arXiv subjects

Hanno Sahlmann

Publications and source records attributed to Hanno Sahlmann.

At least 19 recordsLinked to original sources

arXiv2026

Emergent Thiemann coherent states in the near-kernel sector of quantum reduced loop gravity

We study the near-kernel sector of the Hamiltonian constraint operator in the one-vertex model of quantum reduced loop gravity using variational Monte Carlo methods with neural quantum states. The analysis is based on the symmetric Hamiltonian containing both Euclidean and Lorentzian contributions, and on the variational minimization of the positive quadratic operator $\hat{\mathcal Q}=\hat C \hat C^\dagger$ in truncated Hilbert spaces with spin cutoff up to $j_{\mathrm{max}}=1001$. The resulting near-kernel states are found to organize into three qualitatively distinct classes. At low cutoffs, we find solutions that do not factorize across the three edge degrees of freedom. At larger cutoffs, we find two different factorized branches, both described to very high accuracy by products of one-edge wavefunctions but localized in different spin regimes. One of these branches is matched with near-unit fidelity by reduced Thiemann coherent states, providing evidence for an emergent semiclassical organization of the near-kernel sector. The other is likewise strongly factorized, but its one-edge factors are not well described by the same coherent-state family.

gr-qc↗

arXiv2026

Finding and characterising physical states of Euclidean Abelianized loop quantum gravity using neural quantum states

We study physical (near-kernel of constraints) states of 4-d Euclidean loop quantum gravity in Smolin's weak coupling limit on the complete graph $K_5$ using variational Monte Carlo with neural network quantum states. We investigate the Hamilton constraint $\hat{H}$ in the ordering proposed by Thiemann, as well as $\hat{H}^\dagger$ and $\hat{H}+\hat{H}^\dagger$. We find that the variational optimisation selects distinct solution families for $\hat{H}$ and $\hat{H}^\dagger$ across several considered cutoffs on the kinematical degrees of freedom. The solution family of $\hat{H}$ is flat on all minimal loops and has non-vanishing volume expectation values. Its edge-charge marginals delocalise with increasing cutoff, which indicates they are approximations to solutions that are non-normalisable in the kinematical inner product. The solution family for $\hat{H}^\dagger$ is normalisable, shows non-trivial charge correlations, lies in the kernel of volume and is not flat. $\hat{H}+\hat{H}^\dagger$ turns out to be much harder to solve and yields quasi-solutions combining features of both previous families. We characterise all solutions using chromaticity 1- and 2-point functions, minimal loop holonomies, geometric area and volume observables and show that the two families can be interpreted as, on the one hand, a family of states close to the Ashtekar-Lewandowski vacuum and the Dittrich-Geiller vacuum with some numerical noise on the other hand. We also present some results that link solutions of the truncated theory to solutions of the continuum theory.

gr-qc↗

arXiv2026

Multipartite entanglement in the quantum tetrahedron

The space $\mathrm{Inv}(j_1,j_2,j_3,j_4)$ of SU(2)-invariant four-valent tensors, also known as intertwiners, can be understood as the quantum states of a tetrahedron in Euclidean space with fixed areas. In loop quantum gravity, they are states of the smallest "atom of space" with non-zero volume. At the same time they correspond to four-party tensor product states invariant under global rotations. We consider the multipartite entanglement of states in $\mathrm{Inv}(j_1,j_2,j_3,j_4)$ using the recently proposed entropic fill. Numerically evaluating entropic fill in the case of equal spins between $1/2$ and $11$, we find that the distributions of entanglement are very different for intertwiners as compared to generic tensors, and for coherent intertwiners as compared to generic ones. The peak in the distribution seems to be at the highest entanglement for generic intertwiners and at the lowest for generic tensors, but in terms of average entanglement, the roles are switched: average entanglement is highest in arbitrary tensors and lower in intertwiners, at least in the regime of large $j$. We also find that entanglement depends on the geometric data of coherent intertwiners in a complicated way.

