arXiv subjects

Wolfgang Wieland

Publications and source records attributed to Wolfgang Wieland.

At least 19 recordsLinked to original sources

arXiv2026

Topological field theory plus local Lorentz symmetry is gravity

Four-dimensional gravity admits many equivalent formulations - metric, Einstein-Cartan, teleparallel, McDowell-Mansouri, among others - each offering distinct advantages, particularly, in view of quantization. We propose a new formulation based on Weyl spinor-valued 1-forms, ultimately encoding the frame-field data. Starting from a topological field theory with a global $\mathrm{SL}(2,\mathbb{C})$ symmetry, we show that promoting this symmetry to a local gauge symmetry leads to the emergence of gravity. We analyze the covariant phase space of this theory, its symmetries and charge structure and explore the role of admissible corner terms together with their impact on boundary charges and their algebra. We study several extensions of this framework, including the incorporation of a cosmological constant and a novel $ G \rightarrow 0 $ scaling limit obtained from this model. The presence of the frame field already at the topological level allows point particles to be coupled uniformly in both the topological and gravitational theories. We perform a detailed Hamiltonian analysis of the theory and clarify the implementation of the reality conditions. We argue that this formulation provides structural features that make it particularly well suited for both discretization and quantization.

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arXiv2025

Fermionisation of the Aharonov--Bohm Phase on the Lightfront

We consider the phase space of the Maxwell field as a simplified framework to study the quantisation of holonomies (Wilson line operators) on lightlike (null) surfaces. Our results are markedly different from the spacelike case. On a spacelike surface, electric and magnetic fluxes each form a commuting subalgebra. This implies that the holonomies commute. On a lightlike hypersurfaces, this is no longer true. Electric and magnetic fluxes are no longer independent. To compute the Poisson brackets explicitly, we choose a regularisation. Each path is smeared into a thin ribbon. In the resulting holonomy algebra, Wilson lines commute unless they intersect the same light ray. We compute the structure constants of the holonomy algebra and show that they depend on the geometry of the intersection and the conformal class of the metric at the null surface. Finally, we propose a quantisation. The resulting Hilbert space shows a number of unexpected features. First, the holonomies become anti-commuting Grassmann numbers. Second, for pairs of Wilson lines, the commutation relations can continuously interpolate between fermionic and bosonic relations. Third, there is no unique ground state. The ground state depends on a choice of framing of the underlying paths.

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arXiv2025

Linking Edge Modes and Geometrical Clocks in Linearized Gravity

Reference frames are crucial for describing local observers in general relativity. In quantum gravity, different proposals exist for how to treat reference frames. There are models with either classical or quantum reference frames. Recently, different choices appeared for investigating these possibilities at the level of the classical and quantum algebra of observables. One choice is based on the covariant phase space approach, using gravitational edge modes. In the canonical approach, there is another choice, relational clocks, built from matter or geometry itself. In this work, we extend existing results and show how to relate edge modes and geometrical clocks in linearized gravity. We proceed in three steps. First, we introduce an extension of the ADM (Arnowitt-Deser-Misner) phase space to account for covariant gauge fixing conditions and the explicit time dependence they add to Hamilton's equations. Second, we show how these gauge fixing conditions recover a specific choice of geometrical clocks in terms of Ashtekar-Barbero connection variables. Third, we study the effect of the Barbero-Immirzi parameter on the generators of asymptotic symmetries and the corresponding charges. This parameter, which disappears from metric gravity, affects the generators for angle-dependent asymptotic symmetries and the corresponding super-translation charges, while it has no effect on the global charges.

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arXiv2025

Foundational Structure of Local Amplitudes in Quantum Gravity

There has been recently renewed interest in the quantisation of gravity by considering local subsystems on light-like hypersurfaces. The main purpose of this paper is to present a theory-independent perspective on these developments assuming only basic knowledge of quantum theory and general relativity. In addition, we present a top-down approach to constructing local amplitudes in causal diamonds. The fundamental building block is a slab of light-like geometry (e.g. a segment of a light cone embedded into spacetime). Each slab is a three-dimensional light-like hypersurface bounded by two cuts, its past and future corners. After briefly reviewing the timeless formalism of quantum theory, we equip each null slab a with a kinematical Hilbert space that factorizes into constituents associated to the three-dimensional interior and its two corners. Upon assuming the existence of vacuum states for the bulk and boundary symmetries and a fundamental projector onto physical states, we explain how to introduce local amplitudes by contracting boundary states according to the pattern of a causal diamond. Finally, we show that the resulting local transition amplitudes satisfy Ward identities and charge conservation for the underlying symmetries. The paper closes with a summary and discussion for how different approaches to quantum gravity can realise our proposal in practice.

