I review the lattice approach to quantum gravity, and how it relates to the non-trivial ultraviolet fixed point scenario of the continuum theory. After a brief introduction covering the general problem of ultraviolet divergences in gravity and other non-renormalizable theories, I cover the general methods and goals of the lattice approach. An underlying theme is the attempt at establishing connections between the continuum renormalization group results, which are mainly based on diagrammatic perturbation theory, and the recent lattice results, which apply to the strong gravity regime and are inherently non-perturbative. A second theme in this review is the ever-present natural correspondence between infrared methods of strongly coupled non-abelian gauge theories on the one hand, and the low energy approach to quantum gravity based on the renormalization group and universality of critical behavior on the other. Towards the end of the review I discuss possible observational consequences of path integral quantum gravity, as derived from the non-trivial ultraviolet fixed point scenario. I argue that the theoretical framework naturally leads to considering a weakly scale-dependent Newton's costant, with a scaling violation parameter related to the observed scaled cosmological constant (and not, as naively expected, to the Planck length).
We present a separable version of Loop Quantum Gravity (LQG) based on an inductive system of cubic lattices. We construct semi-classical states for which the LQG operators -- the flux, the area and the volume operators -- have the right classical limits. Also, we present the Hamilton and diffeomorphism constraints as operator constraints and show that they have the right classical limit. Finally, we speculate whether the continuum limit, which these semi-classical states probe, can be defined for the entire construction and thereby restore an action of the diffeomorphism group.
I review recent progress in understanding non-perturbative aspects of string theory, quantum gravity and non-commutative geometry using lattice methods.
We consider quantum gravity model with the squared curvature action. We construct lattice discretization of the model (both on hypercubic and simplicial lattices) starting from its teleparallel equivalent. The resulting lattice models have the actions that are bounded from below while Einstein equations (without matter) appear in their classical limit.
Lattice spinor gravity is a proposal for regularized quantum gravity based on fermionic degrees of freedom. In our lattice model the local Lorentz symmetry is generalized to complex transformation parameters. The difference between space and time is not put in a priori, and the euclidean and Minkowski quantum field theory are unified in one functional integral. The metric and its signature arise as a result of the dynamics, corresponding to a given ground state or cosmological solution. Geometrical objects as the vierbein, spin connection or the metric are expectation values of collective fiel…
A local quantum bosonic model on a lattice is constructed whose low energy excitations are gravitons described by linearized Einstein action. Thus the bosonic model is a quantum theory of gravity, at least at the linear level. We find that the compactification and the discretization of metric tenor are crucial in obtaining a quantum theory of gravity.
The key ingredient for lattice regularized quantum gravity is diffeomorphism symmetry. We formulate a lattice functional integral for quantum gravity in terms of fermions. This allows for a diffeomorphism invariant functional measure and avoids problems of boundedness of the action. We discuss the concept of lattice diffeomorphism invariance. This is realized if the action does not depend on the positioning of abstract lattice points on a continuous manifold. Our formulation of lattice spinor gravity also realizes local Lorentz symmetry. Furthermore, the Lorentz transformations are generalized…
We propose a new, discretized model for the study of 3+1-dimensional canonical quantum gravity, based on the classical $SL(2,\C)$-connection formulation. The discretization takes place on a topological $N^3$- lattice with periodic boundary conditions. All operators and wave functions are constructed from one-dimensional link variables, which are regarded as the fundamental building blocks of the theory. The kinematical Hilbert space is spanned by polynomials of certain Wilson loops on the lattice and is manifestly gauge- and diffeomorphism- invariant. The discretized quantum Hamiltonian $\hat …
We propose a regularized lattice model for quantum gravity purely formulated in terms of fermions. The lattice action exhibits local Lorentz symmetry, and the continuum limit is invariant under general coordinate transformations. The metric arises as a composite field. Our lattice model involves no signature for space and time, describing simultaneously a Minkowski or euclidean theory. It is invariant both under Lorentz transformations and euclidean rotations. The difference between space and time arises from expectation values of composite fields. Our formulation includes local gauge symmetri…
