Some components of the graviton propagator have recently been computed in the context of background independent loop quantum gravity. We extend these results to the non-diagonal terms of the propagator. These involve the intertwiners' dependence of the theory's vertex and therefore allow a detailed evaluation of the forms of the vertex considered so far, and of the possible alternatives.
Some new properties of area operator in loop quantum gravity are introduced and their application in the fluctuations of a black hole horizon are discussed.
The usual complexifier coherent states provide a tool for investigating the semiclassical limit of LQG. However, these states are purely kinematical, i.e. do not satisfy any of the constraints. But if one wants to work on the gauge- or Diff-invariant level, coherent states that satisfy the Gauss- and/or the Diff-constraint are desirable. In this talk I will, as a first step into this direction, describe the projection of the complexifier coherent states on the gauge-invariant Hilbert space and demonstrate their semiclassical properties. This will show that these gauge-invariant coherent states are the appropriate tool for adressing semiclassical issues in the gauge-invariant sector.
In the loop approach to quantum gravity, we study large scale correlations of geometric operators on a semiclassical state peaked on a flat geometry. The correlations found can be compared to the two- and three-point functions computed in perturbative quantum gravity on a flat background. The calculations are done using the Riemannian Barrett-Crane model and involve the boundary amplitude formalism. Perturbative area-Regge-calculus plays a key role as an intermediate step in the comparison between correlations computed from the spinfoam model and from perturbative quantum gravity. The calculation presented is to be considered as a first step towards an effective field theoretical description of the semiclassical regime of loop quantum gravity. The presence of large scale correlation is a key test for spinfoam models for quantum gravity, and the details of the calculation provide constraints and hints for possible improvements of the Barrett-Crane model.
We will describe how the particles of the standard model may be formed of composite structures. When these structures are represented as braids, features such as parity, charge conjugation, hypercharge, and weak interactions emerge naturally. This raises the prospect that braided network states in LQG contain emergent particle states which do not need to be put in "by hand", but instead occur as small-scale topological features of spacetime.
A quantum gravity theory associated with a violation of Lorentz invariance faces serious theoretical challenges. This leads us to consider an alternative line of thought for such phenomenological search. We discuss the underlying viewpoint and present the calculation of the dominant part of the measurable effects. A violation of discrete symmetries is also obtained.
We present a Metropolis algorithm for the computation of dual Yang-Mills theory using spin foam configurations, and describe its implementation for pure Yang-Mills in three dimensions and gauge group SU(2). We report results that are in agreement with conventional computations for a range of coupling constants, and discuss the improvements that will be necessary in treating the weak coupling limit. This work provides evidence that spin foam computations can successfully reproduce known physics while offering a compelling, gauge-invariant picture for the evolution of the physical degrees of freedom. We conclude with an outlook for the application of these spin foam algorithms to matter-coupled spin foam quantum gravity, and to other sectors of the Standard Model.
In this talk I will discuss how computational work involving spin networks and spin foams has lead to further understanding and progress in the areas of quantum gravity and lattice gauge theory. I will discuss positivity of the Barrett-Crane model, the vastly differing behaviour of various versions of it, and new computations of the q-deformed model. Then I will mention the asymptotics of the 10j symbol, the key ingredient of the Barrett-Crane model, and will summarize new results on the graviton propagator. I will end with results showing that spin foam methods can be used as a practical way of performing computations in lattice gauge theory.
In the first part of the talk, we present a recent result on 3d SU(2) lattice Yang-Mills theory, showing that it can be cast in the form of an exact string representation. The derivation starts from the spin foam representation of the lattice gauge theory. We demonstrate that every spin foam can be equivalently described as a worldsheet of strings on a framing of the lattice. Using this correspondence, the expectation value of a Wilson loop is translated into a sum over worldsheets that are bounded by strings along a framing of the loop. In the second part of the talk, we take the worldsheet picture as a motivation to discuss a possible approach to renormalization in SU(2) lattice Yang-Mills theory. The Yang-Mills theory is cast in the form of a lattice BF Yang-Mills theory with both the connection A and a 2-form B as variables. Then, a block spin renormlization is proposed where the block variables are a connection and a B-field on a coarser lattice. We suggest a perturbative scheme for the integration over the UV variables.
