In perturbative gravity, it is straight-forward to characterize the two local degrees of freedom of the gravitational field in terms of a mode expansion of the linearized perturbation. In the non-perturbative regime, we are in a more difficult position. It is not at all obvious how to construct Dirac observables that can separate the gauge orbits. Standard procedures rely on asymptotic boundary conditions or formal Taylor expansions of partial observables along the gauge orbits. In this paper, we lay out a new non-perturbative lattice approach to tackle the problem in terms of Ashtekar's self-dual formulation. Starting from a simplicial decomposition of space, we introduce a local kinematical phase space at the lattice sites. At each lattice site, we introduce a set of constraints that replace the generators of the hypersurface deformation algebra in the continuum. We show that the discretized constraints close under the Poisson bracket. The resulting reduced phase space describes two complex physical degrees of freedom representing the two radiative modes at the discretized level. The paper concludes with a discussion of the key open problems ahead and the implications for quantum gravity.
By virtue of the Noether theorems, the vast gauge redundancy of general relativity provides us with a rich algebra of boundary charges that generate physical symmetries. These charges are located at codimension-2 entangling surfaces called corners. The presence of non-trivial corner symmetries associated with any entangling cut provides stringent constraints on the theory's mathematical structure and a guide through quantization. This report reviews new and recent results for non-perturbative quantum gravity, which are natural consequences of this structure. First, we establish that the corner symmetry derived from the gauge principle encodes quantum entanglement across internal boundaries. We also explain how the quantum representation of the corner symmetry algebra provides us with a notion of quantum geometry. We then focus our discussion on the first-order formulation of gravity and show how many results obtained in the continuum connect naturally with previous results in loop quantum gravity. In particular, we show that it is possible to get, purely from quantization and without discretization, an area operator with discrete spectrum, which is covariant under local Lorentz symmetry. We emphasize that while loop gravity correctly captures some of the gravitational quantum numbers, it does not capture all of them, which points towards important directions for future developments. Finally, we discuss the understanding of the gravitational dynamics along null surfaces as a conservation of symmetry charges associated with a Carrollian fluid.
An analysis is given of the local phase space of gravity coupled to matter to second order in perturbation theory. Working in local regions with boundaries at finite distance, we identify matter, Coulomb, and additional boundary modes. The boundary modes take the role of reference frames for both diffeomorphisms and internal Lorentz rotations. Passing to the quantum level, we identify the constraints that link the bulk and boundary modes. The constraints take the form of a multi-fingered Schrödinger equation, which determines the relational evolution of the quantum states in the bulk with respect to the quantum reference fields at the boundary.
In general relativity, it is difficult to localise observables such as energy, angular momentum, or centre of mass in a bounded region. The difficulty is that there is dissipation. A self-gravitating system, confined by its own gravity to a bounded region, radiates some of the charges away into the environment. At a formal level, dissipation implies that some diffeomorphisms are not Hamiltonian. In fact, there is no Hamiltonian on phase space that would move the region relative to the fields. Recently, an extension of the covariant phase space has been introduced to resolve the issue. On the extended phase space, the Komar charges are Hamiltonian. They are generators of dressed diffeomorphisms. While the construction is sound, the physical significance is unclear. We provide a critical review before developing a geometric approach that takes into account dissipation in a novel way. Our approach is based on metriplectic geometry, a framework used in the description of dissipative systems. Instead of the Poisson bracket, we introduce a Leibniz bracket - a sum of a skew-symmetric and a symmetric bracket. The symmetric term accounts for the loss of charge due to radiation. On the metriplectic space, the charges are Hamiltonian, yet they are not conserved under their own flow.
In loop quantum gravity, not only energy and matter, but space itself consists of smallest quanta. Geometric observables such as area, angles and volumes are quantised. For the area spectrum, we obtain, in fact, \[\text{eigenvalues of area} = 8\pi \gamma\,\hbar G/c^3 \sqrt{j(j+1)},\] where \(\hbar G/c^3=\ell_{\mathrm{P}}^2\) is the Planck area and the half-integrers \(j = 0,1/2,1,3/2,\dots\) label the various area-excitations. The area spectrum depends on an additional coupling constant, namely the Barbero–Immirzi parameter \(\gamma\) that determines the minimal area relative to the fundamental Planck scale.
The geometric origin of the loop gravity discreteness of space is the result of an additional parity odd term in the action, which does not affect the field equations in the bulk, but deforms the boundary symmetries. This is explained in the paper arXiv:2104.05803, where the phase space is derived for the edge mode data and the radiative data on a null surface, if the Barbero–Immirzi \(\gamma\)-parameter is taken into account. The radiative data is encoded into an \(SL(2,\mathbb{R})\) holonomy. This holonomy controls the evolution of the conformal metric on a cross-section of the null surface along its null generators. Upon solving the constraints, the commutation relations for the radiative data are independent of \(\gamma\). For the edge modes the situation is very different: \(\gamma\) has a huge effect on the algebra of boundary observables. If, in fact, \(\gamma=0\), there is a \(U(1)\) normal subgroup of the \(SL(2,\mathbb{R})\) boundary symmetries, whose generator vanishes. If \(\gamma\neq0\), this charge does not vanish, but is proportional to the area of the cross section. Since the orbit is compact, the conjugate variable, which is the area, is quantized.
