The Barrett–Crane model: asymptotic measure factor

Kamiński, Wojciech; Steinhaus, Sebastian

doi:10.1088/0264-9381/31/7/075014

Cited by 9 publications

(11 citation statements)

References 46 publications

(114 reference statements)

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“…Alternatively one can devise measure factors that are local and lead to an approximate invariance near very symmetric configurations. Such measure factors could be taken as a hint for choosing the measure for spin foams, see for instance [39] for a first geometric interpretation of the measure factor for a 4D model.…”

Section: Discussionmentioning

confidence: 99%

Discretization independence implies non-locality in 4D discrete quantum gravity

Dittrich¹,

Kamiński²,

Steinhaus³

2014

Class. Quantum Grav.

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2 the measure to obtain the divergence structure that fits the divergences one expects from diffeomorphism symmetry. This of course does not fully guarantee diffeomorphism symmetry or triangulation independence, as divergences might also arise due to other reasons.In Regge calculus, several measures have been proposed: in [6] Hamber and Williams propose a discretization of the formal continuum path integral, with a local discretization of the (DeWitt) measure [6,7], conflicting with the proposal by Menotti and Peirano [8], who mod out a subgroup of the continuum diffeomorphisms resulting in a highly non-local measure. A different discretization, also leading to a non-local measure due to discretizing first the DeWitt super metric [20] and then forming the determinant, was proposed in [9].In this work we will pick up the suggestion in [21], to choose a measure that, at least for the linearized (Regge) theory, leads to as much discretization invariance as possible. These considerations require to actually integrate out degrees of freedom and thus take the dynamics into account.The requirement of discretisation independence seems to be at odds with interacting theories, which possess local / propagating degrees of freedom. This apparent contradiction can be resolved by allowing a non-local action or nonlocal amplitudes for the quantum theory -which in fact are unavoidable if one coarse grains the theory. Non-local amplitudes are however difficult to deal with. We therefore ask in this paper the question, whether in the quantum theory we can retain as much symmetry as in the classical theory, with a choice of local measure. The classical 4D Regge action is known to be invariant under 5 − 1 moves and 4 − 2 moves, but not under 3 − 3 moves [21]. Here the non-invariance under the 3 − 3 moves -in fact the only move involving bulk curvature for the solution -allows the local Regge action to nevertheless lead to a theory with propagating degrees of freedom. We therefore ask whether it is possible to have a local measure for linearized Regge calculus that leads to invariance under 5 − 1 and / or 4 − 2 moves. Such a measure would therefore reproduce the symmetry properties of the Regge action. We will however show that such a local path integral measure does not exist.Our requirement of (maximal) discretisation independence is motivated by the 'perfect action / discretisation' approach [22] that targets to construct a discretisation, which 'perfectly' encodes the continuum dynamics and has a discrete remnant of the continuum diffeomorphism symmetry. Examples of such 'perfect discretisations' are 3D Regge calculus with and without a cosmological constant [23] and also 4D Regge calculus, if the boundary data impose a flat solution in the bulk. In these examples, the basic building blocks mimic the continuum dynamics, e.g. one takes constantly curved tetrahedra for 3D gravity with a cosmological constant. Such perfect discretisations can be constructed as the fixed point of a coarse graining scheme, see for instance [23][24][25...

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Section: Discussionmentioning

confidence: 99%

Discretization independence implies non-locality in 4D discrete quantum gravity

Dittrich¹,

Kamiński²,

Steinhaus³

2014

Class. Quantum Grav.

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“…• Non-degeneracy of the Hessian: This was checked for specific configurations in the standard EPRL model [12]. It is known that it fails in the Barrett-Crane model for certain non-generic but geometric data [29]. We expect that the Hessian in the EPRL model is always nondegenerate if the reconstructed 4-simplex is non-degenerate.…”

Section: • Boundary Contributionsmentioning

confidence: 99%

Asymptotic analysis of the EPRL model with timelike tetrahedra

Kamiński

Kisielowski

Sahlmann

2018

Class. Quantum Grav.

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We perform the stationary phase analysis of the vertex amplitude for the EPRL spin foam model extended to include timelike tetrahedra. We analyse both, tetrahedra of signature − − − (standard EPRL), as well as of signature + − − (Hnybida-Conrady extension), in a unified fashion. However, we assume all faces to be of signature −−. The stationary points of the extended model are described again by 4-simplices and the phase of the amplitude is equal to the Regge action. Interestingly, in addition to the Lorentzian and Euclidean sectors there appear also split signature 4-simplices.

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“…However, spin foams, and also the discussed intertwiner models, are complex assigning ∼ exp(iS) to the path integral (for one orientation), also for Riemannian signature. Moreover, it is well-known from spin foam asymptotics [101][102][103][104][105][106] that both orientations of a 4-simplex have to be considered, such that one rather finds ∼ cos(S Regge ), where S Regge is the Regge action [107] associated to the 4-simplex. To the author's best knowledge, it is not known whether it is consistent to couple a Wick-rotated matter theory to a non-rotated gravitational theory and, moreover, in which way matter should couple to different orientations.…”

Section: Summary and Discussionmentioning

confidence: 99%

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

Steinhaus

2015

Phys. Rev. D

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The universal coupling of matter and gravity is one of the most important features of general relativity. In quantum gravity, in particular spin foams, matter couplings have been defined in the past, yet the mutual dynamics, in particular if matter and gravity are strongly coupled, are hardly explored, which is related to the definition of both matter and gravitational degrees of freedom on the discretisation. However extracting this mutual dynamics is crucial in testing the viability of the spin foam approach and also establishing connections to other discrete approaches such as lattice gauge theories.Therefore, we introduce a simple 2D toy model for Yang-Mills coupled to spin foams, namely an Ising model coupled to so-called intertwiner models defined for SU(2) k . The two systems are coupled by choosing the Ising coupling constant to depend on spin labels of the background, as these are interpreted as the edge lengths of the discretisation. We coarse grain this toy model via tensor network renormalization and uncover an interesting dynamics: the Ising phase transition temperature turns out to be sensitive to the background configurations and conversely, the Ising model can induce phase transitions in the background. Moreover, we observe a strong coupling of both systems if close to both phase transitions.

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The Barrett–Crane model: asymptotic measure factor

Cited by 9 publications

References 46 publications

Discretization independence implies non-locality in 4D discrete quantum gravity

Discretization independence implies non-locality in 4D discrete quantum gravity

Asymptotic analysis of the EPRL model with timelike tetrahedra

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

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