Algebraic {$q$}-Integration and Fourier Theory on Quantum and Braided Spaces

Abstract

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 $\int x$ 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 $\Delta x=x\otimes 1+1\otimes x$ and braided-antipode $S$ to define a convolution product, and prove a convolution theorem. We prove also that $F^2=S$. 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.

50 pages. Minor changes, added 3 references

References in corpus (8)

• Braided Momentum Structure of the q-Poincare Group
• Quantum and Braided Linear Algebra
• Free Braided Differential Calculus, Braided Binomial Theorem and the Braided Exponential Map
• q-Euclidean space and quantum group wick rotation by twisting
• Quantum and Braided Lie Algebras
• The $S{O}_q(N,\mathbf{R})$-Symmetric Harmonic Oscillator on the Quantum Euclidean Space ${\mathbf{R}}_q^N$ and its Hilbert Space Structure
• On the Addition of Quantum Matrices
• Glueing operation for r-matrices, quantum groups and link-invariants of Hecke type

Cited by in corpus (51)

• Maximal Localisation in the Presence of Minimal Uncertainties in Positions and Momenta
• On Quantum Field Theory with Nonzero Minimal Uncertainties in Positions and Momenta
• Quantum Groups and Noncommutative Geometry
• Noncommutative Harmonic Analysis, Sampling Theory and the Duflo Map in 2+1 Quantum Gravity
• Introduction to Braided Geometry and $q$-Minkowski Space
• *-Structures on Braided Spaces
• q-Integration on Quantum Spaces
• q-Epsilon tensor for quantum and braided spaces
• k-Deformed Fourier Transform
• Fourier transform and the Verlinde formula for the quantum double of a finite group
• Algebraic approach to quantum gravity II: noncommutative spacetime
• Supersymmetry from a braided point of view
• Integration on quantum Euclidean space and sphere in $N$ dimensions
• Quantum groups and deformed special relativity
• q-deformed harmonic and Clifford analysis and the q-Hermite and Laguerre polynomials
• The Quantum Geometry of Spin and Statistics
• Analysis on q-deformed quantum spaces
• Maximally localized states and causality in non commutative quantum theories
• Formulation of Supersymmetry on a Lattice as a Representation of a Deformed Superalgebra
• q-Exponentials on quantum spaces
• Double-Bosonisation and the Construction of {$U_q(g)$}
• Meaning of Noncommutative Geometry and the Planck-Scale Quantum Group
• Quantum Groups, Roots of Unity and Particles on quantized Anti-de Sitter Space
• Duality Principle and Braided Geometry
• A braided interpretation of fractional supersymmetry in higher dimensions
• The $q$-Fourier transform of $q$-distributions
• Some remarks on the {$q$}-Poincare algebra in R-matrix form
• Braided Oscillators
• Towards Non-Commutative Deformations of Relativistic Wave Equations in 2+1 Dimensions
• Generalized Q-Exponentials Related to Orthogonal Quantum Groups and Fourier Transformations of Noncommutative Spaces
• On the hermiticity of q-differential operators and forms on the quantum Euclidean spaces R_q^N
• Many Q-Particles from One: A New Approach to $\ast$-Hopf Algebras?
• Geometry of Quantum Group Twists, Multidimensional Jackson Calculus and Regularization
• Wave Propagation and IR/UV Mixing in Noncommutative Spacetimes
• The Fourier Transform on Quantum Euclidean Space
• On the q-convolution on the line
• A q-analogue of convolution on the line
• On the braided Fourier transform in the n-dimensional quantum space
• Non-relativistic Schroedinger theory on q-deformed quantum spaces I, Mathematical framework and equations of motion
• q-convolution and its q-Fourier transform
• Braided Line and Counting Fixed Points of GL(d,F_q)
• Nonrelativistic one-particle problem on $q$-deformed Euclidean space
• Conservation laws for a $q$-deformed nonrelativistic particle
• Momentum and Position Representations for the q-deformed Euclidean Quantum Space
• Sturm-Liouville Problem in Quantum Calculus
• Quasi-{$\ast$} Structure on {$q$}-Poincare algebras
• Quantum kinematics on q-deformed quantum spaces II, Wave functions on position and momentum space
• Quantum kinematics on q-deformed quantum spaces I, Mathematical Framework
• Non-relativistic Schroedinger theory on q-deformed quantum spaces II, The free non-relativistic particle and its interactions
• Lagrangian Formalism Over Graded Algebras
• Conservation laws for a class of nonlinear equations with variable coefficients on discrete and noncommutative spaces