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246
Quantum field theory on noncommutative spaces
"... A pedagogical and selfcontained introduction to noncommutative quantum field theory is presented, with emphasis on those properties that are intimately tied to string theory and gravity. Topics covered include the WeylWigner correspondence, noncommutative Feynman diagrams, UV/IR mixing, noncommuta ..."
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Cited by 273 (15 self)
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A pedagogical and selfcontained introduction to noncommutative quantum field theory is presented, with emphasis on those properties that are intimately tied to string theory and gravity. Topics covered include the WeylWigner correspondence, noncommutative Feynman diagrams, UV/IR mixing, noncommutative YangMills theory on infinite space and on the torus, Morita equivalences of noncommutative gauge theories, twisted reduced models, and an indepth study of the gauge group of noncommutative YangMills theory. Some of the more mathematical ideas and
Gauge Theory on Noncommutative Spaces
 Eur. Phys. J. C16
"... We introduce a formulation of gauge theory on noncommutative spaces based on the concept of covariant coordinates. Some important examples are discussed in detail. A SeibergWitten map is established We introduce a natural method to formulate a gauge theory on more or less arbitrary noncommutative s ..."
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Cited by 139 (18 self)
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We introduce a formulation of gauge theory on noncommutative spaces based on the concept of covariant coordinates. Some important examples are discussed in detail. A SeibergWitten map is established We introduce a natural method to formulate a gauge theory on more or less arbitrary noncommutative spaces. The starting point is the observation that multiplication of a (covariant) field by a coordinate can in general not be a covariant operation in noncommutative geometry, because the coordinates
Transport equations for elastic and other waves in random media
 Wave Motion
, 1996
"... We derive and analyze transport equations for the energy density ofwaves of any kind in a random medium. The equations take account of nonuniformities of the background medium, scattering by random inhomogeneities, polarization e ects, coupling of di erent types of waves, etc. We also show that di u ..."
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Cited by 119 (33 self)
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We derive and analyze transport equations for the energy density ofwaves of any kind in a random medium. The equations take account of nonuniformities of the background medium, scattering by random inhomogeneities, polarization e ects, coupling of di erent types of waves, etc. We also show that di usive behavior occurs on long time and distance scales and we determine the di usion coe cients. The results are specialized to acoustic, electromagnetic, and elastic waves. The analysis is based on the governing equations of motion and uses the Wigner distribution.
Enveloping algebra valued gauge transformations for nonabelian gauge groups on noncommutative spaces
, 2000
"... ..."
Unified Green’s function retrieval by crosscorrelation; connection with energy principles
"... It has been shown theoretically and observationally that the Green’s function for acoustic and elastic waves can be retrieved by crosscorrelating fluctuations recorded at two locations. We extend the concept of the extraction of the Green’s function to a wide class of scalar linear systems. For sys ..."
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Cited by 40 (21 self)
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It has been shown theoretically and observationally that the Green’s function for acoustic and elastic waves can be retrieved by crosscorrelating fluctuations recorded at two locations. We extend the concept of the extraction of the Green’s function to a wide class of scalar linear systems. For systems that are not invariant under time reversal, the fluctuations must be excited by volume sources in order to satisfy the energy balance (equipartitioning) that is needed to extract the Green’s function. The general theory for retrieving the Green’s function is illustrated with examples that include the diffusion equation, Schrödinger’s equation, a vibrating string, the acoustic wave equation, a vibrating beam, and the advection equation. Examples are also shown of situations where the Green’s function cannot be extracted from ambient fluctuations. The general theory opens up new applications for the extraction of the Green’s function from field correlations that include flow in porous media, quantum mechanics, and the extraction of the response of mechanical structures such as bridges.
SelfAveraging in Time Reversal for the Parabolic Wave Equation
 Stochastics and Dynamics
, 2002
"... We analyze the selfaveraging properties of timereversed solutions of the paraxial wave equation with random coefficients, which we take to be Markovian in the direction of propagation. ..."
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Cited by 35 (16 self)
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We analyze the selfaveraging properties of timereversed solutions of the paraxial wave equation with random coefficients, which we take to be Markovian in the direction of propagation.
