbU=">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</latexit>
<latexit sha1_base64="bIZhG1QqI09pAWDU9o2gQaghXbU=">AAACzXicjVHLSsNAFD2Nr1pfVZdugkVwVZIi6EYounFnrfaBtZQkndahaRKSiVCqbv0Bt/pb4h/oX3hnnIJaRCckOXPuPWfm3utGPk+EZb1mjJnZufmF7GJuaXlldS2/vlFPwjT2WM0L/TBuuk7CfB6wmuDCZ80oZs7Q9VnDHRzLeOOGxQkPgwsxilh76PQD3uOeI4i6rHbsaqdkHprnnXzBKlpqmdPA1qAAvSph/gVX6CKEhxRDMAQQhH04SOhpwYaFiLg2xsTFhLiKM9whR9qUshhlOMQO6NunXUuzAe2lZ6LUHp3i0xuT0sQOaULKiwnL00wVT5WzZH/zHitPebcR/V3tNSRW4JrYv3STzP/qZC0CPRyoGjjVFClGVudpl1R1Rd7c/FKVIIeIOIm7FI8Je0o56bOpNImqXfbWUfE3lSlZufd0bop3eUsasP1znNOgXiraVtE+2yuUj/Sos9jCNnZpnvso4wQV1Mg7wCOe8GycGqlxa9x/phoZrdnEt2U8fADRGJH0</latexit>
<latexit sha1_base64="bIZhG1QqI09pAWDU9o2gQaghXbU=">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</latexit>
<latexit sha1_base64="EQQGZxdPbWiCIMWfS4rffAw+7yo=">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</latexit>
Dimensional parameters
Symplectic form on 2-sphere:
Volume of phase space:
V =
Z
S2
! = 4⇡S
<latexit sha1_base64="kdigxS2y8Wm5xqtlFNneuXSHQ2Q=">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</latexit>
<latexit sha1_base64="kdigxS2y8Wm5xqtlFNneuXSHQ2Q=">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</latexit>
<latexit sha1_base64="kdigxS2y8Wm5xqtlFNneuXSHQ2Q=">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</latexit>
<latexit sha1_base64="SIZSykBRgJFSNceHdzgcKWHXv1c=">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</latexit>
Symplectic form
8. is, therefore, the angular momentum (spin) vector with the norm S.
Employing the above, the su(2) algebra is obtained:
with the following components:
Poisson bracket:
~
S = (Sx, Sy, Sz)
<latexit sha1_base64="m/xY74MHlz+mAfq/RBRpw/5Ip2Q=">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</latexit>
<latexit sha1_base64="m/xY74MHlz+mAfq/RBRpw/5Ip2Q=">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</latexit>
<latexit sha1_base64="m/xY74MHlz+mAfq/RBRpw/5Ip2Q=">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</latexit>
<latexit sha1_base64="TJZ5tBzh19LPZSWaEzYMfbqYv0I=">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</latexit>
~
S
<latexit sha1_base64="llYAybiqQdrYUSZLV8CxWNAxoKM=">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</latexit>
<latexit sha1_base64="llYAybiqQdrYUSZLV8CxWNAxoKM=">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</latexit>
<latexit sha1_base64="llYAybiqQdrYUSZLV8CxWNAxoKM=">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</latexit>
<latexit sha1_base64="TBxPMM/yPmf0T6MMvEYGyXsdj0c=">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</latexit>
One can introduce the vector
{Si, Sj} = ✏ijkSk
<latexit sha1_base64="YX9p7dM1VtrYwrYhlp09dB4VSdM=">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</latexit>
<latexit sha1_base64="YX9p7dM1VtrYwrYhlp09dB4VSdM=">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</latexit>
<latexit sha1_base64="YX9p7dM1VtrYwrYhlp09dB4VSdM=">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</latexit>
<latexit sha1_base64="I7UOLjFAAj//XAWx8T1brYw7Vag=">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</latexit>
9. ! =
m
X
i=1
!i =
m
X
i=1
dpi ^ dqi
<latexit sha1_base64="gMnNUmbUwryRVPRnJll25u4LIuQ=">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</latexit>
<latexit sha1_base64="gMnNUmbUwryRVPRnJll25u4LIuQ=">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</latexit>
<latexit sha1_base64="gMnNUmbUwryRVPRnJll25u4LIuQ=">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</latexit>
<latexit sha1_base64="oqL9Gk9p0S3MMtDwUZms6ml1w+Y=">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</latexit>
The symplectic for the system of m DOFs is:
By performing compactification for every canonical pair the system’s
phase space changes as follows:
R2m
! S2m
<latexit sha1_base64="FXfMmysJO2crOyIiYkyxvrWUq7k=">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</latexit>
