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Published byHaley Rooke Modified over 3 years ago

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

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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 ?

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

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

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

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Quantum gravity G = T G

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Free particles t0t0 t

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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

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Relativistic particle x y t X ( )

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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

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Free string x y t X(,)X(,) X(,)X(,) X(,)X(,) Free particle: Free string:

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

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Quantum String X(,)X(,) E J (light cone gauge)

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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

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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

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

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

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

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

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Fin

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