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Basic LQG questions raised by 1998 "LQG Primer" article
B Crowell asked what would be the best way into LQG and then (on looking over the suggested entry-level "LQG Primer" article by Rovelli and Upadhya http://arxiv.org/abs/gr-qc/9806079 ) made a potentially very helpful list of questions.
I hope others here will help answer these questions. They seem to me to be natural ones that could reasonably be asked after taking a look at any LQG introduction. So exceptionally good ones to discuss. Here they are:
==quote==
What was really unclear to me was both the mathematical meaning and the physical significance of the SU(2) connection.
Mathematically, I understand the connection to be a rule for parallel transport, and I've seen it specified using either a metric or Christoffel symbols. The paper seems to be defining it using a smooth vector field, which is unfamiliar to me. Is this covered somewhere in texts like Wald and MTW? The reason for giving it values in su(2) is also unclear to me, and I guess that's a separate issue. Since they're talking about a three-dimensional space, I would think that the tangent space would be R3, not su(2)...??
Physically, I don't understand the motivation for introducing all the spin algebra. Is it basically because they want to be able to describe fundamental particles as existing at different places on this graph? Would there be spin-2 gravitons?
I also didn't understand the physical motivation for introducing the graphs. Is the manifold primary and the graphs secondary? Or vice-versa? What do the graphs represent physically?
Why is this all done in a three-dimensional space? Does this three-dimensional space relate to the quantum state of spacetime on some surface of simultaneity? Or does it not even relate to three dimensions of actual spacetime? I'd thought that the dimensionality of spacetime was an emergent property in LQG, and there was some difficulty in even showing that something like Minkowski space was a solution.
==endquote==
B Crowell asked what would be the best way into LQG and then (on looking over the suggested entry-level "LQG Primer" article by Rovelli and Upadhya http://arxiv.org/abs/gr-qc/9806079 ) made a potentially very helpful list of questions.
bcrowell said:...
I took a stab at the Rovelli-Upadhya paper...
I hope others here will help answer these questions. They seem to me to be natural ones that could reasonably be asked after taking a look at any LQG introduction. So exceptionally good ones to discuss. Here they are:
==quote==
What was really unclear to me was both the mathematical meaning and the physical significance of the SU(2) connection.
Mathematically, I understand the connection to be a rule for parallel transport, and I've seen it specified using either a metric or Christoffel symbols. The paper seems to be defining it using a smooth vector field, which is unfamiliar to me. Is this covered somewhere in texts like Wald and MTW? The reason for giving it values in su(2) is also unclear to me, and I guess that's a separate issue. Since they're talking about a three-dimensional space, I would think that the tangent space would be R3, not su(2)...??
Physically, I don't understand the motivation for introducing all the spin algebra. Is it basically because they want to be able to describe fundamental particles as existing at different places on this graph? Would there be spin-2 gravitons?
I also didn't understand the physical motivation for introducing the graphs. Is the manifold primary and the graphs secondary? Or vice-versa? What do the graphs represent physically?
Why is this all done in a three-dimensional space? Does this three-dimensional space relate to the quantum state of spacetime on some surface of simultaneity? Or does it not even relate to three dimensions of actual spacetime? I'd thought that the dimensionality of spacetime was an emergent property in LQG, and there was some difficulty in even showing that something like Minkowski space was a solution.
==endquote==
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