# Aspects of extra dimensions and membranes Licentiate Thesis Martin Sundin

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 för avläggande av teknologie licentiatexamen (TeknL) inom ämnesområdet teoretisk 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 membranes. In the part of this thesis related to paper A, we extend the theory of nonrelativistic 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 Schrödinger equation. We embedd the Schwarzschild solution and the Robertson-Walker spacetime 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 supersymmetric 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 Månsson for 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 encouraging discussions. Many thanks also to André, Joel and Sebastian at the Department 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preface iii v Acknowledgments vii Contents ix I 1 Introduction and background material 1 Introduction 1.1 Overview of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . 2 Constrained quantum mechanics 2.1 Riemannian geometry . . . . . . 2.1.1 Manifolds . . . . . . . . . 2.1.2 The metric tensor . . . . 2.2 Submaifolds . . . . . . . . . . . . 2.3 The effective Hamiltonian . . . . 3 5 . . . . . 7 8 8 9 10 12 3 Constrained relativistic fields 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Constrained Klein-Gordon field . . . . . . . . . . . . . . . . . . . . 17 17 17 4 Extrinsic curvature effects in brane-world 4.1 Embedded cosmological models . . . . . . 4.2 Embedded Schwarzschild solution . . . . . 4.3 Embedded Robertson-Walker metric . . . 4.4 A model of the early universe . . . . . . . 4.4.1 The standard case . . . . . . . . . 4.4.2 The extended case . . . . . . . . . 4.5 Discussion . . . . . . . . . . . . . . . . . . 21 21 21 25 27 28 30 32 ix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x Contents 5 Membrane dynamics 5.1 Point particle in the light-cone gauge . . . . . . 5.2 A dynamical symmetry . . . . . . . . . . . . . 5.3 The bosonic membrane . . . . . . . . . . . . . . 5.3.1 Light-Cone Gauge . . . . . . . . . . . . 5.4 Hamiltonian formalism . . . . . . . . . . . . . . 5.5 Mode expansion . . . . . . . . . . . . . . . . . 5.6 A dynamical symmetry for bosonic membranes 5.7 Supermembranes . . . . . . . . . . . . . . . . . 5.8 Poisson brackets . . . . . . . . . . . . . . . . . 5.9 A dynamical symmetry for supermembranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 36 38 40 40 41 43 44 45 47 47 6 Summary and conclusions 51 A The tubular neighbourhood theorem 53 B Calculation of the induced potential B.1 Calculation of γ̃ . . . . . . . . . . . . B.2 Derivation of the induced potential . B.3 The Schwarzschild solution . . . . . B.3.1 Embedding for r > rs . . . . B.3.2 Embedding for 0 < r < rs . . B.4 Robertson-Walker metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 55 56 58 58 59 61 Bibliography 62 II 67 Scientific papers 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 Nordström [3, 4] in 1914. Nordström tried 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. 1 After 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 fourdimensional 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 explanation 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 LaplaceBeltrami 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. 2 This 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 worldvolume. 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 extrinsic curvature effects in cosmological brane-world scenarios, and we investigate the effects of induced potentials the embedded Schwarzschild solution and the embedded Robertson-Walker universe. Material related to paper B is discussed in chapter 5. We give a short introduction to membrane dynamics and derive a dynamical symmetry for bosonic membranes. The dynamical symmetry is then generalized to supersymmetric membranes. 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 Schrödinger [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 Schrödinger does not apply in this case. By considering quantum particles 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 particle (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 1 Many 2 For points. different but equivalent formulations of this postulate exists. a general space we can define a (geodesic) triangle as the shortest lines connecting three 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 neighbourhood 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 homeomorphic 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 manifold 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 ϕβ ◦ ϕ−1 α : 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 transition functions ϕβ ◦ ϕ−1 α . 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 Tp M 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) where u, v are vectors in Tp M. We define the norm ||u|| of a vector u ∈ Rn by the relation g(u, u) = ||u||2 . Given a basis {eµ }nµ=1 of Tp M, we can express the vectors 3 Many of the properties in this section are stated for completness and will not be needed in the following. 10 Chapter 2. Constrained quantum mechanics as linear combinations of the basis vectors4 , u = uµ eµ and v = v µ eµ . Using this, we can write the inner product in component form g(u, v) = g(eµ , eν )uµ v ν = gµν uµ v ν 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 Z q dx dxµ dxν l= dτ = dτ = g (x(τ )) gµν (x(τ ))dxµ dxν µν dτ dτ dτ τ0 C τ0 For this reason, the infinitesimal line element is often written as ds2 = gµν dxµ dxν so that we can write Z l= ds C Similarly, the volume of a manifold can be computed as Z p V ol(M) = dn x |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 gµν (x)uµ uν = ||u||2 = (u0 )2 − (u1 )2 − . . . (un−1 )2 + O(x2 ) then the manifold is said to be Lorentzian 6 . 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. 4 Throughout this thesis we use the Einstein summation convention, i.e. repeted indices are summed over. n−1 5 One often enumerates basis vectors as {e }n µ µ=1 for Riemannian manifolds and as {eµ }µ=0 for Lorentzian manifolds. 6 This 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 running 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 gµν = ηAB ∂µ f A ∂ν f B If M is a n-dimensional Riemannian manifold embedded in Rn+p , then we can (locally) parametrize the submanifold as Z 1 , Z 2 , . . . , Z n+p = r(x) where x = (x1 , x2 , . . . , xn ) and Z 1 , Z 2 , . . . , Z n+p are coordinates in a local chart of M and the inertial coordinates in Rn+p respectivly. The components of the metric tensor gµν are then given by gµν = ∂r ∂Z A ∂Z B ∂r · = δAB µ ν ∂x ∂x ∂xµ ∂xν In constrained quantum mechanics we want to study the dynamics of a quantum particle under the influence of a confining potential which localizes the particle to M ⊂ Rn+p . We therefore need to extend the embedding function r(x) to a map R(x, y) covering an open neighbourhood of the submanifold. We can define R(x, y) by introducing linearly independent normal vectors {ni (x)}n+p i=n+1 to the submanifold M at the point r(x). We choose the normal vectors to be orthogonal ni (x) · nj (x) = δij for all x and construct R(x, y) by setting R(x, y) = r(x) + y i ni (x) where (y n+1 , y n+2 , . . . , y n+p ) ∈ Rp . By the tubular neighbourhood theorem (see Appendix A), there exists a constant > 0 such that F is one-to-one for all ||y|| < . n+p The tangent vectors {tµ = ∂µ r}nµ=1 and the normal vectors {ni }n+p i=n+1 span R at each point. Because of this, any vector can be written as a linear combination of them. Especially we can write ∂µ ni = −αiµν tν − Aiµj nj with αiµν and Aiµj defined by this equation. (2.4) 12 Chapter 2. Constrained quantum mechanics Using (2.4) we can compute the metric tensor in the coordinates (x, y) explicitly. Using that GAB = ∂A R · ∂B R, we find that Gµν Giµ γµν + y i y j Aiµk Ajν l hkl −y k Akµl hlj (GAB ) = (2.5) = Gµj Gij −y k Akν l hli δij where γµν = gµν − 2y k αkµλ gλν + y k y l αkµσ αlν λ gλσ We can also explicitly compute the quantity (see Appendix B) γ̃ = |G| | det(GAB )| = = 1 − 2y i αiλλ + y i y j 2αiλλ αjσσ − αiλσ αjσλ + O(y 3 ) |g| | det(gµν )| which will be useful in later calculations. 