Recent detailed analysis within the Loop Quantum Gravity calculation of black hole entropy show a stair-like structure in the behavior of entropy as a function of horizon area. The non-trivial distribution of the degeneracy of the black hole horizon area eigenstates is at the origin of this behavior. We will analyze this results and comment the possible implications in the black hole radiation spectrum.
Based on the framework of partial and complete observables we introduce a novel perturbative scheme for perturbations around symmetry reduced sectors of general relativity. This scheme allows the calculation of perturbations in a gauge invariant manner to an arbitrary high order. In particular we can consider backreaction effects. Applications to Loop Quantum Cosmology will be discussed.
It has been suggested that the black hole entropy arises from entanglement between fields inside and outside the black hole horizon. We apply this proposal to loop quantum gravity by computing the entanglement entropy of the spin network states. We find that the entanglement entropy is finite, extensive over the horizon, and agrees with the asymptotic value previously computed for isolated horizons. We suggest possible future implications for the classical limit of loop quantum gravity.
Given a Hamiltonian system with emergent degrees of freedom one can ask the question of how such a system would look like when viewed from a purely internal perspective. The search for an answer to this question I have termed Internal Relativity. I will review in this talk how geometric notions arise here and why a Lorentzian structure is to be expected. The main point of the talk will be to argue that in the light limit Newtonian gravity arises showing in particular that the emergent spacetime is intrinsically curved.
On the basis of our recent modifications of the Dirac formalism we generalize the Bargmann-Wigner formalism for higher spins to be compatible with other formalisms for bosons. Relations with dual electrodynamics, with the Ogievetskii-Polubarinov notoph and the Weinberg 2(2J+1) theory are found. Next, we introduce the dual analogues of the Riemann tensor and derive corresponding dynamical equations in the Minkowski space. Relations with the Marques-Spehler chiral gravity theory are discussed. The problem of indefinite metrics, particularly, in quantization of 4-vector fields is clarified.
In this talk, I raise and discuss conceptual questions that arise when relating loop quantum cosmology (LQC) to loop quantum gravity (LQG). I then go on to present two proposals for interpreting LQC states in the context of full LQG: The first is an earlier embedding proposed by Bojowald and Kastrup, and the second is a new embedding using holomorphic representations. Consistency with the existing LQC quantization is reviewed, and a prescription for extending these embeddings to the gauge and diffeomorphism invariant levels is considered.
Recently, a topological field theory of brane-like sources coupled to BF theory in arbitrary spacetime dimensions was proposed by Baez and Perez. In this talk, I will discuss various aspects of the resulting four dimensional theory. Firstly, I will concentrate on the physical interpretation of the algebraic variables attached to the matter sources by showing how the theory relates to the Polyakov string on a degenerate background and to cosmic string solutions of general relativity. Secondly, I will present a prescription to regularize the physical inner product of the canonical theory and show how the resulting transition amplitudes are dual to evaluations of Feynman diagrams coupled to three-dimensional quantum gravity. Finally, I will discuss the removal of the regulator from the physical inner product by showing that the transition amplitudes of the theory only depend on the equivalence classes of spatial manifolds and embedded string spin networks up to homeomorphisms.
General relativity being a covariant theory, one expects that the evolution of quantum gravity systems will be described in relational terms. The use of real clocks and measuring rods in quantum mechanics implies a natural loss of unitarity and entanglement. More precisely, even if one assumes that the complete system, including measuring devices, precisely obeys the laws of quantum mechanics when described in terms of an ideal time, the description of the system in terms of real clocks is not unitary. The loss of entanglement is related with a similar phenomenon when real spatial measurements are taken into account. I discuss the implications of these effects for the measurement problem in quantum mechanics.