quant-ph↗

arXiv2025

Hyperinvariant Spin Network States -- An AdS/CFT Model from First Principles

We study the existence and limitations of hyperinvariant tensor networks incorporating a local SU(2) symmetry. As discrete implementations of the anti de-Sitter/conformal field theory (AdS/CFT) correspondence, such networks have created bridges between the fields of quantum information theory and quantum gravity. Adding SU(2) symmetry to the tensor network allows a direct connection to spin network states, a basis of the kinematic Hilbert space of loop quantum gravity (LQG). We consider a particular situation where the states can be interpreted as kinematic quantum states for three-dimensional quantum gravity. We show that important aspects of the AdS/CFT correspondence are realized in certain quantum states of the gravitational field in LQG, thus justifying, from first principles, a class of models introduced by [F. Pastawski et al., JHEP 06, 149 (2015)]. We provide examples of hyperinvariant tensor networks, but also prove constraints on their existence in the form of no-go theorems that exclude absolutely maximally entangled states as well as general holographic codes from local SU(2)-invariance. We calculate surface areas as expectation values of the LQG area operator and discuss further possible constraints as a consequence of a decay of correlations on the boundary.

quant-ph↗

arXiv2025

Dust shell in effective loop quantum black hole model

In this work, the dynamics of a dust shell in an effective theory of spherically symmetric gravity containing quantum corrections from loop quantum gravity is investigated. To provide a consistent framework for including the dust, we go beyond the standard formulation of the effective theory by introducing an action that includes not only the effective Hamiltonian constraint, but also the diffeomorphism constraint, along with appropriate gauge-fixing and boundary terms. By adding the dust shell action and substituting vacuum solutions for the interior and exterior regions, we derive a reduced action in which only the shell radius $\mathfrak{x}$ and the exterior black hole mass appear as dynamical variables. Varying the reduced action yields the evolution equation for $\mathfrak{x}$, which is then solved numerically to explore the dynamical properties of the dust shell and the continuity properties of the metric. Finally, our approach is compared with previous studies to highlight key differences and improvements.

gr-qc↗

arXiv2025

Bell states for fermions in loop quantum gravity

Fermion fields are fundamental for the description of nature and also fit very naturally into the framework of loop quantum gravity. Motivated partially by proposals to use gravitationally mediated entanglement of matter as a witness for the quantum nature of gravity, we investigate how such entanglement can be defined and investigated in loop quantum gravity. In particular, we ask how a pair of fermions in a Bell state could be described in loop quantum, gravity. We demonstrate that the notion of fermionic entanglement in loop quantum gravity is subtle, by showing that some potential ways to define it fail. We then investigate a kinematical observable involving both, fermionic and gravitational degrees of freedom, the component of the fermion spin normal to a surface. We study its properties, and compare it to the standard operator for components of spin in a given direction in quantum mechanics. Using these normal components of spin, we define a kinematical observable that measures the correlation between space-like separated fermions which closely mirrors the CHSH observable. Finally, we exhibit states of the fermions coupled to quantum geometry that violate the Bell-CHSH inequality.

gr-qc↗

arXiv2025

Deep learning spinfoam vertex amplitudes: the Euclidean Barrett-Crane model

Spinfoam theories propose a well-defined path-integral formulation for quantum gravity and are hoped to provide the dynamics of loop quantum gravity. However, it is computationally hard to calculate spinfoam amplitudes. The well-studied Euclidean Barrett-Crane model provides an excellent setting for testing analytical and numerical tools to probe spinfoam models. We explore a data-driven approach to accelerating spinfoam computations by showing that the vertex amplitude is an object that can be learned from data using deep learning. We divide the learning process into a classification and a regression task: Two networks are independently engineered to decide whether the amplitude is zero or not and to predict the precise numerical value, respectively. The trained networks are tested with several accuracy measures. The classifier in particular demonstrates robust generalisation far outside the training domain, while the regressor demonstrates high predictive accuracy in the domain it is trained on. We discuss limitations, possible improvements, and implications for future work.