2.3 The effective Hamiltonian To derive the effective Hamiltonian of a particle confined to the submanifold M ⊂ Rn+p by a strong potential we start with the Hamiltonian of a free particle in Rn+p . Thereafter we apply a confining potential which localizes the particle to the submanifold. We then study the limit in which the extension of the wavefunction in the normal directions tend to zero. Consider the Schrödinger equation of a free particle in Rn+p H0 Ψ = − dΨ 1 2 ∇ Ψ=i 2m dt where m is the mass of the particle and the wavefunction is normalized in Rn+p Z dn+p Z|Ψ|2 = 1 Rn+p Changing to the coordinate system (x, y) induced by R, the Hamiltonian becomes √ 1 H0 = − √ ∂ A GGAB ∂B 2m G We now apply the confining potential Vconf (y) to confine the particle to the submanifold M. Let Vconf be such that there exists a set {χα (y)}∞ α=0 of real eigenfunctions satisfying − 1 ij δ ∂i ∂j χα (y) + Vconf (y)χα (y) = λα χα (y) 2m 2.3. The effective Hamiltonian 13 and Z dp yχα (y)χβ (y) = δαβ B where we for convenience have chosen the confining potential such that for all α, χα (y) = 0 for ||y|| > . It is therefore sufficient to consider the eigenfunctions in the ball B = {y ∈ Rp : ||y|| < }. We further choose the potential such that the ground state is non-degenerate and · · · ≥ λn+1 ≥ λn ≥ · · · ≥ λ1 > λ0 = 0. We can allow for more general potentials, provided that the solutions have proper decay properties7 . When studying quantum mechanics on M (rather then on Rn+p ) we want the wavefunction to be normalized on M. We cannot localize the particle to the embedded manifold completely because of the uncertainty relation, but we can study it for small but finite > 0. To ensure that the wavefunction remains normalized after the particle has been confined we expand it as Ψ(x, y) = γ̃ −1/4 X ψα (x)χα (y) α We then find that XZ √ dn xdp y G|Ψ|2 = Z 1= M×B α √ dn x g|ψα |2 M To find the effective Hamiltonian acting on the wavefunctions ψα we need to compensate for the scaling factor. We define the Hamiltonian H1 acting on P α ψα (x)χα (y) to be H = H0 + Vconf = γ̃ −1/4 H1 γ̃ 1/4 We find that H1 is given by (for details see Appendix B) √ γ̃ 1/4 √ ∂A GGAB ∂B γ̃ −1/4 + Vconf (y) 2m G √ 1 k = − √ ∂µ + y Akµi ∂i ( gg µν ) ∂ν + y l Alν j ∂j + V (x, y) 2m g 1 ij − δ ∂i ∂j + Vconf (y) + O(y) 2m H1 = γ̃ 1/4 H γ̃ −1/4 = − 7 The precise conditions is that the expectation value of any polynomial in y j is finite and decays as at least a for some a > 0 in the limit → 0. 14 Chapter 2. Constrained quantum mechanics where V (x, y) is the potential V (x, y) = 1 ij 3 δ ∂i ∂j γ̃ − (∂i γ̃)(∂j γ̃) 8m 4 = Vind (x) + O(||y||) with Vind (x) = 1 X λ σ αiλ αiσ − 2αiλσ αiσλ 8m i We find that the ”matrix elements” of the Hamiltonian H1 are Z p (H1 )αβ = dp y |h|χα (y)∗ H1 χβ (y) B = X σ 1 √ − √ (Dµ )ασ ( gg µν ) (Dν )σβ + Vind δαβ + λα δαβ + O() 2m g where we defined the ”covariant derivative” Z p (Dµ )αβ = δαβ ∂µ + (Aµ )αβ = dp y |h|χα (y) ∂µ + y k Akµi ∂i χβ (y) B In the limit → 0, the terms of order can be neglected. The eigenvalues λα scale as λα ∼ 12 and therefore become widely separated in the limit of small . Since we have chosen the smallest eigenvalue to be zero, it is unaffected in this limit. On physical grounds we can therefore assume that only the ground state of the confining potential contributes to the effective dynamics of the constrained particle in the limit → 0. We find that Z Z p p 1 dp y |h|y k Akµi ∂i χ2α (Aµ )αα = dp y |h|χα y k Akµi ∂i χα = 2 B B 1 1 1 k = − δi Akµi = − δ ij ni · ∂µ nj = − δ ij ∂µ (ni · nj ) = 0 2 2 4 This implies that, for a non-degenerate ground state, the ”gauge field” Aµ does not affect the dynamics of the constrained particle. If the ground state is N -degenerate, then one can write the ground state wavefunction as a column vector and interpret Aµ as a static gauge field acting on the components of the vector. We have thus shown that, for a quantum mechanical particle confined to a submanifold of Euclidean space by a confining potential with non-degenerate ground state and small ”width” > 0, the effective Hamiltonian is [11–13, 21–24] 1 1 X λ σ √ Hef f = − √ ∂µ ( gg µν ∂ν ) + αiλ αiσ − 2αiλσ αiσλ 2m g 8m i where we ignored terms of order . If we would have quantized a classical particle confined to the embedded manifold, then we would only obtain the kinetic term 2.3. The effective Hamiltonian 15 of the Hamiltonian and not the induced potential. This shows that confining and quantizing in general do not commute. We end this chapter by considering two examples of constrained quantum mechanics. For a curve c(t) in R3 we can choose the normal vectors to be vectors of the Frenet frame [19] of the curve. We then get that the only non-zero coefficient is α2tt = κ where κ is the curvature of the curve. So the induced potential becomes Vind = − 1 2 κ 8m For a two dimensional surface embedded in R3 , there is only one linearly independent normal vector. We can diagonalize α3µν as (α3 is proportional to the Weingarten map of the surface) κ1 0 (α3µν ) = 0 κ2 where κ1 and κ2 are the principal curvatures of the surface (at the point x). We then find that [11] Vind = i 1 h 1 2 2 (κ1 + κ2 ) − 2 κ21 + κ22 = − (κ1 − κ2 ) 8m 8m So Vind ≤ 0 for a 2-dimensional surface embedded in three dimensional space. For higher dimensional manifolds, this statement is not true in general. 16 Chapter 3 Constrained relativistic fields 3.1 Introduction In this chapter we study the dynamics of constrained relativistic bosons (KleinGordon fields). In the derivation of the induced potential for quantum mechanical particles, an important step was the rescaling of the wavefunction, which allowed us to expand the wavefunction in modes. The wavefunction was rescaled in order to preserve the total probability in the thin layer limit. For relativistic quantum particles, there is no conservation of probability because the relativistic mass-energy equivalence makes any one-particle theory inconsistent. Relativistic quantum mechanics needs to be a theory with a variable number of particles, a quantum field theory. We take a first step towards understanding constrained quantum field theories by considering classical relativistic constrained fields. In particular, we derive the effective action for a classical Klein-Gordon field confined to an embedded manifold of a general manifold. We find that the induced potential, which was originally discovered in the context of quantum mechanics, also appears for semi-classical constrained fields. The derivation resembles the one in chapter 2 but with some subtle differences. 3.2 Constrained Klein-Gordon field Let Φ be a free real-valued scalar field on a (n + p + q)-dimensional Lorentzian manifold N where q is the number of time-like directions on M (N ). The action of Φ is Z p 1 AB 1 2 2 n+p+q S= d Z |η| η ∂A Φ∂B Φ − m Φ 2 2 Rq,n+p where ηAB is the metric of N . 17 18 Chapter 3. Constrained relativistic fields As in the previous chapter, we change to the coordinate system (x, y) given by the function F which extend the embedding function f . The metric in the coordinates (x, y) is 0 GAB = ηA0 B 0 ∂A F A ∂B F B 0 We choose the map F such that (GAB )|y=0 = gµν 0 0 −hij where gµν is the metric on M and hij is constant, non-degenerate and positive definite. To confine the field to the submanifold M ⊂ N we apply a confining potential which localizes the field to the submanifold. Applying the confining potential, the action becomes Z p 1 1 S= dn+q xdp y |G| ∂A Φ∂B Φ − (m2 + Vconf (y))Φ2 2 2 M×B We choose the potential to be such that any solution Φ of the equations of motion fulfils Φ = 0 for ||y|| > 01 . It is therefore sufficient to consider the action in the ball B = {y ∈ Rp : ||y|| < }. We assume that Vconf is such that there exists a complete set of eigenfunctions {χα (y)}∞ α=0 satisfying −hij ∂i ∂j χα + Vconf χα = λα χα with corresponding eigenvalue λα and Z dp y p |h|χα χβ = δαβ B We also choose the potential to be such that the ground state χ0 is non-degenerate with λ0 = 0. We see here why it is important that all normal directions are spacelike. If one direction had been time-like, then the eigenvalues {λα } would not have been bounded from below, and we would not be able to derive the low-energy dynamics of the theory. Naively expanding the field in terms of the eigenfunctions is not appropriate since the volume element in general depends on the normal coordinate y. This is 1 More precisly we choose the potential to be such that, given a Cauchy surface (see e.g. [25] for a review of the subject) of initial conditons for Φ, all solutions satisfy Φ = 0 for ||y|| > in the future domain of dependence. 