Gravity coupled to a massless scalar field with full cylindrical symmetry can be exactly quantized by an extension of the techniques used in the quantization of Einstein-Rosen waves. We obtain the quantum Hamiltonian operator and the unitary evolution operator. This system is a useful testbed to discuss a number of issues in quantum general relativity. In this talk we will study two-point functions and radial wave functions for one-particle states. We will observe some interesting effects such as a large probability to find the particles near the axis or how the null geodesics of an emergent metric appear in the semiclassical limit.
We give a combinatorial description of the bosonic string partition function, where the world-sheet is triangulated. In this case the integral over all metrics on a surface is a combinatorial sum over triangulations of the world-sheet surface. The partition function for each topological world-sheet surface is convergent and related to the matrix-tree theorem of combinatorics. Besides we propose that when summing over different topologies, that is considering the general partition function, we still have a convergent sum where the perturbative expansion is dominated by the genus zero surface world-sheet. Finally we consider a finite field description of strings where by finite field we refer to Galois field extensions. Instead of having the real numbers and the complex numbers we consider the theory over finite fields. The description turns out to be purely combinatorial. The world-sheet is described as a finite symmetric space.
We introduce a new top down approach to canonical quantum gravity, called Algebraic Quantum Gravity (AQG): The quantum kinematics of AQG is determined by an abstract *-algebra generated by a countable set of elementary operators labelled by an algebraic graph. The quantum dynamics of AQG is governed by a single Master Constraint operator. While AQG is inspired by Loop Quantum Gravity (LQG), it differs from it because in AQG there is fundamentally no topology or differential structure. The missing information about the topology and differential structure of the spacetime manifold as well as about the background metric to be approximated is supplied by coherent states. Given such data, the corresponding coherent state defines a sector in the Hilbert space of AQG which can be identified with a usual QFT on the given manifold and background. In particular, by means of the introduction of semiclassical perturbation theory for AQG (and LQG) we can show that AQG admits a semiclassical limit whose infinitesimal gauge symmetry agrees with that of General Relativity.
I will introduce concepts from Braided Ribbon Networks leading to the concept of microlocality and isolated substructures. From this I will demonstrate the general ability to translate isolated substructures through Braided Ribbon Networks.
A new procedure for coarse-graining dynamical triangulations is presented. The procedure provides a meaning for the relevant value of observables when 'probing at large scales', e.g. the average scalar curvature. The scheme may also be useful as a starting point for a new type of renormalisation procedure, suitable for CDT quantum gravity. Random Delaunay triangulations have previously been used to produce discretisations of continuous Euclidean manifolds, and the coarse-graining scheme is an extension of this idea, using random simplicial complexes produced from a dynamical triangulation. In order for the coarse-graining process to be useful, it should preserve the properties of the original dynamical triangulation that are relevant when probing at large scales. Some arguments, and evidence from simulations, are presented for this. Hopefully there will also be some results from 3D CDT simulations.
Within a perturbative cosmological regime of loop quantum gravity corrections to effective constraints are computed. This takes into account all inhomogeneous degrees of freedom relevant for scalar metric modes around flat space and results in explicit expressions for modified coefficients and of higher order terms. It also illustrates the role of different scales determining the relative magnitude of corrections. Our results demonstrate that loop quantum gravity has the correct classical limit, at least in its sector of cosmological perturbations around flat space, in the sense of perturbative effective theory.
We numerically study Barrett-Crane models of Riemannian quantum gravity. We have extended the existing numerical techniques to handle q-deformed models and arbitrary space-time triangulations. We present and interpret expectation values of a few selected observables for each model, including a spin-spin correlation function which gives insight into the behaviour of the models. We find the surprising result that, as the deformation parameter q goes to 1 through roots of unity, the limit is discontinuous. Joint work with Dan Christensen.