<latexit sha1_base64="FXfMmysJO2crOyIiYkyxvrWUq7k=">AAAC8HicjVHLSsNAFD2N73fVpZtgEQShJCLosujGZX30AVZlEsd2MC8mE0VKP8KdO3HrD7jVrxD/QP/CO9MUfCA6IcmZc+85M/deLwlEqhzntWANDY+Mjo1PTE5Nz8zOFecX6mmcSZ/X/DiIZdNjKQ9ExGtKqIA3E8lZ6AW84V3s6HjjkstUxNGhuk74ccjakTgXPlNEnRbXWiFTHc/r7vdOuuthz25J0e4oJmV8ZQ9iB/3YabHklB2z7J/AzUEJ+arGxRe0cIYYPjKE4IigCAdgSOk5ggsHCXHH6BInCQkT5+hhkrQZZXHKYMRe0LdNu6OcjWivPVOj9umUgF5JShsrpIkpTxLWp9kmnhlnzf7m3TWe+m7X9Pdyr5BYhQ6xf+kGmf/V6VoUzrFlahBUU2IYXZ2fu2SmK/rm9qeqFDkkxGl8RnFJ2DfKQZ9to0lN7bq3zMTfTKZm9d7PczO861vSgN3v4/wJ6utl1ym7exulynY+6nEsYRmrNM9NVLCLKmrkfYNHPOHZktatdWfd91OtQq5ZxJdlPXwAzzih0g==</latexit>
<latexit sha1_base64="FXfMmysJO2crOyIiYkyxvrWUq7k=">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</latexit>
<latexit sha1_base64="+cKEhSbu8x1k6ugIVW++eJt129I=">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</latexit>
In consequence, the regularized kinematical Hilbert space for the system is:
Hkin = H⌦m
s
<latexit sha1_base64="krvugiKSYYl+DTn0qt0IhTvyJOY=">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</latexit>
<latexit sha1_base64="krvugiKSYYl+DTn0qt0IhTvyJOY=">AAAC+XicjVHLSsQwFD3W93vUpZviILgaWhF0I4huXI7gjIKjQxqjhumLJhWl9EfcuRO3/oBb/QHxD/QvvIkVfCCa0vbk3HtOcu8N0lAq7XnPfU7/wODQ8Mjo2PjE5NR0bWa2rZI846LFkzDJ9gOmRChj0dJSh2I/zQSLglDsBb0tE987F5mSSbyrL1NxGLHTWJ5IzjRR3dpKJ2L6jLOw2C67RUeLC130ZFyW7rr7OaSOik6iZSSUG5XdWt1reHa5P4FfgTqq1UxqT+jgGAk4ckQQiKEJh2BQ9BzAh4eUuEMUxGWEpI0LlBgjbU5ZgjIYsT36ntLuoGJj2htPZdWcTgnpzUjpYpE0CeVlhM1pro3n1tmwv3kX1tPc7ZL+QeUVEatxRuxfuo/M/+pMLRonWLM1SKoptYypjlcuue2Kubn7qSpNDilxBh9TPCPMrfKjz67VKFu76S2z8RebaViz51VujldzSxqw/32cP0F7ueF7DX9npb6xWY16BPNYwBLNcxUb2EYTLfK+wj0e8OgUzrVz49y+pzp9lWYOX5Zz9wbMVKXx</latexit>
<latexit sha1_base64="krvugiKSYYl+DTn0qt0IhTvyJOY=">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</latexit>
<latexit sha1_base64="Iag4kIhX7mzUoKui3dcbTnqNRPE=">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</latexit>
where , so that:
Hs = span{s, si, ..., |s, si}
<latexit sha1_base64="yPLn8hKWap4K7teOWG7o0fEXUOQ=">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</latexit>
<latexit sha1_base64="yPLn8hKWap4K7teOWG7o0fEXUOQ=">AAADA3icjVFNS91AFD3Gav2o9mmXboY+Cl08QyIF3SjSblwq+FQw8piM4zM4+WBmUpT0LftPuuuudNs/0G0LxX9g/0XvTPOgKlInJDlz7j1n5t6bViozNoquJ4LJJ1PTT2dm5+afLSw+7ywtH5iy1kL2RalKfZRyI1VWyL7NrJJHlZY8T5U8TC/eufjhe6lNVhb79qqSJzkfFtlZJrglatDZSnJuzwVXzc5oYNgmS6y8tI2peDFiSWN6qybRvBgq2WNhGPYY+2B6YyoZDTrdKIz8YvdB3IIu2rVbdn4hwSlKCNTIIVHAElbgMPQcI0aEirgTNMRpQpmPS4wwR9qasiRlcGIv6Duk3XHLFrR3nsarBZ2i6NWkZHhFmpLyNGF3GvPx2js79iHvxnu6u13RP229cmItzon9n26c+Vidq8XiDBu+hoxqqjzjqhOtS+274m7O/qnKkkNFnMOnFNeEhVeO+8y8xvjaXW+5j9/4TMe6vWhza/x2t6QBx3fHeR8crIVxFMZ7b7rbb9tRz2AFL/Ga5rmObexgF33y/oTv+IGfwcfgc/Al+Po3NZhoNS9wawXf/gAmqaex</latexit>
<latexit sha1_base64="yPLn8hKWap4K7teOWG7o0fEXUOQ=">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</latexit>
<latexit sha1_base64="iz+nNgiphiM7VeAF/vg5DllYvtY=">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</latexit>
Ŝ2
|s, szi = s(s + 1)|s, szi
<latexit sha1_base64="No35i7NaID6xXfp8DM5XNqHjGIM=">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</latexit>
<latexit sha1_base64="No35i7NaID6xXfp8DM5XNqHjGIM=">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</latexit>
<latexit sha1_base64="No35i7NaID6xXfp8DM5XNqHjGIM=">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</latexit>
<latexit sha1_base64="R3s46+qWvDocRDolfsve4fnR5BY=">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</latexit>
Ŝz|s, szi = sz|s, szi
<latexit sha1_base64="5yIU9JdBaNxiS272TQL1NZ5ulQA=">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</latexit>
<latexit sha1_base64="5yIU9JdBaNxiS272TQL1NZ5ulQA=">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</latexit>
<latexit sha1_base64="5yIU9JdBaNxiS272TQL1NZ5ulQA=">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</latexit>
<latexit sha1_base64="FtnA9M+wDi07g5/jW3UWXXbNWeE=">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</latexit>