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CAN NATURE BE Q-DEFORMED? Hartmut Wachter May 16, 2009 Contents Introduction Milestones in q-deformation Idea of a smallest length Regularization by q-deformation Multi-dimensional q-analysis Application to quantum physics Outlook Introduction „ … Now it seems that the empirical notions on which the metric determinations of space are based … lose their validity in the infinitely small; one ought to assume this as soon as it permits a simpler way of explaining phenomena …“ (Bernhard Riemann) „I … believe firmly the solution to the present troubles (with divergences) will not be reached without a revision of our general ideas still deeper than that contemplated in the present quantum mechanics.“ (Niels Bohr in a letter to Dirac 1927) Introduction „ … the introduction of space-time continuum may be considered as contrary to nature in view of the molecular structure […] on a small scale … we must give up … the space-time continuum. … human ingenuity will someday find methods … to proceed such a path.“ (Albert Einstein) „One must seek a new relativistic quantum mechanics and one‘s prime concern must be to base it on sound mathematics. … Having decided on the branch of mathematics, one should proceed to develop it along suitable lines at the same time looking for that way in which it appears to lend itself naturally to physical interpretation.“ (P. A. M. Dirac) 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) Idea of a smallest length Plane-waves of different wave-length can have the same effect on a lattice: a Thus, we can restrict attention to wave-lengths larger than twice the lattice spacing: λ λ min 2a A smallest wave-length implies an upper bound in momentum space: h h p pmax λ λ min Regularization by q-deformation 100 q-lattice points are very near roots Transition amplitudesfunction contain q-analogs of q-trigonometrical 50Fourier of transforms: Fq ( f )( p) d q 2 x f ( x) cos q ( x p) 0 1 2 3 -50 5 6 q-deformed trigono-metrical function Jackson-integral y 4 Jackson-integral singles out a lattice: 5 2.5 0 d q 2 x f ( x) k (q 2 1) cq 2k f (cq 2k ) 0 0 0.5 1 1.5 2 2.5 points of q-lattice 3 For suitable c q-deformed trigonometrical functions rapidly diminish on q-lattice points: -2.5 -5 lim cos q (cq 2 n ) 0 n x Regularization by q-deformation Fourier transform converges even for polynomial functions: Fq1.1 ( x10 )(1) 2.23021106 Large values of x·p are „suppressed”: „ x p K“ 2 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: f (ai xi ) 1 f ( xi ) a j j f ( xi ) O(a 2 ) An integral is a solution f to equation i f ( x j ) F ( xk ) Exponentials are eigenfunctions of partial derivatives i exp q ( x j | p k ) exp q ( xi | p k ) (i 1 pi ) q-Deformed partial derivatives on Manin plane: 1 f Dq12 f ( x1 , q 2 x 2 ) 2 f Dq22 f (qx1 , x 2 ) with f (q 2 x i ) f ( x i ) D f (q 2 1) x i i q2 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: f (ai xi ) 1 f ( xi ) a j j f ( xi ) O(a 2 ) Integrals generate solutions to equations j f ( xi ) Exponentials are eigenfunctions of partial derivatives i exp q ( x j | p k ) exp q ( xi | p k ) (i 1 pi ) q-Deformed integrals on Manin plane: 1 x 0 ( 1 ) f | 1 x 0 ( 2 ) f | 0 0 d q 2 x1 f ( x1 , q 2 x 2 ) d q 2 x 2 f (q 1 x1 , x 2 ) with 0 dq2 x f 2 2k 2k ( q 1 ) ( cq ) f ( cq ) k 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: f (ai xi ) 1 f ( xi ) a j j f ( xi ) O(a 2 ) Integrals generate solutions to equations j f ( xi ) Exponentials are eigenfunctions of partial derivatives i exp q ( x j | p k ) exp q ( xi | p k ) (i 1 pi ) q-Deformed exponential on Manin plane: n1 n2 2 n2 1 n1 ( x ) ( x ) ( ) ( ) 1 2 exp q ( x i | p j ) [[ n1 ]]q 2 ![[ n2 ]]q 2 ! n1 , n2 0 with q 2n 1 [[ n]]q 2 2 q 1 [[ n]]q 2 ! [[1]]q 2 [[ 2]]q 2 [[ n]]q 2 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: f (ai xi ) 1 f ( xi ) a j j f ( xi ) O(a 2 ) Integrals generate solutions to equations j f ( xi ) Exponentials are eigenfunctions of partial derivatives i exp q ( x j | p k ) exp q ( xi | p k ) (i 1 pi ) 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 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