Quantum graphity is a model based on a complete graph whose purpose is to describe emergent geometry. Its degrees of freedom are associated with edges of a graph and its Hamiltonian is based on Hamiltonians of systems exhibiting string-net condensation. This talk will address the statistical mechanical properties of the system, describe features of the ground state at various temperatures, and use these features to motivate a particular scenario for the very early universe.
A procedure to explicitly construct a reduced quantum system in a full quantum theory without classical reduction and requantization is presented and applied to LQG to obtain a cosmological sector.
Mathematical properties of the quantum constraint and quantum Hamiltonian operators of LQC are studied. Self-adjointness is proven, spectra are characterized. The exact results are compared with the numerical ones.
The message from gravity is clear: our universe is geometric. But gravity does not exist alone -- it interacts with the whole zoo of standard model fields. A unification of these fields requires the gravitational connection to be absorbed as part of a larger connection, along with the intimately intertwined Higgs field and frame of GR. By extending this connection further, one generation of fermions may be derived as BRST ghosts, and the dynamics described by a modified BF theory. This description meshes well with recent approaches quantizing gravity perturbatively around a topological theory. But where could such a big, messy connection come from? As it turns out, precisely this connection matches the geometry of the largest exceptional Lie group -- giving exactly three complete generations of fermions, linked by triality! In this fully unified picture, gravity and all fields of the standard model are described by the pure geometry of a group manifold, with no strings attached.
A long-standing proposal for a deformed loop representation of q-Quantum Gravity is addressed.
After its reduction by gauge fixing, the linearly polarized Gowdy T3 cosmologies can be described in terms of a scalar field propagating on a flat 1+1 dimensional background with the spatial topology of S1, although in the presence of a time-dependent potential. This reduced model is still subject to a homogeneous constraint, which generates S1-translations. The Gowdy cosmologies can then be quantized by introducing a Fock quantization for this scalar field. We prove that such a Fock quantization is unique under the requirements of unitarity of the dynamics and invariance under the gauge group of S1-translations. Furthermore, we complete this uniqueness result by considering other possible scalar field descriptions, resulting from reasonable field reparametrizations of the induced metric of the reduced Gowdy model. In the reduced phase space, these alternate descriptions can be obtained by a time-dependent scaling of the field, the inverse scaling of its canonical momentum, and the possible addition of a linear contribution of the field to this momentum. We then demonstrate that the alternate canonical pairs of fieldlike variables admit a Fock representation which is invariant under the gauge group and allows a unitary implementation of the field dynamics if and only if the scaling of the field is constant in time. For this case of a constant scaling we finally show that, in fact, there exists essentially a unique Fock quantization.
In this talk we review "loop quantum gravity" and the application of the theory to the black hole singularity problem. We consider a semiclassical analysis of "loop quantum black hole" and we calculate the quantum corrected black hole metric. The semiclassical metric is regular and singularity free in contrast to the classical one. By using the new metric we calculate the Hawking temperature, entropy and the mass evaporation process.
Cartan's first and second structure equations together with first and second Bianchi identities can be interpreted as equations of motion for the tetrad, the connection, the torsion, and the curvature. In this sense these equations define by themselves a field theory. Restricting the analysis to 4-dimensional spacetime manifolds (keeping gravity in mind), it is possible to give an action principle of the BF type from which these equations of motion are obtained. The action turns out to be equivalent to a linear combination of the Nieh-Yan, Pontrjagin, and Euler classes, and so the field theory defined by the action is topological. The inclusion of gravity is also discussed.
The Ponzano-Regge model for 3+0-dimensional quantum gravity, in its naive definition, yields infinite results and thus needs regularizing. In this talk we discuss a certain regularization of the model and give examples where the regularization works and where it is ill-defined.