3.2. Constrained Klein-Gordon field 19 because the y-dependence of the volume element can give rise to an infinite sum of mixed modes ψα ψβ (α 6= β). To solve this problem we instead expand the field as Φ(x, y) = γ̃ −1/4 ∞ X φα (x)χα (y) = γ̃ −1/4 φ α=0 where γ̃ −1/4 = |G| |g||h| . Z We then get that dn+q xdp y p |G|Φ2 = XZ dn+q x p |g|φ2α α The kinetic term in the Lagrangian density becomes Z p 1 dn xdp y |G|GAB ∂A Φ∂B Φ 2 Z p 1 dn xdp y |g||h|γ̃ 1/2 GAB ∂A (γ̃ −1/4 φ)∂B (γ̃ −1/4 φ) = 2 Z p γ̃ −1 AB γ̃ −2 1 dn xdp y |g||h| GAB ∂A φ∂B φ − G (∂A γ̃)φ(∂B φ) + ∂A γ̃∂B γ̃φ2 = 2 2 16 Z p p 1 = dn xdp y |g| |h| GAB ∂A φ∂B φ − V (x, y)φ2 2 where p γ̃ −1 AB 3γ̃ −1 γ̃ −1 V (x, y) = − |g|GAB (∂B γ̃) G ∂A ∂B γ̃ − ∂A γ̃∂B γ̃ − p ∂A 8 4 8 |g| To find the effective action we expand the potential as a series in y and use that Z p p i1 i2 iN d y |h|y y . . . y χα χβ ≤ N B We thus find that, for all α and β, Z p dp y |h|V (x, y)χα (y)χβ (y) = Vind (x)δαβ + O() where Vind (x) = V (x, 0) is the induced potential 1 3 1 Vind = hij ∂i ∂j γ̃ − ∂i γ̃∂j γ̃ − 8 4 8 y=0 ∂i Gij (∂j γ̃) (3.1) y=0 The formula (3.1) expresses the induced potential in terms the reduced volume element γ̃. We belive that (3.1) is equivalent to the obvious generalization of Mitchell’s result [26]. Mitchell’s result expresses the potential in terms the coefficients tµ · ∂ν ni and contractions of the Riemann tensor of N . The formula (3.1) is not as explicit as Mitchells result, but we believe that it is more convenient for practical computations. It is especially useful when the coordinate system is such that the submanifold can be parametrized by letting some coordinates be constant. 20 Chapter 4 Extrinsic curvature effects in brane-world scenarios In this chapter we discuss possible physical implications of the induced potential in different cosmological models. We consider embeddings of the Schwarzschild solution and the Robertson-Walker metric, and we investigate possible physical implications. We choose embeddings which we believe are the simplest possible, but the extension to more complicated embeddings (e.g. motivated by some underlying theory) is straightforward. 4.1 Embedded cosmological models As was derived in chapter 2, the effective action for a Klein-Gordon field confined to an (n + 1)-dimensional embedded submanifold f (M) ⊂ R1,n+p is Z p 1 1 µν n+1 2 S= d x |g| g ∂µ φ∂ν φ − Vind (x)φ 2 2 M where Vind (x) is the induced potential h i 1 Vind (x) = hij αiλλ αjσσ − 2αiλσ αjσλ 8 4.2 Embedded Schwarzschild solution The Schwarzschild metric is a solution to Einstein’s equation’s describing spacetime outside a spherically symmetric mass distribution. The line element of the Schwarzschild solution is rs 2 rs −1 2 ds2 = 1 − dt − 1 − dr − r2 (dθ2 + sin2 (θ)dϕ2 ) r r 21 22 Chapter 4. Extrinsic curvature effects in brane-world scenarios where rs = 2GM/c2 is the Schwarzschild radius with M the total mass and G the gravitational constant. The angular variables θ and ϕ are the usual angular variables used for spherical coordinates with 0 ≤ θ ≤ π and 0 ≤ ϕ < 2π. The variables t and r are the coordinate time and radius. Note that the coordinate time and radius are different from the proper (physical) time and radius, e.g. the proper distance between r0 and r1 is not r1 − r0 (for r1 > r0 > rs ), but Z r1 r r0 1− rs −1 dr r In 1921, Kasner [27] showed that the Schwarzschild metric cannot be embedded in five dimensional Minkowski space. In 1959 Fronsdal [28] found an embedding in six dimensional Minkowski space given by1 Z 0 = 2t0 (1 − rs /r)1/2 sinh(t/2t0 ) Z 1 = 2t0 (1 − rs /r)1/2 cosh(t/2t0 ) Z 2 = g(r) Z 3 = r sin(θ) cos(ϕ) Z 4 = r sin(θ) sin(ϕ) Z 5 = r cos(θ) where g 0 (r)2 = rs (r3 − t20 rs ) r3 (r − rs ) and t0 is an auxiliary parameter. It can be shown that, under certain assumptions, this embedding, for t0 = rs , is a unique embedding of the Schwarzschild solution [29]. For t0 6= rs , this embedding does not cover the entire Schwarzschild solution, but can be thought of as an embedding of space-time corresponding to some spherically symmetric mass distribution. The embedding can be modified to also cover the interior region 0 < r < rs [28] by interchanging the hyperbolic functions, setting t0 = rs and replacing (1 − rs /r)1/2 → (rs /r − 1)1/2 the embedding then becomes well defined for 0 < r < rs . 1 We here introduced the parameters r and t . Fronsdal’s original embedding is obtained by s 0 setting t0 = rs = 1. 4.2. Embedded Schwarzschild solution 23 For t0 = rs the embedding can be written in the simple form (for derivation of the induced potential in the case t0 6= rs see appendix B) Z = 2rs |1 − rs /r|1/2 ev + g(r)e2 + rer (4.1) where we introduced the basis vectors ev = (sinh(t/(2rs )), cosh(t/(2rs )), 0, 0, 0, 0) , rs < r (cosh(t/(2rs )), sinh(t/(2rs )), 0, 0, 0, 0) , 0 < r < rs e2 = (0, 0, 1, 0, 0, 0) er = (0, 0, 0, sin(θ) cos(ϕ), sin(θ) sin(ϕ), cos(θ)) One possible set of normal vectors to the embedding is n4 = g 0 (r)|r/rs − 1|1/2 ev − (rs /r)3/2 e2 n5 = (rs /r)3/2 |1 − rs /r|1/2 ev + g 0 (r)(1 − rs /r)(r/rs )1/2 e2 − (rs /r)1/2 er From this embedding we obtain the non-zero coefficients r(r2 + rs r + rs2 )1/2 2(rs r)3/2 3rs2 =− 2 2 2r (r + rs r + rs2 )1/2 1 rs 3/2 = α511 = − 2rs r r 3/2 1 s = α544 = rs r α400 = − α411 α500 α533 which gives us the induced potential Vind (r) = − (r3 + rs r2 + rs2 r + 9rs3 )(r2 + rs2 )(r + rs ) 16rs2 r4 (r2 + rs r + rs2 ) We see that the induced potential is analytic in the whole region 0 < r < ∞. We believe that this is because Fronsdal’s embedding covers the entire region 0 < r < ∞. 24 Chapter 4. Extrinsic curvature effects in brane-world scenarios The induced potential for t0 = rs = 1 0 −0.1 −0.2 −0.3 Vind −0.4 −0.5 −0.6 −0.7 −0.8 −0.9 −1 1 2 3 4 5 6 7 8 9 r The potential has the asymptotic forms ( Vind (r) = 9r 2 − 16rs4 1 + 1 − 16r 1+ 2 s r 2 9rs + O(r ) rs −2 ) r + O(r , for r ≈ 0 , for r rs 10 4.3. Embedded Robertson-Walker metric 25 1 For r rs , the potential approaches the constant value Vind (r) ≈ − 16r 2. s Translating this into a mass magnitude by the formula ~2 |Vind | = m2 c2 we find that the asymptotic value of the potential corresponds to a mass of ~p ~c 3.321 × 1010 kg GeV m= |Vind | ≈ ≈ c 8GM M c2 A peculiar feature is that the mass is inverse proportional to M , the mass of the gravitating object. This means that, for heavy bodies, the effect of the induced potential is negligible, while for light bodies, the effect could be large. For e.g. the sun (M ≈ 1.988 × 1030 kg) this gives a mass of about m ≈ 1.67 × 10−11 eV /c2 . A super-heavy black hole, which are believed to be in the center of our galaxy [30], with a mass of about 4.1 million solar masses the corresponding mass is m ≈ 4 × 10−18 eV /c2 . We believe that these masses are negligible small. This suggests that the induced potential can only produce measurable physical effects for small black holes. One example are the hypothetical primordial black holes [31], which are believed to have formed in the early universe. If primordial black holes exists, they would offer good conditions for studying Hawking radiation and quantum effects in gravity. Such small black holes are, however, believed to have evaporated because of their Hawking radiation on a time scale of 3 M 64 τ = 10 y M This implies that a primordial black hole with mass M = 1010 kg would have evaporated about 104 years after the Big-Bang. 4.3 Embedded Robertson-Walker metric The Robertson-Walker metric describes a isotropic homogeneous universe. The line element of the Robertson-Walker metric is ds2 = dt2 − a(t)2 dr2 + S(r)2 dθ2 + sin2 θdϕ2 (4.2) where a(t) is a scale factor corresponding to the ”radius” of the universe and S(r) is a function describing the R structure of the universe, i.e. the volume of the universe is proportional to a(t)3 S(r)2 dr where the integral is finite if S is bounded. S(r) depends on a parameter K and is given by √ sin( Kr) √ , for K > 0 K r , for K = 0 S(r) = √ sinh( |K|r) √ , for K < 0 |K| For K > 0, S(r) is bounded so the universe has a finite volume and describes a closed universe. For K ≤ 0, S(r) is unbounded, and the universe is therefore open. 26 Chapter 4. Extrinsic curvature effects in brane-world scenarios Robertson-Walker space-time can be embedded in five dimensional Minkowski space [32] as follows √ b(t)/ 2 K R 1 r Z0 = + r0 a + 2 r 0 a(t)C(r) a(t)C(r) R 1 r2 1 Z = 2 r0 − r0 a + p b(t)/ |K| , for K > 0 dt0 2r0 ȧ , for K = 0 , for K < 0 , for K > 0 dt0 2r0 ȧ , for K = 0 , for K > 0 2 Z = a(t)S(r) sin(θ) cos(ϕ) Z 3 = a(t)S(r) sin(θ) sin(ϕ) Z 4 = a(t)S(r) cos(θ) where C(r) = √ √ cos( p Kr)/ Kp , for K > 0 cosh( |K|r)/ |K| , for K < 0 A normal vector to the embedding is n4 = ȧ(t) √ e0 + ḃ(t) (C(r)e1 + S(r)er ) Kh 2 i 1 1 r + r − ȧ(t) e0 0 2 r0 r0 ȧ(t) i h , for K > 0 , for K = 0 2 1 + 12 ȧ(t) rr0 − r0 − r0 ȧ(t) e1 + ȧ(t)rer ȧ(t) √ e + ḃ(t) (C(r)e0 + S(r)er ) K 1 , for K < 0 Using this we find the induced potential (for details see Appendix B) Vind = 1 4 ä K + ȧ2 ä2 6 +3 − 2 a a K + ȧ2 We note that, even though the expressions for the embedding and the normal vectors are very different for different values of K, the induced potential is described by a single expression for all values of K. For K = 0 we can write the induced potential as Vind = Ḣ + 2H 2 − H2 2 Ḣ 2 = − q + 6q − 3 4H 2 4 where H = ȧa is the Hubble parameter and q = parameter [33–35]. aä ȧ2 = −Ḣ − H 2 is the deceleration 4.4. A model of the early universe 27 In the a model where a(t) go to zero in the limit t → 0, a(t) can be modelled as decaying as a power law a(t) = (t/t0 )x for some constant t0 and x > 0. The induced potential is then Vind = 8x2 − 4x − 1 4t2 (4.3) This means that in the limit t → 0, the potential behaves as −∞ 0 < x < x∗ 0 x = x∗ lim Vind = t→0 +∞ x∗ < x √ with x∗ = 1+4 3 ≈ 0.683. Thus under the condition that the potential is bounded, it predicts a specific value of the exponent x. As we will see later, there is a solution for a scalar field in the RW universe where, if the induced potential is included, the scale factor grows as a(t) ∼ tx∗ short after t = 0. 