The Constraint Algebra of Quantum Gravity in the Loop Representation

Gambini, Rodolfo; Garat, Alcides; Pullin, Jorge

doi:10.1142/s0218271895000417

Cited by 14 publications

(27 citation statements)

References 6 publications

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“…Having a well defined action for the loop derivative opens the possibility of introducing quantum versions of the Hamiltonian constraint and to implement in a concrete well defined setting proposals for the Hamiltonian constraint that have been shown to have the correct algebra at a formal quantum level [6]. We will expand on this and other topics in the companion paper.…”

Section: Discussionmentioning

confidence: 99%

Canonical quantum gravity in the Vassiliev invariants arena: I. Kinematical structure

Bartolo

Gambini²,

Griego³

et al. 2000

Class. Quantum Grav.

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We generalize the idea of Vassiliev invariants to the spin network context, with the aim of using these invariants as a kinematical arena for a canonical quantization of gravity. This paper presents a detailed construction of these invariants (both ambient and regular isotopic) requiring a significant elaboration based on the use of Chern-Simons perturbation theory which extends the work of Kauffman, Martin and Witten to four-valent networks. We show that this space of knot invariants has the crucial property -from the point of view of the quantization of gravity-of being loop differentiable in the sense of distributions. This allows the definition of diffeomorphism and Hamiltonian constraints. We show that the invariants are annihilated by the diffeomorphism constraint. In a companion paper we elaborate on the definition of a Hamiltonian constraint, discuss the constraint algebra, and show that the construction leads to a consistent theory of canonical quantum gravity.CGPG-99/11-2 gr-qc/9911009

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Section: Discussionmentioning

confidence: 99%

Canonical quantum gravity in the Vassiliev invariants arena: I. Kinematical structure

Bartolo

Gambini²,

Griego³

et al. 2000

Class. Quantum Grav.

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“…This opens the hope that a consistent set of constraints, satisfying the appropriate commutator algebra, might be found. One could think of computing the commutator algebra off-shell, very much along the same lines as in the formal computations of [6], except that now the operators involved are also well defined on-shell (i.e. when the wavefunctions are knot invariants) .…”

Section: Discussionmentioning

confidence: 99%

Vassiliev invariants: A new framework for quantum gravity

Gambini¹,

Griego²,

Pullin³

1998

Nuclear Physics B

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We show that Vassiliev invariants of knots, appropriately generalized to the spin network context, are loop differentiable in spite of being diffeomorphism invariant. This opens the possibility of defining rigorously the constraints of quantum gravity as geometrical operators acting on the space of Vassiliev invariants of spin nets. We show how to explicitly realize the diffeomorphism constraint on this space and present proposals for the construction of Hamiltonian constraints.CGPG-98/3-1 gr-qc/9803018 * Associate member of ICTP.

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“…where Θ ✸ (y) is one if y ∈ ✸ and zero otherwise. We now replace in (7) the ǫ abc using formula (8) and we represent u c k A c (v ✸ ) using a holonomy in the fundamental representation along the edge u k of the triangulation,…”

Section: The Hamiltonian Constraintmentioning

confidence: 99%

“…where we have identified v ≡ v ✸ to simplify the notation. Using (8) we replace the ǫ acd and ǫ def in terms of the volume of the elementary regions and the edges u of the tetrahedra, and we join four of the u's with the A's to construct holonomies along the edges of the triangulation,…”

Section: A Functions Of Spin Nets With "Marked Vertices"mentioning

confidence: 99%

“…These functions are not the most natural ones to consider in the context of diffeomorphism invariant theories like general relativity, but our constraints are well defined on them, so we can consider them to be a "habitat" where to test the constraint algebra, at least as a mathematical exercise. On these kinds of functions the diffeomorphism constraints that we have introduced here are known to close the appropriate algebra in the limit in which regulators are removed (see [8], although the regularization is slightly different than the one we use in this paper, it is immediate to check that the same calculations go through). We therefore will not repeat the calculation here.…”

Section: A Habitat I: Holonomies Of Smooth Connectionsmentioning

confidence: 99%

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Canonical quantum gravity in the Vassiliev invariants arena: II. Constraints, habitats and consistency of the constraint algebra

Bartolo¹,

Gambini²,

Griego³

et al. 2000

Class. Quantum Grav.

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In a companion paper we introduced a kinematical arena for the discussion of the constraints of canonical quantum gravity in the spin network representation based on Vassiliev invariants. In this paper we introduce the Hamiltonian constraint, extend the space of states to non-diffeomorphism invariant "habitats" and check that the off-shell quantum constraint commutator algebra reproduces the classical Poisson algebra of constraints of general relativity without anomalies. One can therefore consider the resulting set of constraints and space of states as a consistent theory of canonical quantum gravity.

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The Constraint Algebra of Quantum Gravity in the Loop Representation

Cited by 14 publications

References 6 publications

Canonical quantum gravity in the Vassiliev invariants arena: I. Kinematical structure

Canonical quantum gravity in the Vassiliev invariants arena: I. Kinematical structure

Vassiliev invariants: A new framework for quantum gravity

Canonical quantum gravity in the Vassiliev invariants arena: II. Constraints, habitats and consistency of the constraint algebra

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