gr-qc↗

arXiv2024

Towards quantum gravity with neural networks: Solving the quantum Hamilton constraint of U(1) BF theory

In the canonical approach of loop quantum gravity, arguably the most important outstanding problem is finding and interpreting solutions to the Hamiltonian constraint. In this work, we demonstrate that methods of machine learning are in principle applicable to this problem. We consider $U(1)$ BF theory in 3 dimensions, quantized with loop quantum gravity methods. In particular, we formulate a master constraint corresponding to Hamilton and Gauss constraints using loop quantum gravity methods. To make the problem amenable for numerical simulation we fix a graph and introduce a cutoff on the kinematical degrees of freedom, effectively considering $U_q(1)$ BF theory at a root of unity. We show that the Neural Network Quantum State (NNQS) ansatz can be used to numerically solve the constraints efficiently and accurately. We compute expectation values and fluctuations of certain observables and compare them with exact results or exact numerical methods where possible. We also study the dependence on the cutoff.

gr-qc↗

arXiv2024

Towards quantum gravity with neural networks: Solving quantum Hamilton constraints of 3d Euclidean gravity in the weak coupling limit

We consider 3-dimensional Euclidean gravity in the weak coupling limit of Smolin and show that it is BF-theory with $\text{U(1)}^3$ as a Lie group. The theory is quantised using loop quantum gravity methods. The kinematical degrees of freedom are truncated, on account of computational feasibility, by fixing a graph and deforming the algebra of the holonomies to impose a cutoff on the charge vectors. This leads to a quantum theory related to $\text{U}_q \text{(1)}^3$ BF-theory. The effect of imposing the cutoff on the charges is examined. We also implement the quantum volume operator of 3d loop quantum gravity. Most importantly we compare two constraints for the quantum model obtained: a master constraint enforcing curvature and Gauss constraint, as well as a combination of a quantum Hamilton constraint constructed using Thiemann's strategy and the Gauss master constraint. The two constraints are solved using the neural network quantum state ansatz, demonstrating its ability to explore models which are out of reach for exact numerical methods. The solutions spaces are quantitatively compared and although the forms of the constraints are radically different, the solutions turn out to have a surprisingly large overlap. We also investigate the behavior of the quantum volume in solutions to the constraints.

gr-qc↗

arXiv2023

Revisiting loop quantum gravity with selfdual variables: Hilbert space and first reality condition

We consider the quantization of gravity as an SL(2,C) gauge theory in terms of Ashtekar's selfdual variables and reality conditions for the spatial metric (RCI) and its evolution (RCII). We start from a holomorphic phase space formulation. It is then natural to push for a quantization in terms of holomorphic wave functions. Thus we consider holomorphic cylindrical wave functions over SL(2,C) connections. We use an overall phase ambiguity of the complex selfdual action to obtain Poisson brackets that mirror those of the real theory. We then show that there is a representation of the corresponding canonical commutation relations the space of holomorphic cylindrical functions. We describe a class of cylindrically consistent measures that implements RCI. We show that spin networks with SU(2) intertwiners form a basis for gauge invariant states. They are still mutually orthogonal, but the normalisation is different than for the Ashtekar-Lewandowski measure for SU(2). We do not consider RCII in the present article. Work on RCII is ongoing and will be presented elsewhere.

gr-qc↗

arXiv2023

Revisiting loop quantum gravity with selfdual variables: Classical theory

We review the classical formulation of general relativity as an SL(2,C) gauge theory in terms of Ashtekar's selfdual variables and reality conditions for the spatial metric (RCI) and its evolution (RCII), and we add some new observations and results. We first explain in detail how a connection taking values in the Lie algebra of the complex Lorentz group yields two Spin(3,1) connections, one selfdual and one anti-selfdual, without the need for a spin structure. We then demonstrate that the selfdual part of the complexified Palatini action in Ashtekar variables requires a holomorphic phase space description in order to obtain a non-degenerate symplectic structure. The holomorphic phase space does not allow for the implementation of the reality conditions as additional constraints, so they have to be taken care of "by hand" during the quantisation. We also observe that, due to the action being complex, there is an overall complex phase that can be chosen at will. We then review the canonical formulation and the consequences of the implementation of the reality conditions. We pay close attention to the transformation behaviour of the various fields under (complex) basis changes, as well as to complex analytic properties of the relevant functions on phase space.

