1. CAN NATURE BE Q-DEFORMED? Hartmut Wachter May 16, 2009

2. Contents  Introduction  Milestones in q-deformation  Idea of a smallest length  Regularization by q-deformation  Multi-dimensional q-analysis  Application to quantum physics  Outlook

3. Introduction    

4.    

5. Milestones in q-deformation  q-numbers (Euler) and q-hypergeometric series (Heine)  q-integrals and q-derivatives (Jackson)  quantized universal enveloping algebras (Kulish, Reshetikhin, Drinfeld, Jimbo)  quantum matrix algebras (Woronowicz, Vaksman, Soibelman)  quantum spaces with differential calculi (Manin, Wess, Zumino)  braided groups (Majid)

6. Idea of a smallest length  Plane-waves of different wave-length can have the same effect on a lattice:  Thus, we can restrict attention to wave-lengths larger than twice the lattice spacing:  A smallest wave-length implies an upper bound in momentum space: a

7. Regularization by q-deformation  Transition amplitudes contain q-analogs of Fourier transforms:  Jackson-integral singles out a lattice:  For suitable c q-deformed trigonometrical functions rapidly diminish on q-lattice points: q-deformed trigono-metrical function Jackson-integral points of q-lattice q-lattice points are very near roots of q-trigonometrical function

8. Regularization by q-deformation  Fourier transform converges even for polynomial functions:  Large values of x·p are „suppressed”:

9. Multi-dimensional q-analysis  Star-product realizes non-commutative product of quantum space on a commutative coordinate algebra.  Braided Hopf-structure of quantum space gives law for vector addition.  Partial derivatives generate infinitesimal translations on quantum space:  An integral is a solution f to equation  Exponentials are eigenfunctions of partial derivatives

10. q-Deformed partial derivatives on Manin plane: with

11. Multi-dimensional q-analysis  Star-product realizes non-commutative product of quantum space on a commutative coordinate algebra.  Braided Hopf-structure of quantum space gives law for vector addition.  Partial derivatives generate infinitesimal translations on quantum space:  Integrals generate solutions to equations  Exponentials are eigenfunctions of partial derivatives

12. q-Deformed integrals on Manin plane: with

13. Multi-dimensional q-analysis  Star-product realizes non-commutative product of quantum space on a commutative coordinate algebra.  Braided Hopf-structure of quantum space gives law for vector addition.  Partial derivatives generate infinitesimal translations on quantum space:  Integrals generate solutions to equations  Exponentials are eigenfunctions of partial derivatives

14. q-Deformed exponential on Manin plane: with

15. Multi-dimensional q-analysis  Star-product realizes non-commutative product of quantum space on a commutative coordinate algebra.  Braided Hopf-structure of quantum space gives law for vector addition.  Partial derivatives generate infinitesimal translations on quantum space:  Integrals generate solutions to equations  Exponentials are eigenfunctions of partial derivatives

16. Applications to quantum physics  q-analog of Schrödinger equation in three-dimensional q-deformed Euclidean space  plane-wave solutions of definite momentum and energy  propagator of q-deformed free particle  q-analog of Lippmann Schwinger equation and Born series

17. Outlook  discretization of space-time without lack of space-time symmetries  construction of q-deformed supersymmetry  q-deformed Minkowski space as most realistic quantum space  construction of q-deformed wave equations  calculation of quantum processes