String Theory { A Postgraduate Course for Physicists and Mathematicians Thomas Mohaupt November 23, 2017 ii Contents 1 Introduction 1 I From particles to strings 3 2 Classical relativistic point particles 5 2.1 Minkowski Space . 5 2.2 Particles . 8 2.3 A non-covariant action principle for relativistic particles . 11 2.4 Canonical momenta and Hamiltonian . 14 2.5 Length, proper time and reparametrizations . 15 2.6 A covariant action for massive relativistic particles . 17 2.7 Particle Interactions . 19 2.8 Canonical momenta and Hamiltonian for the covariant action . 20 2.9 A covariant action for massless and massive particles . 22 3 Classical relativistic strings 27 3.1 The Nambu-Goto action . 27 3.2 The Polyakov action . 32 3.2.1 Action, symmetries, equations of motion . 32 3.2.2 Interpretation as a two-dimension field theory . 34 3.2.3 The conformal gauge . 36 3.2.4 Lightcone coordinates . 39 3.2.5 From symmetries to conservation laws . 40 3.2.6 Explicit solutions . 44 3.2.7 Open strings with Dirichlet boundary conditions . 48 iii iv CONTENTS 3.2.8 Non-oriented strings . 48 4 Quantized relativistic particles and strings 49 4.1 Quantized relativistic particles . 49 4.2 Field quantization and Quantum Field Theory . 53 4.3 Quantized relativistic strings . 58 A Physical Units 61 B Dirac δ-functions, Fourier analysis and bra-ket notation 63 B.1 Dirac δ-functions . 63 B.2 Fourier sums and Fourier integrals . 64 B.3 Bra-ket formalism . 65 Chapter 1 Introduction 1 2 CHAPTER 1. INTRODUCTION Part I From particles to strings 3 Chapter 2 Classical relativistic point particles In this chapter we revise classical relativistic point particles and set out our conventions and notation. Readers familiar with this material can skip through the chapter and use it as a reference when needed. 2.1 Minkowski Space According to Einstein's theory of special relativity, space and time are com- bined into spacetime, which is modelled by Minkowski space M.1 The elements P; Q; : : : 2 M are events, which combine a moment of time with a position. We leave the dimension D of spacetime unspecified. D-dimensional Minkowski space M = MD is the D-dimensional affine space associated with the D-dimensional vector space RD. The difference between an affine space (or point space) and a vector space is that an affine space has no distinguished origin. In contrast in the vector space RD we have the distinguished zero vector ~0 2 RD, which is the neutral element of vector addition. Vectors x 2 RD do not naturally correspond to points P 2 MD, but to displacements relating a point P to another point Q −−! x = PQ: 1 For brevity's sake we will use `Minkowksi space' instead of `Minkowski spacetime.' 5 6 CHAPTER 2. CLASSICAL RELATIVISTIC POINT PARTICLES But we can use the vector space RD to introduce linear coordinates on the point space MD by chosing a point O 2 MD and declaring it the origin of our coordinate system. Then points are in one-to-one correspondence with position vectors: −−! xP = OP ; and displacements correspond to differences of position vectors: −−! −−! −−! PQ = OQ − OP : While position vectors depend on a choice of origin, displacements are indepen- dent of this choice. This reflects that Minkowski space has translation symmetry. Having fixed an origin O, we can refer to points in Minkowski space in terms of their position vectors. The components xµ, µ = 0; 1;:::;D = 1 of vectors x 2 RD provide linear coordinates on M. The coordinate x0 is related to the time t, measured by an `inertial', that is force-free, or freely falling observer, by x0 = ct. We set the speed of light c to unity, c = 1, so that x0 = t. The remain- ing coordinates xi, i = 1;:::;D = 1 can be combined into a (D − 1)-component vector ~x = (xi). The components xi parametrize space, as seen by the inertial observer. In special relativity, the vector space RD is endowed with the indefinite scalar product µ ν x · y = ηµν x y ; with Gram matrix ! −1 0 η = (ηµν ) = : (2.1) 0 1D−1 Inertial observes are required to use linear coordinates which are orthonormal with respect to this scalar product, that is, inertial frames are η-orthonormal frames. Observe that we have chosen the `mostly plus' convention, while part of the literature uses the `mostly minus' convention, where the Gram matrix is multi- plied by −1. We are using the index notation common in the physics literature, including Einstein's summation convention. It is understood that we have iden- tified the vector space V = RD with its dual V ∗ using the Minkowski metric. µ Thus a vector x has contravariant coordinates x and covariant coordinates xµ ν µ µν which are related by `raising and lowering indices' xµ = ηµν x , and x = η xµ. 2.1. MINKOWSKI SPACE 7 For reference, we also list the corresponding line element. 