gr-qc↗

arXiv2023

No invariant perfect qubit codes

Perfect tensors describe highly entangled quantum states that have attracted particular attention in the fields of quantum information theory and quantum gravity. In loop quantum gravity, the natural question arises whether SU(2) invariant tensors, which are fundamental ingredients of the basis states of spacetime, can also be perfect. In this work, we present a number of general constraints for the layout of such invariant perfect tensors (IPTs) and further describe a systematic and constructive approach to check the existence of an IPT of given valence. We apply our algorithm to show that no qubit encoding of valence 6 can be described by an IPT and close a gap to prove a no-go theorem for invariant perfect qubit encodings. We also provide two alternative proofs for the non-existence of 4-valent qubit IPTs which has been shown in [1,2].

quant-ph↗

arXiv2023

A Fock space structure for the diffeomorphism invariant Hilbert space of loop quantum gravity and its applications

Loop quantum gravity (LQG) is a quantization program for gravity based on the principles of QFT and general covariance of general relativity. Quantum states of LQG describe gravitational excitations based on graphs embedded in a spatial slice of spacetime. We show that, under certain assumptions on the class of diffeomorphisms, the space of diffeomorphism invariant states carries a Fock space structure. The role of one-particle excitations for this structure is played by the diffeomorphism invariant states based on graphs with a single (linked) component. This means, however, that a lot of the structure of the diffeomorphism invariant Hilbert space remains unresolved by this structure. We show how the Fock structure allows to write at least some condensate states of group field theory as diffeomorphism invariant coherent states of LQG in a precise sense. We also show how to construct other interesting states using this Fock structure. We finally explore the quantum geometry of single- and multi-particle states and tentatively observe some resemblance to geometries with a single or multiple components, respectively.

gr-qc↗

arXiv2023

Emergence of Riemannian Quantum Geometry

In this chapter we take up the quantum Riemannian geometry of a spatial slice of spacetime. While researchers are still facing the challenge of observing quantum gravity, there is a geometrical core to loop quantum gravity that does much to define the approach. This core is the quantum character of its geometrical observables: space and spacetime are built up out of Planck-scale quantum grains. The interrelations between these grains are described by spin networks, graphs whose edges capture the bounding areas of the interconnected nodes, which encode the extent of each grain. We explain how quantum Riemannian geometry emerges from two different approaches: in the first half of the chapter we take the perspective of continuum geometry and explain how quantum geometry emerges from a few principles, such as the general rules of canonical quantization of field theories, a classical formulation of general relativity in which it appears embedded in the phase space of Yang-Mills theory, and general covariance. In the second half of the chapter we show that quantum geometry also emerges from the direct quantization of the finite number of degrees of freedom of the gravitational field encoded in discrete geometries. These two approaches are complimentary and are offered to assist readers with different backgrounds enter the compelling arena of quantum Riemannian geometry.

gr-qc↗

arXiv2022

Chiral loop quantum supergravity and black hole entropy

Recent work has shown that local supersymmetry on a spacetime boundary in $\mathcal{N}$-extended AdS supergravity in chiral variables implies coupling to a boundary $\mathrm{OSp}(\mathcal{N}|2)_{\mathbb{C}}$ super Chern-Simons theory. We propose a way to calculate the entropy $S$ for the boundary, in the supersymmetric version of loop quantum gravity, for the minimal case $\mathcal{N}=1$. We calculate the dimensions of the quantum state spaces of $\mathrm{UOSp}(1|2)$ super Chern-Simons theory with punctures, and analytically continue, for fixed quantum super area of the surface, to $\mathrm{OSp}(1|2)_{\mathbb{C}}$. We find $S = a_H/4$ for large areas and determine the subleading correction.

