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Non-compact Hopf Maps, Quantum Hall Effect, and Twistor Theory. Supersymmetries and Quantum Symmetries, 29 Jul. ~ 4.Aug. 2009. Kazuki Hasebe. Takuma N.C.T. Takuma N.C.T. arXiv: 0902.2523, 0905.2792. Introduction. 1. Twistor Theory. (Mathematical Physics: - PowerPoint PPT Presentation
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Takuma N.C.T.
Supersymmetries and Quantum Symmetries, 29 Jul. ~ 4.Aug. 2009
Non-compact Hopf Maps,
Quantum Hall Effect,
and Twistor Theory
Takuma N.C.T.
Kazuki Hasebe
arXiv: 0902.2523, 0905.2792
Introduction
There are remarkable close relations between these two independently developed fields !
2. Quatum Hall Effect
Novel Quantum State of Matter
(Condensed matter: Non-relativistic Quantum Mechanics)
Quatum Spin Hall Effect, Quantum Hall Effect in Graphene etc.
1. Twistor Theory
Quantization of Space-Time
(Mathematical Physics: Relativistic Quantum Mechanics)
ADHM Construction, Integrable Models. Twistor String etc.
Light has special importance.
Monopole plays an important role.
Brief Introduction of Twistors
Twistor ProgramRoger Penrose (1967)
Quantization of Space-Time What is the fundamental variables ?
Light (massless-paticle) will play the role !
Space-Time Twistor Space``moduli space of light’’
Quantized space–time will be induced.
Quantize not .Philosophy
Massless Free Particle
Massless particle
Free particle
Gauge symmetry
:
: Incidence Relation
Twistor Description
Fuzzy twistor space
Massless limit
Fundamental variable
Helicity:
Hopf Maps and QHE
Landau Quantization
2D - plane
Magnetic Field
Landau levels
LLL
1st LL
2nd LL
LLL projection ``massless limit’’
Cyclotron frequency
Lev Landau (1930)
To keep finite,is kept finite.
Quantum Hall Effect and Monopole
Stereographic projection
F.D.M. Haldane (1983)
Many-body state on a sphere in a monopole b.g.d. SO(3) symmetry
R. Laughlin (1983)
Dirac Monopole and 1st Hopf Map
The 1st Hopf map
P.A.M. Dirac (1931)
Dirac Monopole
Connection of fibre
Explicit Realization of 1st Hopf Map
Hopf spinor
One-particle Mechanics
LLL Lagrangian
Constraint
Constraint
Lagrangian
LLL
Fundamental variable
LLL PhysicsEmergence of Fuzzy Geometry
Holomorphic wavefunctions
No
Fuzzy Sphere
Many-body state
Laughlin-Haldane wavefunction
The groundstate is invariant under SU(2) isometry of , and does not include complex conjugations.
: SU(2) singlet combination of Hopf spinors
QHE with Higher Symmetry
Hopf Maps Topological maps from sphere to sphere with different dimensions.
Heinz Hopf (1931,1935)
1st
2nd
3rd
(Complex number)
(Quaternion)
(Octonion)
ONLY THREE !
The 2nd Hop Map & SU(2) Monopole
C.N. Yang (1978)
Yang MonopoleThe 2nd Hopf map
SO(5) global symmetry
4D QHE and Twistor
D. Mihai, G. Sparling, P. Tillman (2004)
S.C. Zhang, J.P. Hu (20
01)
Many-body problem on a four-sphere in a SU(2) monopole b.g.d.
In the LLL
Point out relations to Twistor theory
In particular, Sparling and his coworkers suggested the use of the ultra-hyperboloid
G. Sparling (2002)
D. Karabali, V.P. Nair (2002,2003) S.C. Zhang (2002)
Short Summary
QHE Hopf Map Monopole
2D
4D
8D
1st
2nd
3rd
U(1)
SU(2)
SO(8)
LLLTwistor ??
QHE with SU(2,2) symmetry
Noncompact Version of the Hopf Map
Hopf maps
Non-compact groups
Non-compact Hopf maps !
Split-Complex number
Split-Quaternions
Split-Octonions
Complex number
Quaternions
Octonions
James Cockle (1848,49)
Non-compact Hopf Maps: Ultra-Hyperboloid with signature (p,q)
p q+1
1st
2nd
3rd
(Split-complex number)
(Split-quaternion)
(Split-octonion) NO OTHER !
Non-compact 2nd Hopf Map
SO(3,2) gamma matrices
The fibre : (c.f.)
SO(3,2) Hopf spinor SO(3,2) Hopf spinor
Incidence Relation
generators
Stereographic coordinates
SO(3,2) symmetry
SU(1,1) monopole
One-particle action
One-particle Mechanics on Hyperboloid
constraint
LLL projection
SU(2,2) symmetry
Symmetry is Enhanced from SO(3,2) to SU(2,2)!
LLL-limit
Fundamental variable
constraint
Realization of the fuzzy geometry
The space(-time) non-commutativity comes from that of the more fundamental space.
Then, the hyperboloid also becomes fuzzy.
This demonstrates the philosophy of Twistor !
First. the Hopf spinor space becomes fuzzy.
satisfy SU(2,2) algebra.
Analogies
Complex conjugation = Derivative
Twistor QHE
More Fundamental Quantity than Space-Time
Massless Condition
Noncommutative Geometry,
SU(2,2) Enhanced Symmetry
Holomorphic functions
Quantize and rather than !
Table Non-compact 4D QHE Twistor Theory
Fundamental Quantity
Quantized value Monopole charge Helicity
Base manifold Hyperboloid Minkowski space
Original symmetry
Hopf spinor Twistor
Fuzzy Hyperboloid Fuzzy Twistor space Noncommutative Geometry
Emergent manifold
Enhanced symmetry
Special limit
Poincare
LLL zero-mass
Physics of the non-compact 4D QHE
One-particle ProblemLandau problem on a ultra-hyperboloid
: fixedThermodynamic limit
Many-body Groudstate
Higher D. Laughlin-Haldane wavefunction
On the QH groundstate, particles are distributed uniformly on the basemanifold.
The groundstate is invariant under SO(3,2) isometry of , and does not include complex conjugations.
Topological Excitations Topological excitations are generated by flux penetrations.
Membrane-like excitations !
The flux has SU(1,1) internal structures.
Particular Features
Uniqueness
Everything is uniquely determined by the geometry of the Hopf map !
Base manifold
Gauge Symmetry
Fundamental space
(For instance)n-c. 2nd Hopf map
Global symmetry :
Extra-Time Physics ?
Sp(2,R) gauge symmetry is required to eliminate the negative norms.
Base manifold
Gauge Symmetry
2T
The present model geometrically fulfills this requirement ! There may be some kind of ``duality’’ ??
This set-up exactly corresponds to 2T physics developed by I. Bars !
Hull, Khuri (98,00), Andrade, Rojas, Toppan (01)
Magic Dimensions of Space-Time ? Compact Hopf maps Non-compact maps
1st
2nd
3rd
Exotic Math and Physical Concepts
Split-algebras
Higher D. quantum liquid
Membrane-like excitation
Non-compact Hopf Maps
Non-commutative Geometry
Twistor Theory
Uniqueness
Extra-time physics Magic Dimensions
The Entire Picture is still a Mystery. END
We have seen close relations between QHE and Twistors.
Immediate Questions
Deeper reasons for the analogies?
Supertwistors and Super Landau models ? (Prof. Mezincescu’s talk) Noncompact Super Hopf Maps?
(Prof. Toppan’s talk) etc. etc.