CAN NATURE BE Q-DEFORMED? Hartmut Wachter May 16, 2009
Can Nature be q-Deformed ?
Jan 26, 2016
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. Introduction. - PowerPoint PPT Presentation
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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
Related Documents