We propose a model with quantum bosons on the fcc lattice, which has a stable algebraic Bose liquid phase at low energy. We show that this phase is described by emergent quantum gravity at the Gaussian z = 3 Lifshitz fixed point in 3+1 dimensions. The stability of this algebraic Bose liquid phase is guaranteed by the gauge symmetry of gravitons and self-duality of the low energy field theory. By tuning one parameter in the lattice boson model we can drive a phase transition between the z = 3 Lifshitz gravity and another algebraic Bose liquid phase, described by gravity at the z = 2 Lifshitz po…
In the context of group field theory condensate cosmology, we clarify the extraction of cosmological variables from the microscopic quantum gravity degrees of freedom. We show that an important implication of the second quantized formalism is the dependence of cosmological variables and equations on the quantum gravitational atomic number N (number of spin network vertices/elementary simplices). We clarify the relation of the effective cosmological equations with loop quantum cosmology, understood as an effective (hydrodynamic-like) approximation of a more fundamental quantum gravity theory. B…
We study the behavior of holonomy spin foam partition functions, a form of lattice gauge gravity, on generic 4d-triangulations using micro local analysis. To do so we adapt tools from the renormalization theory of quantum field theory on curved space times. This allows us, for the first time, to study the partition function without taking any limits on the interior of the triangulation. We establish that for many of the most widely used models the geometricity constraints, which reduce the gauge theory to a geometric one, introduce strong accidental curvature constraints. These limit the curva…
We construct an ionic lattice background in the framework of Einstein-Maxwell-dilaton theory in four dimensional space time. The optical conductivity of the dual field theory on the boundary is investigated. Due to the lattice effects, we find the imaginary part of the conductivity is manifestly suppressed in the zero frequency limit, while the DC conductivity approaches a finite value such that the previous delta function reflecting the translation symmetry is absent. Such a behavior can be exactly fit by the Drude law at low frequency. Moreover, we find that the modulus of the optical conduc…
We investigate a lattice model for Euclidean quantum gravity based on discretization of the Palatini formulation of General Relativity. Using Monte Carlo simulation we show that while a naive approach fails to lead to a vacuum state consistent with the emergence of classical spacetime, this problem may be evaded if the lattice action is supplemented by an appropriate counter term. In this new model we find regions of the parameter space which admit a ground state which can be interpreted as (Euclidean) de Sitter space.
I propose the Langevin equation for 3-geometries in the Ashtekar's formalism to describe 4D Euclidean quantum gravity, in the sense that the corresponding Fokker-Planck hamiltonian recovers the hamiltonian in 4D quantum gravity exactly. The stochastic time corresponds to the Euclidean time in the gauge, N=1 and $N^i=0$. In this approach, the time evolution in 4D quantum gravity is understood as a stochastic process where the quantum fluctuation of ` ` triad \rq\rq is characterized by the curvature at the one unit time step before. The lattice regularization of 4D quantum gravity is presented i…
Topological gravity is the reduction of general relativity to flat space-times. A lattice model describing topological gravity is developed starting from a Hamiltonian lattice version of $B\w F$ theory. The extra symmetries not present in gravity that kill the local degrees of freedom in $B\wedge F$ theory are removed. The remaining symmetries preserve the geometrical character of the lattice. Using self-dual variables, the conditions that guarantee the geometricity of the lattice become reality conditions. The local part of the remaining symmetry generators, that respect the geometricity-real…
We present an elegant and simple dynamical model of symmetric, non-degenerate (n x n) matrices of fixed signature defined on a n-dimensional hyper-cubic lattice with nearest-neighbor interactions. We show how this model is related to General Relativity, and discuss multiple ways in which it can be useful for studying gravity, both classical and quantum. In particular, we show that the dynamics of the model when all matrices are close to the identity corresponds exactly to a finite-difference discretization of weak-field gravity in harmonic gauge. We also show that the action which defines the …
The improved lattice regularization method of the Ashtekar connection holonomy representation in loop quantum gravity is described in this article. The approach is based on the geometric expansion of holonomies into power series up to the quadratic order terms in the regularization parameter. As a result, a more accurate procedure than the currently established approach to the canonical lattice quantization of gravity is obtained. Moreover, if holonomies are defined along linear links, this procedure becomes exact. Furthermore, in the improved method, the symmetry of holonomies assigned to lin…
Borrowing techniques from cosmology, I compute the power spectrum of quantum fluctuations in (2+1)-dimensional causal dynamical triangulations, a promising discrete path integral approach to quantum gravity. The results agree with those of canonical quantization to a high degree of precision, providing strong evidence for the equivalence of the two approaches and for the validity of the discrete method.