4.4 A model of the early universe The induced potential depends on the scale factor a(t) and its derivatives and becomes large in some limits when a(t) becomes small. This suggests that the induced potential could be important in the early universe. One model of the early universe is the theory of cosmological inflation (see e.g. [33–35]). In the usual Big Bang scenario [34], the expansion of the universe is determined by the Friedmann equations ρ̇ = −3H(ρ + P ) ρ K H2 = − 2 3MP2 l a 1 ä = Ḣ + H 2 = − (ρ + 3P ) a 6MP2 l where ρ is the (energy) density and P is the pressure. In the following we will set K = 0. It is natural to assume that these equations together with an equation of state P = wρ gives the expanding universe we observe today. There are, however, some problems which this model does not solve: • The Flatness problem. If the curvature of the universe was non-zero in the early universe, then the universe becomes more curved with time. The universe we observe today is measured to be flat with small uncertainty. This means that there should have been some mechanism in the early universe, making the universe more flat. 28 Chapter 4. Extrinsic curvature effects in brane-world scenarios • The Horizon problem. As the universe expands, different regions separate faster then they can exchange information. This implies that different regions should have different temperature since they cannot come into thermal equilibrium fast enough. This contradicts observations, so this tendency should have been suppressed in the early universe. • Unwanted relics (or monopole problem). In the Big Bang, different exotic particles are believed to have been produced. But since we do not observe them today, they should have been diluted by some mechanism in the early universe. Cosmological inflation solves the above problems by postulating that a rapid expansion occurred in the early universe [34]. This solves the problem of unwanted relics since they then became strongly diluted in the expansion, the horizon problem since all wavelength become red-shifted and the flatness problem since the rapid expansion suppresses the curvature of the universe. 4.4.1 The standard case In the standard case, where the induced potential is not present, the Friedmann equations of the expanding universe can be obtained from Einstein’s equations for the Robertson-Walker metric (4.2) with the Energy-Momentum tensor (Tµν ) = diag(−ρ, p, p, p). If matter consists of an isotropic Klein-Gordon field (φ = φ(t)), then 1 2 φ̇ + V(φ) 2 1 P = φ̇2 − V(φ) 2 ρ= where V(φ) contains all interactions of the Klein-Gordon field. In this case the equations of motion can also be derived from the action Z p 1 µν 2 4 S = d x |g| MP l (R − 2Λ) + g (∂µ φ)(∂ν φ) − V(φ) 2 where MP2 l = ~c 8πG is the reduced Planck mass and R = −6 ä ȧ2 + a a2 is the Ricci (scalar) curvature of the Robertson-Walker universe and Λ is the cosmological constant. For concreteness we set V(φ) = 1 2 2 λ 4 m φ + φ 2 4 4.4. A model of the early universe 29 which is a potential often studied in the literature [34]. Since both R and φ only depends on the time variable, we can integrating over the variables r, θ and ϕ and write the action as Z S = const dtL where L = MP2 l (−3aȧ2 − Λa3 ) + a3 2 φ̇2 − m2 φ2 − λ 4 φ 2 It follows that the equations of motion for a and φ are φ̈ + 3H φ̇ + m2 φ + λφ3 = 0 1 λ 4 2 2 2Ḣ + 3H 2 − Λ + φ̇ − m φ − φ =0 2MP2 l 2 (4.4) (4.5) Since L does not depend explicitly on time we have the conservation law d ∂L ∂L ∂L ∂L − ä + ȧ + φ̇ − L ∂ ȧ dt ∂ä ∂ä ∂ φ̇ 1 λ 4 Λ 2 2 2 φ̇ + m φ + φ = 3a3 MP2 l H 2 − − 3 6MP2 l 2 I= (4.6) from which we obtain the Friedmann equations by setting I = 0. To solve the equations, one often uses the ”slow-roll” approximation where one approximates (4.4) and (4.6) with 3H φ̇ + m2 φ + λφ3 = 0 1 λ 4 2 2 H − m φ+ φ =0 6MP2 l 2 The approximation is valid when φ̈ 3H φ̇ φ̇2 m2 φ2 Under this approximation, one can calculate a(t2 )/a(t1 ) exactly (for more details see e.g. [34]). The third equation (4.5) is then automatically solved since it follows from the other two equations. 30 Chapter 4. Extrinsic curvature effects in brane-world scenarios We here use an alternative method to solve the equations. We expand the solutions in series φ(t) = ty φ0 + φ1 t + φ2 t2 + φ3 t3 + O(t4 ) x H(t) = + H0 + H1 t + H2 t2 + H3 t3 + O(t4 ) t and solve for each coefficient. We find the solution φ21 φ1 t+ 3H0 φ1 + m2 φ0 + λφ30 t2 + O(t3 ) 2 2 2MP l 2MP l 1 3H0 φ1 + m2 φ0 + λφ30 t2 + O(t3 ) φ(t) = φ0 + φ1 t − 2 s 1 Λ 2 + m 2 φ2 + λ φ3 H0 = + φ 1 0 3 6MP2 l 2 0 H(t) = H0 − The solution has two free parameters, φ0 = φ(0) and φ1 = φ̇(0), which describe the initial amplitude and speed of the scalar Rfield φ. We see that the solution has t x = y = 0, implying that a(t) = a(0) exp 0 H(t0 )dt0 and φ(t) are analytic at t = 0. 4.4.2 The extended case In the extended case where the induced potential is present, we do not know how to derive the Einstein equations since it is unclear how the induced potential changes when the metric tensor is varied. We therefore, in analogy with the standard case, postulate that the equations of motion for a(t) and φ(t) are the equations of motion following from the action p 1 1 λ d4 x |g| m2P l (R − 2Λ) + g µν (∂µ φ)(∂ν φ) − (m2 + Vind )φ2 − φ4 2 2 4 Z = const dtL Z S= with the condition that the conservation law for this action satisfies I = 0. The Lagrangian is L = MP2 l (−3aȧ2 − Λa3 ) + a3 2 1 ä ȧ2 ä2 λ φ̇2 − m2 + 6 +3 2 − 2 φ2 − φ4 4 a a ȧ 2 4.4. A model of the early universe 31 The equations of motion for a(t), φ(t) and the condition that I = 0 gives the extended Friedmann equations ! Ḣ 2 2 φ̈ + 3H φ̇ + m + Ḣ + 2H − φ + λφ3 = 0 4H 2 1 1 2 1 2 2 λ 4 2 2Ḣ + 3H − Λ + 2 φ̇ − m φ − φ MP l 2 2 4 " ... # 3 2 1 Ḧ Ḣ H Ḣ Ḧ 5Ḣ 3Ḣ 2 H + 2 + − − + − + φ2 MP l 2H 4H 4 2 3H 3 12 8H 2 12H 2 ! ! # 2 Ḧ Ḣ 2 Ḣ 1 Ḣ 2 − H+ − φφ̇ + − φφ̈ + φ̇ =0 + H 3 3H 2 2H 3 6H 2 3 Λ 1 λ 4 2 2 2 2 H − − φ̇ + m φ + φ 3 6MP2 l 2 ! # " ! 1 Ḣ 1 Ḧ 1 2 Ḣ 2 1 φ2 = 0 + 2 − H φφ̇ + Ḣ + − H − MP l 6H 3 4 12H 6 8H 2 where H = ȧ a as usual. For this case the conservation law is I = − 3a + 3 1 λ 4 Λ 2 2 2 2 φ̇ + m φ + φ H − − 3 6MP2 l 2 ! ! # Ḧ Ḣ H2 Ḣ 2 1 Ḣ H + − − φ2 + 2 − φφ̇ = 0 12H 4 6 8H 2 MP l 6H 3 MP2 l 1 MP2 l Using MAPLE we find that the equations have one solution with x = 43 , y = 1 and no free parameters r 2 2m 126λMP2 l 2 1 3 φ(t) = − − t + O(t ) λ 4t 15 65 2 2 m 594λMP2 l H(t) = + − t + O(t3 ) 4t 5 65 Another solution with x = (1 + √ 3)/4 ≈ 0.683, y = 0 and no free parameters is φ(t) = MP l 1.5 − 0.228m2 + 0.402λMP2 l + 0.0883Λ t2 + O(t4 ) x H(t) = − 0.0142m2 + 0.120λMP2 l − 0.0690Λ t + O(t3 ) t The exact values of the coefficients can easily be calculated, but are not very illuminating. 32 Chapter 4. Extrinsic curvature effects in brane-world scenarios There is also a solution with x = y = 0 1 H12 2 2 2 φ(t) = φ0 + φ1 t − φ0 t2 + O(t3 ) 3H0 φ1 + m + λφ0 + H1 + 2H0 − 2 4H02 3H12 λ 2 3 2 2 H(t) = H0 + H1 t + m + φ0 + H0 − H1 H0 + 2 2 4H0 φ H 1 0 +(2H02 − H1 ) + (φ21 + 2ΛMP2 l − 6H02 MP2 l ) 2 t2 + O(t3 ) φ0 φ0 which has four free parameters, φ0 6= 0, φ1 , H0 and H1 . In contrast with the other two solutions, the solution is analytical at t = 0. We believe that each of these solutions has a finite radius of convergence, i.e. the exact solution can be well approximated by these expansions (truncated at some finite order) in some finite interval. We can therefore expand the solutions around a initial point given initial values of φ and H. The solution can thereafter be extended by expanding around another point in the interval to obtain an approximative solution in a larger interval. For small t, the second solution has a power-law behaviour a(t) ∼ tx∗ with x∗ ≈ 0.683 precisely the exponent required for the potential (4.3) to have a finite value in the limit t → 0. 