1997
DOI: 10.1103/physrevd.55.6099
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Graphical evolution of spin network states

Abstract: The evolution of spin network states in loop quantum gravity can be defined with respect to a time variable, given by the surfaces of constant value of an auxiliary scalar field. We regulate the Hamiltonian, generating such an evolution, and evaluate its action both on edges and on vertices of the spin network states. The analytical computations are carried out completely to yield a finite, diffeomorphism invariant result. We use techniques from the recoupling theory of colored graphs with trivalent vertices t… Show more

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Cited by 20 publications
(31 citation statements)
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“…This formulation has been used in the context of continuum and lattice gauge theory 3], and it has found a particularly e ective application in quantum gravity 2,4], because it allows a description of the di eomorphism invariant quantum states in terms of knot theory 2,5], and, at the same time, because it partially diagonalizes the quantum dynamics of the theory, leading to the discovery of solutions of the dynamical constraints 2,6]. Recent results in quantum gravity based on the loop representation include the construction of a nite physical Hamiltonian operator for pure gravity 7] and fermions 8], the computation of the physical spectra of area 9] and volume 10], and the developement of a perturbation scheme that may allow transition amplitudes to be explicitely computed 7,11,12]. A mathematically rigorous formulation of quantum eld theories whose con guration space is a space of connections, inspired by the loop representation, has been recently developed 13,14] and the kinematics of the theory is now on a level of rigor comparable to that of constructive quantum eld theory 15].…”
Section: Introductionmentioning
confidence: 99%
See 1 more Smart Citation
“…This formulation has been used in the context of continuum and lattice gauge theory 3], and it has found a particularly e ective application in quantum gravity 2,4], because it allows a description of the di eomorphism invariant quantum states in terms of knot theory 2,5], and, at the same time, because it partially diagonalizes the quantum dynamics of the theory, leading to the discovery of solutions of the dynamical constraints 2,6]. Recent results in quantum gravity based on the loop representation include the construction of a nite physical Hamiltonian operator for pure gravity 7] and fermions 8], the computation of the physical spectra of area 9] and volume 10], and the developement of a perturbation scheme that may allow transition amplitudes to be explicitely computed 7,11,12]. A mathematically rigorous formulation of quantum eld theories whose con guration space is a space of connections, inspired by the loop representation, has been recently developed 13,14] and the kinematics of the theory is now on a level of rigor comparable to that of constructive quantum eld theory 15].…”
Section: Introductionmentioning
confidence: 99%
“…The fact that it diagonalizes the operator that measures the volume of a spatial slice 10] gives us a physical picture of a discrete quantum geometry and also makes the spin network basis useful for perturbation expansions of the dynamics of general relativity, as described in 7,11,12]. It has also played a role in the mathematically rigorous investigations of refs.…”
Section: Introductionmentioning
confidence: 99%
“…Instead, we seek the simplest algorithm for a transition amplitude between spin network states that is consistent with some discrete microscopic form of causality. The reason for this is that attempts to follow the procedure of canonical quantization, although having led to partial success [21,22,19,23,24], face both conceptual and technical problems that it is not clear can be resolved successfully. Besides the problem of causal structure mentioned above, there is the whole problem of time and observables in quantum cosmology.…”
Section: Introductionmentioning
confidence: 99%
“…Besides the problem of causal structure mentioned above, there is the whole problem of time and observables in quantum cosmology. In addition, while it seems to have been possible to construct well defined finite diffeomorphism invariant operators that represent the hamiltonian and hamiltonian constraint [21,22,19,23,24], these suffer from problems related to both the algebra of quantum constraints and the existence of a good continuum limit [25].…”
Section: Introductionmentioning
confidence: 99%
“…We close this section with a short aside concerning the definition of evolution laws of 'spin networks' by Markopoulou, Smolin and Borissov (see [21] or [22]). As in our case there are more or less two possibilities: treating evolution laws within an integrated space-time formalism or regard the network as representing space alone with the time evolution being implanted via some extra principle ( which is the way we have chosen above).…”
Section: Definition (General Local Law On Cellular Network)mentioning
confidence: 99%