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The 26 dimensions of string theory

The 26 dimensions of string theory. Why string theory? What is string theory? Why 26 dimensions?. General relativity. Special relativity. Newtonian mechanics. Statistical mechanics. Particle physics. Non linear dynamics. Solid state physics. Biology. A theory of everything.

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The 26 dimensions of string theory

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  1. The 26 dimensions of string theory • Why string theory? • What is string theory? • Why 26 dimensions?

  2. General relativity Special relativity Newtonian mechanics Statistical mechanics Particle physics Non linear dynamics Solid state physics Biology A theory of everything Quantum Gravity ? Quantum mechanics

  3. Quantum particle Quantization Classical particle:  p(t), x(t)  <P(t)>, <X(t)>

  4. <j(X,t)> For a point source at origin For a constant electric field Quantization of fields  <P(t)>, <X(t)> Maxwell equations: <r> j

  5. m0c2 <j(X,t)> + E = e- a m0c2 = + mrc2 = Quantum Electro Dynamics The self energy problem.

  6. <G> Quantum gravity G = T G <T> Self energy problem

  7. Free particles t0 t

  8. Symmetries Galilean group Boosts: Rotations: X → X - Vt X → Cos(q)X + Sin(q)Y K1,K2,K3 [Ki,Kj]=0 R1,R2,R3 [Ri,Rj]=eijkRk Translations: X → X + a [Pi,Pj]=0 P1,P2,P3

  9. Relativistic particle l0 l x Xm(l) y t

  10. Symmetries Poincare group +Diffeomorphism Boosts: Rotations: X → g(X – Vt) X → Cos(q)X + Sin(q)Y K1,K2,K3 [Ki,Kj]= eijkKk R1,R2,R3 [Ri,Rj]=eijkRk Diffeomorphism: Translations: l → f(l) X → X + a [Pi,Pj]=0 P1,P2,P3

  11. t s Free string Free string: Free particle: Xm(t,s0) Xm(t0,s) Xm(t0,s0) t y x

  12. Symmetries Poincare group +Diffeomorphism +Weyl Rotations Translations Boosts Diffeomorphism Weyl

  13. Quantum String (light cone gauge)  Xm(s,t) <Xm(s,t)>   <E> E  J <J>

  14. M2 a†1|0> = 0 S2 a†1|0> = 0 String theory Tachyon Tachyon Photon Graviton Massive particle a†2|0> or a†1a†1|0> M2 a†2|0> = 1 a†1|0> |0> M2 |0> = -1 b†1|0> M2 b†1|0> = 0 S2 b†1|0> = 2

  15. Quantum String (light cone gauge) [Pi,Pj]=0  [Ji,H]=0  [Ki,Kj]=ifijkJk  [Pi,H]=0 [Ki,Pj]=-iHdij [Ji,Jj]=ifijkJk  [Ji,Kj]=ifijkKk [Ji,Pj]=ifijkPk  [Ki,H]=-iPi [H,H]=0 

  16. Gauge symmetries Maxwell equations Gauge symmetries Gauge fixing (Lorentz gauge) (Coulomb gauge) String equations Gauge fixing X0+X1=t (Light cone gauge)

  17. S→S’ = S Dx(t)→Dx(t)’ = Dx(t) Anomalies Feynmans approach: ~ħ t=tf t=t0 ?

  18. String theory • String theory is a consistent theory of gravitons. • The dimension of space time is fixed by a quantum anomaly. D=26

  19. Possibilities • String theory is incorrect. • 26 (10) dimensions • ???

  20. Fin

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