The basic message is that relative to metric gravity, loop gravity alters the algebra of gravitational edge modes. This alteration is subtle, because it does not change the commutation relations for the radiative data, but only affects the orbits of the edge modes. This observation is important, because it is a strong hint that we will recover the same low energy \(S\)-matrix elements as in perturbative gravity. Various strategies to test this scenario are discussed in the conclusions to the paper.
When a system emits gravitational radiation, the Bondi mass decreases. If the Bondi energy is Hamiltonian, it can thus only be a time dependent Hamiltonian. In this paper, we show that the Bondi energy can be understood as a time-dependent Hamiltonian on the covariant phase space. Our derivation starts from the Hamiltonian formulation in domains with boundaries that are null. We introduce the most general boundary conditions on a generic such null boundary, and compute quasi-local charges for boosts, energy and angular momentum. Initially, these domains are at finite distance, such that there is a natural IR regulator. To remove the IR regulator, we introduce a double null foliation together with an adapted Newman--Penrose null tetrad. Both null directions are surface orthogonal. We study the falloff conditions for such specific null foliations and take the limit to null infinity. At null infinity, we recover the Bondi mass and the usual covariant phase space for the two radiative modes at the full non-perturbative level. Apart from technical results, the framework gives two important physical insights. First of all, it explains the physical significance of the corner term that is added in the Wald--Zoupas framework to render the quasi-conserved charges integrable. The term to be added is simply the derivative of the Hamiltonian with respect to the background fields that drive the time-dependence of the Hamiltonian. Secondly, we propose a new interpretation of the Bondi mass as the thermodynamical free energy of gravitational edge modes at future null infinity. The Bondi mass law is then simply the statement that the free energy always decreases on its way towards thermal equilibrium.
It is arguably one of the main achievements of loop quantum gravity to have demonstrated that space itself may have an atomic structure. One of the key open problems of the theory is to reconcile the fundamental loop quantum gravity discreteness of space with general relativity in the continuum. In this paper, it is shown, in fact, that the loop gravity discreteness of space can be understood from a conventional Fock quantisation of gravitational boundary modes on a null surface. These boundary modes are found by considering a quasi-local Hamiltonian analysis, where general relativity is treated as a Hamiltonian system in domains with inner null boundaries. The presence of such null boundaries requires then additional boundary terms in the action. Using Ashtekar’s original \(SL(2,\mathbb{C})\) self-dual variables, it is demonstrated that the natural such boundary term is nothing but a kinetic term for a spinor (defining the null flag of the boundary) and a spinor-valued two-form, which are both intrinsic to the boundary. Finally, it is shown that in quantum theory, the cross-sectional area two-form turns into the difference of two number operators (unless the Barbero–Immirzi parameter is sent to infinity). The resulting area spectrum is discrete without ever introducing spin networks or triangulations of space.
This is an important result, because it addresses some objections often raised. It was often argued, in fact, that the LQG discreteness of space is unphysical, because it relies on working on a discrete spin network graph (whence the discreteness is brought in from the onset), or can be traced back to using \(SU(2)\) gauge covariant variables, see e.g. arXiv:hep-th/0501114. The derivation given here is manifestly Lorentz invariant and does not require spin networks. An analogous argument also holds for quantum gravity in three dimensions, see arXiv:1804.08643.
The diffeomorphism-invariant LQG Hilbert space can be decomposed into an infinite sum of finite-dimensional Hilbert spaces associated to a graph. Each of these graph Hilbert spaces can be understood as the quantisation of a finite-dimensional phase space. What was missing, however, was a good proposal for how to formulate the dynamics using these finite-dimensional phase spaces. A consistent such proposal was developed here. The basic idea is to build a curved manifold by a gluing procedure, where one cuts Minkowski space along null surfaces and glues them together such that the resulting manifold satisfies Einstein’s equations in a distributional sense. This is related to certain results from twistor theory, where one builds a curved manifold by glueing together flat patches of spacetime along null surfaces.
This article appeared as an Editors' Choice
article, an honor that is reserved for the best papers published in General Relativity and Gravitation.
This paper is about three-dimensional quantum gravity, and devlops a derivation of the Ponzano–Regge spinfoam model from an ordinary one-dimensional path integral for a worldline action. The underlying worldlines are simply the edges of the spinfoam. This paper proved influential in connecting causal sets and spinfoams, see arXiv:1308.2206.
In this and the next paper below, the phase space for self-dual holonomy flux variables on a given spin network graph is parametrised in terms of twistors. This approach led to a new understanding of spinfoam amplitudes in terms of integrals in twistor space.
This article has been selected by the Editorial Board of Classical and Quantum Gravity (CQG) to be one of the journal’s Highlights of 2011–2012.
In this article on spinfoam fermions, we developed a minimal coupling of Dirac spinors to the EPRL spinfoam model.
This project was about the relation between the covariant and canonical formulations of LQG. The basic idea is to map the simplicity constraints that are crucial for the defintiion of the covariant spinfoam approach to the reality conditions in canonical loop quantum gravity.