Post on 20-Dec-2015
Takuma N.C.T.
Graduate School of Mathematics, Nagoya University, 23 Jul. 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. R. Laughlin (1983)
R. Penrose (1967)
Landau Quantization
2D - plane
Magnetic Field
Landau levels
LLL
1st LL
2nd LL
LLL projection ``massless limit’’
Cyclotron frequency Lev Landau (1930)
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) global symmetry
``Edge’’ breaks translational sym.
Heuristic Observation: Why Light & MonopoleMassless particle with helicity
To satisfy the SU(2) algebra
Massless particle ``sees’’ a charge monopole in p-space !
spin
momentum
The position of a massless particle with definite helicityis uncertain !
Bacry (1981)
Atre, Balachandran,Govindarajan (1986)
If
Brief Introduction to Twistor
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’’
Massless Free Particle
Massless particle
Free particle
:
Gauge symmetry
Twistor Description
Suggests a fuzzy space-time.
Massless limit
Fundamental variable
Helicity:
: Incidence Relation
Hopf Maps and QHE
Dirac Monopole and 1st Hopf Map
The 1st Hopf map
P.A.M. Dirac (1931)
Dirac Monopole
Connection of bundle
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
Many-body Groudstate
Laughlin-Haldane wavefunction
On the QH groundstate, particles are distributed uniformly on the basemanifold.
The groundstate is invariant under SU(2) isometry of , and does not include complex conjugations.
: SU(2) singlet combination of Hopf spinors
Higher D. Hopf Maps
Topological maps from sphere to sphere with different dimensions.
Heinz Hopf (1931,1935)
1st
2nd
3rd
(Complex number)
(Quaternion)
(Octonion)
Quaternion and 2nd Hopf Map
Willian R. Hamilton (1843)
Quaternion
2nd Hopf map
Unit
1st Hopf map
Unit
C
C
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 Division algebra
2D
4D
8D
1st
2nd
3rd
complex numbers
quaternions
octonions
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)
Split-Algebras Split-Complex number
Split-Quaternions
Non-compact Hopf Maps
1st
2nd
3rd
Ultra-Hyperboloid with signature (p,q)
:
p q+1
Non-compact 2nd Hopf Map
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
(c.f.)
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 out from that of the more fundamental space.
First, the Hopf spinor space becomes fuzzy.
This demonstrates the philosophy of Twistor !
Then, the hyperboloid also becomes fuzzy.
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 !
Summary 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 Problem
Landau problem on a ultra-hyperboloid
: fixedThermodynamic limit
Many-body Groudstate
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.
Perspectives
Uniqueness
Everything is uniquely determined by the geometry of the Hopf map !
Base manifold
Gauge Symmetry
Global Symmetry in LLL
(For instance)n-c. 2nd Hopf map
Extra-Time Physics ?
Sp(2,R) gauge symmetry is required to eliminate the negative norms.
Base manifold
Gauge Symmetry
2T
The present model fulfills this requirement from the very beginning ! There may be some kind of ``duality’’ ??
This set-up exactly corresponds to 2T physics developed by I. Bars !
C.M. Hull (99)
Magic Dimensions of Space-Time ? Compact Hopf maps Non-compact maps
1st
2nd
3rd
After ALL
Split-algebras
Higher D. quantum liquid
Membrane-like excitation
Non-compact Hopf Maps
Non-commutative Geometry Twistor Theory
Uniqueness
Extra-time physics Magic Dimensions
END
Deep mathematical structure exists behind the model !
Entire picture is still Mystery !