4.5 Discussion Since the extrinsic and intrinsic curvature of embedded submanifolds affects the particles of our world through the induced potential, it can provide a means to indirectly explore extra dimensions. If Schwarzschild space-time is embedded in the way discussed in this chapter, then the induced potential could be present and give measurable physical effects. Since the contribution is inverse proportional to the mass of the gravitating object, the effect will probably be negligible for massive objects like the sun and the earth. Only for small black holes, such as hypothetical primordial black holes, could the induced potential give measurable effects. The embedded Robertson-Walker universe produces an induced potential which depends on the scale factor a(t) and its derivatives. This suggests that the presence of the induced potential interacting with a scalar field could influence the expansion of the universe. Especially, the potential affects the expansion of the early universe where a(t) is small. The embeddings considered in this chapter where chosen because they are (we believe) the simplest possible. It is possible that other theories with extra dimensions, like e.g. string theory, suggest more complicated embeddings of the considered space-times. For such cases, the induced potential can be calculated in a similar fashion. 4.5. Discussion 33 Rosen lists in his paper [32] embeddings of many solutions of general relativity. An interesting future project would be to calculate the induced potentials for these embeddings and to derive possible physical consequences. To calculate the induced potential for more complicated embeddings motivated by an underlying physical theory is another interesting project which could improve our understanding of extra dimensions. 34 Chapter 5 Membrane dynamics One often considers the fundamental particles of nature to be point particles. One possible generalization of point particles can be obtained by considering higher dimensional extended objects. One motivation for this is that one-dimensional extended objects, so called strings, are believed to describe gravity and other fundamental forces (see e.g. the textbook by Polchinski [14]). Two kinds of string theories are usually considered. Bosonic string theory, which contains bosons, and supersymmetric string theory, which contains bosons and fermions. String theory has the peculiar feature that the quantum theory requires a certain number of space-time dimensions to be consistent1 . Bosonic string theory requires D = 26 space-time dimensions, and supersymmetric string theory requires D = 10. Five different kinds of supersymmetric string theory are known, and there are hopes that the different supersymmetric string theories actually are limits of a single underlying theory called M -theory [36]. It is not known what M -theory exactly is, but it is known that it requires D = 11 space-time dimensions to be consistent, and it is believed to contain quantum membranes. Since M -theory and string theory contains quantum membranes, a better understanding of the dynamics of membranes is desired. Today, more is known about quantum strings than about the quantum theory of higher dimensional extended objects. It is, for example, not known if membranes can be quantized or what their mass spectrum is. By examining symmetries of the theory, we hope to gain greater insight and understanding of the theory [37–40]. Dynamical symmetries are certain symmetries related to the dynamics of a system and are sometimes called ”hidden” symmetries. In this chapter we first review the light-cone gauge in a simple example and then describe a well known dynamical symmetry. We then introduce the bosonic membrane and study its Hamiltonian formulation in the light-cone gauge. The dynamical symmetry of the bosonic membrane is described. In the final sections 1 This is because of an anomaly related to the Lorentz invariance of the theory. 35 36 Chapter 5. Membrane dynamics we introduce the supersymmetric membrane and study a dynamical symmetry for the superembrane. 5.1 Point particle in the light-cone gauge It is resonable to assume that the motion of a relativistic point particle in a Ddimensional Lorentzian manifold is such that the length of the particles world-line is minimized (we here follow [41] and [14]). The action can be written as2 Z τ1 S = −m dτ q gαβ (x)ẋα ẋβ (5.1) τ0 where x = x(τ ), ẋ = ∂τ x, gαβ is the metric tensor of the ambient space and we let greek letters denote the indices 0, 1, 2, . . . , D. By varying the action we find the geodesic equation ẍµ + Γµνσ ẋν ẋσ = 0 We can naively try to calculate the Hamiltonian of the relativistic point particle in flat Minkowski space, where gαβ = ηαβ . The conjugate momenta is pµ = ∂L mẋµ = −√ µ ∂ ẋ ẋα ẋα which gives us the Hamiltonian q mẋµ ẋµ H = pµ ẋµ − L = − √ + m ẋβ ẋβ = 0 ẋα ẋα Thus the Hamiltonian vanishes. This means that the variables of the theory are not dynamical, but are determined by the conservation law pµ pµ − m2 = 0 (5.2) which follows from the definition of the canonical momenta or from Noether’s theorem [42]. The Hamiltonian vanishes because the action is reparametrization invariant [43], i.e. the time variable, which is treated as a special variable in the Hamiltonian formalism, can be substituted by a function of other variables without changing the action. To obtain dynamical variables and a non-zero Hamiltonian we make 2 We use the metric convention (+, −, −, . . . , −). 5.1. Point particle in the light-cone gauge 37 a gauge choice, i.e. we choose a specific time variable. One possible choice is the light-cone gauge where we set x+ = x0 + xD−1 =τ 2 and x− = ζ = x0 − xD−1 . The action (5.1) then becomes Z q 2 S = −m 2ζ̇ − ẋ dτ where x = (x1 , x2 , . . . , xD−2 ). In this gauge, the metric is η++ η+− η+j 0 1 0 0 (ηµν ) = η−+ η−− η−j = 1 0 ηi+ ηi− ηij 0 0 −δij where we use latin letters to denote indices 1, 2, . . . , D − 2. We find that (pi ) = p = π= ∂L mẋ =q 2 ∂ ẋ 2ζ̇ − ẋ −m ∂L =q 2 ∂ ζ̇ 2ζ̇ − ẋ mζ̇ p2 + m2 H = p · ẋ + π ζ̇ − L = q = −2π 2 2ζ̇ − ẋ An important tool in the Hamiltonian formalism is the Poisson bracket. For two function F (xµ , pν ) and G(xµ , pν ), the Poisson bracket is defined as {F, G} = D−1 X µ=0 ∂F ∂G ∂G ∂F − ∂xµ ∂pµ ∂xµ ∂pµ = D−2 X ∂F ∂G ∂F ∂G ∂G ∂F ∂G ∂F − + − ∂ζ ∂π ∂ζ ∂π ∂xi ∂pi ∂xi ∂pi i=1 We have the Poisson brackets {xi , pj } = δij {ζ, π} = 1 Since the Hamiltonian is canonically conjugate to x− = x+ = τ (by that 1 = τ̇ = {τ, H}) we set p− = p+ = H One notes that π is constant since π̇ = {π, H} = − ∂H =0 ∂ζ 38 Chapter 5. Membrane dynamics The other variables have the time dependence pi π H ζ̇ = {ζ, H} = − π ẋi = {xi , H} = − with all other brackets zero. The generators of Lorentz transformations in the light-cone gauge are Mij = xi pj − xj pi Mi− = xi H − ζpi We find that the generators fulfil the Poisson bracket relations {Mij , Mkl } = δik Mjl − δil Mjk − δjk Mil + δjl Mik {Mi− , Mj− } = 0 {Mi− , Mkl } = −δik Ml− + δil Mk− In the quantum theory of strings, the critical dimension is determined from the requirement that the second relation holds also at the quantum level. 5.2 A dynamical symmetry In classical mechanics, symmetries are closely related to conserved quantities by Noethers theorem. For example, translation invariance is related to momentum conservation, time independence to energy conservation and rotational symmetry to conservation of angular momentum. In many cases, the existence of symmetries allows us to reduce the number of degrees of freedom of a problem and possibly obtain an exact solution. A symmetry which depends on the dynamics of the system is called a dynamical symmetry. An important example of a dynamical symmetry is the Kepler problem for which the Hamiltonian is3 H= p2 GM m − 2m r where G is the gravitational constant, m is the mass of the particle and M is the mass of a heavy particle often considered to be stationary. The generators of angular momentum, Lij = xi pj − xj pi , generate the symmetry group SO(3) of 3 A quantum analogue is the Hydrogen atom model which can be obtained by substituting GM m → e2 and replacing variables with operators. 