I will discuss an extension of analysis of isotropic universe with massless scalar field to the case with nonvanishing cosmological constant (Lambda). As in Lambda=0 case the constrained system can be treated as free one evolving in scalar field, which thus becomes an emergent time. For negative value of Lambda classical recollapse together with quantum geometric effects (causing bounce at Planckian densities) leads to periodic evolution. For Lambda>0 quantum corrections also lead to bounce. In this case however classical universe reaches an infinite volume for finite emergent time. This leads to question about extensions of evolution and uniqueness of it, answer to which depends on essential self-adjointness of an evolution operator.
This talk is complementary to the one given in plenary by Carlo Rovelli.
Spinfoam theories are hoped to provide the dynamics of non-perturbative loop quantum gravity. But a number of their features remain elusive. The best studied one -the euclidean Barrett-Crane model- does not have the boundary state space needed for this, and there are recent indications that, consequently, it may fail to yield the correct low-energy $n$-point functions. These difficulties can be traced to the SO(4) -> SU(2) gauge fixing and the way certain second class constraints are imposed, arguably incorrectly, strongly. We present an alternative model, that can be derived as a bona fide quantization of a Regge discretization of euclidean general relativity, and where the constraints are imposed weakly. Its state space is a natural subspace of the SO(4) spin-network space and matches the SO(3) hamiltonian spin network space. The model provides a long sought SO(4)-covariant vertex amplitude for loop quantum gravity.
By avoiding the Legendre transform we show that the canonical constraint algebra can be computed without primary constraints, and without gauge fixing. The diffeomorphism and Lorentz algebras are realized exactly, and the Hamiltonian constraint gains extra degrees of freedom. The full constraint algebra is a deformation of the four dimensional de Sitter algebra together with three dimensional diffeomorphisms. We discuss the implications of this formalism for the quantum theory, and in particular for the existence of the Kodama state.
A set of differential operators acting by continuous deformations on path dependent functionals of open and closed curves is introduced. Geometrically, these path operators are interpreted as infinitesimal generators of curves in the base manifold of the gauge theory. They furnish a representation with the action of the group of loops having a fundamental role. We show that the path derivative, which is covariant by construction, satisfies the Ricci and Bianchi identities. Also, we provide a geometrical derivation of covariant Taylor expansions based on particular deformations of open curves. The formalism includes, as special cases, other path dependent operators such as end point derivatives and area derivatives.
something about diffeomorphism invariant functionals for scalar- and gauge-fields
Braided quantum field theories proposed by Oeckl can provide a framework for defining quantum field theories with Hopf algebra symmetries. In quantum field theories, symmetries lead to non-perturbative relations among correlation functions, which are generally called Ward-Takahashi identities. We discuss Hopf algebra symmetries and Ward-Takahashi identities in braided quantum field theories. We give the four algebraic conditions between Hopf algebra symmetries and braided quantum field theories, which are required for Ward-Takahashi identities to hold. As concrete examples, we apply our discussions to the Poincare symmetries of two examples of noncommutative field theories. One is the effective quantum field theory of three-dimensional quantum gravity coupled with spinless particles given by Freidel and Livine, and the other is noncommutative field theory on Moyal plane. We also comment on quantum field theory on kappa-Minkowski spacetime.
The Unruh-DeWitt particle detector is a valuable tool for probing the physics of quantum fields in curved backgrounds. In this talk we provide a definition of the transition rate of the detector which can be applied foe general background spacetimes, general Hadamard states of the field and general detector trajectories. As an example the transition rate of a detector at rest in a static Newtonian gravitational field is calculated: the excitation rate is zero but there are small corrections to the Minkowskian de-excitation rate.
The Lorentzian spacetime metric is replaced by an area metric which naturally emerges as a generalized geometry from the canonical quantization of gauge theories, string theory and gravity. Employing the area metric curvature scalar, the gravitational Einstein-Hilbert action is re-interpreted as dynamics for an area metric. Without the need for dark energy or fine-tuning, area metric cosmology explains the observed small acceleration of the late Universe.
