_k63-deformed [kappa-deformed] gauge theory and {_h63-deformed [theta-deformed] gravity [Elektronische Ressource] / vorgelegt von Marija Dimitrijević
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_k63-deformed [kappa-deformed] gauge theory and {_h63-deformed [theta-deformed] gravity [Elektronische Ressource] / vorgelegt von Marija Dimitrijević

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-deformed gauge theory and-deformed gravityDissertationan der Fakult at fur Physikder Ludwig{Maximilians{Universit atMunc henvorgelegt vonMarija Dimitrijevicaus JagodinaMunc hen, den 12. Oktober 2005Erstgutachter: Prof. Dr. Julius WessZweitgutachter: Prof. Dr. Ivo SachsTag der mundlic hen Prufung: 14.12.2005AbstractNoncommutative (deformed, quantum) spaces are deformations of the usual commutativespace-time. They depend on parameters, such that for certain values of parameters theybecome the usual space-time. The symmetry acting on them is given in terms of a deformedquantum group symmetry. In this work we discuss two special examples, the -deformedspace and the -deformed space.In the case of the -deformed space we construct a deformed theory of gravity. In the rst step the deformed di eomorphism symmetry is introduced. It is given in terms of theHopf algebra of deformed di eomorphisms. The algebra structure is unchanged (as com-pared to the commutative symmetry), but the comultiplication changes. Inthe commutative limit we obtain the Hopf algebra of undeformed di eomorphisms. Basedon this deformed symmetry a covariant tensor calculus is constructed and concepts such asmetric, covariant derivative, curvature and torsion are de ned. An action that is invariantunder the deformed di eomorphisms is constructed.

