University of Wrocław Department of Physics and Astronomy Institute for Theoretical Physics PhD Dissertation "κ- Minkowski spacetime: mathematical formalism and applications in Planck scale physics" by Anna Pachoł arXiv:1112.5366v1 [math-ph] 22 Dec 2011 Supervisor Dr hab. Andrzej Borowiec Wrocław, 2011 . "To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature ... If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in." Richard Phillips Feynman Preface This dissertation is based on research done at the Institute for Theoretical Physics University of Wrocław between October 2007 and June 2011 in collaboration with Dr hab. Andrzej Borowiec and in Part II also with Prof. Stjepan Meljanac (Rudjer Boskovic Institute, Zagreb, Croatia) and Prof. Kumar S. Gupta (Theory Division, Saha Institute of Nuclear Physics, Calcutta, India). This thesis contains material which has been published in the following papers: • Andrzej Borowiec and Anna Pachoł "kappa-Minkowski spacetimes and DSR algebras: fresh look and old problems" SIGMA 6, 086 (2010) [arXiv:1005.4429]. • Andrzej Borowiec, Kumar S. Gupta, Stjepan Meljanac and Anna Pachoł "Constraints on the quantum gravity scale from kappa - Minkowski spacetime" EPL 92, 20006 (2010) [arXiv:0912.3299]. • Andrzej Borowiec and Anna Pachoł "The classical basis for kappa-Poincare algebra and doubly special relativity theories" J. Phys. A: Math. Theor. 43, 045203 (2010) [arXiv:0903.5251]. • Andrzej Borowiec and Anna Pachoł "kappa-Minkowski spacetime as the result of Jordanian twist deformation" Phys. Rev. D 79, 045012 (2009) [arXiv:0812.0576]. • Andrzej Borowiec and Anna Pachoł "On Heisenberg doubles of quantized Poincare algebras" Theoretical and Mathematical Physics, 169(2), 1611 (2011) Acknowledgements I am deeply grateful to my supervisor Dr A. Borowiec, for his help and guidance, for the infinitely many discussions and his endless patience. Special thanks go to Prof. S. Meljanac and his group for helpful discussions and their hospitality during my visits at Rudjer Boskovic Institute in Zagreb (Croatia). This PhD dissertation has been also supported by The National Centre for Science Research Grant (N N202 238540) and the scholarship within project: "Rozwój potencjału i oferty edukacyjnej Uniwersytetu Wrocławskiego szansa zwiekszenia konkurencyjnosci Uczelni." v Contents Introduction xi 1 Noncommutative Spacetimes and Their Symmetries 1 1.1 Preliminaries . .2 1.2 Twist-deformation . 11 2 κ-Minkowski spacetime from Drinfeld twist 17 3 The κ-Poincare quantum group 25 3.1 h-adic κ-Poincaré quantum group in classical basis . 27 3.2 Different algebraic form of κ-Poincaré Hopf algebra . 30 4 κ-Minkowski as covariant quantum spacetime and DSR algebras 33 4.1 “h-adic” universal κ-Minkowski spacetime and “h-adic” DSR algebra. 34 4.2 κ-Minkowski spacetime in q-analog version and canonical DSR algebra . 40 5 Possible experimental predictions from κ -Minkowski spacetime 47 5.1 Constraints on the quantum gravity scale . 48 5.2 Dispersion relations . 50 vii viii Contents 5.3 Time delay formulae for different realizations of DSR algebra . 52 Conclusions and perspectives 59 A h-adic topology 65 B Bicrossproduct construction versus Weyl-Heisenberg algebras 69 B.1 Preliminaries and notation: Weyl-Heisenberg algebras . 69 B.2 Crossed product and coproduct . 72 B.3 Bicrossproduct construction . 74 B.4 Classical basis for κ -Poincare Hopf algebra as bicrossproduct basis . 75 B.5 The case of Weyl-Heisenberg algebras . 77 C Heisenberg Doubles of quantized Poincare algebras 79 Bibliography 81 Abstract The dissertation presents possibilities of applying noncommutative spacetimes descrip- tion , particularly kappa-deformed Minkowski spacetime and Drinfelds deformation the- ory, as a mathematical formalism for Doubly Special Relativity theories (DSR), which are thought as phenomenological limit of quantum gravity theory. Deformed relativis- tic symmetries are described within Hopf algebra language. In the case of (quantum) kappa-Minkowski spacetime the symmetry group is described by the (quantum) kappa- Poincare Hopf algebra. Deformed relativistic symmetries were used to construct the DSR algebra, which unifies noncommutative coordinates with generators of the sym- metry algebra. It contains the deformed Heisenberg-Weyl subalgebra. It was proved that DSR algebra can be obtained by nonlinear change of generators from undeformed algebra. We show that the possibility of applications in Planck scale physics is con- nected with certain realizations of quantum spacetime, which in turn leads to deformed dispersion relations. x Abstract Introduction One of the most intriguing theoretical problems nowadays is the search for fundamen- tal theory describing