The 26 dimensions of string theory Why string theory? What is string theory? Why 26 dimensions?

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

Quantization p(t), x(t)  Classical particle:,  Quantum particle

Quantization of fields,  For a point source at origin For a constant electric field Maxwell equations: <><>  

Quantum Electro Dynamics  The self energy problem. e-e- + m0c2m0c2 E = m0c2m0c2 + = mrc2mrc2 = a

Quantum gravity G = T G <G><G>  Self energy problem

Free particles t0t0 t

Symmetries Rotations: X → Cos  )X + Sin(  Y R 1, R 2,R 3 [R i,R j ]=  ijk R k Translations: X → X + a P 1,P 2,P 3 [P i,P j ]=0 Boosts: X → X - Vt K 1, K 2,K 3 [K i,K j ]=  Galilean group

Relativistic particle x y t  X  ( )

Symmetries Rotations: X → Cos  )X + Sin(  Y R 1, R 2,R 3 [R i,R j ]=  ijk R k Translations: X → X + a P 1,P 2,P 3 [P i,P j ]=0 Boosts: X →  X – Vt) K 1, K 2,K 3 [K i,K j ]=  ijk K k Poincare group Diffeomorphism: → f( ) +Diffeomorphism

Free string x y t   X(,)X(,) X(,)X(,) X(,)X(,) Free particle: Free string:

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

Quantum String X(,)X(,)   E  J  (light cone gauge)

String theory Tachyon Photon Massive particle Tachyon Graviton a † 1 |0>a † 2 |0> or a † 1 a † 1 |0> M 2 a † 2 |0> = 1 |0> M 2 |0> = -1 b † 1 |0> M 2 b † 1 |0> = 0 S 2 b † 1 |0> = 2 M 2 a † 1 |0> = 0 S 2 a † 1 |0> = 0

Quantum String (light cone gauge) [P i,P j ]=0 [J i,J j ]=i f ijk J k [J i,K j ]=if ijk K k [K i,K j ]=if ijk J k [J i,P j ]=if ijk P k [K i,P j ]=-iH  ij [J i,H]=0 [P i,H]=0 [H,H]=0 [K i,H]=-i P i

Gauge symmetries Maxwell equations Gauge symmetries (Lorentz gauge) (Coulomb gauge) String equations (Light cone gauge) X 0 +X 1 =  Gauge fixing

Anomalies Feynmans approach: t=t 0 t=t f ~ħ ? S→S’ = S Dx(t)→Dx(t)’ = Dx(t)

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

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

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