Under very mild assumptions we are led to an exactly solvable model in LQC that can be rigorously compared to the Wheeler-DeWitt theory. We show that LQC and Wheeler-DeWitt can be constructed from the solutions of a 1+1 dimensional Klein-Gordon theory. Like any exactly solvable model the non-triviality lies in the mapping between the simple common description and the physical theories. Working with a complete set of solutions we provide an explicit formula for the evolution of the volume operator -- the only non-trivial Dirac observable in the theory and show that: (1) Quantum Bounce is not restricted to semi-classical states in the simple models studied so far, (2) There is a precise sense in which Wheeler-DeWitt theory is not a continuum limit of LQC and (3) LQC is a fundamentally discrete theory.
Spin networks provide a basis for the Hilbert space of loop quantum gravity and give rise to quantized 3-geometries with discrete spectra for areas and volumes. I suggest here that there is a natural simplicial approximation to a smooth spacetime which corresponds to any spin network. This approximation sheds light on some otherwise non-obvious results from loop quantum gravity and helps address the issue of locality in spin networks. It provides additional input to other approaches to discrete geometry such as Regge calculus and dynamical triangulation, and offers a new approach to quantum geometrodynamics from spin networks. Finally it naturally suggests that fundamental matter might be related to distributional 1-dimensional objects -- i.e. be stringlike.
I will talk about a recently proposed perturbation scheme, that allows to calculate cosmological perturbations up to arbitrary order. This framework is based on the concept of relational observables and approximates observables of full GR around some symmetry reduced sector. I will compare this framework to the standard theory of cosmological perturbations and will illustrate how to calculate backreaction effects using Bianchi-I as a simple cosmological model.
We re-examine the CGHS model from a nonperturbative Hamiltonian perspective. We present a framework for describing the quantum geometry of spacetime. In this framework a pure quantum state on left-scri-minus evolves to a pure state on right-scri-plus, thus there is no information loss. A truncation to 1st order (mean field approximation) recovers the Hawking effect and there is still a singularity at this approximation even though the quantum geometry is perfectly fine. A 2nd order truncation yields semiclassical equations including the backreaction. An asymptotic analysis of the metric near right-scri-plus yields the back-reaction corrected semiclassical metric. Summarily, although the truncated theories yield a consistent semiclassical picture, there is no information loss in the full quantum theory since the quantum spacetime is larger than the classically singular spacetime.
In the framework of loop quantum gravity we generalize our previous boundary state counting to a full bulk state counting. After a suitable gauge fixing we are able to compute the bulk entropy of a bounded region with fixed boundary. This allows us to study the relationship between the entropy and the boundary area and identify the holographic regime of LQG, where the leading order of the entropy scales with the area. In a BF theory the constraint can be fully implemented and lead to the distributional states. We construct a regularized bulk entropy which results in a natural form of the renormalized entropy in the presence of non-trivial holonomies. The upper bound on this bulk entropy is proportional to the number of edges that intersect the boundary.
I will give the motivation for and the construction of a causal spin foam model, which is to be considered as the analogue of the Feynman propagator on the superspace of geometries. Time allowing, I will discuss regularization of the resulting dual-face amplitudes, coupling with matter, and an apparently new symmetry which seems to be present only in the discrete quantum case.
We discuss dynamics of the Schwarzschild interior using an effective semi-classical description of the loop quantization. We will consider the effects of an improved loop quantization using techniques from loop quantum cosmology.
The study of particle-like excitations of quantum gravitational fields in loop quantum gravity is extended to the case of four valent graphs and the corresponding natural evolution moves based on the dual Pachner moves. This makes the results applicable to spin foam models. We find that some braids propagate on the networks and they can interact with each other, by joining and splitting. The chirality of the braid states determines the motion and the interactions, in that left handed states only propagate to the left, and vise versa.
I will review recent work on computing Lorentzian 10J symbols and related functions.