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Quantum isotropy and dynamical quantum symmetry reduction
Title
Quantum isotropy and dynamical quantum symmetry reduction.
Creator
Hogan, Matthew, Beetle, Christopher, Engle, Jonathan S., Graduate College, Mendonca, P.
Abstract/Description
We give a diffeomorphism and gauge covariant condition equivalent to homogeneity and isotropy which can be quantized, yielding a definition of a diffeomorphism-invariant, homogeneous isotropic sector of loop quantum gravity without fixing a graph. We then specialize this condition to Bianchi I cosmologies, in which case it becomes a condition for isotropy. We show how, by quantizing and imposing this condition in Bianchi I loop quantum cosmology, one exactly recovers isotropic loop quantum...
Show moreWe give a diffeomorphism and gauge covariant condition equivalent to homogeneity and isotropy which can be quantized, yielding a definition of a diffeomorphism-invariant, homogeneous isotropic sector of loop quantum gravity without fixing a graph. We then specialize this condition to Bianchi I cosmologies, in which case it becomes a condition for isotropy. We show how, by quantizing and imposing this condition in Bianchi I loop quantum cosmology, one exactly recovers isotropic loop quantum cosmology, including the usual ‘improved dynamics.’ We will also discuss how this reduction sheds light on which operator ordering to use when defining operators corresponding to directional Hubble rates, expansion, and shear quantities relevant for discussing the resolution of the initial singularity.
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Date Issued
2014
PURL
http://purl.flvc.org/fau/fd/FA00005821
Format
Document (PDF)
Exploring the stability of an eigenvalue problem approximation technique used to define the angular momentum of almost spherical black holes
Title
Exploring the stability of an eigenvalue problem approximation technique used to define the angular momentum of almost spherical black holes.
Creator
Wilder, Shawn M., Beetle, Christopher, Graduate College
Date Issued
2011-04-08
PURL
http://purl.flvc.org/fcla/dt/3164804
Subject Headings
Eigenvalues, Black holes (Astronomy), Deformations (Mechanics)
Format
Document (PDF)
Approximate Isometries as an Eigenvalue Problem and Angular Momentum
Title
Approximate Isometries as an Eigenvalue Problem and Angular Momentum.
Creator
Wilder, Shawn M., Beetle, Christopher, Graduate College
Date Issued
2013-04-12
PURL
http://purl.flvc.org/fcla/dt/3361373
Subject Headings
Black holes (Astronomy), Eigenvalues
Format
Document (PDF)
Periodic standing-wave approximation: Eigenspectral computations for linear gravity and nonlinear toy models
Title
Periodic standing-wave approximation: Eigenspectral computations for linear gravity and nonlinear toy models.
Creator
Beetle, Christopher, Bromley, Benjamin, Price, Richard H.
Date Issued
2006-07-13
PURL
http://purl.flvc.org/fau/flvc_fau_islandoraimporter_10.1103_PhysRevD.74.024013_1632234668
Format
Citation
Generic isolated horizons in loop quantum gravity
Title
Generic isolated horizons in loop quantum gravity.
Creator
Beetle, Christopher, Engle, Jonathan
Date Issued
2010-11-15
PURL
http://purl.flvc.org/fau/flvc_fau_islandoraimporter_10.1088_0264-9381_27_23_235024_1632236016
Format
Citation
General relativistic quasi-local angular momentum continuity and the stability of strongly elliptic eigenvalue problems
Title
General relativistic quasi-local angular momentum continuity and the stability of strongly elliptic eigenvalue problems.
Creator
Wilder, Shawn M., Beetle, Christopher, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Physics
Abstract/Description
In general relativity, angular momentum of the gravitational field in some volume bounded by an axially symmetric sphere is well-defined as a boundary integral. The definition relies on the symmetry generating vector field, a Killing field, of the boundary. When no such symmetry exists, one defines angular momentum using an approximate Killing field. Contained in the literature are various approximations that capture certain properties of metric preserving vector fields. We explore the...