5.2. A dynamical symmetry 39 spatial rotations and are conserved, i.e. {Lij , H} = 0. Another conserved quantity in the problem is the Laplace-Runge-Lenz vector [44] which has the components Ci = 1 GM m pj Lij − xi m r The Lenz-vector and the generators of angular momentum have the Poisson brackets {Lij , Lkl } = δik Ljl − δil Ljk − δjk Lil + δjl Lik 2H {Ci , Cj } = − Lij m {Ci , Lkl } = −δik Cl + δil Ck The fact that the algebra is closed under Poisson brackets shows that Lij and Ci are generators of a symmetry group of the Hydrogen atom. The symmetry group is larger since the group of spatial rotations and depends on the Hamiltonian. This shows that the problem has a dynamical symmetry. To simplify the expressions we restrict ourselves to solution where H < 0 (bounded orbits) and set r m Ck Ak = − 2H This gives the relations {Lij , Lkl } = δik Ljl − δil Ljk − δjk Lil + δjl Lik {Ai , Aj } = Lij {Ai , Lkl } = −δik Al + δil Ak By defining generators Iij and Jij 1 (Lij + ijk Ak ) 2 1 Jij = (Lij − ijk Ak ) 2 Iij = we find that Iij and Jij have the Poisson brackets {Iij , Ikl } = δik Ijl − δil Ijk − δjk Iil + δjl Iik {Jij , Jkl } = δik Jjl − δil Jjk − δjk Jil + δjl Jik {Iij , Jkl } = 0 These relations show that Iij and Jij together generate the symmetry group SU (2) × SU (2) ∼ = SO(4). This shows that the Kepler problem atom has a larger ”hidden” SO(4) symmetry. For the quantum mechanical Hydrogen atom, this dynamical symmetry can be used to calculate the spectrum [45]. 40 Chapter 5. Membrane dynamics 5.3 The bosonic membrane As a generalization of point particles, one can consider higher dimensional extended objects moving in a Lorentzian manifold (for details see e.g. [38]). In analogy with the point particle, we consider membranes which moves in a way that minimizes the world volume of the membrane M as it moves in a Lorentzian manifold N . The corresponding action is the Dirac-Nambu-Goto action4 Z √ (5.3) S = −V ol(M) = − dϕ0 dM ϕ G where G is the absolute value of the determinant of the induced metric Gαβ = ηµν ∂α xµ ∂β xν of the membrane, xµ = xµ (ϕ) for ϕ = (ϕ0 , ϕ1 , . . . , ϕM ) and ηµν is the metric tensor of N . The equations of motion for the membrane, which follows from (5.3), are √ 1 √ ∂α GGαβ ∂β xµ + Gαβ ∂α xν ∂β xλ Γµνλ (x) = 0 G In the following we will only consider the case where the embedding space is Minkowski space, i.e. N = R1,D−1 and Γµνλ (x) = 0. 5.3.1 Light-Cone Gauge Just as for the point particle, we consider the membrane in the light-cone gauge where we set x0 + xD−1 =τ 2 ζ = x0 − xD−1 ϕ0 = The matrix Gαβ is then (Gαβ ) = 2ζ̇ − ẋ u1 .. . 2 u1 . . . uM −grs uM where x = (x1 , x2 , . . . , xD−1 ) grs = ∂r x · ∂s x ur = ∂r ζ − ẋ · ∂r x 4 The action is often called the Nambu-Goto action in the string theory literature. In the context of membranes one often also credits Dirac since he formulated a model where the electron was modelled as a membrane [46]. 5.4. Hamiltonian formalism 41 We thus find that 2 1 up 2ζ̇ − ẋ + ur g rs us G = | det(G)| = det 0 −grp −g pq uq 2 = g 2ζ̇ − ẋ + ur g rs us = gΓ 0 δsp with Γ defined by this relation. 5.4 Hamiltonian formalism As discussed earlier, membranes are an essential part of M -theory which is believed to unify the different supersymmetric string theories in a single theory. M -theory is a quantum mechanical theory, and the understanding of quantum mechanical membranes is therefore important. One way to formulate a quantum theory is in the operator formalism based on the Hamiltonian formulation of classical mechanics. We here work towards a quantum theory of membranes by investigating the Hamiltonian structure of classical membranes [41]. Let us first (as before) naively try to find the Hamiltonian of the theory. The canonical momenta are pµ = √ ∂L = − GG0α ηµν ∂α xν µ ∂ ẋ and the Hamiltonian (density) is found to be √ √ √ √ H = pµ ẋµ − L = − GG0α ηµν ∂α xν ẋµ + G = − GG0α Gα0 + G = 0 i.e. the variables of the theory are not dynamical, but they are instead determined by the constraints [47] Ja := pµ ∂a xµ = 0 1 H̃ := (η µν pµ pν + det(µab )) = 0 2 i = 1, 2, . . . where µab = ηµν ∂xµ ∂xν ∂ϕa ∂ϕb for a, b, = 1, 2, . . . , M . The constraints follow from the definition of the canonical momenta. As for the point particle, we choose a specific time variable to obtain a theory with dynamical variables. We then lose the full Lorentz symmetry of the theory, 42 Chapter 5. Membrane dynamics but obtain a (possibly) non-zero Hamiltonian and thereby dynamical variables. In the light-cone gauge, the canonical momenta are r g ∂L = p= ẋ Γ ∂ ẋ r g ∂L =− π= ∂ζ Γ where we made the gauge choice ua = ∂r ζ − ẋ · ∂a x = 0. One finds that the Hamiltonian density is r g H = p · ẋ + π ζ̇ − L = ζ̇ Γ Like for the point particle, we can write the Hamiltonian density as H= p2 + g −2π We see that π is time-independent since π̇ = − ∂H =0 ∂ζ We therefore write π = π(ϕ) = −ηρ(ϕ), where ρ(ϕ) is a non-dynamical density which fulfils Z ρ(ϕ)dM ϕ = 1 M Just as for the point particle, we have that p− = H p+ = −π which gives us that the Lorentz invariant mass squared is [41] 2 Z M = 2P+ P− − P · P = p2 + g M d ϕ− ρ where Z Pµ = is the zero-mode of pµ . pµ dM ϕ Z 2 pi d ϕ M 5.5. Mode expansion 5.5 43 Mode expansion We introduce a complete orthonormal set of (non-constant) basis functions {Yα (ϕ)}α∈I defined on the membrane satisfying Z Yα (ϕ)Yβ (ϕ)ρ(ϕ)dM ϕ = δαβ M ∆Yα (ϕ) = −µα Yα (ϕ) and expand the dynamical variables in terms of the basis functions X x=X+ xα Yα (ϕ) (5.4) α6=0 X p pα Yα (ϕ) =P + ρ (5.5) α6=0 where X and P are the zero-modes Z X= Z P = x ρdM ϕ p dM ϕ of the position and momentum variables. In the following we will use the convention that α denotes indices different from zero and we sum over repeated indices. For functions F (xµ (ϕ), pν (ϕ)) and G(xµ (ϕ0 ), pν (ϕ0 )) we define the Poisson bracket as Z δF δG δG δF M {F, G} = d ϕ̃ − δxµ (ϕ̃) δpµ (ϕ̃) δxµ (ϕ̃) δpµ (ϕ̃) We find the Poisson bracket {xi (ϕ), pj (ϕ̃)} = δij δ(ϕ, ϕ̃) which gives us that {xiα , pjβ } = δij δαβ (5.6) {Xi , Pj } = δij (5.7) and all other brackets zero. Using the basis functions, we can construct the ”Green’s function” G(ϕ, ϕ̃) = X −1 α µα Yα (ϕ)Yα (ϕ̃) 44 Chapter 5. Membrane dynamics satisfying δ(ϕ, ϕ̃) ∆ϕ̃ G(ϕ, ϕ̃) = − 1, ρ(ϕ) Z G(ϕ, ϕ̃)ρ(ϕ)dM ϕ = 0 From the equations for ζ p · ∂a x ρ p2 + g 2η 2 ζ̇ = ρ2 η∂a ζ = we can reconstruct ζ using the Green’s function. We get that [48] Z 1 p ˜ a ˜ ζ(ϕ) = ζ0 + G(ϕ, ϕ̃)∇ · ∇a x (ϕ̃)ρ(ϕ̃)dM ϕ̃ η ρ 5.6 A dynamical symmetry for bosonic membranes As for the Kepler problem, there is a dynamical symmetry for the membranes [1, 39]. For the bosonic membrane, the generators of Lorentz transformations in the light-cone gauge are Z Mij = (xi pj − xj pi ) dM ϕ Z Mi− = (xi H − ζpi ) dM ϕ The generators fulfil the same commutation relations as the Lorentz generators for the point particle [49] {Mij , Mkl } = δik Mjl − δil Mjk − δjk Mil + δjl Mik (5.8) {Mi− , Mj− } = 0 (5.9) {Mi− , Mkl } = δik Ml− − δil Mk− (5.10) Using the expansions (5.4) and (5.5) we expand the angular momentum generators as Mij = Xi Pj − Xj Pi + Mij where Mij = xiα pjα − xjα piα only contains internal degrees of freedom. By using the relation (5.6) we get that {Mij , Mkl } = δik Mjl − δil Mjk − δjk Mil + δjl Mik Thus the purely internal angular momentum generate a symmetry group. (5.11) 5.7. Supermembranes 45 We can expand the Lorentz generators Mi− in the same way using that 2ηζ(ϕ) = 2ηζ0 + 2P · xα Yα (ϕ) + (internal modes) 2 2ηH(ϕ) = ρP + 2ρP · pα Yα (ϕ) + (internal modes) which gives us that 2 2ηMi− = Xi P − 2ηζ0 Pi + 2Pj Mij + 2ηMi− (5.12) where Mi− only contains internal modes. We find that ηMi− satisfy {ηMi− , ηMj− } = M2 Mij and also that {ηMi− , Mkl } = −ηδki Mj− + ηδkj Mi− (5.13) From (5.11), (5.12) and (5.13) we see that the purly internal Lorentz generators Mij and ηMi− generate a dynamical symmetry similar to the one of the Kepler problem in section 5.2. This shows that the bosonic membrane has a dynamical symmetry. 