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Curvature function and coarse graining · Around

2011

Curvature function and coarse graining

Diaz-Marin, Homero G., Zapata, Jose A.

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A classic theorem in the theory of connections on principal fiber bundles states that the evaluation of all holonomy functions gives enough information to characterize the bundle structure (among those sharing the same structure group and base manifold) and the connection up to a bundle equivalence map.

This result and other important properties of holonomy functions has encouraged their use as the primary ingredient for the construction of families of quantum gauge theories.
However, in these applications often the set of holonomy functions used is a discrete proper subset of the set of holonomy functions needed for the characterization theorem to hold.
We show that the evaluation of a discrete set of holonomy functions does not characterize the bundle and does not constrain the connection modulo gauge appropriately.

Built on

Robion C. Kirby and Laurence C. Siebenmann, Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, Ann. of Math. Stud. vol. 88, Princeton University Press, Princeton, NJ (1977) With notes by John Milnor and Michael Atiyah
1977
Earlier work this paper cites.
R. Sorkin, “The Electromagnetic Field On A Simplicial Net,” J. Math. Phys. 16
Original
1978
Earlier work this paper cites.
R. Fritsch and R. A. Piccinini, “Cellular Structures in Topology” Cambridge studies in advanced mathematics. Cambridge University Press (1990)
1990
Earlier work this paper cites.

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A. Ashtekar and J. Lewandowski, “Projective Techniques And Functional Integration For Gauge Theories,” J. Math. Phys. 36
1995
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Then

J. W. Barrett, “Holonomy and path structures in general relativity and Yang-Mills theory,” Int. J. Theor. Phys. 30
1996
Later among the works it cites.
J. Martinez, C. Meneses and J. A. Zapata, “Geometry of C-flat connections, coarse graining and the continuum limit,” J. Math. Phys. 46
2005
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D. Oriti, C. Rovelli and S. Speziale, Class. Quant. Grav. 22
2005
Later among the works it cites.

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