2 µ ν 2 2 ds = ηµν dx dx = −dt + d ~x : (2.2) Mink_Line_Element One can, and in many applications does, make use of other, curvilinear coordi- nate systems, such as spherical or cylindrical coordinates. But η-orthonormal coordinates are distinguished by the above standard form of the metric. The most general class of transformations which preserve this form are the Poincar´e transformations µ µ ν µ x ! Λ ν x + a ; µ where Λ = (Λ ν ) is an invertible D × D matrix satisfying ΛT ηΛ = η ; and where a = (aµ) 2 RD. The matrices Λ describe Lorentz transforma- tions, which are the most general linear transformations preserving the met- ric. They contain rotations together with `Lorentz boosts', which relate iner- tial frames travelling a constant velocity relative to each other. The Lorentz transformations form a six-dimensional Lie group, called the Lorentz group O(1;D−1). Elements Λ 2 O(1;D−1) have determinant det Λ = ±1, and satisfy 0 jLambda 0j ≥ 1. The matrices with det Λ = 1 form a subgroup SO(1;D − 1). 0 0 This subgroup still has two connected component, since Λ 0 ≥ 1 or Λ 0 ≤ −1. The component containing the unit matrix 1 2 O(1;D − 1) is connected and denoted SO0(1;D − 1). This is the Lie group obtained by exponentiating the corresponding Lie algebra so(1;D − 1). The Lorentz group and translation group combine into the Poincar´egroup, or inhomogeneous Lorentz group, IO(1;D − 1), which is a ten-dimensional Lie group. Note that since Lorentz transformations and translations do not com- mute, the Poincar´egroup is not a direct product. The composition law (Λ; a) ◦ (Λ0; a0) = (ΛΛ0; a + Λa0) shows that the Lorentz group operates on the translation group. More precisely, the translation group RD happens to be a vector space, on which O(1;D − 1) acts by its fundamental representation. The Poincar´egroup is the semi-direct product of the Lorentz and translation group, D IO(1;D − 1) = O(1;D − 1) n R : 8 CHAPTER 2. CLASSICAL RELATIVISTIC POINT PARTICLES Mink_Line_Element Since the Minkowski metric (2.2 is defined by an indefinite scalar product, the distance-squared between events can be positive, zero or negative. This carries information about the causal structure of spacetime. Vectors are classified as time-like, light-like (or null), or space-like according to their norm-squared: x time-like , x · x < 0 ; x light-like , x · x = 0 ; x space-like , x · x > 0 : −−! Then, if x = PQ is the displacement between between two events, then these events are called time-like, light-like or space-like relative to each other, depend- ing on x. The zero-component of x carries the information whether P is later than Q (x0 > 0), or simultanous with Q (x0 = 0) or earlier than Q (x0 < 0), relative to a given Lorentz frame. If we fix the orientation of space and the direction of space, Lorentz frames are related by proper orthochronous Poincar´e D transformations, (Λ; a) 2 SO0(1;D − 1) n R . Under these transformations the temporal order of events is invariant, if and only if the events are time-like or light-like relative to each other. In contrast, the events that are relatively space-like can be put in any relative order by applying these transformations. This reflects that signals propagating with superluminal speed are not consis- tent with `causality', that is the ideat that the past can influence the future but not the other way round: events which are time-like, light-like and space-like relative to one another are connected by straight lines corresponding to signals propagating with subluminal, luminal and superluminal speed, respectively. 2.2 Particles The particle concept assumes that there are situations where matter can be modelled (possibly only approximately) as a system of extensionless mathemat- ical points. The motion of such a point particle, or particle for short, is described by a parametrized curve, called the world-line. If we restrict ourselves to iner- tial frames, it is natural to choose the coordinate time t as the curve parameter. Then the worldline of the particle is a parametrized curve C : I ! M : t 7! x(t) = (xµ(t)) = (t; ~x(t)) ; (2.3) 2.2. PARTICLES 9 where I ⊂ R is the interval in time for which the particle is observed (I = R is included as a limiting case). The velocity of the particle relative to the given inertial frame is d~x ~v = : (2.4) dt Particles are equipped with an inertial mass m, which characterizes their resis- tance against a change of velocity.