Holographic description of boundary gravitons in (3+1) dimensions

Gravity is uniquely situated in between classical topological field theories and standard local field theories. This can be seen in the the quasi-local nature of gravitational observables, but is nowhere more apparent than in gravity's holographic formulation. Holography holds promise for simplifying computations in quantum gravity. While holographic descriptions of three-dimensional spacetimes and of spacetimes with a negative cosmological constant are well-developed, a complete boundary description of zero curvature, four-dimensional spacetime is not currently available. Building on previous work in three-dimensions, we provide a new route to four-dimensional holography and its boundary gravitons. Using Regge calculus linearized around a flat Euclidean background with the topology of a solid hyper-torus, we obtain the effective action for a dual boundary theory which describes the dynamics of the boundary gravitons. Remarkably, in the continuum limit and at large radii this boundary theory is local and closely analogous to the corresponding result in three-dimensions. The boundary effective action has a degenerate kinetic term that leads to singularities in the one-loop partition function that are independent of the discretization. These results establish a rich boundary dynamics for four-dimensional flat holography.Comment: 43 pages, 3 figures, 1 tabl

A perturbative approach to Dirac observables and their space-time algebra

We introduce a general approximation scheme in order to calculate gauge invariant observables in the canonical formulation of general relativity. Using this scheme we will show how the observables and the dynamics of field theories on a fixed background or equivalently the observables of the linearized theory can be understood as an approximation to the observables in full general relativity. Gauge invariant corrections can be calculated up to an arbitrary high order and we will explicitly calculate the first non--trivial correction. Furthermore we will make a first investigation into the Poisson algebra between observables corresponding to fields at different space--time points and consider the locality properties of the observables.Comment: 23 page

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

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

Algebraic Quantum Gravity (AQG) III. Semiclassical Perturbation Theory

In the two previous papers of this series we defined a new combinatorical approach to quantum gravity, Algebraic Quantum Gravity (AQG). We showed that AQG reproduces the correct infinitesimal dynamics in the semiclassical limit, provided one incorrectly substitutes the non -- Abelean group SU(2) by the Abelean group $U(1)^3$ in the calculations. The mere reason why that substitution was performed at all is that in the non -- Abelean case the volume operator, pivotal for the definition of the dynamics, is not diagonisable by analytical methods. This, in contrast to the Abelean case, so far prohibited semiclassical computations. In this paper we show why this unjustified substitution nevertheless reproduces the correct physical result: Namely, we introduce for the first time semiclassical perturbation theory within AQG (and LQG) which allows to compute expectation values of interesting operators such as the master constraint as a power series in $\hbar$ with error control. That is, in particular matrix elements of fractional powers of the volume operator can be computed with extremely high precision for sufficiently large power of $\hbar$ in the $\hbar$ expansion. With this new tool, the non -- Abelean calculation, although technically more involved, is then exactly analogous to the Abelean calculation, thus justifying the Abelean analysis in retrospect. The results of this paper turn AQG into a calculational discipline

Quantum versus Classical Dynamics in a driven barrier: the role of kinematic effects

We study the dynamics of the classical and quantum mechanical scattering of a wave packet from an oscillating barrier. Our main focus is on the dependence of the transmission coefficient on the initial energy of the wave packet for a wide range of oscillation frequencies. The behavior of the quantum transmission coefficient is affected by tunneling phenomena, resonances and kinematic effects emanating from the time dependence of the potential. We show that when kinematic effects dominate (mainly in intermediate frequencies), classical mechanics provides very good approximation of quantum results. Moreover, in the frequency region of optimal agreement between classical and quantum transmission coefficient, the transmission threshold, i.e. the energy above which the transmission coefficient becomes larger than a specific small threshold value, is found to exhibit a minimum. We also consider the form of the transmitted wave packet and we find that for low values of the frequency the incoming classical and quantum wave packet can be split into a train of well separated coherent pulses, a phenomenon which can admit purely classical kinematic interpretation

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