Can Nature be q-Deformed ?

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. Introduction

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 q-lattice points are very near roots of q-trigonometrical function • 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

8. Regularization by q-deformation • Fourier transform converges even for polynomial functions: • Large values of x·pare „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