developments of q.f.t. & string theory
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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 - PowerPoint PPT PresentationTRANSCRIPT
Developments of Q.F.T. & String Theory
Geometrical Construction of Supertwistor Theory
Jul.28 - Aug.1 2008
Kazuki Hasebe
Takuma National College of Technology
Shikoku-Seminar
arXiv:0805.2644
Introduction: Twistor ProgramRoger Penrose (1967)
From ``The Road to Reality’’
Space-time is taken to be a secondary construction from the more primitive twistor notions.
Space-Time Event Twistor Space
Incidence Relation
Incidence Relation
Non-local transformation
Light
Projective complex-line
(Null-line)
4D Minkowski-space Twistor-space
Massless particle and Twistor
Massless particle
Free particle
Pauli-Lubanski spin-vector
Helicity
Hopf Map: Template of Twistor
Topological map from sphere to sphere in different dimensions.
Heinz Hopf (1931)
1st Hopf map
2nd Hopf map
3rd Hopf map
1st Hopf Map
Incidence Relation
Hopf spinor
1st Hopf Map
2nd Hopf Map
2nd Hopf map
S.C. Zhang & J.P. Hu (2001)
2nd Hopf spinor
Direct Relation to Twistor
Incidence Relation
Constraint
Null Twistor Helicity zero
is Hermitian (space-time is real)
Idea of Supertwistor
Complex space-time is postulated.
Fermion coordinates
Complexified space-time
Super-twistor
Incidence relation
A. Ferber (1978)
Non-Hermitian
Fermion number can be even or odd integer.
The SUSY Hopf MapC. Bartocci, U. Bruzzo, G. Landi (1987)
The SUSY Hopf map
Supertwistor VariablesSuper Incidence Relation
Supertwistor variables
Not-complexified
Even number
: Super-Hermitian
:null-supertwistor
Super Incidence Relation
Minkowski-superspace Supertwistor-space
Non-local super-transformation
Supertwistor action and Quantization
wave-function for mass-less particle
should be even integer.
Supertwistor action
Twistor function
Helicity
Relation to Lowest Landau Level
U(1) connection
LLL-limit
One-particle action
Dirac monopole
Analogies between Twistor and LLL
Complex conjugation = Derivative
Twistor LLL
More Fundamental Quantity than Space-Time
Massless Condition
Noncommutative Geometry
Holomorphicity, Incidence Relations
Enhanced Symmetry
Conclusion Geometrical construction of the supertwistor based on the SUSY Hopf map.
Properties of this construction
Close Analogy between LLL physics and Twistor
1. Space-time is not complexified.
2. Even number of fermionic components of twistor is automatically incorporated.
Does it suggest something deeper??