Chen Integrals, Generalized Loops and Loop Calculus

Details Chen Integrals, Generalized Loops and Loop Calculus Chen Integrals, Generalized Loops and Loop Calculus

1993-05-29

arxiv.org/abs/hep-th/9305173v1

Int. J. Mod. Phys. A9 (1994) 4511-4548

Physics

High Energy Physics

High Energy Physics - Theory

43p., Latex. arXiv admin NOTE: large portions of pages 8-10 of this submission are taken verbatim and without attribution from

Scientific paper

We use Chen iterated line integrals to construct a topological algebra ${\cal A}_p$ of separating functions on the {\it Group of Loops} ${\bf L}{\cal M}_p$. ${\cal A}_p$ has an Hopf algebra structure which allows the construction of a group structure on its spectrum. We call this topological group, the group of generalized loops $\widetilde {{\bf L}{\cal M}_p}$. Then we develope a {\it Loop Calculus}, based on the {\it Endpoint} and {\it Area Derivative Operators}, providing a rigorous mathematical treatment of early heuristic ideas of Gambini, Trias and also Mandelstam, Makeenko and Migdal. Finally we define a natural action of the "pointed" diffeomorphism group $Diff_p({\cal M})$ on $ \widetilde {{\bf L}{\cal M}_p}$, and consider a {\it Variational Derivative} which allows the construction of homotopy invariants. This formalism is useful to construct a mathematical theory of {\it Loop Representation} of Gauge Theories and Quantum Gravity. Figures available by request.

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