5.7 Supermembranes One objection to bosonic string theory as a physical theory is that it only contains bosons. A string theory with fermions is supersymmetric string theory (or superstring theory) which is a string theory with bosonic and fermionic variables related by supersymmetry. Because of the different statistics of bosons and fermions, bosonic variables commute, xµ xν = xν xµ , while fermionic variables anti-commute θα θβ = −θβ θα . The action for supersymmetric strings is called the Green-Schwarz action. The generalization of the Green-Schwarz action to p-dimensional supermembranes, p-branes, is [50] Z q 1 1 µ αβγ µ ν ν |det(Gαβ )| + ∂α x Eβ + θΓ ∂α θθΓ ∂β θ θΓµν ∂γ θ dp ϕdϕ0 S=− 2 6 where Gαβ = Eα · Eβ = ηµν Eαµ Eβν Eαµ = ∂α xµ + θΓµ ∂α θ and the matrices Γµ are the D-dimensional gamma matrices which satisfy Γµ Γν + Γν Γµ = 2η µν Since the theory of classical bosonic membranes can be formulated in spacetime of arbitrary dimension, it is only at the quantum level that restrictions on 46 Chapter 5. Membrane dynamics the number of space-time dimensions (like for the quantum bosonic string) might occur. The supersymmetric membrane, however, must be formulated in such a way that the bosonic and fermionic degrees of freedom match [51]. This means that, even at the classical level, a p-dimensional supermembrane can only be formulated in D space-time dimensions, where D depends on p. The possible supermembrane theories are p=1 2 3 4 5 D=3 X 4 X X 5 X 6 X X 7 X 8 X 9 X 10 X X 11 X where we have put a X to indicate that the dynamics is consistent. Note that p = 1 corresponds to the string. The classical superstring can be formulated in D = 3, 4, 6 or 10 dimensions, while the quantum superstring requires D = 10 dimensions. All different superstring theories are believed to be unified in M -theory which requires D = 11 space-time dimensions. Since this is also the dimension required for the 2-dimensional supermembrane, 2-dimensional supermembranes is the most probable candidate for M -theory [52]. In the light-cone gauge, the generalized Green-Schwarz Lagrangian is q √ 2 L = − g 2ζ̇S + ẋ + 2θΓ− θ̇ + rs ∂r xa θΓ− Γa ∂s θ g = det(grs ) grs = ∂r x · ∂s x We find that r ∂L g p= = ẋ − ur g rs ∂s x Γ ∂ ẋ r ∂L g π= = Γ ∂ ζ̇ r ∂L g − S= =− Γ θ Γ ∂ θ̇ p2 + g H = p · ẋ + π ζ̇ + S θ̇ − L = − rs ∂r xi θΓ− Γi ∂s θ −2π By super symmetry, we can reduce the number of fermionis variables by imposing the constraint (Γ0 + ΓD )θ = 0 5.9. A dynamical symmetry for supermembranes 47 It is convenient to factorize the gamma matrices as √ 0 2 √0 0 ⊗ I Γ+ = ⊗ I Γ− = 2 0 0 0 Γi = 0 1 1 0 ⊗ γi where I is the identity matrix and γ a are the 9-dimensional (SO(9)) gamma matrices (for details see e.g. [52]). 5.8 Poisson brackets In order for the Poisson bracket to be formulated consistently5 we need the special property that for fermionic variables [53] {θα , θβ } = {θβ , θα } {θα θβ , θγ } = θα {θβ , θγ } − {θα , θγ }θβ The bosonic and fermionic variables have the non-zero Poisson brackets {xi (ϕ), pj (ϕ)} = δij δ(ϕ, ϕ̃) iδαβ δ(ϕ, ϕ̃) {θα (ϕ), θβ (ϕ̃)} = − ρ Expanding the variables in terms of the basis functions xi (ϕ) = Xi + xiA YA (ϕ) pi (ϕ)/ρ = Pi + piA YA (ϕ) θα (ϕ) = θα0 + θαA YA (ϕ) we get the Poisson brackets {Xi , Pj } = δij {θα0 , θβ0 } = −iδαβ 5.9 {xiA , pjB } = δij δAB {θαA , θβB } = −iδαβ δAB A dynamical symmetry for supermembranes In analogy with the bosonic membrane, there is a corresponding dynamical symmetry for the supermembrane [40]. We find the symmetry by calculating the Poisson brackets of the generators of Lorentz transformations. 5 We want the Poisson bracket to be such that e.g. F (xµ , pν , θα ). Ḟ = {F, H} for any function F = 48 Chapter 5. Membrane dynamics The generators of Lorentz transformations for the supermembrane are Z i Jij = xi pj − xj pi − θγ ij θρ d2 ϕ 4 Z i i Ji− = xi HS − ζS pi − θγ ik θpk − rs ∂r xj ∂s xk θγ ijk θ d2 ϕ 2η 8η with Z i p ˜ 1 Gr (ϕ, ϕ̃) · ∂r x + θ∂˜r θ ρ(ϕ̃)d2 ϕ̃ η ρ 2 2 p +g i HS = − θγ i rs ∂r xi ∂s θ 2ηρ 2η 1 γ ij = γ [i γ j] = [γ i , γ j ] 2 1 i j k γ ijk = γ [i γ j γ k] = γ γ γ + γj γk γi + γk γiγj − γj γiγk − γk γj γi − γiγk γj 3! ζS = ζ0 − It can be shown that (see [54, 55]) the generators fulfil the Poisson bracket relations {Jij , Jkl } = δik Jjl − δil Jjk − δjk Jil + δjl Jik {Ji− , Jk− } = 0 We separate the zero modes by defining (0) Jij = Jij + J ij (0) Ji− = Ji− + J˜i− + J i− where i (0) Jij = Xi Pj − Xj Pi − θ0 γ ij θ0 4 i (0) Ji− = (Xi HS − ζ0 Pi ) − θ0 γ ik θ0 Pk 4η 1 i J˜i− = J ij Pk − θ0 γ i Q η 2η Z 1 rs ij i i Qβ = pi γβα θα + ∂r xi ∂s xj γβα θα d2 ϕ − Pi γβα θ0α 2 We find that {ηJ i− , ηJ k− } = M2 J ik − Qik where i ik Qik = − Qα γαβ Qβ 4 5.9. A dynamical symmetry for supermembranes 49 Calculating the Poisson brackets we find that {Qij , Qkl } = (−δjk Qil + δik Qjl − δil Qjk + δjl Qik ) M2 {J ik , ηJ l− } = −δkl ηJ i− + δil J k− {Qik , ηJ l− } = 0 {J ij , Qkl } = −δjk Qil + δik Qjl − δil Qjk + δjl Qik This gives us that J ik , ηJ i− and Mik = M2 J ik − Qik generate the dynamical symmetry {Mij , Mkl } = −δjk Mil + δik Mjl − δil Mjk + δjl Mik M2 {η J¯i− , η J¯k− } = Mik {Mik , η J¯l− } = −δkl η J¯i− + δil η J¯k− M2 This shows that the supermembrane has a dynamical symmetry similar to the dynamical symmetry of the bosonic membrane. We hope that this symmetry, in the future, will help us to better understand the dynamics of both classical and quantum supermembranes. 50 Chapter 6 Summary and conclusions Extra dimensions have been proposed in many theories. Since we do not observe extra dimensions in our daily life, it is plausible to assume that some mechanism prevents particles from moving in the extra dimensions (if they exists). For such scenarios, extrinsic curvature effects could result in physical effects, which might be accessible to experiments. In paper A, we derived the induced potential for a relativistic scalar field (Klein-Gordon field). This generalizes the notion of constrained quantum mechanics to relativistic fields. We also investigated embeddings of certain cosmological models, which gave indications about how extrinsic curvature effects can produce measurable physical effects. We first considered the Fronsdal embedding of the Schwarzschild solution and found that the induced potential is continuous for all values 0 < r < ∞. We believe that this is a consequence of that Fronsdal’s embedding covers the entire Schwarzschild space-time. Thereafter we calculated the induced potential for a simple embedding of the Robertson-Walker space-time, which describes the evolution of a homogeneous isotropic universe. The induced potential appears to be negligible for measured values of the Hubble constant and deceleration parameter, but it could have been important in the early universe. We formulated a model from which we deduced equations extending the usual Friedman equations. It was found that the extended model allows a scaling solution not possible in the original model. This implies that the induced potential can give rise to physical effects in this scenario. If, as suggested by many theories, we live on a membrane embedded in some higher dimensional ambient space, then the membrane should be dynamical, i.e. it should be affected by particles on the membrane. A simple model is to assume that the dynamics of a membrane is analogous to that of a point-particle, i.e. a membrane moves in a way that minimizes its world volume. By understanding the symmetries of the theory we hope to gain insight into the dynamics of such membranes. 51 52 Chapter 6. Summary and conclusions In paper B, we derived a dynamical symmetry of a supermembrane. We found that both the bosonic and fermionic variables satisfy Poisson relations like the ones for the bosonic membrane. This shows that the supermembrane has a dynamical symmetry with richer structure then that of the bosonic membrane. Appendix A The tubular neighbourhood theorem In this appendix we prove the tubular neighbourhood theorem which proves that the embedding function can always be extended in the way done in chapter 2 and 3. All maps and manifolds considered will be assumed to be smooth. Theorem [Tubular neighbourhood theorem] Let f : M → N be a smooth embedding of a smooth n-dimensional manifold M in a smooth (n + p)-dimensional manifold N and let Ba = {y ∈ Rp : ||y|| < a}. Then there exists an extension F : M × Rp → N of f in the sense that F (x, 0) = f (x) for all x ∈ M and such that for every compact set U ⊂ M there exists a constant > 0 such that F restricted to U × B is one-to-one. Proof At each point q ∈ M, the tangent space Tf (q) N can be decomposed as Tf (q) N = Tq M ⊕ Nq M Nq M is called the normal bundle of M at f (q) and is isomorphic to Rp . Let n+1 n+2 {ni }n+p ,y , . . . , y n+p ) ∈ Nq M. Denote i=n+1 be a basis of Nq M and set y = (y n+p by ϕq : Wq → R a local chart of N , where Wq is an open neighbourhood of f (q) ∈ N . Define the map F to be F : M × Rp → N −1 i F (x, y) = ϕ−1 q (ϕq (f (x)) + dϕq (y ni )) where q is such that x ∈ f −1 (Wq ∩ f (M)). This shows the existence of F . It remains to show that there exists some > 0 such