Coupled intertwiner dynamics: A toy model for coupling matter to spin foam models

Steinhaus, Sebastian

doi:10.1103/physrevd.92.064007

Cited by 12 publications

(20 citation statements)

References 121 publications

(285 reference statements)

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“…Thus the spectra of observables, which preserve the triangulation, are discrete. This also avoids divergencies that appear in SU(2) based spin foam models [23][24][25][26] and allows for (tensor network) coarse graining schemes [27][28][29][30][31][32][33].…”

Section: Jhep05(2017)123mentioning

confidence: 99%

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(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

Dittrich

2017

J. High Energ. Phys.

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We apply the recently suggested strategy to lift state spaces and operators for (2 + 1)-dimensional topological quantum field theories to state spaces and operators for a (3 + 1)-dimensional TQFT with defects. We start from the (2 + 1)-dimensional TuraevViro theory and obtain a state space, consistent with the state space expected from the Crane-Yetter model with line defects.This work has important applications for quantum gravity as well as the theory of topological phases in (3 + 1) dimensions. It provides a self-dual quantum geometry realization based on a vacuum state peaked on a homogeneously curved geometry. The state spaces and operators we construct here provide also an improved version of the Walker-Wang model, and simplify its analysis considerably.We in particular show that the fusion bases of the (2 + 1)-dimensional theory lead to a rich set of bases for the (3 + 1)-dimensional theory. This includes a quantum deformed spin network basis, which in a loop quantum gravity context diagonalizes spatial geometry operators. We also obtain a dual curvature basis, that diagonalizes the Walker-Wang Hamiltonian.Furthermore, the construction presented here can be generalized to provide state spaces for the recently introduced dichromatic four-dimensional manifold invariants.

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Section: Jhep05(2017)123mentioning

confidence: 99%

“…To restore this symmetry one would have to consider a continuum limit [19,20,[104][105][106], which can be constructed via an auxiliary coarse graining flow [58]. For coarse graining schemes along the lines of [27][28][29][30][31][32][33] the finiteness of the state spaces constructed here facilitates a numerical implementation.…”

Section: Jhep05(2017)123mentioning

confidence: 99%

(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

Dittrich

2017

J. High Energ. Phys.

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“…See in particular [46][47][48][49][50] for algorithms involving 2D models with a quantum group symmetry and [51,52] for algorithms for 3D (generalized) lattice gauge models. Let us note that for non-Abelian lattice gauge models as well as the models considered here one has to generally expect that the flow does not preserve the coupling conditions (4), see ref.…”

Section: The Turaev-viro Partition Function and Related Modelsmentioning

confidence: 99%

“…This is important for numerical coarse graining techniques, such as tensor network algorithms for lattice gauge theories [51,52,[116][117][118][119]. The finiteness of such q-deformed models has also been used to investigate certain two-dimensional models, designed to reproduce key dynamical mechanisms of spin foams [46][47][48][49][50]. The fusion basis [18,65], which we will also make use of here, is a useful tool for canonical and covariant coarse graining schemes [120].…”

Section: √λmentioning

confidence: 99%

Cosmological Constant from Condensation of Defect Excitations

Dittrich

2018

Universe

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A key challenge for many quantum gravity approaches is to construct states that describe smooth geometries on large scales. Here we define a family of (2 + 1)-dimensional quantum gravity states which arise from curvature excitations concentrated at point like defects and describe homogeneously curved geometries on large scales. These states represent therefore vacua for three-dimensional gravity with different values of the cosmological constant. They can be described by an anomaly-free first class constraint algebra quantized on one and the same Hilbert space for different values of the cosmological constant. A similar construction is possible in four dimensions, in this case the curvature is concentrated along string-like defects and the states are vacua of the Crane-Yetter model. We will sketch applications for quantum cosmology and condensed matter.

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“…First, it allows for a numerical implementation of so-called tensor network algorithms for coarse graining [40,41,78,79]. For this reason, quantum group models are used in [39,80,81]. Second, spin foam models based on the undeformed ( ) SU 2 group feature divergencies due to the unbounded summation over spins (see [82] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Quantum gravity kinematics from extended TQFTs

2017

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In this paper, we show how extended topological quantum field theories (TQFTs) can be used to obtain a kinematical setup for quantum gravity, i.e. a kinematical Hilbert space together with a representation of the observable algebra including operators of quantum geometry. In particular, we consider the holonomy-flux algebra of (2+1)-dimensional Euclidean loop quantum gravity, and construct a new representation of this algebra that incorporates a positive cosmological constant. The vacuum state underlying our representation is defined by the Turaev-Viro TQFT. This vacuum state can be thought of as being peaked on connections with homogeneous curvature. We therefore construct here a generalization, or more precisely a quantum deformation at root of unity, of the previously introduced SU(2) BF representation. The extended Turaev-Viro TQFT provides a description of the excitations on top of the vacuum. These curvature and torsion excitations are classified by the Drinfeld center category of the quantum deformation of SU(2), and are essential in order to allow for a representation of the holonomies and fluxes. The holonomies and fluxes are generalized to ribbon operators which create and interact with the excitations. These excitations agree with the ones induced by massive and spinning particles, and therefore the framework presented here allows automatically for a description of the coupling of such matter to + ( ) 2 1 -dimensional gravity with a cosmological constant. The new representation constructed here presents a number of advantages over the representations which exist so far. In particular, it possesses a very useful finiteness property which guarantees the discreteness of spectra for a wide class of quantum (intrinsic and extrinsic) geometrical operators. Also, the notion of basic excitations leads to a so-called fusion basis which offers exciting possibilities for the construction of states with interesting global properties, as well as states with certain stability properties under coarse graining. In addition, the work presented here showcases how the framework of extended TQFTs, as well as techniques from condensed matter, can help design new representations, and construct and understand the associated notion of basic excitations. This is essential in order to find the best starting point for the construction of the dynamics of quantum gravity, and will enable the study of possible phases of spin foam models and group field theories from a new perspective.Several such realizations of quantum geometry are now known. Each realization is based on and can be characterized by a different kinematical vacuum state. So far, all representations have in common that this vacuum is totally squeezed, i.e. sharply peaked either on the flux operators or on the holonomy operators. The operators of quantum geometry act on this vacuum state, and generate thereby all the excitations of the Hilbert space underlying the representation of the holonomy-flux algebra. The vacuum is therefore cyclic with respect to the set of ...

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Coupled intertwiner dynamics: A toy model for coupling matter to spin foam models

Cited by 12 publications

References 121 publications

(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

Cosmological Constant from Condensation of Defect Excitations

Quantum gravity kinematics from extended TQFTs

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