We show that the algebra of discretized spatial diffeomorphism constraints in Hamiltonian lattice quantum gravity closes without anomalies in the limit of small lattice spacing. The result holds for arbitrary factor-ordering and for a variety of different discretizations of the continuum constraints, and thus generalizes an earlier calculation by Renteln.
Using the Cartan formulation of General Relativity, we construct a well defined lattice-regularized theory capable to describe large non-perturbative quantum fluctuations of the frame field (or the metric) and of the spin connection. To that end we need to present the tetrad by a composite field built as a bilinear combination of fermion fields. The theory is explicitly invariant under local Lorentz transformations and, in the continuum limit, under general covariant transformations, or diffeomorphisms. Being well defined for large and fast varying fields at the ultraviolet cutoff, the theory …
The Weak Gravity Conjecture postulates the existence of superextremal charged particles, i.e. those with mass smaller than or equal to their charge in Planck units. We present further evidence for our recent observation that in known examples a much stronger statement is true: an infinite tower of superextremal particles of different charges exists. We show that effective Kaluza-Klein field theories and perturbative string vacua respect the Sublattice Weak Gravity Conjecture, namely that a finite index sublattice of the full charge lattice exists with a superextremal particle at each site. In …
Lattice regularization is a standard technique for the nonperturbative definition of a quantum theory of fields. Several approaches to the construction of a quantum theory of gravity adopt this technique either explicitly or implicitly. A crucial complement to lattice regularization is the process of renormalization through which a continuous description of the quantum theory arises. I provide a comprehensive conceptual discussion of the renormalization of lattice-regularized quantum gravity models. I begin with a presentation of the renormalization group from the Wilsonian perspective. I then…
We advocate lattice methods as the tool of choice to constructively define a background-independent theory of Lorentzian quantum gravity and explore its physical properties in the Planckian regime. The formulation that arguably has most furthered our understanding of quantum gravity (and of various pitfalls present in the nonperturbative sector) uses dynamical triangulations to regularize the nonperturbative path integral over geometries. Its Lorentzian version in terms of Causal Dynamical Triangulations (CDT) - in addition to having a definite quantum signature on short scales - has been show…
We investigate some properties of geometric operators in canonical quantum gravity in the connection approach à la Ashtekar, which are associated with volume, area and length of spatial regions. We motivate the construction of analogous discretized lattice quantities, compute various quantum commutators of the type [area,volume], [area,length] and [volume,length], and find they are generally non-vanishing. Although our calculations are performed mostly within a lattice-regularized approach, some are -- for special, fixed spin-network configurations -- identical with corresponding continuum com…
We apply a recent argument of Verlinde to loop quantum gravity, to conclude that Newton's law of gravity emerges in an appropriate limit and setting. This is possible because the relationship between area and entropy is realized in loop quantum gravity when boundaries are imposed on a quantum spacetime.
Lattice universes are spatially closed space-times of spherical topology in the large, containing masses or black holes arranged in the symmetry of a regular polygon or polytope. Exact solutions for such spacetimes are found in 2+1 dimensions for Einstein gravity with a non-positive cosmological constant. By means of a mapping that preserves the essential nature of geodesics we establish analogies between the flat and the negative curvature cases. This map also allows treatment of point particles and black holes on a similar footing.