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Publié le 01 janvier 2005
Nombre de lectures 46
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-deformed gauge theory and
-deformed gravity
Dissertation
an der Fakult at fur Physik
der Ludwig{Maximilians{Universit at
Munc hen
vorgelegt von
Marija Dimitrijevic
aus Jagodina
Munc hen, den 12. Oktober 2005Erstgutachter: Prof. Dr. Julius Wess
Zweitgutachter: Prof. Dr. Ivo Sachs
Tag der mundlic hen Prufung: 14.12.2005Abstract
Noncommutative (deformed, quantum) spaces are deformations of the usual commutative
space-time. They depend on parameters, such that for certain values of parameters they
become the usual space-time. The symmetry acting on them is given in terms of a deformed
quantum group symmetry. In this work we discuss two special examples, the -deformed
space and the -deformed space.
In the case of the -deformed space we construct a deformed theory of gravity. In the
rst step the deformed di eomorphism symmetry is introduced. It is given in terms of the
Hopf algebra of deformed di eomorphisms. The algebra structure is unchanged (as com-
pared to the commutative symmetry), but the comultiplication changes. In
the commutative limit we obtain the Hopf algebra of undeformed di eomorphisms. Based
on this deformed symmetry a covariant tensor calculus is constructed and concepts such as
metric, covariant derivative, curvature and torsion are de ned. An action that is invariant
under the deformed di eomorphisms is constructed. In the zeroth order in the deformation
parameter it reduces to the commutative Einstein-Hilbert action while in higher orders cor-
rection terms appear. They are given in terms of the commutative elds (metric, vierbein)
and the deformation parameter enters as the coupling constant. One special example of this
deformed symmetry, the -deformed global Poincare symmetry, is also discussed.
In the case of the space our aim is the construction of noncommutative
gauge theories. Starting from the algebraic de nition of the -deformed space, derivatives
and the deformed Lorentz generators are introduced. Choosing one particular set of deriva-
tives, the -Poincare Hopf algebra is de ned. The algebraic setting is then mapped to the
space of commuting coordinates. In the next step, using the enveloping algebra approach
and the Seiberg-Witten map, a general nonabelian gauge theory on this deformed space is
constructed. As a consequence of the deformed Leibniz rules for the derivatives used in
the construction, the gauge eld is derivative-valued. As in the -deformed case, in the
zeroth order of the deformation parameter the theory reduces to its commutative analog
and the higher order corrections are given in terms of the usual (commutative) elds. In
this way the eld content of the theory is unchanged, but new interactions appear. The
deformation parameter takes the role of the coupling constant. For the special case of U(1)
gauge theory the action for the gauge eld coupled to fermionic matter is formulated and
the equations of motion and the conserved current(s) are calculated. The ambiguities in the
Seiberg-Witten map are discussed and partially xed, and an e ectiv e action (up to rst
order in the deformation parameter) which is invariant under the usual Poincare symmetry
is obtained.iiAcknowledgements
I would like to thank Professor Julius Wess for many things, rst of all for letting me join his
group in autumn 2002. I am grateful to him for not referring me to a book or a paper to read
and learn form, but including me in discussions form the very beginning and in that way
making it possible to learn directly from him and his enormous knowledge and experience.
For interesting discussions, for ideas, comments, critics and for his patience, thanks. Also,
he is "responsible" for providing the nancial support from the Max-Plank Institute during
my staying in Munich.
Many thanks to the whole group for accepting me the way I am and for including me in
the group from the very beginning. Specially, I would like to thank the people with whom I
worked most of the time and from whom I learnt very much. Thanks to Larisa Jonke, Frank
Meyer, Lutz M oller and Efrossini Tsouchnika for making the work more understandable,
less di cult and very often fun. Also, I thank them and Claudia Jambor for all their help
concerning (not-so)ordinary life in Munich. Parts of the work presented here were done
together with Paolo Aschieri, Christian Blohmann, Peter Schupp and Michael Wohlgenannt
and I thank them for the fruitful collaboration.
To Wolfgang Behr, Maja Buric, Larisa Jonke and Frank Meyer I am grateful for reading
some parts (or the whole) of this manuscript and for their comments concerning both physics
and English.
All the words I can think of (in Serbian as well) would not be enough to express my
gratitude to my parents Milica and Zivorad Dimitrijevic and my sister Aleksandra Dimitri-
jevic. Their love, support and encouragements made this work possible.
Finally, I would like to thank all my friends in Serbia. Specially, to Slavica Maletic,
Katarina and Andrija Matic, Zorica Pajovic and Novica Paunovic many thanks for their
support and for all the emails they wrote during this three years.ivContents
Introduction 1
1 Noncommutative spaces 5
1.1 De nition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Representation on the space of commuting coordinates . . . . . . . . . . . . 11
1.5 An example, the -deformed space . . . . . . . . . . . . . . . . . . . . . . . . 14
2 The -deformed space 17
2.1 Quantum space and derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Symmetry generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Dirac derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Representation on the space of commuting coordinates . . . . . . . . . . . . 26
2.5 Fields and equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5.1 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5.2 Covariant equations of motion . . . . . . . . . . . . . . . . . . . . . . 31
3 Construction of gauge theories on the -deformed space 33
3.1 Commutative theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 Noncommutative gauge theory, setting . . . . . . . . . . . . . . . . . . . . . 36
3.3 Enveloping algebra approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4 Seiberg-Witten map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.5 Covariant derivative and the gauge eld . . . . . . . . . . . . . . . . . . . . 42
3.6 Gauge covariant Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 U(1) gauge theory on the -deformed space 49
4.1 Integral and the variational principle . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Modi ed Seiberg-Witten map . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 The action for the -deformed electrodynamics . . . . . . . . . . . . . . . . . 56
4.3.1 Matter eld action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.2 Gauge eld action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3.3 Conserved currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.4 Seiberg-Witten map and the gauge symmetry . . . . . . . . . . . . . . . . . 62vi Contents
5 Gravity on the -deformed space 67
5.1 Commutative di eomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.2 Deformed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2.1 Inversion of the ?-product . . . . . . . . . . . . . . . . . . . . . . . . 70
5.2.2 Hopf algebra of deformed di eomorphisms . . . . . . . . . . . . . . . 72
5.2.3 Consequences of the deformed coproduct . . . . . . . . . . . . . . . . 75
5.3 Tensor calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3.1 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3.2 Metric tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.4 Curvature and torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.4.1 Covariant derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.4.2 Curvature tensor, Ricci tensor and scalar curvature . . . . . . . . . . 80
5.5 Deformed Einstein-Hilbert action . . . . . . . . . . . . . . . . . . . . . . . . 81
5.6 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.7 Expansion in the deformation parameter . . . . . . . . . . . . . . . . . . . . 85
5.8 The -deformed Poincare algebra . . . . . . . . . . . . . . . . . . . . . . . . 86
5.9 Noncommutative gauge theory, revisited . . . . . . . . . . . . . . . . . . . . 89
A Vector elds in the -deformed space 95
B The -deformed symmetry from the inversion of the ?-product 99
C The general -deformed space 103
C.1 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
C.2 Deformed symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
C.3 Dirac derivative, invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
C.4 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Bibliography 109Introduction<

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