4,235 research outputs found

    (Broken) Gauge Symmetries and Constraints in Regge Calculus

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    We will examine the issue of diffeomorphism symmetry in simplicial models of (quantum) gravity, in particular for Regge calculus. We find that for a solution with curvature there do not exist exact gauge symmetries on the discrete level. Furthermore we derive a canonical formulation that exactly matches the dynamics and hence symmetries of the covariant picture. In this canonical formulation broken symmetries lead to the replacements of constraints by so--called pseudo constraints. These considerations should be taken into account in attempts to connect spin foam models, based on the Regge action, with canonical loop quantum gravity, which aims at implementing proper constraints. We will argue that the long standing problem of finding a consistent constraint algebra for discretized gravity theories is equivalent to the problem of finding an action with exact diffeomorphism symmetries. Finally we will analyze different limits in which the pseudo constraints might turn into proper constraints. This could be helpful to infer alternative discretization schemes in which the symmetries are not broken.Comment: 32 pages, 15 figure

    Curved planar quantum wires with Dirichlet and Neumann boundary conditions

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    We investigate the discrete spectrum of the Hamiltonian describing a quantum particle living in the two-dimensional curved strip. We impose the Dirichlet and Neumann boundary conditions on opposite sides of the strip. The existence of the discrete eigenvalue below the essential spectrum threshold depends on the sign of the total bending angle for the asymptotically straight strips.Comment: 7 page

    On the two-magnon bound states for the quantum Heisenberg chain with variable range exchange

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    The spectrum of finite-difference two-magnon operator is investigated for quantum S=1/2 chain with variable range exchange of the form h(jk)sinh2a(jk)h(j-k)\propto \sinh^{-2}a(j-k). It is found that usual bound state appears for some values of the total pseudomomentum of two magnons as for the Heisenberg chain with nearest-neighbor spin interaction. Besides this state, a new type of bound state with oscillating wave function appears at larger values of the total pseudomomentum.Comment: 8 pages, latex, no figure

    Spectral correlations in systems undergoing a transition from periodicity to disorder

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    We study the spectral statistics for extended yet finite quasi 1-d systems which undergo a transition from periodicity to disorder. In particular we compute the spectral two-point form factor, and the resulting expression depends on the degree of disorder. It interpolates smoothly between the two extreme limits -- the approach to Poissonian statistics in the (weakly) disordered case, and the universal expressions derived for the periodic case. The theoretical results agree very well with the spectral statistics obtained numerically for chains of chaotic billiards and graphs.Comment: 16 pages, Late

    From the discrete to the continuous - towards a cylindrically consistent dynamics

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    Discrete models usually represent approximations to continuum physics. Cylindrical consistency provides a framework in which discretizations mirror exactly the continuum limit. Being a standard tool for the kinematics of loop quantum gravity we propose a coarse graining procedure that aims at constructing a cylindrically consistent dynamics in the form of transition amplitudes and Hamilton's principal functions. The coarse graining procedure, which is motivated by tensor network renormalization methods, provides a systematic approximation scheme towards this end. A crucial role in this coarse graining scheme is played by embedding maps that allow the interpretation of discrete boundary data as continuum configurations. These embedding maps should be selected according to the dynamics of the system, as a choice of embedding maps will determine a truncation of the renormalization flow.Comment: 22 page

    From covariant to canonical formulations of discrete gravity

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    Starting from an action for discretized gravity we derive a canonical formalism that exactly reproduces the dynamics and (broken) symmetries of the covariant formalism. For linearized Regge calculus on a flat background -- which exhibits exact gauge symmetries -- we derive local and first class constraints for arbitrary triangulated Cauchy surfaces. These constraints have a clear geometric interpretation and are a first step towards obtaining anomaly--free constraint algebras for canonical lattice gravity. Taking higher order dynamics into account the symmetries of the action are broken. This results in consistency conditions on the background gauge parameters arising from the lowest non--linear equations of motion. In the canonical framework the constraints to quadratic order turn out to depend on the background gauge parameters and are therefore pseudo constraints. These considerations are important for connecting path integral and canonical quantizations of gravity, in particular if one attempts a perturbative expansion.Comment: 37 pages, 5 figures (minor modifications, matches published version + updated references

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    Regge calculus from a new angle

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    In Regge calculus space time is usually approximated by a triangulation with flat simplices. We present a formulation using simplices with constant sectional curvature adjusted to the presence of a cosmological constant. As we will show such a formulation allows to replace the length variables by 3d or 4d dihedral angles as basic variables. Moreover we will introduce a first order formulation, which in contrast to using flat simplices, does not require any constraints. These considerations could be useful for the construction of quantum gravity models with a cosmological constant.Comment: 8 page
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