A lattice quantum gravity model in 4 dimensional Riemannian spacetime is constructed based on the SU(2) Ashtekar formulation of general relativity. This model can be understood as one of the family of models sometimes called ``spin foam models.'' A version of the action of general relativity in continuum is introduced and its lattice version is defined. A dimensionless ``(inverse) coupling'' constant is defined so that the value of the action of the model is finite per lattice point. The path integral of the model is expanded in the characters and shown to be written as a sum over surface-like…
We study scalar fields propagating on Euclidean dynamical triangulations (EDT). In this work we study the interaction of two scalar particles, and we show that in the appropriate limit we recover an interaction compatible with Newton's gravitational potential in four dimensions. Working in the quenched approximation, we calculate the binding energy of a two-particle bound state, and we study its dependence on the constituent particle mass in the non-relativistic limit. We find a binding energy compatible with what one expects for the ground state energy by solving the Schrödinger equation for …
Just as for non-abelian gauge theories at strong coupling, discrete lattice methods are a natural tool in the study of non-perturbative quantum gravity. They have to reflect the fact that the geometric degrees of freedom are dynamical, and that therefore also the lattice theory must be formulated in a background-independent way. After summarizing the status quo of discrete covariant lattice models for four-dimensional quantum gravity, I describe a new class of discrete gravity models whose starting point is a path integral over Lorentzian (rather than Euclidean) space-time geometries. A number…
In this work nonperturbative aspects of quantum gravity are investigated using the lattice formulation, and some new results are presented for critical exponents, amplitudes and invariant correlation functions. Values for the universal scaling dimensions are compared with other nonperturbative approaches to gravity in four dimensions, and specifically to the conjectured value for the universal critical exponent $ν=1 /3$. It is found that the lattice results are generally consistent with gravitational anti-screening, which would imply a slow increase in the strength of the gravitational couplin…
We introduce a new basis for the gauge-invariant Hilbert space of lattice gauge theory and loop quantum gravity in $(2+1)$ dimensions, the fusion basis. In doing so, we shift the focus from the original lattice (or spin-network) structure directly to that of the magnetic (curvature) and electric (torsion) excitations themselves. These excitations are classified by the irreducible representations of the Drinfel'd double of the gauge group, and can be readily "fused" together by studying the tensor product of such representations. We will also describe in detail the ribbon operators that create …
I describe two phenomenological windows on quantum gravity that seem promising to me. I argue that we already have important empirical inputs that should orient research in quantum gravity.
The nonlinear dynamics of preheating after early-Universe inflation is often studied with lattice simulations. In this work I present a new lattice code HLATTICE. It differs from previous public available codes in the following three aspects: (i) A much higher accuracy is achieved with a modified sixth-order symplectic integrator; (ii) scalar, vector, and tensor metric perturbations in synchronous gauge and their feedback to the dynamics of scalar fields are all included; (iii) the code uses a projector that completely removes the scalar and vector components defined by the discrete spatial de…
We discuss the possible appearance of several rather exotic phenomena in quantum gravity, including UV/IR mixing, novel modifications of infrared phenomenology that extend effective field theory approaches, and the relaxation of the usual notions of locality. We discuss the relevance of such concepts in quantum gravity for quantum information science, cosmology and general quantum gravity phenomenology.
The causal dynamical triangulations approach aims to construct a quantum theory of gravity as the continuum limit of a lattice-regularized model of dynamical geometry. A renormalization group scheme--in concert with finite size scaling analysis--is essential to this aim. Formulating and implementing such a scheme in the present context raises novel and notable conceptual and technical problems. I explored these problems, and, building on standard techniques, suggested potential solutions in the first paper of this two-part series. As an application of these solutions, I now propose a renormali…
Any approach to pure quantum gravity must eventually face the question of coupling quantum matter to the theory. In the past, several ways of coupling matter to spin foam quantum gravity have been proposed, but the dynamics of the coupled matter-gravity system is challenging to explore. To take first steps towards uncovering the influence quantum matter has on spin foam models, we couple free, massive scalar lattice field theory to a restricted, semi-classical 4d spin foam model, called quantum cuboids. This model can be understood as a superposition of hypercuboidal (and thus irregular) latti…
I review recent progress in simplicial quantum gravity in three and four dimensions, in particular new results on the phase structure of modified models of dynamical triangulations, the application of a strong-coupling expansion, and the benefits provided by including degenerate triangulations. In addition, I describe some recent numerical and analytical results on anisotropic crystalline membranes.
We propose a spin foam model for pure gauge fields coupled to Riemannian quantum gravity in four dimensions. The model is formulated for the triangulation of a four-manifold which is given merely combinatorially. The Riemannian Barrett--Crane model provides the gravity sector of our model and dynamically assigns geometric data to the given combinatorial triangulation. The gauge theory sector is a lattice gauge theory living on the same triangulation and obtains from the gravity sector the geometric information which is required to calculate the Yang--Mills action. The model is designed so that…
Topological gravity is the reduction of Einstein's theory to spacetimes with vanishing curvature, but with global degrees of freedom related to the topology of the universe. We present an exact Hamiltonian lattice theory for topological gravity, which admits translations of the lattice sites as a gauge symmetry. There are additional symmetries, not present in Einstein's theory, which kill the local degrees of freedom. We show that these symmetries can be fixed by choosing a gauge where the torsion is equal to zero. In this gauge, the theory describes flat space-times. We propose two methods to…
I review the present theoretical attempts to understand the quantum properties of spacetime. In particular, I illustrate the main achievements and the main difficulties in: string theory, loop quantum gravity, discrete quantum gravity (Regge calculus, dynamical triangulations and simplicial models), Euclidean quantum gravity, perturbative quantum gravity, quantum field theory on curved spacetime, noncommutative geometry, null surfaces, topological quantum field theories and spin foam models. I also briefly review several recent advances in understanding black hole entropy and attempt a critica…
We argue that quantum gravity should be a super-quantum theory, that is, a theory whose non-local correlations are stronger than those of canonical quantum theory. As a super-quantum theory, quantum gravity should display distinct experimentally observable super-correlations of entangled stringy states.
