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Juan F Pedraza - One of the best experts on this subject based on the ideXlab platform.

  • bit threads einstein s equations and bulk locality
    Journal of High Energy Physics, 2021
    Co-Authors: Cesar A Agon, Elena Caceres, Juan F Pedraza
    Abstract:

    In the context of holography, entanglement entropy can be studied either by i) extremal surfaces or ii) bit threads, i.e., divergenceless vector fields with a norm bound set by the Planck length. In this paper we develop a new method for metric reconstruction based on the latter approach and show the advantages over existing ones. We start by studying general Linear perturbations around the vacuum state. Generic thread configurations turn out to encode the information about the metric in a highly nonlocal way, however, we show that for boundary regions with a local modular Hamiltonian there is always a canonical choice for the perturbed thread configurations that exploits bulk locality. To do so, we express the bit thread formalism in terms of Differential forms so that it becomes manifestly background independent. We show that the Iyer-Wald formalism provides a natural candidate for a canonical local perturbation, which can be used to recast the problem of metric reconstruction in terms of the inversion of a particular Linear Differential Operator. We examine in detail the inversion problem for the case of spherical regions and give explicit expressions for the inverse Operator in this case. Going beyond Linear order, we argue that the Operator that must be inverted naturally increases in order. However, the inversion can be done recursively at different orders in the perturbation. Finally, we comment on an alternative way of reconstructing the metric non-perturbatively by phrasing the inversion problem as a particular optimization problem.

Dmitri Vassiliev - One of the best experts on this subject based on the ideXlab platform.

  • analytic definition of spin structure
    arXiv: Differential Geometry, 2016
    Co-Authors: Zhirayr Avetisyan, Yanlong Fang, Nikolai Saveliev, Dmitri Vassiliev
    Abstract:

    We work on a parallelizable time-orientable Lorentzian 4-manifold and prove that in this case the notion of spin structure can be equivalently defined in a purely analytic fashion. Our analytic definition relies on the use of the concept of a non-degenerate two-by-two formally self-adjoint first order Linear Differential Operator and gauge transformations of such Operators. We also give an analytic definition of spin structure for the 3-dimensional Riemannian case.

  • A non-geometric representation of the Dirac equation in curved spacetime
    2016
    Co-Authors: Yanlong Fang, Dmitri Vassiliev
    Abstract:

    The talk is an attempt at developing a relativistic field theory based on the concepts from the analysis of partial Differential equations as opposed to geometric concepts. The long-term goal is to recast quantum electrodynamics in curved spacetime in such ‘non-geometric ’ terms. The potential advantage of formulating a field theory in ‘analytic ’ terms is that there might be a chance of describing the interaction of different physical fields in a more consistent, and, hopefully, non-perturbative manner. Consider a formally self-adjoint first order Linear Differential Operator act-ing on pairs (two-columns) of complex-valued scalar fields over a four-manifold without boundary. We examine the geometric content of such an Operator and show that it implicitly contains a Lorentzian metric, Pauli matrices, connection coefficients for spinor fields and an electromagnetic covector potential. This ob-servation allows us to give a simple representation of the massive Dirac equation as a system of four scalar equations involving an arbitrary two-by-two matrix Operator as above and its adjugate. The point of the talk is that in order to write down the Dirac equation in the physically meaningful four-dimensional hy-perbolic setting one does not need any geometric constructs. All the geometry required is contained in a single analytic object — an abstract formally self-adjoint first order Linear Differential Operator acting on pairs of complex-valued scalar fields. The talk is based on the paper [1]

  • analysis as a source of geometry a non geometric representation of the dirac equation
    Journal of Physics A, 2015
    Co-Authors: Yanlong Fang, Dmitri Vassiliev
    Abstract:

    Consider a formally self-adjoint first order Linear Differential Operator acting on pairs (two-columns) of complex-valued scalar fields over a four-manifold without boundary. We examine the geometric content of such an Operator and show that it implicitly contains a Lorentzian metric, Pauli matrices, connection coefficients for spinor fields and an electromagnetic covector potential. This observation allows us to give a simple representation of the massive Dirac equation as a system of four scalar equations involving an arbitrary two-by-two matrix Operator as above and its adjugate. The point of the paper is that in order to write down the Dirac equation in the physically meaningful four-dimensional hyperbolic setting one does not need any geometric constructs. All the geometry required is contained in a single analytic object—an abstract formally self-adjoint first order Linear Differential Operator acting on pairs of complex-valued scalar fields.

James R Fienup - One of the best experts on this subject based on the ideXlab platform.

  • holography with extended reference by autocorrelation Linear Differential operation
    Optics Express, 2007
    Co-Authors: Manuel Guizarsicairos, James R Fienup
    Abstract:

    We introduce a generalization of Fourier transform holography that allows the use of the boundary waves of an extended object to act as a holographic-like reference. By applying a Linear Differential Operator on the field autocorrelation, we use a sharp feature on the extended reference to reconstruct a complex-valued image of the object of interest in a single-step computation. We generalize the approach of Podorov et al. [Opt. Express 15, 9954 (2007)] to a much wider class of extended reference objects. Effects of apertures in Fourier domain and imperfections in the reference object are analyzed. Realistic numerical simulations show the feasibility of our approach and its robustness against noise.

Rajkumar Roychoudhury - One of the best experts on this subject based on the ideXlab platform.

  • position dependent mass schrodinger equation and isospectral potentials intertwining Operator approach
    Journal of Mathematical Physics, 2010
    Co-Authors: Bikashkali Midya, Barnana Roy, Rajkumar Roychoudhury
    Abstract:

    Here, we have studied first- and second-order intertwining approaches to generate isospectral partner potentials of position dependent (effective) mass Schrodinger equation. The second-order intertwiner is constructed directly by taking it as second-order Linear Differential Operator with position dependent coefficients, and the system of equations arising from the intertwining relationship is solved for the coefficients by taking an ansatz. A complete scheme for obtaining general solution is obtained, which is valid for any arbitrary potential and mass function. The proposed technique allows us to generate isospectral potentials with the following spectral modifications: (i) to add new bound state(s), (ii) to remove bound state(s), and (iii) to leave the spectrum unaffected. To explain our findings with the help of an illustration, we have used point canonical transformation to obtain the general solution of the position dependent mass Schrodinger equation corresponding to a potential and mass function. ...

  • position dependent mass schroedinger equation and isospectral potentials intertwining Operator approach
    arXiv: Mathematical Physics, 2010
    Co-Authors: Bikashkali Midya, Barnana Roy, Rajkumar Roychoudhury
    Abstract:

    Here we have studied first and second-order intertwining approach to generate isospectral partner potentials of position-dependent (effective) mass Schroedinger equation. The second-order intertwiner is constructed directly by taking it as second order Linear Differential Operator with position depndent coefficients and the system of equations arising from the intertwining relationship is solved for the coefficients by taking an ansatz. A complete scheme for obtaining general solution is obtained which is valid for any arbitrary potential and mass function. The proposed technique allows us to generate isospectral potentials with the following spectral modifications: (i) to add new bound state(s), (ii) to remove bound state(s) and (iii) to leave the spectrum unaffected. To explain our findings with the help of an illustration, we have used point canonical transformation (PCT) to obtain the general solution of the position dependent mass Schrodinger equation corresponding to a potential and mass function. It is shown that our results are consistent with the formulation of type A N-fold supersymmetry [14,18] for the particular case N = 1 and N = 2 respectively.

Sinei Takahasi - One of the best experts on this subject based on the ideXlab platform.