Planck scale physics. The so called "Planck scale" is the scale at which gravitational effects are equivalently strong as quantum ones and it relates to either a very big energy scale or equivalently to a tiny size scale. A new theory which would consistently describe this energy region is necessary to understand the real nature of the Universe. It would allow, for example, to give a meaning to the Big Bang and black holes, physical situations were both quantum mechanics and general relativity are relevant, but cannot be applied consistently at the same time. Such theory, in the weak gravity limit, should give back already known Quantum Mechanics and in the limit of Planck’s constant going to zero it should reduce to Einstein’s Gravity. The quantum field theory description at the Planck scale breaks down due to the nonrenormalisability of Einstein’s theory of gravity. Therefore the search for a theory of Quantum Gravity, describing gravitational interactions at the quantum level is one of the most important problems in modern physics. There are few theories trying to solve that problem, like String Theory, Loop Quantum Gravity, Noncommutative Geometry and many more. The topic of this thesis is connected with noncommutative spacetimes as one of the approaches to the description of Planck scale physics, particularly the description of geometry from the quantum mechanical point of view. At the Planck scale the idea of size or distance in classical terms is not valid any more, because one has to take into account quantum uncertainty. The Compton wavelength of any photon sent to probe the realm at this energy scale will be of the order of Schwarzschild radius. Therefore a −35 photon sent to probe object of the Planck size (Planck length Lp = 1:62 × 10 m) will be massive enough to create a black hole, so it will not be able to carry any informa- tion. The research on the structure of spacetime, at the scale where quantum gravity effects take place, is one of the most important questions in fundamental physics. One of the possibilities is to consider noncommutative spacetimes. Since in Quantum Me- chanics and Quantum Field Theory the classical variables become noncommutative in the quantization procedure and in the General Theory of Relativity spacetime is xii Introduction a dynamical variable itself, one could assume that noncommutative spacetime will be one of the properties of physics at the Planck scale. In noncommutative spacetime the quantum gravity effects modify the coordinate relations in the quantum phase space and, besides Heisenberg relations between coordinates and momenta, one has to in- troduce noncommutativity of coordinates themselves. This leads to new uncertainty relations for coordinates, which exclude the existence of Planck particles (of the size of the Schwarzschild radius) from the theory. The first attempts of constructing a field theory on noncommutative spacetimes are related with Heisenberg’s ideas from the 30’s. In the 1940’s, Snyder proposed the model of Lorentzian invariant discrete spacetime as a first example of noncommuta- tive spacetime [1]. Noncommutative coordinates lead to a modification of relativistic symmetries, which are described by the Quantum Groups and Quantum Lie algebras. However the Theory of Quantum Groups has been developed independently at first. The mathematical formalisms connected with Hopf algebras, were introduced and used in mathematical physics by V. Drinfeld, L.D. Faddeev, M. Jimbo, S. Woronowicz and Y. Manin in the 80’s. The term "quantum group" first appeared in the theory of quantum integrable systems and later was formalized by V. Drinfeld and M. Jimbo as a particular class of Hopf algebras with connection to deformation theory (strictly speaking, as deformation of universal enveloping Lie algebra). Such deformations are classified in terms of classical r-matrix satisfying the classical Yang-Baxter equation. However Noncommutative Geometry as an independent area of mathematics appeared together with papers by Alain Connes and since then it found applications in many theories as String Theory, Noncommutative Quantum Field Theory (NC QFT) and NC Gauge Theories, NC Gravity, as well as in noncommutative version of the Stan- dard Model. Nonetheless the Hopf algebras (Quantum Groups) formalism is strictly connected with noncommutative spacetimes and Quantum Groups are thought as gen- eralizations of symmetry groups of the underlying spacetime. Mathematically speaking such noncommutative spacetime is the Hopf module algebra (a.k.a. covariant quantum space) and stays invariant under Quantum Group transformations. The first example of deformed coordinate space as a background for the unifica- tion of gravity and