Developments of Q.F.T. & String Theory
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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
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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
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Incidence Relation
Non-local transformation
Light
Projective complex-line
(Null-line)
4D Minkowski-space Twistor-space
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Massless particle and Twistor
Massless particle
Free particle
Pauli-Lubanski spin-vector
Helicity
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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
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1st Hopf Map
Incidence Relation
Hopf spinor
1st Hopf Map
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2nd Hopf Map
2nd Hopf map
S.C. Zhang & J.P. Hu (2001)
2nd Hopf spinor
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Direct Relation to Twistor
Incidence Relation
Constraint
Null Twistor Helicity zero
is Hermitian (space-time is real)
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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.
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The SUSY Hopf MapC. Bartocci, U. Bruzzo, G. Landi (1987)
The SUSY Hopf map
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Supertwistor VariablesSuper Incidence Relation
Supertwistor variables
Not-complexified
Even number
: Super-Hermitian
:null-supertwistor
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Super Incidence Relation
Minkowski-superspace Supertwistor-space
Non-local super-transformation
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Supertwistor action and Quantization
wave-function for mass-less particle
should be even integer.
Supertwistor action
Twistor function
Helicity
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Relation to Lowest Landau Level
U(1) connection
LLL-limit
One-particle action
Dirac monopole
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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
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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??