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arXiv2025

Evidence for Planck Luminosity Bound in Quantum Gravity

Recently, we introduced a non-perturbative quantization of impulsive gravitational null initial data. In this note, we investigate an immediate physical implication of the model. One of the quantum numbers is the total luminosity carried to infinity. We show that a transition happens when the luminosity reaches the Planck power $\mathcal{L}_{\mathrm{P}}$. Below $\mathcal{L}_{\mathrm{P}}$, the spectrum of the radiated power is discrete. Above the Planck power, the spectrum is continuous and contains caustics that can spoil the semi-classical interpretation of the resulting quantum states of geometry.

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arXiv2025

Quantum Geometry of the Light Cone: Fock representation and Spectrum of Radiated Power

Starting from the symplectic potential for the $γ$-Palatini--Holst action on a null hypersurface, we identify an auxiliary conformal field theory (CFT), which carries a representation of the constraint algebra of general relativity on a null surface. The radiative data, which is encoded into the shear of each null generator, is mapped into an $SU(1,1)$ current algebra on each light ray. We study the resulting quantum theory for both bosonic and fermionic representations. In the fermionic representation, the central charge on each null ray is positive, for bosons it is negative. A negative central charge implies a non-unitary CFT, which has negative norm states. In the model, there is a natural $SU(1,1)$ Casimir. For the bosonic representations, the $SU(1,1)$ Casimir can have either sign. For the fermionic representations, the $SU(1,1)$ Casimir is always greater or equal to zero. To exclude negative norm states, we restrict ourselves to the fermionic case. To understand the physical implications of this restriction, we express the $SU(1,1)$ Casimir in terms of the geometric data. In this way, the positivity bound on the $SU(1,1)$ Casimir translates into an upper bound for the shear of each null generator. In the model, this bound must be satisfied for all three-dimensional null hypersurfaces. This in turn suggests to apply it to an entire null foliation in an asymptotically flat spacetime. In this way, we obtain a bound on the radiated power of gravitational waves in the model.

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arXiv2024

Quantum Reference Frames at the Boundary of Spacetime

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.

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arXiv2024

Quantum geometry of the null cone

We present a non-perturbative quantization of gravitational null initial data. Our starting point is the characteristic null initial problem for tetradic gravity with a parity-odd Holst term in the bulk. After a basic review about the resulting Carrollian boundary field theory, we introduce a specific class of impulsive radiative data. This class is defined for a specific choice of relational clock. The clock is chosen in such a way that the shear of the null boundary follows the profile of a step function. The angular dependence is arbitrary. Next, we solve the residual constraints, which are the Raychaudhuri equation and a Carrollian transport equation for an $SL(2,\mathbb{R})$ holonomy. We show that the resulting submanifold in phase space is symplectic. Along each null generator, we end up with a simple mechanical system. The quantization of this system is straightforward. Our basic strategy is to start from an auxiliary Hilbert space with constraints. The physical Hilbert space is the kernel of a constraint, which is a combination of ladder operators. The constraint and its hermitian conjugate are second-class. Solving the constraint amounts to imposing a simple recursion relation for physical states. On the resulting physical Hilbert space, the $SL(2,\mathbb{R})$ Casimir is a Dirac observable. This observable determines the spectrum of the two radiative modes. The area at the initial and final cross sections are Dirac observables as well. They have a discrete spectrum, which agrees with earlier results on this topic.

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arXiv2023

Simplicial Graviton from Selfdual Ashtekar Variables

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.

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arXiv2023

Corner symmetry and quantum geometry

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.

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arXiv2022

Metriplectic geometry for gravitational subsystems

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.

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arXiv2022

Canonical transformations generated by the boundary volume: Unimodular and non-Abelian teleparallel gravity

Recently, a new choice of variables was identified to understand how the quantum group structure appeared in three-dimensional gravity [1]. These variables are introduced via a canonical transformation generated by a boundary term. We show that this boundary term can actually be taken to be the volume of the boundary and that the new variables can be defined in any dimension greater than three. In addition, we study the associated metric and teleparallel formalisms. The former is a variant of the Henneaux--Teitelboim model for unimodular gravity. The latter provides a non-Abelian generalization of the usual Abelian teleparallel formulation.

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arXiv2021

Barnich-Troessaert Bracket as a Dirac Bracket on the Covariant Phase Space

The Barnich--Troessaert bracket is a proposal for a modified Poisson bracket on the covariant phase space for general relativity. The new bracket allows us to compute charges, which are otherwise not integrable. Yet there is a catch. There is a clear prescription for how to evaluate the new bracket for any such charge, but little is known how to extend the bracket to the entire phase space. This is a problem, because not every gravitational observable is also a charge. In this paper, we propose such an extension. The basic idea is to remove the radiative data from the covariant phase space. This requires second-class constraints. Given a few basic assumptions, we show that the resulting Dirac bracket on the constraint surface is nothing but the BT bracket. A heuristic argument is given to show that the resulting constraint surface can only contain gravitational edge modes.

hep-th↗

arXiv2021

Null infinity as an open Hamiltonian system

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.

