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An Outline of String Theory

An Outline of String Theory. Miao Li Institute of Theoretical Physics Beijing, China. Contents Background Elements of string theory Branes in string theory Black holes in string theory-holography-Maldacena ’ s conjecture. I. Background

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An Outline of String Theory

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  1. An Outline of String Theory Miao Li Institute of Theoretical Physics Beijing, China

  2. Contents • Background • Elements of string theory • Branes in string theory • Black holes in string theory-holography-Maldacena’s conjecture

  3. I. Background • The world viewed by a reductionist • Let’s start from where Feynman’s lecture starts • A drop of water enlarged 10^9 • times H O

  4. Feynman was able to deduce a lot of things • from a single sentence: • All forms of matter consist of atoms. • Qualitative properties of gas, liquid… • Evaporation, heat transport (to cool your • Soup, blow it) • 3. Understanding of sounds, waves…

  5. Electron, point-like Atomic structure H: 10^{-8}cm Theory: QED (including Lamb shift) Interaction strength: Nucleus 10^{-13} cm

  6. Dirac: QED explains all of chemistry and most of physics. Periodic table of elements, chemical reactions, superconductors, some of biology.

  7. Sub-atomic structure Nucleus of H=proton u=2/3 U(1), d=-1/3 U(1), in addition, colors of SU(3) u u d

  8. Neutron: Interaction strengths QED Size of H=Compton length of electron/α= d u d

  9. Strong interaction Size of proton=Compton length of quark/ So the strong interactions are truly strong, perturbative methods fail. QCD is Still unsolved

  10. Another subatomic force: weak interaction β-decay How strong (or how weak) is weak interaction? Depends on the situation. For quarks: -mass of u-quark -mass of W-boson

  11. Finally, gravity, the weakest of all four interactions -mass of proton -Planck mass (so )

  12. Summary: Strong interaction-SU(3) Yang-Mills Electromagnetic Weak interaction SU(2)XU(1) Gravity

  13. To asses the possibility of unification, let’s Take a look at 2. A brief history of amalgamation of physical theories. Movement of earthly bodies. Movement of celestial bodies. Newtonian mechanics + universal gravitation. 17th century.

  14. Mechanics Heat, thermodynamics Atomic theory, statistical mechanics of Maxwell, Boltzmann, Gibbs, 19th century. Electrodynamics Magnetism Light, X-rays, γ-rays Faraday, Maxwell, 19th century.

  15. Quantum electrodynamics Weak interaction Semi-unification, Weinberg-Salam model. The disparity between 10^{-2} and 10^{-6} is solved by symmetry breaking in gauge theory. 1960’s-1970’s (`t Hooft, Veltman, Nobel prize in 1999, total Five Nobel medals for this unification.)

  16. Although eletro-weak, strong interaction appear as different forces, they are governed by the same universal principle: Quantum mechanics or better Qantum field theory valid up to

  17. Further, there is evidence for unification of • 3 forces: • In 4 dimensions, • goes up with E • goes down with E • (b) runs as powers of E if there are large compact dimensions ( )

  18. 3. Difficulty with gravity Gravity, the first ever discovered interaction, has resisted being put into the framework of quantum field theory. So, we have a great opportunity here! Why gravity is different? There are many aspects, here is a few. (a) The mediation particle has spin 2.

  19. Thus amplitude= The next order to the Born approximation amplitude=

  20. (b) According to Einstein theory, gravity is geometry. If geometry fluctuates violently, causal structure is lost. (c) The existence of black holes. (c1) The failure of classical geometry. singularity

  21. (c2) A black hole has a finite entropy, or a state of a black hole can not be specified by what is observed outside. Hawking radiation, is quantum coherence lost? Curiously, the interaction strength at the horizon is not . The larger the BH, the weak the interaction.

  22. GR predicts the surface gravity be Curiously, Size of black hole=Compton length/ or

  23. To summarize, the present day’s accepted picture of our fundamental theory is

  24. 4. The emergence of string theory A little history Strong interaction is described by QCD, however, the dual resonance model was invented to describe strong interaction first, and eventually became a candidate of theory of quantum gravity. Initially, there appeared infinitely many resonant states ( π,ρ,ω…)

  25. None of the resonant states appears more fundamental than others. In calculating an amplitude, we need to sum up all intermediate states: ππππ = Σ n π πππ Denote this amplitude by A(s,t) : (a)

  26. (b) Analytically extend A(s,t) to the complex plane of s, t, we must have Namely Σn = Σ n This is the famous s-t channel duality.

  27. A simple formula satisfying (a) and (b) is the famous Veneziano amplitude polynomial in t: Σ t^J, J-spin of the intermediate state linear trajectory

  28. This remarkable formula leads us to String theory For simplicity, consider open strings (to which Veneziano amplitude corresponds) Ground state v=c v=c An excited state v=c v=c

  29. To calculate the spectrum of the excited states, We look at a simple situation (Neuman->Dirichlet) x σ x σ

  30. Let the tension of the string be T, according to Heisenberg uncertainty relation Now or

  31. If , then Casimir effect The above derivation ignores factors such as 2’s, π’s. More generally, there can be We discovered the linear trajectory.

  32. Morals: • There are infinitely many massive states resulting from a single string (Q.M. is essential) • If we have only “bosonic strings”, no internal colors, we can have only integral spins. • spin 1: gauge bosons • spin 2: graviton • To have a massless gauge boson, a=-1. To have a • massless graviton, a=-2 (need to use closed strings).

  33. II. Elements of string theory • First quantized strings, Feynman rules • Particle analogue • Action

  34. A classical particle travels along the shortest path, while a quantum particle can travel along different paths simultaneously, so we would like to compute

  35. Generalization to a string T tension of the string dS Minkowski area element dS

  36. Curiously, string can propagate consistently only when the dimension of spacetime is D=26 Why is it so? We have the string spectrum

  37. Each physical boson on the world sheet contributes to the Casimir energy an amount a=-1/24. When n=1, we obtain a spin vector field with # of degrees D-2 For A tachyon! This breaks Lorentz invariance, so only for D=26, Lorentz invariance is maintained.

  38. But there is a tachyon at n=0, bosonic string theory is unstable. Unstable mode if E is complex For a closed string (There are two sets of D-2 modes, left moving and right moving: )

  39. For n=2, we have a spin 2 particle, there are however only ½ D(D-3) such states, it ought to be massless to respect Lorentz invariance, again D=26. Interactions In case of particles, use Feynman diagram to describe physical process perturbatively: ++ + …

  40. Associated to each type of vertex more legs there is a coupling constant The only constraint on these couplings is renormalizability. Associated with each propagator =

  41. Or By analogy, for string interaction + +… The remarkable fact is that for each topology there is only one diagram.

  42. While for particles, this is not the case, for example = + + + +…

  43. Surely, this is the origin of s-t channel duality. One can trace this back to the fact that there is unique string interaction vertex: = Rejoining or splitting

  44. The contribution of a given diagram is n=# of vertices = genus of the world sheet. In case of the closed strings +

  45. Again, there is a unique diagram for each topology, the vertex is also unique = The open string theory must contain closed Strings =

  46. The intermediate state is a closed string, unitarity requires closed strings be in the spectrum. There is a simple relation between the open string and the closed string couplings. Emission vertex=

  47. Now Emission vertex= Thus,

  48. 2.Gauge interaction and gravitation = massless open strings = massless closed strings Define the string scale

  49. Yang-Mills coupling = by dimensional analysis. Gravitational coupling

  50. So If there is a compact space D=4+d =volume of the compactspace We have

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