Fundamental physics, year 1 (Teknisk fysik, årskurs 4)

String theory — Lecture notes

by Christian von Schultz

Lecturer: Bengt E W Nilsson

This is an introductory course in string theory/M theory. The course home page can be found at http://fy.chalmers.se/~tfebn/StringTheoryMSc2009.html.

Tuesday 2009-03-17

Introduction to the course. Unifications in the history of physics: Earthly and celestial mechanics; Maxwell's equations (light, electricity, magnetism); Special relativity (space and time unified, mechanics and electromagnetism unified, mass and energy unified); Nordström; Kaluza; Klein; Electroweak theory; Grand Unified Theories (GUT's). Group theory: SO(3), SU(2), SU(n). The classical matrix groups, and five exceptional cases. Quantum Mechanics as a framework. Many string theories, unified into M-theory. Theory of everyting — or a frame work. Forces and particles are vibrational modes. Quantum gravity. No free dimensionless parameters. Many solutions. Extra dimensions are looked for at CERN. The size of a string.

Wednesday 2009-03-18

Special relativity and extra dimensions. Units: The SI system, Gaussian units, natural units. Lorentz transformations. Light-cone coordinates. Four-velocity and four-momentum. Extra dimensions. Compact extra dimensions. Fundamental domain. Coordinates on a circle. Fixpoints. Making a cone by indentifying points. Square well potential with an extra circle dimension.

Thursday 2009-03-19

Electromagnetism and gravity in various dimensions. Maxwell's equations in Lorentz-Heaviside units. The electromagnetic force (the Lorentz force) in Lorentz-Heaviside units. The victor potential A. The scalar potential φ. The role of the potentials in classical and quantum mechanics, respectively. Gauge transformations. Different potentials in different patches of a manifold. Electromagnetism in 2+1 dimensions. Manifestly relativistic electromagnetism. Spheres and balls in d dimensions. Electric fields in higher dimensions. Gravitation and the Planck length. Gravitational waves. Planck time, Planck mass, Planck energy. The gravitational potential. The D dimensional analogue to Newton's constant G. The Planck length in various dimensions.

Monday 2009-03-23

Gravitational potentials. The Planck length in various dimensions. The gravitational constant and compactification. The compactification scale l_c. Relating l_c to the Planck length. Large extra dimensions (Brane World scenarios). Non-relativistic strings. A fundamental string cannot oscillate longitudinally. Tension and mass density (independent in the non-relativistic case, dependent relativistically). The wave velocity. Equation of motion for a non-relativistic string. Boundary conditions: Dirichlet and Neumann. The coordinates x^±. The Lagrangian. The Action. Variational calculus, with bulk term and boundary term. Deriving Newton's second law from a variational principle and a Lagrangian. The non-relativistic string Lagrangian. Lagrangian versus Lagrangian density.

Tuesday 2009-03-24

Non-relativistic strings. The relativistic point particle: The Lagrangian, action, canonical momenta and Hamiltonian. Reparametrisation invariance. Equations of motion. A 4-vector in Minkowski space that is a scalar on the world line. Relativistic particles with charge. Relativistic strings. The soap film. World surface and target space. The metric g_{i j}. Area element and metric determinant.

Thursday 2009-03-26

Th relativistic point particle. Spatial soap films. Induced metric (pull-back). The invariant measure (volume) from general relativity. Wedge products. Area functional for space-time surfaces. String coordinate. Any small but finite section of the string moves with velocity v < c. The Numbu–Goto action. Equations of motion. Free end condition becomes Neumann after gauge fixing. D-branes and boundary conditions. The static gauge. Tension and energy. Action in terms of transverse velocity. The endpoints of open strings.

Monday 2009-03-30

(Images not yet uploaded. My camera broke down.) String parametrisation and string motion. Static gauge. The free end condition. σ parametrisation. 7.2. ...

Tuesday 2009-03-31

Electric charge conservation. Conserved currents. Lagrangian symmetries and Noether's theorem. Current on the string world sheet. The complete momentum current.

Friday 2009-04-03

The light-cone relativistic string.

Monday 2009-04-20

Light-cone fields and particles.

Tuesday 2009-04-21

The relativistic quantum point particle.

Thursday 2009-04-23

The relativistic quantum open string.

Monday 2009-04-27

The relativistic quantum open string, continuation.

Tuesday 2009-04-28

Closed strings.

Monday 2009-05-04

Superstrings.

Tuesday 2009-05-05

Superstrings. D-branes and gauge fields.

Thursday 2009-05-07

String charge and electric charge.

Monday 2009-05-11

T-duality of closed strings

Tuesday 2009-05-12

T-duality of open strings

Monday 2009-05-18

String theory and particle physics.

Tuesday 2009-05-19

The connection between type IIA string theory and M theory.

Course book

Barton Zwiebach. A First Course in String Theory, second edition. Cambridge university press, 2009. ISBN 978-0-521-88032-9. An older edition is available as an e-book through Chalmers library.