MultiPhase Computations Of The Semiclassical Limit Of The Schrödinger Equation And Related Problems: Whitham Vs Wigner
 Wigner, Physica D
"... We present and compare two different techniques to obtain the multiphase solutions for the Schrödinger equation in the semiclassical limit. The first is Whitham's averaging method, which gives the modulation equations governing the evolution of multiphase solutions. The second is the Wigner transf ..."
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Cited by 33 (14 self)
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We present and compare two different techniques to obtain the multiphase solutions for the Schrödinger equation in the semiclassical limit. The first is Whitham's averaging method, which gives the modulation equations governing the evolution of multiphase solutions. The second is the Wigner transform, a convenient tool to derive the semiclassical limit equation in the phase space (the Vlasov equation) for the linear Schrödinger equation. Motivated by the linear superposition principle, we derive and prove the multiphase ansatz for the Wigner function by the stationary phase method, and then use the ansatz to close the moment equations of the Vlasov equation and obtain the multiphase equations in the physical space. We show that the multiphase equations so derived agree with those derived by Whitham's averaging method, which can be proved using different arguments. Generic way of obtaining and computing the multiphase equations by the Wigner function is given, and kinetic schemes are introduced to solve the multiphase equations. The numerical schemes are purely Eulerian and only operate in the physical space. Several numerical examples are given to explore the validity of this approach. Similar studies are conducted for the linearized Kortewegde Vries equation and the linear wave equation.
The spectral action for Moyal planes
 J. Math. Phys
"... Extending a result of D. V. Vassilevich [50], we obtain the asymptotic expansion for the trace of a spatially regularized heat operator LΘ (f)e−t△Θ, where △Θ is a generalized Laplacian defined with Moyal products and LΘ (f) is Moyal left multiplication. The Moyal planes corresponding to any skewsymm ..."
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Cited by 31 (6 self)
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Extending a result of D. V. Vassilevich [50], we obtain the asymptotic expansion for the trace of a spatially regularized heat operator LΘ (f)e−t△Θ, where △Θ is a generalized Laplacian defined with Moyal products and LΘ (f) is Moyal left multiplication. The Moyal planes corresponding to any skewsymmetric matrix Θ being spectral triples [24], the spectral action introduced in noncommutative geometry by A. Chamseddine and A. Connes [6] is computed. This result generalizes the ConnesLott action [15] previously computed by Gayral [23] for symplectic Θ.
4D frequency analysis of computational cameras for depth of field extension
 MIT CSAIL TR
, 2009
"... latticefocal lens. The defocus kernel of this lens is designed to preserve high frequencies over a wide depth range. Right: An allfocused image processed from the latticefocal lens input. Since the defocus kernel preserves high frequencies, we achieve a good restoration over the full depth range. ..."
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Cited by 27 (5 self)
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latticefocal lens. The defocus kernel of this lens is designed to preserve high frequencies over a wide depth range. Right: An allfocused image processed from the latticefocal lens input. Since the defocus kernel preserves high frequencies, we achieve a good restoration over the full depth range. Depth of field (DOF), the range of scene depths that appear sharp in a photograph, poses a fundamental tradeoff in photography— wide apertures are important to reduce imaging noise, but they also increase defocus blur. Recent advances in computational imaging modify the acquisition process to extend the DOF through deconvolution. Because deconvolution quality is a tight function of the frequency power spectrum of the defocus kernel, designs with high spectra are desirable. In this paper we study how to design effective extendedDOF systems, and show an upper bound on the maximal power spectrum that can be achieved. We analyze defocus kernels in the 4D light field space and show that in the frequency domain, only a lowdimensional 3D manifold contributes to focus. Thus, to maximize the defocus spectrum, imaging systems should concentrate their limited energy on this manifold. We review several computational imaging systems and show either that they spend energy outside the focal manifold or do not achieve a high spectrum over the DOF. Guided by this analysis we introduce the latticefocal lens, which concentrates energy at the lowdimensional focal manifold and achieves a higher power spectrum than previous designs. We have built a prototype latticefocal lens and present extended depth of field results.