The holonomy-flux algebra in low dimensions - Frembs, Markus; Sahlmann, Hanno (2013)
We derive the commutation relations of the holonomy-flux algebra in general for any dimension and gauge group. We explicitly calculate those for the case of two dimensions and one dimension. We generalize the intersection number which occurs for every commutator of holonomies and fluxes to certain equivalence classes of loops resembling homotopy classes before we start with the discussion of representations for holonomy-flux-algebras. In the case of two dimensions we discover some interesting parallels to the canonical commutation relations. We find the dual Ashtekar-Lewandowski-representation and also define the transformation rule which, just as the Fourier transformation in quantum mechanics, translates between the Ashtekar- Lewandowski-representation and its dual. We finally show that in two dimensions and for the abelian case, holonomy-flux-algebras can be related to Weyl algebras in a simple way.