A framework for quantum field theory coupled to three-dimensional quantum gravity is proposed. The coupling with quantum gravity regulates the Feynman diagrams. One recovers the usual Feynman amplitudes in the limit as the cosmological constant tends to zero.
Results for the gravitational Wilson loop, in particular the area law for large loops in the strong coupling region, and the argument for an effective positive cosmological constant discussed in a previous paper, are extended to other proposed theories of discrete quantum gravity in the strong coupling limit. We argue that the area law is a generic feature of almost all non-perturbative lattice formulations, for sufficiently strong gravitational coupling. The effects on gravitational Wilson loops of the inclusion of various types of light matter coupled to lattice quantum gravity are discussed…
Due to seminal works of Weinberg, Cachazo and Strominger we know that tree level quantum gravity amplitudes satisfy three factorization constraints. Building on previous works which relate two of these constraints to symmetries of quantum gravity at null infinity, we present rather strong evidence that the third constraint is also equivalent to a new set of symmetries of (perturbative) quantum gravity. Our analysis implies that the symmetry group of quantum gravity may be even richer than the BMS group (or infinite dimensional extension thereof) previously considered.
We consider a nonlocal gravity model motivated by nonperturbative Quantum Gravity studies. This model, if correct, suggests the existence of strong IR relevant effects which can lead to an interesting late time cosmology. We implement the IR modification directly in the effective action. We show that, upon some assumptions on initial conditions, this model describes an observationally viable background cosmology being also consistent with local gravity tests.
Form factors constitute the key building block when organising the gravitational dynamics in terms of a curvature expansion. They generalise the concept of momentum-dependent couplings to curved spacetime. Moreover, they may capture modifications of the gravitational interactions at high energy scales in a non-perturbative way. We briefly review the role of form factors in various quantum gravity programmes with special emphasis on non-local, ghost-free gravity and asymptotically safe gravity. In this context, we introduce a quantum gravity motivated scalar toy model, which demonstrates that L…
It is proposed that gravity may arise in the low energy limit of a model of matter fields defined on a special kind of a dynamical random lattice. Time is discretized into regular intervals, whereas the discretization of space is random and dynamical. A triangulation is associated to each distribution of the spacetime points using the flat metric of the embedding space. We introduce a diffeomorphism invariant, bilinear scalar action, but no ``pure gravity'' action. Evidence for the existence of a non-trivial continuum limit is provided by showing that the zero momentum scalar excitation has a …
We present an implementation of the loop representation of quantum gravity on a square lattice. Instead of starting from a classical lattice theory, quantizing and introducing loops, we proceed backwards, setting up constraints in the lattice loop representation and showing that they have appropriate (singular) continuum limits and algebras. The diffeomorphism constraint reproduces the classical algebra in the continuum and has as solutions lattice analogues of usual knot invariants. We discuss some of the invariants stemming from Chern--Simons theory in the lattice context, including the issu…
Spin foam models for quantum gravity are derived from lattice path integrals. The setting involves variables from both lattice BF theory and Regge calculus. The action consists in a Regge action, which depends on areas, dihedral angles and includes the Immirzi parameter. In addition, a measure is inserted to ensure a consistent gluing of simplices, so that the amplitude is dominated by configurations which satisfy the parallel transport relations. We explicitly compute the path integral as a sum over spin foams for a generic measure. The Freidel-Krasnov and Engle-Pereira-Rovelli models corresp…
Entanglement entropy is a valuable tool for characterizing the correlation structure of quantum field theories. When applied to gauge theories, subtleties arise which prevent the factorization of the Hilbert space underlying the notion of entanglement entropy. Borrowing techniques from extended topological field theories, we introduce a new definition of entanglement entropy for both Abelian and non-Abelian gauge theories. Being based on the notion of excitations, it provides a completely relational way of defining regions. Therefore, it naturally applies to background independent theories, e.…
