Algebraic q-Integration and Fourier Theory on Quantum and Braided Spaces

We introduce an algebraic theory of integration on quantum planes and other braided spaces. In the one dimensional case we obtain a novel picture of the Jackson q-integral as indefinite integration on the braided group of functions in one variable x. Here x is treated with braid statistics q rather than the usual bosonic or Grassmann ones. We show 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 our algebraic integration with a representation then leads to ordinary numbers for the integral. We also use our integration to develop a full theory of q-Fourier transformation F . We use the braided addition ∆x = x⊗ 1 + 1⊗x and braided-antipode S to define a convolution product, and prove a convolution theorem. We prove also that F = S. We prove the analogous results on any braided group, including integration and Fourier transformation on quantum planes associated to general R-matrices, including q-Euclidean and q-Minkowski spaces.