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. String theory is based on the simple assumption that, in the same way as a relativistic particle moves in a way that minimizes the length of its world-line, a string moves in a way that minimizes the area of its world-sheet. Unlike other theories with extra dimensions, string theory requires a specific number of space-time dimensions to be consistent. Bosonic string theory requires D = 26 dimensions and supersymmetric string theory requires D = 10.
2This model is often known as the RS1 model. 1.1. Overview of the thesis 5
One way to generalize string theory is to consider higher dimensional extended objects, membranes, which (like strings) moves in a way that minimizes their world- volume. The extra degrees of freedom and non-linearities of the theory makes the equations of motion hard to solve explicitly. Because of this, it is not clear how the quantum theory of membranes works. By understanding the symmetries of the theory, one hopes to reduce the degrees of freedom. This might eventually lead to an explicit solution of the classical theory and thereby also the quantum theory.
1.1 Overview of the thesis
Chapters 2,3 and 4 discusses material related to paper A. In chapter 2 we give a short introduction to differential geometry and introduce the concept of constrained quantum mechanics. This is further developed in chapter 3 where we derive the induced potential for constrained Klein-Gordon field. In chapter 4 we discuss ex- trinsic curvature effects in cosmological brane-world scenarios, and we investigate the effects of induced potentials the embedded Schwarzschild solution and the em- bedded Robertson-Walker universe. Material related to paper B is discussed in chapter 5. We give a short in- troduction to membrane dynamics and derive a dynamical symmetry for bosonic membranes. The dynamical symmetry is then generalized to supersymmetric mem- branes. We summarize part I of the thesis in chapter 6 where we state and discuss conclusions of the thesis. The scientific papers are in part II. 6 Chapter 2
Constrained quantum mechanics
In classical mechanics, the dynamics of a particle is usually unaffected by whether or not there exists an ambient space in which the particle cannot move. This is because the extra dimensions are no degrees of freedom for the particle. How does constraints of this type affect quantum particles? It was first proposed by Schr¨odinger[10] in 1926 that the Hamiltonian of a quantum particle on a manifold is proportional to the Laplace-Beltrami operator of the manifold. Almost 50 years later that Jensen and Koppe [11] and independently da Costa [12, 13] examined if this was also true for constrained particles. They found that the assumption made by Schr¨odinger does not apply in this case. By considering quantum parti- cles confined to a thin layer around a surface they found that constrained quantum particles are affected by the intrinsic and extrinsic geometry of the surface. The geometry of the surface induces a potential which affects the dynamics of the parti- cle (this was found earlier by Marcus [15] in the context of quantum chemistry). In short, this implies that constrained quantum particles is not the same as quantized constrained particles. The existence of an induced potential in the thin layer limit lead Exner and Seba [16] to investigate if bound states exists also for layers of finite width. They found that a strip of finite width does support bound states under certain conditions on the curvature and thickness of the strip. It was later shown by Goldstone and Jaffe [17] that any non-straight strip supports bound states. This has later been confirmed experimentally (see e.g. [18]). We begin this chapter by giving a short introduction to Riemannian geometry. We then proceed to use Riemannian geometry to describe the confinement of a quantum particle to a manifold embedded in Euclidean space. Thereafter we derive the effective low-energy Hamiltonian for a non-relativistic constrained quantum particle.
7 8 Chapter 2. Constrained quantum mechanics
2.1 Riemannian geometry
Classical geometry (see e.g. [19]) was studied in order to understand relations between e.g. length, area, volume and angles of geometrical shapes in the plane or in space. It was found that by assuming a set of postulates, many geometrical relations could be proven rigorously. One postulate is the parallel postulate which states that1 the angles α, β, γ of a triangle always satisfy
α + β + γ = π (2.1) where the angles are measured in radians. Many attempts were made to prove the postulate, but none were successful. It was later realized that the postulate could be omitted and that one can study spaces where the postulate does not hold. One such example is the sphere (of radius 1) where the angles of a triangle2 does not add up to π as in (2.1) but instead
α + β + γ = π + A (2.2) where A is the area enclosed by the triangle. By considering spaces where the parallel postulate is omitted, the notion of geometry was generalized to what is now known as differential geometry. To measure the deviation from ”flatness”, one can use the notion of curvature. By using curvature, the formulas in (2.1) and (2.2) can be generalized to Z α + β + γ = π + KdA (2.3) C for triangles on arbitrary surfaces. Here K is the (Gaussian) curvature of the surface and C is the region enclosed by the triangle. The formula (2.3) is a generalization of (2.1) and (2.2) since the curvature of the plane is K = 0 and the curvature of the unit sphere is K = 1. The formula (2.3) is actually a special case of a more general theorem, the Gauss-Bonnet theorem [19].