that F |U ×B is one-to-one. For each x ∈ M, F has full rank at y = 0. By the inverse function theorem there exists an open neighbourhood Wx ⊂ N of F (x, 0) = f (x) such that F restricted 53 54 Appendix A. The tubular neighbourhood theorem to F −1 (Wx ) is one-to-one. Since Wx is open, there is a constant x > 0 such that F (Ux × Bx ) ⊂ Wx , where Ux = f −1 (Wx ∩ f (M)). Let U ⊂ M be compact. For every x ∈ U there is an x > 0 such that F |Ux ×Bεx is one-to-one. Since U is compact, there is a finite set of points x1 , x2 , . . . , xm in U such that U ⊂ ∪m i=1 Uxi . Setting = min{x1 , x2 , . . . , xm } we get that F |U ×B is one-to-one. This proves the theorem. If M is compact, then there is a constant > 0 such that F is one-to-one for all (x, y) ∈ M × B . If M is non-compact, there is in general no constant > 0 such that the above statement holds for all x ∈ M. A simple counterexample is the manifold {(x, y) ∈ R2 : x2 y 2 = 1}. For non-compact manifolds one can use Urysohn’s lemma to construct a contionous function (x) > 0 such that F is one-to-one for all (x, y) ∈ M × Rp such that ||y|| < (x). Appendix B Calculation of the induced potential We here present details of the calculations of the induced potential. B.1 Calculation of γ̃ The expression (2.5) is an explicit expression for the metric tensor GAB . In order to calculate γ̃ we use the block-matrix relation γµν + y a y b Aaµk Abν l hkl −y a Aaµk hkj Gµν Gµj (GM N ) = = Giν Gij −y b Abν l hli −hij λ σ a k 0 δν −y Aaµ δµ γλσ 0 = 0 −hkl 0 δi k −y b Abν l δj l which gives us that G = | det(GM N )| = | det(γµν )|| det(hij )| = |γ||h| where |γ| = | det(γµν )| and |h| = det(hij ). This gives us that γ̃ = |G| = | det(γµν )|/|g| |g||h| = | det(gµν − 2y i αiµλ gλν + y i y j αiµλ αjν σ gλσ )|/|g| = det(δν β − 2y i αiν β + y i y j αiγ λ αjν σ gλσ g γβ ) (B.1) A useful relation for determinants is det(A) = etr(log(A)) 55 (B.2) 56 Appendix B. Calculation of the induced potential To simplify calculation when expanding (B.1) we introduce the matrices I, Ai and Bij with components I = (δν β ) Ai = (−2αiν β ) Bij = (αiγ λ αjν σ gλσ g γβ ) = (αiσλ αjν σ gλγ g γβ ) = (αiσβ αjν σ ) Using these expressions and (B.2) we find that det(I + y i Ai + y i y j Bij ) = exp tr(log(I + y i Ai + y i y j Bij )) 1 i j i i j 3 = exp tr(y Ai + y y Bij − y y Ai Aj + O(y )) 2 1 1 i i j = 1 + y tr(Ai ) + y y tr(Bij ) − tr(Ai Aj ) + tr(Ai )tr(Aj ) + O(y 3 ) 2 2 1 1 λ λ i λ i j σ 2 σ 2 λ σ = 1 − 2y αiλ + y y αiλ αjσ − (−2) αiλ αjσ + (−2) αiλ αjσ + O(y 3 ) 2 2 = 1 − 2y i αiλλ + y i y j 2αiλλ αjσσ − αiλσ αjσλ + O(y 3 ) B.2 Derivation of the induced potential In the derivation of the induced potential, we expanded the scalar field as Φ = γ̃ −1/4 φ, where φ(x, y) = X α φα (x)χα (y) B.2. Derivation of the induced potential We get that the Hamiltonian H1 is √ γ̃ 1/4 H1 = γ̃ 1/4 H γ̃ −1/4 = − √ ∂A GGAB ∂B γ̃ −1/4 + Vconf (y) 2 G γ̃ −1/4 √ 1/2 µν −1/4 γ̃ −1/4 √ 1/2 µσ k gγ̃ γ ∂µ γ̃ − √ ∂µ gγ̃ γ y Akσj ∂j γ̃ −1/4 = − √ ∂µ 2 g 2 g −1/4 γ̃ γ̃ −1/4 √ 1/2 ij −1/4 √ 1/2 νσ k − √ ∂i gγ̃ γ y Akσi ∂ν γ̃ −1/4 − √ ∂i gγ̃ b ∂j γ̃ 2 g 2 g γ̃ −1/4 √ 1/2 k l gγ̃ y y Akσi Alρj γ σρ ∂j γ̃ −1/4 + Vconf (y) − √ ∂i 2 g √ 1 1 √ gg µσ y k Akσj ∂j = − √ ∂µ ( gg µν ∂µ ) − √ ∂µ 2 g 2 g 1 1 √ √ − √ y k Akσi ∂i ( gg νσ ∂ν ) − √ y k Akσi ∂i ( gg σρ )y l Alρj ∂j 2 g 2 g 3 1 1 ij + δ ∂i ∂j γ̃ − (∂i γ̃)(∂j γ̃) − δ ij ∂i ∂j + Vconf (y) + O(y) 8 4 2 1 √ = − √ ∂µ + y k Akµi ∂i ( gg µν ) ∂ν + y l Alν j ∂j + V (x, y) 2 g 1 ij − δ ∂i ∂j + Vconf (y) + O(y) 2 where V (x, y) is the potential 1 ij 3 V (x, y) = δ ∂i ∂j γ̃ − (∂i γ̃)(∂j γ̃) 8 4 3 1X λ σ σ λ λ σ = 2 2αiλ αiσ − αiλ αiσ − (−2αiλ )(−2αiσ ) + O(y) 8 i 4 X 1 = αiλλ αiσσ − 2αiλσ αiσλ + O(y) = Vind (x) + O(y) 8 i using that Z Z p p dp y |h|y i1 y i2 . . . y iN χ∗α χβ ≤ dp y |h||y i1 ||y i2 | . . . |y iN ||χ∗α ||χβ | Z p ≤ N dp y |h||χ∗α ||χβ | ≤ N gives us that XZ α,β dp y p |h|V (x, y)χα (y)χβ (y) = Vind (x)δαβ + O() 57 58 Appendix B. Calculation of the induced potential B.3 The Schwarzschild solution We here describe the details of the derivation of the induced potential for the embedded Schwarzschild solution for general t0 (not necessarily t0 = rs ). We first discuss the case r > rs and then the case 0 < r < rs . B.3.1 Embedding for r > rs The embedding for r > rs can be written as Z 0 = 2t0 (1 − rs /r)1/2 sinh(t/2t0 ) Z 1 = 2t0 (1 − rs /r)1/2 cosh(t/2t0 ) Z 2 = g(r) Z 3 = r sin(θ) cos(ϕ) Z 4 = r sin(θ) sin(ϕ) Z 5 = r cos(θ) where g 0 (r)2 = rs (r3 − t20 rs ) r3 (r − rs ) To simplify calculations we introduce the basis vectors et = (cosh(t/2t0 ), sinh(t/2t0 ), 0, 0, 0, 0) eτ = (sinh(t/2t0 ), cosh(t/2t0 ), 0, 0, 0, 0) eg = (0, 0, 1, 0, 0, 0) er = (0, 0, 0, sin(θ) cos(ϕ), sin(θ) sin(ϕ), cos(θ)) So that the embedding can be written as Z = f = 2t0 (1 − rs /r)1/2 eτ + g(r)eg + rer A set of normal vectors is s r3 − t20 rs (1 − rs /r)1/2 eτ + f (r)eg 2 r (r − rs ) r t20 rs r 1/2 n5 = (1 − r /r) e + h(r)e − e s τ g r r3 t0 n4 = B.3. The Schwarzschild solution 59 where s f (r) = − s h(r) = t20 rs (r − rs ) r(r3 − t20 rs ) (r3 − t20 rs )(r − rs ) t20 rs r From the embedding we can calculate the second derivatives of the embedding function, tµν = ∂µ ∂ν Z, and use them to find the components αiµν = αiµλ gλν = ni · tµν which in turn allows us to calculate αiµν = g νλ αiµλ . A useful way to organize the coefficients αiµν is to define matrices αi = (αiµν ). With these conventions we find that −1/2t0 0 0 0 s 3t0 rs (r−rs ) r3 − t20 rs 0 − 2r(r 0 0 3 −t2 r ) 0 s α4 = r2 (r − rs ) 0 0 0 0 0 0 0 0 and − 2t10 2 t0 rs 0 3 r 0 0 0 r α5 = − 2t10 0 0 0 0 1 t0 0 0 0 0 1 t0 This finally gives us the induced potential Vind = B.3.2 2 1X (r4 − 4t2 rs r + 3t20 rs2 )2 rs − 3 tr(αi )2 − 2tr(αi2 ) = − 2 4 0 2 3 4 i=1 16t0 r (r − rs )(r − t0 rs ) r Embedding for 0 < r < rs For 0 < r < rs the embedding can be written as Z 0 = 2t0 (rs /r − 1)1/2 cosh(t/2t0 ) Z 1 = 2t0 (rs /r − 1)1/2 sinh(t/2t0 ) Z 2 = g(r) Z 3 = r sin(θ) cos(ϕ) Z 4 = r sin(θ) sin(ϕ) Z 5 = r cos(θ) 60 Appendix B. Calculation of the induced potential As before we can simplify calculations by introducing the basis vectors so that Z = 2t0 (rs /r − 1)1/2 et + g(r)eg + rer A set of normal vectors to the embedding is 1/2 −1/2 r t0 rs rs 0 n4 = 1 − −1 g (r)et − 2 eg rs r r −1/2 rs − r t0 rs rs rs 0 −1 er n5 = √ et − g (r)eg + rs r r2 r r − rs Using this we find that (α4µν ) = − 2t10 q t20 rs −r 3 r 2 (rs −r) 0 − 3t2r0 r2s 0 0 0 q 0 rs −r t20 rs −r 3 0 0 0 0 0 0 0 0 0 and p rs 1 − 2r r 0 (α5µν ) = 0 0 0p rs 1 − 2r 0 0 r 1 r 0 0 p rs r 0 1 r 0 0 0 p rs r So we find that the induced potential for 0 < r < rs is Vind = − (r4 − 4t20 rs r + 3t20 rs2 )2 rs − 3 2 2 4 3 4(2t0 ) r (t0 rs − r )(rs − r) r Note that the expression for the induced potential is is the same as for r > rs . The parameter t0 needs to be chosen such that g 0 (r)2 is non-negative. When t0 6= rs one can study the regions 0 < r < t0 < rs and rs < t0 < r separately, but for t0 = rs the embedding covers the entire region 0 < r < ∞. Setting t0 = rs we can simplify the expression for the potential as Vind = − (r + rs )(r2 + rs2 )(r3 + rs r2 + rs2 r + 9rs3 ) 16rs2 r4 (r2 + rs r + rs2 ) Which is the expression given in chapter 4. B.4. Robertson-Walker metric B.4 61 Robertson-Walker metric The Robertson-Walker metric can be embedded in five dimensional Minkowski space by setting [32] √ b(t)/ 2 K R 1 r Z0 = 2 r0 + r0 a + a(t)C(r) a(t)C(r) R 1 r2 Z1 = 2 r0 − r0 a + p b(t)/ |K| , for K > 0 dt0 2r0 ȧ , for K = 0 , for K < 0 , for K > 0 dt0 2r0 ȧ , for K = 0 , for K > 0 2 Z = a(t)S(r) sin(θ) cos(ϕ) Z 3 = a(t)S(r) sin(θ) sin(ϕ) Z 4 = a(t)S(r) cos(θ) where ḃ(t)2 = K + ȧ(t)2 √ √ cos( p Kr)/ Kp C(r) = cosh( |K|r)/ |K| √ sin( Kr) √ , for K K r √ , for K S(r) = |K|r) sinh( √ , for K |K| , for K > 0 , for K < 0 >0 =0 <0 For the different values of K, a normal vector is n4 = ȧ(t) √ e0 + ḃ(t) (C(r)e1 + S(r)er ) Kh 2 i 1 r 1 ȧ(t) + r − e0 0 2 r0 h r0 ȧ(t) i 2 1 + 12 ȧ(t) rr0 − r0 − r0 ȧ(t) e1 + ȧ(t)rer ȧ(t) √ e + ḃ(t) (C(r)e0 + S(r)er ) K 1 , for K > 0 , for K = 0 , for K < 0 where e0 = (1, 0, 0, 0, 0) e1 = (0, 1, 0, 0, 0) er = (0, 0, sin(θ) cos(ϕ), sin(θ) sin(ϕ), cos(θ)) 62 Appendix B. 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Phys. 98, 485 (1997). Part II Scientific papers 67 68 Paper A Edwin Langmann and Martin Sundin Extrinsic curvature effects in brane-world scenario arXiv: 1103.3230 Paper B Jonas de Woul, Douglas Lundholm, Jens Hoppe and Martin Sundin A dynamical symmetry for supermembranes arXiv:1004.0266, accepted for publication in the Journal of High Energy Physics

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