- #36
Ambitwistor
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Yes, connections are geometrically intuitive sorts of structures, and spin networks are similarly intuitive ways of describing functions on the Hilbert space: they are not far from lattice gauge theory, with edge representations literally representing parallel transport, etc.
You can find Baez writing about gauge connections in various places, explaining how (in the semi-classical limit) they describe the phase transformation of dragging a quantum particle around in a classical background field given by a connection. With respect to spin networks, the Baez paper I cited in the other thread (gr-qc/9504036) is the clearest I've seen.
Now, physical interpretation is another matter. With Yang-Mills theory, the connection has the above interpretation. With the Levi-Civita connection of GR, it has the usual interpretation of parallel transport, obviously being an SO(3,1) connection to relate different tangent spaces by Lorentz transformations.
When it come to the original Ashtekar variables, like the Sen variables they were based on, they describe the propagation of left-handed neutrinos (spinors) -- that too can be described intuitively, though spinors and complex spin connections are harder to get across. The problem comes when we go to the real connection variables: as nonunitary noted in the other thread, you lose the nice geometric interpretation that the complex Ashtekar variables have (and gain the ease that comes with not having to worry about reality conditions). The real connection, being a connection, does give rise to a geometry and thus should have a geometric interpretation, but I have never heard of any intuitive way of understanding the geometry of this particular connection. I don't think anyone else has either, which is why you hear people saying that the real variables lose the geometric interpretation of the original complex variables.
Incidentally, the Wilson who proposed traces of holonomies as observables is THE Wilson: namely, Kenneth Wilson, the 1982 Nobel laureate who was behind renormalization group theory and lattice gauge theory.
You can find Baez writing about gauge connections in various places, explaining how (in the semi-classical limit) they describe the phase transformation of dragging a quantum particle around in a classical background field given by a connection. With respect to spin networks, the Baez paper I cited in the other thread (gr-qc/9504036) is the clearest I've seen.
Now, physical interpretation is another matter. With Yang-Mills theory, the connection has the above interpretation. With the Levi-Civita connection of GR, it has the usual interpretation of parallel transport, obviously being an SO(3,1) connection to relate different tangent spaces by Lorentz transformations.
When it come to the original Ashtekar variables, like the Sen variables they were based on, they describe the propagation of left-handed neutrinos (spinors) -- that too can be described intuitively, though spinors and complex spin connections are harder to get across. The problem comes when we go to the real connection variables: as nonunitary noted in the other thread, you lose the nice geometric interpretation that the complex Ashtekar variables have (and gain the ease that comes with not having to worry about reality conditions). The real connection, being a connection, does give rise to a geometry and thus should have a geometric interpretation, but I have never heard of any intuitive way of understanding the geometry of this particular connection. I don't think anyone else has either, which is why you hear people saying that the real variables lose the geometric interpretation of the original complex variables.
Incidentally, the Wilson who proposed traces of holonomies as observables is THE Wilson: namely, Kenneth Wilson, the 1982 Nobel laureate who was behind renormalization group theory and lattice gauge theory.