2.1.1 Manifolds A manifold is the mathematical notion of a curved space. By a space we mean something that locally is like Rn and by curved we mean something which in general is not like Rn globally [20]. To make the definition precise we need some definitions. Definition A function f : U → V between two sets is a homeomorphism if f is a continuous and invertible function such that f −1 is also continuous. Two sets are said to be homeomorphic if there exists a homeomorphism between the sets. The homeomorphism f is said to be a diffeomorphism if both f and f −1 are differentiable. Two sets are diffeomorphic if there exists a diffeomorphism between
1Many different but equivalent formulations of this postulate exists. 2For a general space we can define a (geodesic) triangle as the shortest lines connecting three points. 2.1. Riemannian geometry 9 the sets. Sets being homeomorphic is more a statement about the topologies of the sets, rather than of the sets themselves. In this sense, diffeomorphism is a stronger condition then homeomorphism since, for differentiable functions we often use the metric topology. With these definitions, we are ready to define what a manifold is. Definition An n-dimensional topological manifold is a second countable set M with a Hausdorff topology3 and such that every point p ∈ M has an open neigh- bourhood homeomorphic to an open subset of Rn. If every point of a manifold has an open neighbourhood diffeomorphic to an open subset of Rn, then the manifold is a differentiable manifold. Somewhat simplified, one can say that the definition of a manifold states that a manifold is not ”too big” (second countable), finite dimensional (locally homeo- morphic to Rn) and that the limit of a converging sequence is unique (Hausdorff topology). By the theorem of Invariance of Domain [20] the dimension of a man- ifold is constant, hence a manifold cannot intersect itself. Intersecting ”manifold like” spaces are sometimes called pseudomanifolds. A homeomorphism ϕ : U → V ⊂ Rn is often called a chart. From the definition of a manifold we find that, if Uα ∩ Uβ 6= ∅ and ϕα : Uα → Vα, ϕβ : Uβ → Vβ are two homeomorphisms (diffeomorphisms), then
−1 ϕβ ◦ ϕα : ϕα(Uα ∩ Uβ) → ϕβ(Uα ∩ Uβ) is also a homomorphism (diffeomorphism). One often does not need to work with the underlying manifold explicitly, but can define the manifold in terms the tran- −1 sition functions ϕβ ◦ ϕα . A manifold can be given more structure by considering transition functions belonging to different function classes. All manifolds considered in this thesis are smooth manifolds, i.e. manifolds with C∞ transitions functions.
2.1.2 The metric tensor There are several ways to define vectors on differentiable manifolds (see e.g. [19] for three different definitions). One way is to define a vector at a point p as the evaluation (at the point p) of the derivative of a curve running through the point. The tangent space TpM is the set of all vectors at the point p and is diffeomorphic to Rn when the manifold is n-dimensional. We can define an inner product of vectors in the tangent space in the same way as for vectors in Rn by defining an inner product as a bilinear function g : Rn × Rn → R such that g(u, v) = g(v, u)
n where u, v are vectors in TpM. We define the norm ||u|| of a vector u ∈ R by the 2 n relation g(u, u) = ||u|| . Given a basis {eµ}µ=1 of TpM, we can express the vectors
3Many of the properties in this section are stated for completness and will not be needed in the following. 10 Chapter 2. Constrained quantum mechanics
4 µ µ as linear combinations of the basis vectors , u = u eµ and v = v eµ. Using this, we can write the inner product in component form
µ ν µ ν g(u, v) = g(eµ, eν )u v = gµν u v where gµν = g(eµ, eν ) is the metric tensor. The basis vectors and the metric tensor on a manifold are in general position dependent. The metric tensor can be used to calculate e.g. the length of a curve x(τ) on a manifold as
Z τ1 Z τ1 r µ ν Z q dx dx dx µ ν l = dτ = gµν (x(τ)) dτ = gµν (x(τ))dx dx τ0 dτ τ0 dτ dτ C For this reason, the infinitesimal line element is often written as
2 µ ν ds = gµν dx dx so that we can write Z l = ds C Similarly, the volume of a manifold can be computed as Z V ol(M) = dnxp|g| M where |g| is the absolute value of the determinant of (gµν ). A manifold is said to be Riemannian if the inner product is positive definite. If the inner product is indefinite and there exists a coordinate system and basis such that5
µ ν 2 0 2 1 2 n−1 2 2 gµν (x)u u = ||u|| = (u ) − (u ) − ... (u ) + O(x ) then the manifold is said to be Lorentzian6.
2.2 Submaifolds
A submanifold is a subset of a manifold which itself is also a manifold. Submanifolds can be considered as embedded in their ambient manifold. An embedding of a manifold M in a manifold N is a map f : M → N such that f restricted to its image is a homeomorphism.
4Throughout this thesis we use the Einstein summation convention, i.e. repeted indices are summed over. 5 n n−1 One often enumerates basis vectors as {eµ}µ=1 for Riemannian manifolds and as {eµ}µ=0 for Lorentzian manifolds. 6This is the metric convention used throughout the thesis. 2.2. Submaifolds 11
In this chapter we use capital latin letters (A, B, C, . . . ) to denote indices run- ning over 1, 2, . . . , n + p, greek letters (µ, ν, λ, σ, . . . ) to denote indices running over 1, 2, . . . n and lower case latin letters (i, j, k, l, . . . ) to denote indices running over n + 1, n + 2, . . . , n + p. If η is the metric tensor on N , then the metric on M is given by the pullback of η by f, g = f ∗η, i.e. in components
A B gµν = ηAB∂µf ∂ν f
If M is a n-dimensional Riemannian manifold embedded in Rn+p, then we can (locally) parametrize the submanifold as