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Jal has called attention to this paper of Reuter and Schwindt
http://arxiv.org/abs/hep-th/0511021
A Minimal Length from the Cutoff Modes in Asymptotically Safe Quantum Gravity
Martin Reuter, Jan-Markus Schwindt
(Submitted on 2 Nov 2005)
"Within asymptotically safe Quantum Einstein Gravity (QEG), the quantum 4-sphere is discussed as a specific example of a fractal spacetime manifold. The relation between the infrared cutoff built into the effective average action and the corresponding coarse graining scale is investigated. Analyzing the properties of the pertinent cutoff modes, the possibility that QEG generates a minimal length scale dynamically is explored. While there exists no minimal proper length, the QEG sphere appears to be 'fuzzy' in the sense that there is a minimal angular separation below which two points cannot be resolved by the cutoff modes."
You may have a different viewpoint on this. From my perspecitve, the general background for this is that a very interesting quantum theory of spacetime (geometry and matter) could come from the convergence of several quantum geometric approaches
LQG, Reuter QED, CDT, Spinfoam, Groupfieldtheory.
Of these AFAIK only LQG has a minimal length. Also as far as I know all these approaches start with a distanceless CONTINUUM---a smooth 4D differentiable manifold on which there is no pre-established metric distance-function.
This is because they are all children of Einstein GR, which is built on a smooth distanceless continuum where the metric function is not pre-established but instead grows out of the situation.
If these approaches were to converge there would be the issue of a minimal length. Notice that only LQG has it. Also notice that in LQG the minimal length is not necessarily implemented IN THE FABRIC, logically all one can say is that it's a limitation on THE BUSINESS OF MEASUREMENT that emerges from the theory. E.g. you can't measure area and get a smaller answer than such and such because the spectrum of the quantum observable, when it is calculated, turns out to have a smallest nonzero eigen-number. An area operator (called an "observable") corresponds to the act of measuring the area of something. In all kinds of Quantum Mechanics there is this tantalizing interplay between measurement and reality---we aren't going to resolve that here.
If there is going to be a convergence of approaches then we should educate ourselves about either possibility. On the one hand, it could turn out that somehow there was some mistake and LQG does not after all have a minimal length: then it would agree with the others. Or on the other hand it could turn out that some of the other approaches, e.g. Reuter QEG, unexpectedly HAVE a minimal length. or anyway "kind of".
So maybe we should study this Reuter Schwindt paper just in case it is right and there is (unexpectedly) a kind of minimal length measurement in QEG. Again I should stress that this could be merely a constraint ON THE BUSINESS OF MEASUREMENT and it might not be implemented in some ultimate reality presumed to have absolute existence.
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OK AFTER THAT LONG BUILD-UP I finally read the paper and it doesn't work for me! Isnt that a letdown?
Maybe someone else will take a look at it and come up with another take. But at present I don't understand how QEG can yield a minimum length
as i understand one.
http://arxiv.org/abs/hep-th/0511021
A Minimal Length from the Cutoff Modes in Asymptotically Safe Quantum Gravity
Martin Reuter, Jan-Markus Schwindt
(Submitted on 2 Nov 2005)
"Within asymptotically safe Quantum Einstein Gravity (QEG), the quantum 4-sphere is discussed as a specific example of a fractal spacetime manifold. The relation between the infrared cutoff built into the effective average action and the corresponding coarse graining scale is investigated. Analyzing the properties of the pertinent cutoff modes, the possibility that QEG generates a minimal length scale dynamically is explored. While there exists no minimal proper length, the QEG sphere appears to be 'fuzzy' in the sense that there is a minimal angular separation below which two points cannot be resolved by the cutoff modes."
You may have a different viewpoint on this. From my perspecitve, the general background for this is that a very interesting quantum theory of spacetime (geometry and matter) could come from the convergence of several quantum geometric approaches
LQG, Reuter QED, CDT, Spinfoam, Groupfieldtheory.
Of these AFAIK only LQG has a minimal length. Also as far as I know all these approaches start with a distanceless CONTINUUM---a smooth 4D differentiable manifold on which there is no pre-established metric distance-function.
This is because they are all children of Einstein GR, which is built on a smooth distanceless continuum where the metric function is not pre-established but instead grows out of the situation.
If these approaches were to converge there would be the issue of a minimal length. Notice that only LQG has it. Also notice that in LQG the minimal length is not necessarily implemented IN THE FABRIC, logically all one can say is that it's a limitation on THE BUSINESS OF MEASUREMENT that emerges from the theory. E.g. you can't measure area and get a smaller answer than such and such because the spectrum of the quantum observable, when it is calculated, turns out to have a smallest nonzero eigen-number. An area operator (called an "observable") corresponds to the act of measuring the area of something. In all kinds of Quantum Mechanics there is this tantalizing interplay between measurement and reality---we aren't going to resolve that here.
If there is going to be a convergence of approaches then we should educate ourselves about either possibility. On the one hand, it could turn out that somehow there was some mistake and LQG does not after all have a minimal length: then it would agree with the others. Or on the other hand it could turn out that some of the other approaches, e.g. Reuter QEG, unexpectedly HAVE a minimal length. or anyway "kind of".
So maybe we should study this Reuter Schwindt paper just in case it is right and there is (unexpectedly) a kind of minimal length measurement in QEG. Again I should stress that this could be merely a constraint ON THE BUSINESS OF MEASUREMENT and it might not be implemented in some ultimate reality presumed to have absolute existence.
================================
OK AFTER THAT LONG BUILD-UP I finally read the paper and it doesn't work for me! Isnt that a letdown?
Maybe someone else will take a look at it and come up with another take. But at present I don't understand how QEG can yield a minimum length
as i understand one.
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