Algebraic q-integration and Fourier theory on quantum and braided spaces

An algebraic theory of integration on quantum planes and other braided spaces is introduced. In the one-dimensional case a novel picture of the Jackson q- integral as indefinite integration on the braided group of functions in one variable x is obtained. Here x is treated with braid statistics q rather than the usual bosonic or Grassmann ones. It is shown that the definite integral ∫x∞−x∞ can also be evaluated algebraically as multiples of the integral of a q-Gaussian, with x remaining as a bosonic scaling variable associated with the q-deformation. Further composing the algebraic integration with a representation then leads to ordinary numbers for the integral. Integration is also used to develop a full theory of q-Fourier transformation ℱ The braided addition Δx=x⊗1+1⊗x and braided-antipode S is used to define a convolution product, and prove a convolution theorem. It is also proven that ℱ2=S. The analogous results are proven on any braided group, including integration and Fourier transformation on quantum planes associated to general R matrices, including q-Euclidean and q-Minkowski spaces.