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Holography in LQG--new result by Gambini and Pullin
http://arxiv.org/abs/0708.0250
Holography in spherically symmetric loop quantum gravity
Rodolfo Gambini, Jorge Pullin
5 pages
(Submitted on 2 Aug 2007)
"We show that holography arises naturally in the context of spherically symmetric loop quantum gravity. The result is not dependent on detailed assumptions about the dynamics of the theory being considered. It ties strongly the amount of information contained in a region of space to the tight mathematical underpinnings of loop quantum geometry, at least in this particular context."
=====exerpts from introduction and conclusions sections======
Any successful theory of quantum gravity that incorporates holography should be able to derive it as a consequence of its framework. We would like to argue that holography does indeed follow from the framework of loop quantum gravity in spherical symmetry and that the result is robust: it does not depend on the details of the dynamics of the theory nor the type of matter included but rather on its kinematical structure and elementary dynamical considerations independent of the details of the Hamiltonian or its potential regularization ambiguities. In a nutshell, holography follows from the dependence of the volume operator in spherical loop quantum gravity on the radial distance, yielding an uncertainty in the determination of volumes that grows radially. Such a dependence for the uncertainty of spatial measurements had already been postulated in heuristic treatments relating limitations of space-time measurements to holography by Ng [3] and with alternative reasonings by Ng and Lloyd [4]). In this article we show that such a dependence can be derived from the kinematical structure of spherical loop quantum gravity. That holography in its simple and straightforward spatial form is materialized in the spherical case is appropriate, since it is known that in non-spherical cases more care is needed (in particular involving spatiotemporal regions) in its definition in order not to run into counterexamples (see [1] for details).
Loop quantum gravity is emerging as a viable candidate for a theory of quantum gravity. Recent general discussions of the approach can be found in [5]. The kinematical setting for loop quantum gravity in spherically symmetric situations is well established and was discussed in detail by Bojowald and Swiderski [6]...
...
We have therefore established that the kinematical structure of loop quantum gravity in spherical symmetry, together with elementary dynamical considerations independent of the details of the Hamiltonian, implies holography. It is therefore a very general result. It stems from the fact that the elementary volume that any dynamical operator may involve goes as [radial distance times the square of Planck length] (as suggested by previous heuristic estimates [3]). We have assumed a finite amount of information per elementary volume, as is usually argued in this context [1]. This is usually justified by thinking that the fields are collections of harmonic oscillators and the energy in each oscillator is bounded by the Planck energy and therefore has a finite number of states. Although a complete quantum gravity analysis has not been done, studies of the harmonic oscillator [13] and of linearized gravity [14] suggest that this bound is even tighter in loop quantum gravity. The fact that elementary volumes grow with the distance to the black hole implies that the information in a spatial region is bounded by the area, contrary to what happens if one assumes the elementary volumes go as [the cube of the Planck length].
Holography is therefore naturally built into the elementary framework of loop quantum gravity with spherical symmetry. The calculation we showed also implies for the first time a derivation from first principles of equation (15) which had been heuristically proposed [3] as a fundamental limit on the measurement of space and time and the ultimate limits of computability in nature and which may even be tested observationally in the near future in astronomical settings [15]. These conjectured limits could imply limitations in the unitarity of quantum mechanics, with far ranging implications [16]. We therefore see that in spite of the limitations of the spherically symmetric minisuperspace, attractive physical conclusions that have been conjectured for the full theory are verified.
==endquote==
http://arxiv.org/abs/0708.0250
Holography in spherically symmetric loop quantum gravity
Rodolfo Gambini, Jorge Pullin
5 pages
(Submitted on 2 Aug 2007)
"We show that holography arises naturally in the context of spherically symmetric loop quantum gravity. The result is not dependent on detailed assumptions about the dynamics of the theory being considered. It ties strongly the amount of information contained in a region of space to the tight mathematical underpinnings of loop quantum geometry, at least in this particular context."
=====exerpts from introduction and conclusions sections======
Any successful theory of quantum gravity that incorporates holography should be able to derive it as a consequence of its framework. We would like to argue that holography does indeed follow from the framework of loop quantum gravity in spherical symmetry and that the result is robust: it does not depend on the details of the dynamics of the theory nor the type of matter included but rather on its kinematical structure and elementary dynamical considerations independent of the details of the Hamiltonian or its potential regularization ambiguities. In a nutshell, holography follows from the dependence of the volume operator in spherical loop quantum gravity on the radial distance, yielding an uncertainty in the determination of volumes that grows radially. Such a dependence for the uncertainty of spatial measurements had already been postulated in heuristic treatments relating limitations of space-time measurements to holography by Ng [3] and with alternative reasonings by Ng and Lloyd [4]). In this article we show that such a dependence can be derived from the kinematical structure of spherical loop quantum gravity. That holography in its simple and straightforward spatial form is materialized in the spherical case is appropriate, since it is known that in non-spherical cases more care is needed (in particular involving spatiotemporal regions) in its definition in order not to run into counterexamples (see [1] for details).
Loop quantum gravity is emerging as a viable candidate for a theory of quantum gravity. Recent general discussions of the approach can be found in [5]. The kinematical setting for loop quantum gravity in spherically symmetric situations is well established and was discussed in detail by Bojowald and Swiderski [6]...
...
We have therefore established that the kinematical structure of loop quantum gravity in spherical symmetry, together with elementary dynamical considerations independent of the details of the Hamiltonian, implies holography. It is therefore a very general result. It stems from the fact that the elementary volume that any dynamical operator may involve goes as [radial distance times the square of Planck length] (as suggested by previous heuristic estimates [3]). We have assumed a finite amount of information per elementary volume, as is usually argued in this context [1]. This is usually justified by thinking that the fields are collections of harmonic oscillators and the energy in each oscillator is bounded by the Planck energy and therefore has a finite number of states. Although a complete quantum gravity analysis has not been done, studies of the harmonic oscillator [13] and of linearized gravity [14] suggest that this bound is even tighter in loop quantum gravity. The fact that elementary volumes grow with the distance to the black hole implies that the information in a spatial region is bounded by the area, contrary to what happens if one assumes the elementary volumes go as [the cube of the Planck length].
Holography is therefore naturally built into the elementary framework of loop quantum gravity with spherical symmetry. The calculation we showed also implies for the first time a derivation from first principles of equation (15) which had been heuristically proposed [3] as a fundamental limit on the measurement of space and time and the ultimate limits of computability in nature and which may even be tested observationally in the near future in astronomical settings [15]. These conjectured limits could imply limitations in the unitarity of quantum mechanics, with far ranging implications [16]. We therefore see that in spite of the limitations of the spherically symmetric minisuperspace, attractive physical conclusions that have been conjectured for the full theory are verified.
==endquote==
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