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Aspects of Extra Dimensions and Membranes Licentiate Thesis Aspects of extra dimensions and membranes Martin Sundin Mathematical Physics, Department of Theoretical Physics, School of Engineering Sciences Royal Institute of Technology, SE-106 91 Stockholm, Sweden Stockholm, Sweden 2011 Typeset in LATEX Akademisk avhandling f¨or avl¨aggande av teknologie licentiatexamen (TeknL) inom ¨amnesomr˚adetteoretisk fysik. Scientific thesis for the degree of Licentiate of Engineering (Lic Eng) in the subject area of Theoretical physics. ISBN 978-91-7415-934-9 TRITA-FYS-2011:14 ISSN 0280-316X ISRN KTH/FYS/--11:14--SE c Martin Sundin, April 2011 Printed in Sweden by Universitetsservice US AB, Stockholm April 2011 Abstract This thesis is about thwo papers related to extra dimensions. Paper A discusses extrinsic curvature effects, and paper B treats symmetries of supersymmetric mem- branes. In the part of this thesis related to paper A, we extend the theory of non- relativistic quantum particles confined to submanifolds to relativistic boson fields. We show that a Klein-Gordon field constrained to a submanifold of a Lorentzian manifold experiences an induced potential similar to the one for the Schr¨odinger equation. We embedd the Schwarzschild solution and the Robertson-Walker space- time and derive the induced potentials. Possible physical consequences of these induced potentials are also discussed. The second part is related to paper B, we study the dynamics of supersym- metric membranes, which are higher dimensional generalizations of supersymmetric strings. We derive a supersymmetric analogue of a dynamical symmetry for bosonic membranes. Key words: Extra dimensions, brane world scenarios, supermembranes iii iv Preface This thesis is the result of my research at the Department of Theoretical Physics during the time period April 2009 to April 2011. The first part of the thesis contains background material and results on the subjects constrained quantum mechanics, constrained relativistic fields and membrane dynamics. The second part consists of the scientific papers listed below. List of papers [A] Edwin Langmann and Martin Sundin Extrinsic curvature effects in brane-world scenarios arxiv:1103.3230, submitted for publication. [B] Jonas de Woul, Jens Hoppe, Douglas Lundholm and Martin Sundin A dynamical symmetry of supermembranes arXiv:1004.0266, accepted for publication in the Journal of High Energy Physics (JHEP). The thesis author's contribution to the papers [A] I contributed ideas to most parts of the paper, and I wrote a first draft. The results and final version of the paper weres obtained in collaboration of both authors. [B] This paper was a development of earlier results in [1]. The calculations and writing of the paper was done in cooperation with the co-authors. v vi Acknowledgments I want to thank my supervisor Edwin Langmann for giving me the opportunity to do research in theoretical physics and my assistant supervisor Teresia M˚anssonfor her help and support during my work. I am also very grateful to the collaborators of paper B, Jens Hoppe, Jonas de Woul and Douglas Lundholm. Further I would like to thank the other members of the Department of Theoretical physics for making my stay enjoyable. I am especially grateful to Erik Duse for many interesting, inspiring and encour- aging discussions. Many thanks also to Andr´e,Joel and Sebastian at the Depart- ment of Mathematics. Most of all I want to thank my family for their encouragement and support during the work of this thesis. vii viii Contents Abstract . iii Preface v Acknowledgments vii Contents ix I Introduction and background material 1 1 Introduction 3 1.1 Overview of the thesis . 5 2 Constrained quantum mechanics 7 2.1 Riemannian geometry . 8 2.1.1 Manifolds . 8 2.1.2 The metric tensor . 9 2.2 Submaifolds . 10 2.3 The effective Hamiltonian . 12 3 Constrained relativistic fields 17 3.1 Introduction . 17 3.2 Constrained Klein-Gordon field . 17 4 Extrinsic curvature effects in brane-world scenarios 21 4.1 Embedded cosmological models . 21 4.2 Embedded Schwarzschild solution . 21 4.3 Embedded Robertson-Walker metric . 25 4.4 A model of the early universe . 27 4.4.1 The standard case . 28 4.4.2 The extended case . 30 4.5 Discussion . 32 ix x Contents 5 Membrane dynamics 35 5.1 Point particle in the light-cone gauge . 36 5.2 A dynamical symmetry . 38 5.3 The bosonic membrane . 40 5.3.1 Light-Cone Gauge . 40 5.4 Hamiltonian formalism . 41 5.5 Mode expansion . 43 5.6 A dynamical symmetry for bosonic membranes . 44 5.7 Supermembranes . 45 5.8 Poisson brackets . 47 5.9 A dynamical symmetry for supermembranes . 47 6 Summary and conclusions 51 A The tubular neighbourhood theorem 53 B Calculation of the induced potential 55 B.1 Calculation ofγ ~ ............................. 55 B.2 Derivation of the induced potential . 56 B.3 The Schwarzschild solution . 58 B.3.1 Embedding for r > rs ..................... 