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 • 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

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…

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

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

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

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

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

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

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

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

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.

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

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.)

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

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 ( )

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.

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

(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

(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.

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

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

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…)

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)

(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.

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

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

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

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

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.

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).

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

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

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

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

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.

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: )

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: ++ + …

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 =

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

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

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

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

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

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=

Now Emission vertex= Thus,

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

Yang-Mills coupling = by dimensional analysis. Gravitational coupling

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

An Outline of String Theory

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