Show moreIn general relativity, angular momentum of the gravitational field in some volume bounded by an axially symmetric sphere is well-defined as a boundary integral. The definition relies on the symmetry generating vector field, a Killing field, of the boundary. When no such symmetry exists, one defines angular momentum using an approximate Killing field. Contained in the literature are various approximations that capture certain properties of metric preserving vector fields. We explore the continuity of an angular momentum definition that employs an approximate Killing field that is an eigenvector of a particular second-order differential operator. We find that the eigenvector varies continuously in Hilbert space under smooth perturbations of a smooth boundary geometry. Furthermore, we find that not only is the approximate Killing field continuous but that the eigenvalue problem which defines it is stable in the sense that all of its eigenvalues and eigenvectors are continuous in Hilbert space. We conclude that the stability follows because the eigenvalue problem is strongly elliptic. Additionally, we provide a practical introduction to the mathematical theory of strongly elliptic operators and generalize the above stability results for a large class of such operators.
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Date Issued
2014
PURL
http://purl.flvc.org/fau/fd/FA00004235
Subject Headings
Boundary element methods, Boundary value problems, Differential equations, Elliptic -- Numerical solutions, Differential equations, Partial -- Numerical solutions, Eigenvalues, Spectral theory (Mathematics)
Format
Document (PDF)
Improving In Vivo Two Photon Microscopy Without Adaptive Optics
Title
Improving In Vivo Two Photon Microscopy Without Adaptive Optics.
Creator
Estrada, Gerardo, Beetle, Christopher, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Physics
Abstract/Description
Two photon microscopy is one of the fastest growing methods of in-vivo imaging of the brain. It has the capability of imaging structures on the scale of 1μm. At this scale the wavelength of the imaging field (usually near infra-red), is comparable to the size of the structures being imaged, which makes the use of ray optics invalid. A better understanding is needed to predict the result of introducing different media into the light path. We use Wolf's integral, which is capable of fulfilling...
Show moreTwo photon microscopy is one of the fastest growing methods of in-vivo imaging of the brain. It has the capability of imaging structures on the scale of 1μm. At this scale the wavelength of the imaging field (usually near infra-red), is comparable to the size of the structures being imaged, which makes the use of ray optics invalid. A better understanding is needed to predict the result of introducing different media into the light path. We use Wolf's integral, which is capable of fulfilling these needs without the shortcomings of ray optics. We predict the effects of aberrating media introduced into the light path like glass cover-slips and then correct the aberration using the same method. We also create a method to predict aberrations when the interfaces of the media in the light-path are not aligned with the propagation direction of the wavefront.
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Date Issued
2015
PURL
http://purl.flvc.org/fau/fd/FA00004495
Subject Headings
Cellular signal transmission -- Measurement, Image analysis, Imaging systems in medicine, Membranes (Biology) -- Imaging, Neurons -- Imaging, Optics, Adaptive
Format
Document (PDF)
ON THE PARAXIAL APPROXIMATION IN QUANTUM OPTICS
Title
ON THE PARAXIAL APPROXIMATION IN QUANTUM OPTICS.
Creator
Jongewaard, Maria Fernanda de Boer, Beetle, Christopher, Florida Atlantic University, Department of Physics, Charles E. Schmidt College of Science
Abstract/Description
We examine how best to associate quantum states of a single particle to modes of a narrowly collimated beam of classical radiation modeled in the paraxial approximation, both for scalar particles and for photons. Our analysis stresses the importance of the relationship between the inner product used to define orthogonal modes of the paraxial beam, on the one hand, and the inner product underlying the statistical interpretation of the quantum theory, on the other. While several candidates for...
Show moreWe examine how best to associate quantum states of a single particle to modes of a narrowly collimated beam of classical radiation modeled in the paraxial approximation, both for scalar particles and for photons. Our analysis stresses the importance of the relationship between the inner product used to define orthogonal modes of the paraxial beam, on the one hand, and the inner product underlying the statistical interpretation of the quantum theory, on the other. While several candidates for such an association have been proposed in the literature, we argue that one of them is uniquely well suited to the task. Specifically, the mapping from beam modes to ”henochromatic” fields on spacetime is unique within a large class of similar mappings in that it is unitary in a mathematically precise sense. We also show that the single-particle quantum states associated to the orthogonal modes of a classical beam in the henochromatic approach are not only orthogonal, but also complete in the quantum Hilbert space.
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Date Issued
2023
PURL
http://purl.flvc.org/fau/fd/FA00014212
Subject Headings
Quantum optics, Hilbert space, Quantum theory
Format
Document (PDF)