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marcus

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In the Physics forum, jby started a LQG thread which at one point included some discussion of Connections. It might be helpful to have a thread focused on the connection which is taking over the central role from the metric for some purposes.

GR used to be the study of the metric on a smooth manifold. A metric would give rise to an idea of curvature and an idea of parallel transport of vectors along a path from one point to another. Imagine a sphere and a tangent vector at the equator pointing in some direction and imagine scooting that vector up to the north pole. What direction does it point then?

The simplest way to imagine a connection, I believe, is as a machine to parallel translate vectors along paths from one point to another. If you have a metric and derive a connection from it and then your disk crashes and you have forgotten the metric, then you can RECOVER the metric from the connection (except for scale). So the connection-----the definite proceedure for parallel transport----has the same information in it, including an idea of curvature.

Rovelli begins his brief history of LQG with 1986 when it occured to someone to quantize the connection instead of the metric

http://www.livingreviews.org/Articles/Volume1/1998-1rovelli/index.html

in the index of Rovelli's review of LQG you see section 3: History.

Just jump to "3 History of Loop Quantum Gravity" and you find the first few sentences are about this 1986 move to connections.

[edit: the 1986 paper was Ashtekar's seminal reformulation

of GR in terms of connections, also called the "new variables", if interested, look at the brief history in section 3 for yourself]

Anyone is welcome to take over the job of telling the story. I am just summarizing from Rovelli's LivingReviews article at this point

GR used to be the study of the metric on a smooth manifold. A metric would give rise to an idea of curvature and an idea of parallel transport of vectors along a path from one point to another. Imagine a sphere and a tangent vector at the equator pointing in some direction and imagine scooting that vector up to the north pole. What direction does it point then?

The simplest way to imagine a connection, I believe, is as a machine to parallel translate vectors along paths from one point to another. If you have a metric and derive a connection from it and then your disk crashes and you have forgotten the metric, then you can RECOVER the metric from the connection (except for scale). So the connection-----the definite proceedure for parallel transport----has the same information in it, including an idea of curvature.

Rovelli begins his brief history of LQG with 1986 when it occured to someone to quantize the connection instead of the metric

http://www.livingreviews.org/Articles/Volume1/1998-1rovelli/index.html

in the index of Rovelli's review of LQG you see section 3: History.

Just jump to "3 History of Loop Quantum Gravity" and you find the first few sentences are about this 1986 move to connections.

[edit: the 1986 paper was Ashtekar's seminal reformulation

of GR in terms of connections, also called the "new variables", if interested, look at the brief history in section 3 for yourself]

Anyone is welcome to take over the job of telling the story. I am just summarizing from Rovelli's LivingReviews article at this point

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