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 infi-nitely 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 pre-sent 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 me-chanics 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:

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

a2λλ min

maxminλλ

phh

p

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

)(cos)())((0

2 pxxfxdpfF qqq

k

kk

qcqfcqqxfxd )()1()( 222

02

points of q-lattice

0)(coslim 2

nq

ncq

1 2 3 4 5 6

-50

50

100

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

Regularization by q-deformation

Fourier transform converges even for polynomial functions:

Large values of x·p are „suppressed”:

6101.1 1023021.2)1)(( xFq

“2„ Kpx

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

)()()(1)( 2aOxfaxfxaf ij

jiii

)()( kji xFxf

)i()|(exp)|(exp 1 ikiq

kjq

i ppxpx

q-Deformed partial derivatives on Manin plane:

),(

),(

2122

22111

2

2

xqxfDf

xqxfDf

q

q

with

i

iii

q xq

xfxqffD

)1(

)()(2

2

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:

Integrals generate solutions to equations

Exponentials are eigenfunctions of partial derivatives

)()()(1)( 2aOxfaxfxaf ij

jiii

)( ij xf

)i()|(exp)|(exp 1 ikiq

kjq

i ppxpx

q-Deformed integrals on Manin plane:

),(|)(

),(|)(

211

0

20

12

221

0

10

11

2

2

xxqfxdf

xqxfxdf

qx

qx

with

k

kk

qcqfcqqfxd )()()1( 222

02

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

)()()(1)( 2aOxfaxfxaf ij

jiii

)( ij xf

)i()|(exp)|(exp 1 ikiq

kjq

i ppxpx

q-Deformed exponential on Manin plane:

0, 21

2112

21 22

2112

!]][[!]][[

)()()()()|(exp

nn qq

nnnnji

q nn

xxpx

with

2222

2

]][[]]2[[]]1[[!]][[

1

1]][[

2

2

qqqq

n

q

nn

q

qn

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

)()()(1)( 2aOxfaxfxaf ij

jiii

)( ij xf

)i()|(exp)|(exp 1 ikiq

kjq

i ppxpx

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