Spin Foam Perturbation Theory for Three-Dimensional Quantum Gravity

Martins, João Faria; Mikovic, Aleksandar

doi:10.1007/s00220-009-0776-6

Cited by 6 publications

(25 citation statements)

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“…The latter observation points to a possible solution. First of all, any manifold admits a cubulation, that is, a triangulation by embedded hypercubes 14 [42]. We assume that M has a countable cover by open sets O α .…”

Section: Cubulations Versus Simplicial Triangulationsmentioning

confidence: 99%

The Holst spin foam model via cubulations

Baratin¹,

Flori²,

Thiemann³

2012

New J. Phys.

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Spin foam models are an attempt at a covariant or path integral formulation of canonical loop quantum gravity. The construction of such models usually relies on the Plebanski formulation of general relativity as a constrained BF theory and is based on the discretization of the action on a simplicial triangulation, which may be viewed as an ultraviolet regulator. The triangulation dependence can be removed by means of group field theory techniques, which allows one to sum over all triangulations. The main tasks for these models are the correct quantum implementation of the Plebanski constraints, the existence of a semiclassical sector implementing additional 'Regge-like' constraints arising from simplicial triangulations and the definition of the physical inner product of loop quantum gravity via group field theory. Here we propose a new approach to tackle these issues stemming directly from the Holst action for general relativity, which is also a proper starting point for canonical loop quantum gravity. The discretization is performed by means of a 'cubulation' of the manifold rather than a triangulation. We give a direct interpretation of the resulting spin foam model as a generating functional for the n-point functions on the physical Hilbert space at finite regulator. This paper focuses on ideas and tasks to be performed before the model can be taken seriously. However, our analysis reveals some interesting features of this model: firstly, the structure of its amplitudes differs Content from this work may be used under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. 9 In fact, as shown in recent work [8,33,34], SFM defined as constrained BF models take the form of noncommutative discrete path integrals making use of a star product on functionals of the B variables. It can also be shown that the generating group field theories are just a particular class of non-commutative field theories [35]. 10 See, however, [8,33,37] for discussions of this point. 11 See, however,[33] for a recent critical review of the various arguments raised against the Barrett-Crane model.

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Section: Cubulations Versus Simplicial Triangulationsmentioning

confidence: 99%

The Holst spin foam model via cubulations

Baratin¹,

Flori²,

Thiemann³

2012

New J. Phys.

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“…is negative, can one choose maximally regular cubulations as to minimise the error in our assumption of globally regular cubulations? In 3D some results on that issue seem to exist [98].…”

Section: Cubulations Versus Simplicial Triangulationsmentioning

confidence: 99%

“…One straightforward way of possibly repairing this situation is to generalise spin foam models to allow for arbitrary-in particular, cubic-triangulations, as suggested in [98,75].…”

Section: Expectation Value Of the Volume Operatormentioning

confidence: 99%

“…Or, if that turns out to be impossible, one should accept this result and conclude that, in order that the boundary Hilbert space of spin foam models has a semiclassical sector, one should generalise them to more general than simplicial triangulations of the four manifold, as advocated in [98,75].…”

Section: Iσσ3mentioning

confidence: 99%

“…First of all any manifold admits a cubulation, that is a triangulation by embedded hypercubes 18 [98]. We now assume that M has a countable cover by open sets O α .…”

Section: Cubulations Versus Simplicial Triangulationsmentioning

confidence: 99%

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Approaches to Quantum Gravity

Oriti

2009

215

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One of the main challenges in theoretical physics over the last five decades has been to reconcile quantum mechanics with general relativity into a theory of quantum gravity. However, such a theory has been proved to be hard to attain due to i) conceptual difficulties present in both the component theories (General Relativity (GR) and Quantum Theory); ii) lack of experimental evidence, since the regimes at which quantum gravity is expected to be applicable are far beyond the range of conceivable experiments. Despite these difficulties, various approaches for a theory of Quantum Gravity have been developed.In this thesis we focus on two such approaches: Loop Quantum Gravity and the Topos theoretic approach. The choice fell on these approaches because, although they both reject the Copenhagen interpretation of quantum theory, their underpinning philosophical approach to formulating a quantum theory of gravity are radically different. In particular LQG is a rather conservative scheme, inheriting all the formalism of both GR and Quantum Theory, as it tries to bring to its logical extreme consequences the possibility of combining the two. On the other hand, the Topos approach involves the idea that a radical change of perspective is needed in order to solve the problem of quantum gravity, especially in regard to the fundamental concepts of 'space' and 'time'. Given the partial successes of both approaches, the hope is that it might be possible to find a common ground in which each approach can enrich the other.This thesis is divided in two parts: in the first part we analyse LQG, paying particular attention to the semiclassical properties of the volume operator. Such an operator plays a pivotal role in defining the dynamics of the theory, thus testing its semiclassical limit is of uttermost importance. We then proceed to analyse spin foam models (SFM), which are an attempt at a covariant or path integral formulation of canonical Loop Quantum Gravity (LQG). In particular, in this thesis we propose a new SFM, whose path integral is defined in terms of the Holst action rather than the Plebanski action (used in current SFM). This departure from current SFM has enabled us to solve, explicitly, certain constraints which seem rather problematic in the current SFM.In the second part of this thesis we introduce Topos theory and how it has been utilised to reformulate quantum theory in a way that a consistent quantum logic can be defined. Moreover, we also define a Topos formulation of history quantum theory. The striking difference of this approach and the current consistent-history approach is that, in the former no fundamental role is played by the notion of a consistent sets (set of histories which do not interfere with each other) while, in the latter, such notions are central. This is an exciting departure since one of the main difficulty in the consistenthistory approach is how to choose the correct consistent set of history propositions, since there are many sets, most of which incompatible. However, we have s...

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Anisotropic spinfoam cosmology

Rennert¹,

Sloan²

2013

Class. Quantum Grav.

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The dynamics of a homogeneous, anisotropic universe are investigated within the context of spinfoam cosmology. Transition amplitudes are calculated for a graph consisting of a single node and three links -the 'Daisy graph' -probing the behaviour a classical Bianchi I spacetime. It is shown further how the use of such single node graphs gives rise to a simplification of states such that all orders in the spin expansion can be calculated, indicating that it is the vertex expansion that contains information about quantum dynamics.

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Spin Foam Perturbation Theory for Three-Dimensional Quantum Gravity

Cited by 6 publications

References 15 publications

The Holst spin foam model via cubulations

The Holst spin foam model via cubulations

Approaches to Quantum Gravity

Anisotropic spinfoam cosmology

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