Download Can nature be q-deformed?

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:
Fq1.1 ( x10 )(1)  2.23021106

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