58 B.3.2 Embedding for 0 < r < rs ................... 59 B.4 Robertson-Walker metric . 61 Bibliography 62 II Scientific papers 67 Part I Introduction and background material 1 2 Chapter 1 Introduction Nature is full of geometry. The yellow disc flowers of the Ox-Eye Daisy are organised in spirals and the planets move in elliptic orbits. With the idea of Einstein, that space itself is geometrical, geometry has become an important part of all aspects of nature. Einstein himself expressed this as [2] Geometry [...] is evidently a natural science; we may in fact regard it as the most ancient branch of physics. One way to solve certain problems in physics is to consider scenarios where the four dimensional world we experience is embedded in a higher dimensional space. The first theory with extra dimensions was probably the theory proposed by Gunnar Nordstr¨om[3, 4] in 1914. Nordstr¨omtried to unify Newtonian gravity with electromagnetism by introducing an additional dimension. The theory, however, was unable to explain certain phenomena such as light deflection. The theory was therefore considered as non-physical by many. Three years later, Hermann Weyl [3, 4] proposed a theory which is a classical predecessor of both Kaluza-Klein and Yang-Mills theory. Weyl proposed that there existed an additional degree of freedom in space. He also introduced a field which "measured" the change of this degree of freedom and called it a "gauge field”1. Weyl's theory unified gravity with electromagnetism, but received critique from Einstein and others who objected that although the theory was mathematically beautiful it implied that measurements dependes on the history of the measurement device. Despite the shortcomings of the theory, it was eventually published in a journal (with Einsteins objections as an appendix). It was later hypothesised that the additional degree of freedom could actually be an extra dimension, as in Kaluza-Klein theory [5, 6], or a quantum mechanical phase, as in Yang-Mills theory. Today, there is a renewed interest in extra dimensions and many theories have been proposed that use extra dimensions to solve certain physical problems (see e.g. 1After the verb "gauge", meaning "to measure" or "to estimate". 3 4 Chapter 1. Introduction [7] for references). One problem in physics is why the gravitational force is much weaker then the other forces in nature. Extra dimensions might solve this problem by allowing gravity to spread freely in the extra dimensions. This makes gravity appear weaker to someone on a lower dimensional submanifold of the ambient space. This idea underlies the ADD model (after Arkani-Hamed-Dimopoulos-Dvali [8]) considered in particle physics. Another model is the Randall-Sundrum (RS) model [9]. The model assumes that the universe is five dimensional with two four- dimensional membranes (branes), the "Planck" and "TeV" brane2. In the theory, the branes are separated in space. We live on the TeV brane while gravity is localized on the Planck brane. According to the RS model, we experience gravity as weaker than the other forces since it is localized on the other brane. There are hopes that, in the future, experiments will either support or falsify the ADD and Randall-Sundrum model. Since extra dimensions have not been discovered to this day, there should be some mechanism preventing us from detecting them (if they exists). One explana- tion is that the extra dimensions are so small that the energies needed to detect them are very high. Another explanation is that some particles are confined to our four-dimensional universe by some mechanism, e.g. a strong potential. Classically, extra dimensions usually do not affect the dynamics of a particle constrained to a lower dimensional space. It is therefore natural to assume that the hamiltonian for a quantum particle on a manifold is proportional to the Laplace- Beltrami operator [10]. For a long time this was believed also to be true for quantum mechanical particles. In the 1970's Jensen and Koppe [11], and independently da Costa [12, 13] in the 1980's, derived the effective Hamiltonian for a non-relativistic quantum particle confined by a strong potential to a surface embedded in three dimensional Euclidean space. They found that, unlike classical particles, quantum particles are affected by both the intrinsic and extrinsic geometry of a surface. This is because the curvatures induce an additional potential. One theory with extra dimensions is string theory (see e.g. the textbook by Polchinski [14]). In string theory, point particles are replaced by vibrating strings. The theory was first proposed in the context of the strong interaction, where it was believed that quarks were bounded together by strings making them hard to separate. This idea was later abandoned in favour of QCD, but string theory lived on despite of this. The quantum theory of strings predicts an infinite number of particles of different masses, spin and interactions.
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