Quantum theory of geometry: III. Non-commutativity of Riemannian structures

In this work we investigate the canonical quantization of 2+1 gravity with cosmological constant Λ > 0 in the canonical framework of loop quantum gravity. The unconstrained phase space of gravity in 2+1 dimensions is coordinatized by an SU(2) connection A and the canonically conjugate triad field e. A natural regularization of the constraints of 2+1 gravity can be defined in terms of the holonomies of A ± = A± √ Λe. As a first step towards the quantization of these constraints we study the canonical quantization of the holonomy of the connection A λ = A + λe (for λ ∈ R) on the kinematical Hilbert space of loop quantum gravity. The holonomy operator associated to a given path acts non trivially on spin network links that are transversal to the path (a crossing). We provide an explicit construction of the quantum holonomy operator. In particular, we exhibit a close relationship between the action of the quantum holonomy at a crossing and Kauffman's q-deformed crossing identity (with q = exp(i λ/2)). The crucial difference is that (being an operator acting on the kinematical Hilbert space of LQG) the result is completely described in terms of standard SU(2) spin network states (in contrast to q-deformed spin networks in Kauffman's identity). We discuss the possible implications of our result.

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“…factor ordering ambiguities (operators associated to e are non commuting in the quantum theory [15]). …”

Section: Jhep10(2011)036mentioning

confidence: 99%

Canonical quantization of non-commutative holonomies in 2 + 1 loop quantum gravity

Noui

Pérez

Pranzetti

2011

J. High Energ. Phys.

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“…Using this basis they showed how to construct operators corresponding to observables such as the areas of surfaces and volumes of regions [28]. This work was soon made rigorous by Ashtekar, Lewandowski, Baez and others, and spin networks quickly became a standard tool in this field [1,2,3,4,5].…”

Section: Physics Applicationsmentioning

confidence: 99%

Integrability for relativistic spin networks

Baez

Barrett

2001

Class. Quantum Grav.

113

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Abstract. The evaluation of relativistic spin networks plays a fundamental role in the Barrett-Crane state sum model of Lorentzian quantum gravity in 4 dimensions. A relativistic spin network is a graph labelled by unitary irreducible representations of the Lorentz group appearing in the direct integral decomposition of the space of L 2 functions on three-dimensional hyperbolic space. To 'evaluate' such a spin network we must do an integral; if this integral converges we say the spin network is 'integrable'. Here we show that a large class of relativistic spin networks are integrable, including any whose underlying graph is the 4-simplex (the complete graph on 5 vertices). This proves a conjecture of Barrett and Crane, whose validity is required for the convergence of their state sum model.

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“…The reason for this "anomaly" is due to the fact that we did not properly smear the φ's as explained in [26] : we can expect the correspondence {., .} → 1 ih [., .]…”

mentioning

confidence: 99%

Kinematical Hilbert spaces for fermionic and Higgs quantum field theories

Thiemann

1998

Class. Quantum Grav.

162

330

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We extend the recently developed kinematical framework for diffeomorphism invariant theories of connections for compact gauge groups to the case of a diffeomorphism invariant quantum field theory which includes besides connections also fermions and Higgs fields. This framework is appropriate for coupling matter to quantum gravity.The presence of diffeomorphism invariance forces us to choose a representation which is a rather non-Fock-like one : the elementary excitations of the connection are along open or closed strings while those of the fermions or Higgs fields are at the end points of the string.Nevertheless we are able to promote the classical reality conditions to quantum adjointness relations which in turn uniquely fixes the gauge and diffeomorphism invariant probability measure that underlies the Hilbert space.Most of the fermionic part of this work is independent of the recent preprint by Baez and Krasnov and earlier work by Rovelli and Morales-Tecótl because we use new canonical fermionic variables, so-called Grassman-valued half-densities, which enable us to to solve the difficult fermionic adjointness relations.

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Quantum theory of geometry: III. Non-commutativity of Riemannian structures

Cited by 139 publications

References 27 publications

Canonical quantization of non-commutative holonomies in 2 + 1 loop quantum gravity

Canonical quantization of non-commutative holonomies in 2 + 1 loop quantum gravity

Integrability for relativistic spin networks

Kinematical Hilbert spaces for fermionic and Higgs quantum field theories

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