gr-qc↗

arXiv2022

Towards black hole entropy in chiral loop quantum supergravity

Recently, many geometric aspects of $\mathcal{N}$-extended AdS supergravity in chiral variables have been encountered and clarified. In particular, if the theory is supposed to be invariant under SUSY transformations also on boundaries, the boundary term has to be the action of a $\mathrm{OSp}(\mathcal{N}|2)_{\mathbb{C}}$ super Chern-Simons theory, and particular boundary conditions must be met. Based on this, we propose a way to calculate an entropy $S$ for surfaces, presumably including black hole horizons, in the supersymmetric version of loop quantum gravity for the minimal case $\mathcal{N}=1$. It proceeds in analogy to the non-supersymmetric theory, by calculating dimensions of quantum state spaces of the super Chern-Simons theory with punctures, for fixed quantum (super) area of the surface. We find $S = a_H/4$ for large areas and determine the subleading correction. Due to the non-compactness of $\mathrm{OSp}(1|2)_{\mathbb{C}}$ and the corresponding difficulties with the Chern-Simons quantum theory, we use analytic continuation from the Verlinde formula for a compact real form, $\mathrm{UOSp}(1|2)$, in analogy to work by Noui et al. This also entails studying some properties of $\mathrm{OSp}(1|2)_{\mathbb{C}}$ representations that we have not found elsewhere in the literature.

gr-qc↗

arXiv2021

Holst-MacDowell-Mansouri action for (extended) supergravity with boundaries and super Chern-Simons theory

In this article, the Cartan geometric approach toward (extended) supergravity in the presence of boundaries will be discussed. In particular, based on new developments in this field, we will derive the Holst variant of the MacDowell-Mansouri action for $\mathcal{N}=1$ and $\mathcal{N}=2$ pure AdS supergravity in $D=4$ for arbitrary Barbero-Immirzi parameters. This action turns out to play a crucial role in context of boundaries in the framework of supergravity if one imposes supersymmetry invariance at the boundary. For the $\mathcal{N}=2$ case, it follows that this amounts to the introduction of a $θ$-topological term to the Yang-Mills sector which explicitly depends on the Barbero-Immirzi parameter. This shows the close connection between this parameter and the $θ$-ambiguity of gauge theory. We will also discuss the chiral limit of the theory, which turns out to possess some very special properties such as the manifest invariance of the resulting action under an enlarged gauge symmetry. Moreover, we will show that demanding supersymmetry invariance at the boundary yields a unique boundary term corresponding to a super Chern-Simons theory with $\mathrm{OSp}(\mathcal{N}|2)$ gauge group. In this context, we will also derive boundary conditions that couple boundary and bulk degrees of freedom and show equivalence to the results found in the D'Auria-Fré approach in context of the non-chiral theory. These results provide a step towards of quantum description of supersymmetric black holes in the framework of loop quantum gravity.

gr-qc↗

arXiv2021

Kinematics of arbitrary spin matter fields in loop quantum gravity

Loop quantum gravity envisions a small scale structure of spacetime that is markedly different from that of the classical spacetime continuum. This has ramifications for the excitation of matter fields and for their coupling to gravity. There is a general understanding of how to formulate scalar fields, spin $\frac{1}{2}$ fields and gauge fields in the framework of loop quantum gravity. The goal of the present work is to investigate kinematical aspects of this coupling. We will study implications of the Gauß and diffeomorphism constraint for the quantum theory: We define and study a less ambiguous variant of the Baez-Krasnov path observables, and investigate symmetry properties of spin network states imposed by diffeomorphism group averaging. We will do this in a setting which allows for matter excitations of spin $\frac{1}{2}$ and higher. In the case of spin $\frac{1}{2}$, we will also discuss extensions of it by introducing an electromagnetic field and antiparticles. We finally discuss in how far the picture with matter excitations of higher spin can be obtained from classical actions for higher spin fields.

gr-qc↗