A First Course in String Theory

Transcription

1 A First Course in String Theory BARTON ZWIEBACH Massachusetts Institute of Technology CAMBRIDGE UNIVERSITY PRESS

2 Contents Parti Basics Foreword Preface Acknowledgements 1 A brief introduction 1.1 The road to unification 1.2 String theory as a unified theory of physics 1.3 String theory and its verification 2 Special relativity and extra dimensions 2.1 Units and parameters 2.2 Intervals and Lorentz transformations 2.3 Light-cone coordinates 2.4 Relativistic energy and momentum 2.5 Light-cone energy and momentum 2.6 Lorentz invariance with extra dimensions 2.7 Compact extra dimensions 2.8 Quantum mechanics and the square well 2.9 Square well with an extra dimension Problems 3 Electromagnetism and gravitation in various dimensions 3.1 Classical electrodynamics 3.2 Electromagnetism in three dimensions 3.3 Manifestly relativistic electrodynamics 3.4 An aside on spheres in higher dimensions 3.5 Electric fields in higher dimensions 3.6 Gravitation and Planck's length 3.7 Gravitational potentials 3.8 The Planck length in various dimensions 3.9 Gravitational constants and compactification 3.10 Large extra dimensions Problems vil

3 viii Contents Ш- 4 Nonrelativistic strings Equations of motion for transverse oscillations Boundary conditions and initial conditions Frequencies of transverse oscillation More general oscillating strings A brief review of Lagrangian mechanics The nonrelativistic string Lagrangian 72 Problems 76 5 The relativistic point particle Action for a relativistic point particle Reparameterization invariance Equations of motion Relativistic particle with electric charge 86 Problems 87 6 Relativistic strings Area functional for spatial surfaces Reparameterization invariance of the area Area functional for spacetime surfaces The Nambu-Goto string action Equations of motion, boundary conditions, and D-branes The static gauge Tension and energy of a stretched string Action in terms of transverse velocity Motion of open string endpoints 111 Problems String parameterization and classical motion Choosing a a parameterization Physical interpretation of the string equation of motion Wave equation and constraints General motion of an open string 122 Problems World-sheet currents Electric charge conservation Conserved charges from Lagrangian symmetries Conserved currents on the world-sheet The complete momentum current Lorentz symmetry and associated currents The slope parameter a Problems Light-cone relativistic strings A class of choices for r The associated a parameterization 152

4 Contents 9.3 Constraints and wave equations Wave equation and mode expansions Light-cone solution of equations of motion 160 Problems Light-conefieldsand particles Introduction An action for scalar fields Classical plane-wave solutions Quantum scalar fields and particle states Maxwell fields and photon states Gravitational fields and graviton states 180 Problems The relativistic quantum point particle Light-cone point particle Heisenberg and Schrödinger pictures Quantization of the point particle Quantum particle and scalar particles Light-cone momentum generators Light-cone Lorentz generators 200 Problems Relativistic quantum open strings Light-cone Hamiltonian and commutators Commutation relations for oscillators Strings as harmonic oscillators Transverse Virasoro operators Lorentz generators Constructing the state space Equations of motion Tachyons and D-brane decay* 238 Problems Relativistic quantum closed strings Mode expansions and commutation relations Closed string Virasoro operators Closed string state space String coupling and the dilaton Abrief look at superstring theories* 262 Problems 268 Part II Developments D-branes and gauge fields Dp-branes and boundary conditions Quantizing open strings on Dp-branes 277

5 X Contents 14.3 Open strings between parallel Dp-branes Strings between parallel Dp- and Dg-branes Intersecting D6-branes 295 Problems String charge, electric charge, and particle physics Fundamental string charge Visualizing string charge Strings ending on D-branes D-brane charges D-branes and the Standard Model gauge group Open strings and the Standard Model fermions* The Standard Model on intersecting D6-branes* String theory and particle physics* 344 Problems String thermodynamics and black holes A review of statistical mechanics Partitions and the quantum violin string Hagedorn temperature Relativistic particle partition function Single string partition function Black holes and entropy Counting states of a black hole* A string theory/gauge theory correspondence* 377 Problems T-duality of closed strings Duality symmetries and Hamiltonians Winding closed strings Left movers and right movers Quantization and the double strike Constraint and mass formula State space of compactified closed strings A striking spectrum coincidence Duality as a full quantum symmetry 403 Problems T-duality of open strings T-duality and D-branes (/(1) gauge transformations Wilson lines on circles Open strings and Wilson lines 420 Problems 423

6 XI Contents t 19 Electromagneticfieldson D-branes Maxwell fields coupling to open strings D-branes with electric fields D-branes with magnetic fields 432 Problems Nonlinear and Born-Infeld electrodynamics The framework of nonlinear electrodynamics Born-Infeld electrodynamics Born-Infeld theory and T-duality 451 Problems Covariant string quantization Introduction Open string Virasoro operators Selecting the quantum constraints Lorentz covariant state space Closed string Virasoro operators The Polyakov string action 472 Problems String interactions and Riemann surfaces Introduction Interactions and observables String interactions and global world-sheets World-sheets as Riemann surfaces Schwarz-Christoffel map and three-string interaction Moduli spaces of Riemann surfaces Four open string interaction Veneziano amplitude 510 Problems Loop amplitudes in string theory Loop diagrams and ultraviolet divergences Annuli and one-loop open strings Annuli and electrostatic capacitance Nonplanar open string diagrams Four closed string interactions The moduli space of tori 534 Problems 543 References Index