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arXiv2021

Gravitational $SL(2,\mathbb{R})$ Algebra on the Light Cone

In a region with a boundary, the gravitational phase space consists of radiative modes in the interior and edge modes at the boundary. Such edge modes are necessary to explain how the region couples to its environment. In this paper, we characterise the edge modes and radiative modes on a null surface for the tetradic Palatini--Holst action. Our starting point is the definition of the action and its boundary terms. We choose the least restrictive boundary conditions possible. The fixed boundary data consists of the radiative modes alone (two degrees of freedom per point). All other boundary fields are dynamical. We introduce the covariant phase space and explain how the Holst term alters the boundary symmetries. To infer the Poisson brackets among Dirac observables, we define an auxiliary phase space, where the $SL(2\mathbb{R})$ symmetries of the boundary fields is manifest. We identify the gauge generators and second-class constraints that remove the auxiliary variables. All gauge generators are at most quadratic in the fundamental $SL(2,\mathbb{R})$ variables on phase space. We compute the Dirac bracket and identify the Dirac observables on the light cone. Finally, we discuss various truncations to quantise the system in an effective way.

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arXiv2020

Twistor representation of Jackiw-Teitelboim gravity

Jackiw-Teitelboim (JT) gravity is a 1+1-dimensional toy model for quantum gravity in four spacetime dimensions. In the absence of matter, JT gravity is a topological field theory and there are no local observables. The introduction of a boundary changes the situation. What was a un-physical gauge direction before turns into a physical boundary degree of freedom (a gravitational edge mode). Starting from the BF formulation of JT gravity, we develop a twistor representation for the boundary charges of JT gravity. We introduce point sources in the bulk and study the coupled gravity plus matter system at the quantum level. Eigenstates of quasi-local energy are built from the tensor product of unitary irreducible representations of SL(2,R). Empty patches of AdS2 are characterised by the continuous series representations, point sources are related to the discrete series. Physical states are constructed by fusing the SL(2,R) representations into an SL(2,R) invariant singlet. The singlet lies in the kernel of the constraints, which are the analogue of the Wheeler-DeWitt equations in the presence of distributional matter sources.

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arXiv2020

First-order formulation of teleparallel gravity and dual loop gravity

There are at least two ways to encode gravity into geometry: Einstein's general theory of relativity (GR) for the metric tensor, and teleparallel gravity, where torsion as opposed to curvature encodes the dynamics of the gravitational degrees of freedom. The main purpose of the paper is to explore the relation between loop gravity and teleparallel gravity. We argue that these two formulations of gravity are related to two different discretizations of the Einstein--Cartan action, which were studied recently in the literature. The first discretization leads to the \emph{loop gravity} kinematical phase space where the zero torsion condition is enforced {first} and the other is the \emph{dual loop gravity} kinematical phase space where curvature is imposed to vanish {first}. Our argument is based on the observation that the GR first-order Einstein--Cartan action can also be seen as a first-order action for teleparallel gravity up to a boundary term. The results of our paper suggest that the \emph{dual loop gravity} framework is a natural discretization of teleparallel gravity, whereas \emph{loop gravity} is naturally related to the standard GR metric description.

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arXiv2019

Deformed Heisenberg charges in three-dimensional gravity

In this paper, we consider the bulk plus boundary phase space for three-dimensional gravity with negative cosmological constant for a particular choice of conformal boundary conditions: the conformal class of the induced metric at the boundary is kept fixed and the mean extrinsic curvature is constrained to be one. Such specific conformal boundary conditions define so-called Bryant surfaces, which can be classified completely in terms of holomorphic maps from Riemann surfaces into the spinor bundle. To study the observables and gauge symmetries of the resulting bulk plus boundary system, we introduce an extended phase space, where these holomorphic maps are now part of the gravitational bulk plus boundary phase space. The physical phase space is obtained by introducing two sets of Kac-Moody currents, which are constrained to vanish. The constraints are second-class and the corresponding Dirac bracket yields an infinite-dimensional deformation of the Heisenberg algebra for the spinor-valued surface charges. Finally, we compute the Poisson algebra among the generators of conformal diffeomorphisms and demonstrate that there is no central charge. Although the central charge vanishes and the boundary CFT is likely non-unitary, we will argue that a version of the Cardy formula still applies in this context, such that the entropy of the BTZ black hole can be derived from the degeneracy of the eigenstates of quasi-local energy.

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