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Developments of Q.F.T. & String Theory

Geometrical Construction of Supertwistor Theory. Shikoku-Seminar. Developments of Q.F.T. & String Theory. Jul.28 - Aug.1 2008. Kazuki Hasebe. Takuma National College of Technology. arXiv:0805.2644. Space-time is taken to be a secondary

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Developments of Q.F.T. & String Theory

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  1. Geometrical Construction of Supertwistor Theory Shikoku-Seminar Developments of Q.F.T. & String Theory Jul.28 - Aug.1 2008 Kazuki Hasebe Takuma National College of Technology arXiv:0805.2644

  2. Space-time is taken to be a secondary construction from the more primitive twistor notions. Introduction: Twistor Program Roger Penrose (1967) From ``The Road to Reality’’ Space-Time Event Twistor Space Incidence Relation

  3. 4D Minkowski-space Twistor-space Incidence Relation Light (Null-line) Projective complex-line Non-local transformation

  4. Free particle Massless particle Helicity Massless particle and Twistor Pauli-Lubanski spin-vector

  5. 1st Hopf map 2nd Hopf map 3rd Hopf map Hopf Map: Template of Twistor Heinz Hopf (1931) Topological map from sphere to sphere in different dimensions.

  6. 1st Hopf Map Hopf spinor Incidence Relation 1st Hopf Map

  7. 2nd Hopf spinor 2nd Hopf Map 2nd Hopf map S.C. Zhang & J.P. Hu (2001)

  8. Incidence Relation Constraint is Hermitian (space-time is real) Direct Relation to Twistor Null Twistor Helicity zero

  9. Incidence relation • Fermion number can be even or odd integer. Idea of Supertwistor A. Ferber (1978) Complexified space-time Fermion coordinates Non-Hermitian Super-twistor • Complex space-time is postulated.

  10. The SUSY Hopf map The SUSY Hopf Map C. Bartocci, U. Bruzzo, G. Landi (1987)

  11. Super Incidence Relation Supertwistor variables Supertwistor Variables Not-complexified : Super-Hermitian Even number :null-supertwistor

  12. Minkowski-superspace Supertwistor-space Super Incidence Relation Non-local super-transformation

  13. Supertwistor action Helicity Twistor function should be even integer. Supertwistor action and Quantization wave-function for mass-less particle

  14. Relation to Lowest Landau Level Dirac monopole U(1) connection One-particle action LLL-limit

  15. Twistor LLL Analogies between Twistor and LLL • Massless Condition Enhanced Symmetry • More Fundamental Quantity than Space-Time Holomorphicity, Incidence Relations • Complex conjugation = Derivative Noncommutative Geometry

  16. Conclusion • Geometrical construction of the supertwistor based on the SUSY Hopf map. • Properties of this construction 1. Space-time is not complexified. 2. Even number of fermionic components of twistor is automatically incorporated. • Close Analogy between LLL physics and Twistor Does it suggest something deeper??

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