Chen Integrals, Generalized Loops and Loop Calculus

Abstract

We use Chen iterated line integrals to construct a topological algebra A p of separating functions on the Group of Loops LM p . 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 LM p .Then we develope a Loop Calculus, based on the Endpoint and 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 Dif f p (M) on LM p , and consider a Variational Derivative which allows the construction of homotopy invariants.This formalism is useful to construct a mathematical theory of Loop Representation of Gauge Theories and Quantum Gravity.

Generalized loop space and TMDs

“…The loop space can be turned, using the Gel'fand spectrum, into a topological group which is Banach, Hopf, Hausdorff, Tychonov, commutative, nuclear and multiplicative convex. In [17] it is shown 02011-p.2…”

“…The solution we used to assess these problems is to use the generalized loop space introduced in [17], which makes use of Chen's algebraic paths [18]. Here the usual loop space is endowed with an extra equivalence relation realized by the Wilson loop functionals:…”

“…Among the differential operators that are allowed on generalized loop space we have the path-and areaderivatives [17,20,21]. Both of them were used by Makeenko and Migdal in combination with the Dyson-Schwinger approach to derive their Makeenko-Migdal loop equations:…”

Gauge-Invariant Gluon TMD and Evolution in the Coordinate Space

“…They are integral-differential equations which the elements of this space (generic Wilson loops) obey, if the underlying contours γ i on which the path-ordered exponentials of the gauge fields are defined experience certain variations. The variations of the contours give rise to the variations of the exponentials themselves, the latter being described by the infinite set of the Makeenko-Migdal loop equations [16,17,18,19,20]. More specifically, the elements of the loop space are the vacuum matrix elements of products of the Wilson loops, that is…”

Evolution of light-like Wilson loops with a self-intersection in loop space

“…In this work, we consider some symmetrical combinations of two quadrilateral Wilson loops on the light-cone, for which we test conjecture (4). Put in a Wilson loop variable language: we are calculating W 1 [Γ], with Γ = Γ 1 Γ 2 , where the product between the loops is defined in generalized loop space [12]. Important is that these Wilson loop configurations exhibit intricacies, associated with the self-intersection and overlap, that usually cause problems in a loop space approach.…”