How I learned to stop worrying and love the tachyon

Transcription

1 love the tachyon Max Planck Institute for Gravitational Physics Potsdam 6-October-2008

2 Historical background Open string field theory Closed string field theory

3 Experimental Hadron physics Mesons mass spectrum obeys the Regge trajectory m 2 = α 0 + α 1 J Four-point functions have an s/t duality The Veneziano amplitude has both properties String scattering gives the Veneziano amplitude There is only one free parameter the string tension QCD turned out as a better model for hadronic physics Yet string theory predicts gravity

4 Strings are the fundamental objects (instead of point particles) Strings span a worldsheet (instead of a worldline) The string action is proportional to the worldsheet area L = 1 2πα dσ 2 h h ab g µν a X µ b X ν a, b = 0, 1 µ, ν = 0,..., 25(D 1) String equation of motion ( X µ 2 = 2 τ 2 2 σ ) X µ = 0

5 Classic relativistic massless strings Open string spectrum (Neuman boundary conditions) p A µ 1, p Aµ 2, p Closed string spectrum (periodical boundary conditions) p A (L)µ 1, A (R)ν 1, p A (L)µ 2, A (R)ν 2, p

6 Quantum relativistic massless strings Open string spectrum 0, p a µ 1 0, p a µ 2 0, p m 2 = 1 m 2 = 0 m 2 = 1 Closed string spectrum 0, p a (L)µ 1 a (R)ν 1 0, p a (L)µ 2 a (R)ν 2 0, p m 2 = 2 m 2 = 0 m 2 = 2

7 Vacuum state Vacuum energy 0 a 1 2 (D 2) n=1 D 1 n=1 µ=2 0 µ n n = D 2 24 D=26 1 Using zeta function regularization n=1 n = 1 Regge spectrum m 2 = a spin m 2 = a + 1 spin 1 a µ 1 0 (D 2) m 2 = a + 2 spin 2 a µ 1 a ν (D 2)(D 1) m 2 = a + 2 spin 1 a µ (D 2) The last two sum-up to a spin-2 representation of SO(D 1) 12

8 Current inventory Meson spectrum, string tension related to Regge slope T s Ms 2 = MQCD 2 (100MeV )2 Gravitons, string tension related to gravitational constant G 1 N = M2 Planck = g sm 2 s (10 19 GeV ) 2 Gauge bosons, string tension related to regularization scale M 2 s > M 2 GUT (1016 GeV ) 2 g s = g 2 YM dimensions No fermions A tachyon

9 Add supersymmetry to the string worldsheet Down to 10 dimension The tachyon is projected out Space-time fermions Consistent chiral models Space-time supersymmetry D-branes mutli-dimensional non-perturbative objects

10 String theory is defined perturbatively The perturbative vacuum is not stable (SFT) is a non-perturbative formulation of String theory We need an action principle for the string field instead of the worldsheet action

11 s worldline action is its proper time L = ds = dτ g µν Ẋ µẋ ν Equation of motion E 2 = p 2 + m 2 First quantization gives the Klein-Gordon equation Φ + m 2 Φ = 0 This is a classical scalar field

12 Scalar field with Higgs-like potential example L = 1 2 µφ µ Φ m2 Φ λφ4 Feynman diagrams Equation of motion Φ m 2 Φ + λφ 3 = 0 Static solutions Φ = 0 Φ = 0 or Φ = ± m λ

13 Open Bosonic Cubic The string field is a functional from the string configuration to a scalar value Ψ[X µ (σ)] In mode-expansion we get Ψ = dp T (p) 0, p + A µ (p)a µ 1 0, p +... The action is Equation of motion S = 1 2 Ψ (p2 + m 2 ) Ψ + g s 3 Ψ 3 (p 2 + m 2 )Ψ + Ψ 2 = 0

14 Open strings must end on a D-brane has a space-time filling D25-brane The tachyon signifies the instability of the D25-brane A solution to the E.O.M would correspond to annihilation of the D25-brane The potential height would be exactly the D25-brane energy E = V 25 T 25 There would be no open-string degrees of freedom around the solution

15 Closed string field theory Seems redundant open string field theory already contains closed strings at one-loop Seems necessarily because of the closed string tachyon No equivalent to

16 Subjects not covered Black hole entropy Holography AdS/CFT correspondence Super Open questions Is the tachyon related to the Higgs boson? Is the tachyon related to the inflaton field? Does string theory describe hadron physics after all? Is string theory the theory of everything?