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kodama
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- TL;DR Summary
- another approach
there are theoretical problems with loop quantum gravity based on directly quantizing Ashtekar variables,
for example,
"One can pinpoint the technical error in LQG explicitly:To recall, the starting point of LQG is to encode the Riemannian metric in terms of the parallel transport of the affine connection that it induces. This parallel transport is an assignment to each smooth curve in the manifold between points x
and y of a linear isomorphism TxX→TyY
between the tangent spaces over these points.This assignment is itself smooth, as a function on the smooth space of smooth curves, suitably defined. Moreover, it satisfies the evident functoriality conditions, in that it respects composition of paths and identity paths.It is a theorem that smooth (affine) connections on smooth manifolds are indeed equivalent to such smooth functorial assignments of parallel transport isomorphisms to smooth curves. This theorem goes back to Barrett, who considered it for the case that all paths are taken to be loops. For the general case it is discussed in arxiv.org/0705.0452, following suggestion by John Baez.So far so good. The idea of LQG is now to use this equivalence to equivalently regard the configuration space of gravity as a space of parallell transport/holonomy assignments to paths (in particular loops, whence the name "LQG").But now in the next step in LQG, the smoothness condition on these parallel transport assignments is dropped. Instead, what is considered are general functions from paths to group elements, which are not required to be smooth or even to be continuous, hence plain set-theoretic functions. In the LQG literature these assignments are then called "generalized connections". It is the space of these "generalized connections" which is then being quantized.The trouble is that there is no relation left between "generalized connections" and the actual (smooth) affine connections of Riemanniann geometry. The passage from smooth to "generalized connections" is an ad hoc step that is not justified by any established rule of quantization. It effectively changes the nature of the system that is being quantized."
ref https://physics.stackexchange.com/q...lly-not-listed-as-a-theory-of-e/360010#360010
since Ashtekar variables are standard general relativity rewritten from metric to connections formulation, Anti de Sitter spacetime could be formulated in terms of Ashtekar variables
with the advantage that here, Anti de Sitter spacetime is expressed as smooth (affine) connections on smooth manifolds are indeed equivalent to such smooth functorial assignments of parallel transport isomorphisms to smooth curves rather than a metric in standard GR. it is expressed as a SU(2) gauge theory similar to used in the standard model.
according to the AdS/CFT correspondence the bulk is an , Anti de Sitter spacetime here in 4 dimensions, there is a corresponding conformal field theories in one less dimension.
so Ashtekar variables in 4 dimensions and Anti de Sitter spacetime should be dual to a conformal field theories, and so working with conformal field theories could answer many quantum gravity questions as LQG appears to be incorrect.
and then work with the CFT to answer any quantum gravity questions
to date, supersymmetry has not been confirmed by the LHC and may not exist in nature. there is also no evidence of additional Kaluza Klein dimensions as required by string theory.
in summary, one form of quantum gravity is the "bulk" of Anti de Sitter spacetime in 4 dimensions expressed as Ashtekar variables or spinfoam, without supersymmetry. It is dual to a 3 dimensional conformal field theory according to the the AdS/CFT correspondence
so any quantum gravity calculations can be formed by a CFT in 3 dimensions, and theoertical issues of quantum gravity can be respond by the corresponding CFT.
what makes this different from LQG, is that there is no direct quantiziation of Ashtekar variables, only use of CFT via holographic duality
this isn't LQG but holographic Ashtekar theory, or holographic spinfoam theory.I have in mind a spinfoam theory in 4 dimensions completely independent of LQG but satisfies the AdS/CFT correspondence
since the bulk is Anti de Sitter spacetime in 4 dimensions expressed as Ashtekar variables smooth (affine) connections on smooth manifolds are indeed equivalent to such smooth functorial assignments of parallel transport isomorphisms to smooth curves
the CFT could help show how it's quantum gravity properties
for example,
"One can pinpoint the technical error in LQG explicitly:To recall, the starting point of LQG is to encode the Riemannian metric in terms of the parallel transport of the affine connection that it induces. This parallel transport is an assignment to each smooth curve in the manifold between points x
and y of a linear isomorphism TxX→TyY
between the tangent spaces over these points.This assignment is itself smooth, as a function on the smooth space of smooth curves, suitably defined. Moreover, it satisfies the evident functoriality conditions, in that it respects composition of paths and identity paths.It is a theorem that smooth (affine) connections on smooth manifolds are indeed equivalent to such smooth functorial assignments of parallel transport isomorphisms to smooth curves. This theorem goes back to Barrett, who considered it for the case that all paths are taken to be loops. For the general case it is discussed in arxiv.org/0705.0452, following suggestion by John Baez.So far so good. The idea of LQG is now to use this equivalence to equivalently regard the configuration space of gravity as a space of parallell transport/holonomy assignments to paths (in particular loops, whence the name "LQG").But now in the next step in LQG, the smoothness condition on these parallel transport assignments is dropped. Instead, what is considered are general functions from paths to group elements, which are not required to be smooth or even to be continuous, hence plain set-theoretic functions. In the LQG literature these assignments are then called "generalized connections". It is the space of these "generalized connections" which is then being quantized.The trouble is that there is no relation left between "generalized connections" and the actual (smooth) affine connections of Riemanniann geometry. The passage from smooth to "generalized connections" is an ad hoc step that is not justified by any established rule of quantization. It effectively changes the nature of the system that is being quantized."
ref https://physics.stackexchange.com/q...lly-not-listed-as-a-theory-of-e/360010#360010
since Ashtekar variables are standard general relativity rewritten from metric to connections formulation, Anti de Sitter spacetime could be formulated in terms of Ashtekar variables
with the advantage that here, Anti de Sitter spacetime is expressed as smooth (affine) connections on smooth manifolds are indeed equivalent to such smooth functorial assignments of parallel transport isomorphisms to smooth curves rather than a metric in standard GR. it is expressed as a SU(2) gauge theory similar to used in the standard model.
according to the AdS/CFT correspondence the bulk is an , Anti de Sitter spacetime here in 4 dimensions, there is a corresponding conformal field theories in one less dimension.
so Ashtekar variables in 4 dimensions and Anti de Sitter spacetime should be dual to a conformal field theories, and so working with conformal field theories could answer many quantum gravity questions as LQG appears to be incorrect.
if spinfoam can reproduce Anti de Sitter spacetime in 4 dimensions, that it should also be dual to a CFT,and then work with the CFT to answer any quantum gravity questions
to date, supersymmetry has not been confirmed by the LHC and may not exist in nature. there is also no evidence of additional Kaluza Klein dimensions as required by string theory.
in summary, one form of quantum gravity is the "bulk" of Anti de Sitter spacetime in 4 dimensions expressed as Ashtekar variables or spinfoam, without supersymmetry. It is dual to a 3 dimensional conformal field theory according to the the AdS/CFT correspondence
so any quantum gravity calculations can be formed by a CFT in 3 dimensions, and theoertical issues of quantum gravity can be respond by the corresponding CFT.
what makes this different from LQG, is that there is no direct quantiziation of Ashtekar variables, only use of CFT via holographic duality
this isn't LQG but holographic Ashtekar theory, or holographic spinfoam theory.I have in mind a spinfoam theory in 4 dimensions completely independent of LQG but satisfies the AdS/CFT correspondence
since the bulk is Anti de Sitter spacetime in 4 dimensions expressed as Ashtekar variables smooth (affine) connections on smooth manifolds are indeed equivalent to such smooth functorial assignments of parallel transport isomorphisms to smooth curves
the CFT could help show how it's quantum gravity properties
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