Curved planar quantum wires with Dirichlet and Neumann boundary conditions

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

(Broken) Gauge Symmetries and Constraints in Regge Calculus

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

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

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

Gauge invariant perturbations around symmetry reduced sectors of general relativity: applications to cosmology

We develop a gauge invariant canonical perturbation scheme for perturbations around symmetry reduced sectors in generally covariant theories, such as general relativity. The central objects of investigation are gauge invariant observables which encode the dynamics of the system. We apply this scheme to perturbations around a homogeneous and isotropic sector (cosmology) of general relativity. The background variables of this homogeneous and isotropic sector are treated fully dynamically which allows us to approximate the observables to arbitrary high order in a self--consistent and fully gauge invariant manner. Methods to compute these observables are given. The question of backreaction effects of inhomogeneities onto a homogeneous and isotropic background can be addressed in this framework. We illustrate the latter by considering homogeneous but anisotropic Bianchi--I cosmologies as perturbations around a homogeneous and isotropic sector.Comment: 39 pages, 1 figur

Regge calculus from a new angle

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

Lamm, Valluri, Jentschura and Weniger comment on "A Convergent Series for the QED Effective Action" by Cho and Pak [Phys. Rev. Lett. vol. 86, pp. 1947-1950 (2001)]

Complete results were obtained by us in [Can. J. Phys. 71, 389 (1993)] for convergent series representations of both the real and the imaginary part of the QED effective action; these derivations were based on correct intermediate steps. In this comment, we argue that the physical significance of the "logarithmic correction term" found by Cho and Pak in [Phys. Rev. Lett. 86, 1947 (2001)] in comparison to the usual expression for the QED effective action remains to be demonstrated. Further information on related subjects can be found in Appendix A of hep-ph/0308223 and in hep-th/0210240.Comment: 1 page, RevTeX; only "meta-data" update

Manifestly Gauge-Invariant General Relativistic Perturbation Theory: II. FRW Background and First Order

In our companion paper we identified a complete set of manifestly gauge-invariant observables for general relativity. This was possible by coupling the system of gravity and matter to pressureless dust which plays the role of a dynamically coupled observer. The evolution of those observables is governed by a physical Hamiltonian and we derived the corresponding equations of motion. Linear perturbation theory of those equations of motion around a general exact solution in terms of manifestly gauge invariant perturbations was then developed. In this paper we specialise our previous results to an FRW background which is also a solution of our modified equations of motion. We then compare the resulting equations with those derived in standard cosmological perturbation theory (SCPT). We exhibit the precise relation between our manifestly gauge-invariant perturbations and the linearly gauge-invariant variables in SCPT. We find that our equations of motion can be cast into SCPT form plus corrections. These corrections are the trace that the dust leaves on the system in terms of a conserved energy momentum current density. It turns out that these corrections decay, in fact, in the late universe they are negligible whatever the value of the conserved current. We conclude that the addition of dust which serves as a test observer medium, while implying modifications of Einstein's equations without dust, leads to acceptable agreement with known results, while having the advantage that one now talks about manifestly gauge-invariant, that is measurable, quantities, which can be used even in perturbation theory at higher orders.Comment: 51 pages, no figure

Spectral Statistics in Chaotic Systems with Two Identical Connected Cells

Chaotic systems that decompose into two cells connected only by a narrow channel exhibit characteristic deviations of their quantum spectral statistics from the canonical random-matrix ensembles. The equilibration between the cells introduces an additional classical time scale that is manifest also in the spectral form factor. If the two cells are related by a spatial symmetry, the spectrum shows doublets, reflected in the form factor as a positive peak around the Heisenberg time. We combine a semiclassical analysis with an independent random-matrix approach to the doublet splittings to obtain the form factor on all time (energy) scales. Its only free parameter is the characteristic time of exchange between the cells in units of the Heisenberg time.Comment: 37 pages, 15 figures, changed content, additional autho

Scattering through a straight quantum waveguide with combined boundary conditions

Scattering through a straight two-dimensional quantum waveguide Rx(0,d) with Dirichlet boundary conditions on (-\infty,0)x{y=0} \cup (0,\infty)x{y=d} and Neumann boundary condition on (-infty,0)x{y=d} \cup (0,\infty)x{y=0} is considered using stationary scattering theory. The existence of a matching conditions solution at x=0 is proved. The use of stationary scattering theory is justified showing its relation to the wave packets motion. As an illustration, the matching conditions are also solved numerically and the transition probabilities are shown.Comment: 26 pages, 3 figure

Signature of Chaotic Diffusion in Band Spectra

We investigate the two-point correlations in the band spectra of spatially periodic systems that exhibit chaotic diffusion in the classical limit. By including level pairs pertaining to non-identical quasimomenta, we define form factors with the winding number as a spatial argument. For times smaller than the Heisenberg time, they are related to the full space-time dependence of the classical diffusion propagator. They approach constant asymptotes via a regime, reflecting quantal ballistic motion, where they decay by a factor proportional to the number of unit cells. We derive a universal scaling function for the long-time behaviour. Our results are substantiated by a numerical study of the kicked rotor on a torus and a quasi-one-dimensional billiard chain.Comment: 8 pages, REVTeX, 5 figures (eps