Non-compact Hopf Maps, Quantum Hall Effect, and Twistor Theory Graduate School of Mathematics, Nagoya University, 23 Jul. 2009 Kazuki Hasebe Takuma N.C.T. arXiv: 0902.2523, 0905.2792 Takuma N.C.T.
Introduction 1. Twistor Theory (Mathematical Physics: Relativistic Quantum Mechanics) Quantization of Space-Time Light has special importance. R. Penrose (1967) ADHM Construction, Integrable Models. Twistor String etc. 2. Quatum Hall Effect (Condensed matter: Non-relativistic Quantum Mechanics) Novel Quantum State of Matter Monopole plays an important role. R. Laughlin (1983) Quatum Spin Hall Effect, Quantum Hall Effect in Graphene etc. There are remarkable close relations between these two independently developed fields !
Landau levels LLL projection ``massless limit’’ Landau Quantization Magnetic Field 2D - plane Cyclotron frequency Lev Landau (1930) 2nd LL 1st LL LLL
Quantum Hall Effect and Monopole ``Edge’’ breaks translational sym. Stereographic projection F.D.M. Haldane (1983) SO(3) global symmetry Many-body state on a sphere in a monopole b.g.d.
spin momentum If Massless particle ``sees’’ a charge monopole in p-space ! Heuristic Observation: Why Light & Monopole Massless particle with helicity The position of a massless particle with definite helicity is uncertain ! To satisfy the SU(2) algebra Bacry (1981) Atre, Balachandran, Govindarajan (1986)
Space-Time Twistor Space ``moduli space of light’’ Twistor Program Roger Penrose (1967) Quantization of Space-Time What is the fundamental variables ? Light (massless-paticle) will play the role !
Free particle Massless particle : Massless Free Particle Gauge symmetry
Twistor Description Massless limit Fundamental variable Suggests a fuzzy space-time. Incidence Relation : Helicity:
Dirac Monopole and 1st Hopf Map The 1st Hopf map Dirac Monopole P.A.M. Dirac (1931)
Explicit Realization of 1st Hopf Map Hopf spinor Connection of bundle
LLL Fundamental variable One-particle Mechanics Lagrangian Constraint LLL Lagrangian Constraint
Emergence of Fuzzy Geometry Holomorphic wavefunctions LLL Physics
Many-body Groudstate Laughlin-Haldane wavefunction The groundstate is invariant under SU(2) isometry of , and does not include complex conjugations. On the QH groundstate, particles are distributed uniformly on the basemanifold. : SU(2) singlet combination of Hopf spinors
Topological maps from sphere to sphere with different dimensions. Heinz Hopf(1931,1935) 1st (Complex number) 2nd (Quaternion) 3rd (Octonion) Higher D. Hopf Maps
C 1st Hopf map Unit C 2nd Hopf map Unit Quaternion Quaternion and 2nd Hopf Map Willian R. Hamilton(1843)
The 2nd Hop Map & SU(2) Monopole The 2nd Hopf map Yang Monopole SO(5) global symmetry C.N. Yang (1978)
In the LLL 4D QHE and Twistor Many-body problem on a four-sphere in a SU(2) monopole b.g.d. S.C. Zhang, J.P. Hu (2001) Point out relations to Twistor theory D. Karabali,V.P. Nair(2002,2003) S.C. Zhang (2002) In particular, Sparling and his coworkers suggested the use of the ultra-hyperboloid G. Sparling (2002) D. Mihai, G. Sparling, P. Tillman(2004)
QHE Hopf Map Division algebra 1st complex numbers 2D quaternions 4D 2nd LLL Twistor ?? octonions 8D 3rd Short Summary
Complex number Quaternions Octonions Noncompact Version of the Hopf Map Hopf maps Non-compact Hopf maps ! Non-compact groups James Cockle (1848,49) Split-Complex number Split-Quaternions Split-Octonions
Split-Complex number Split-Quaternions Split-Algebras
p q+1 : 1st 2nd 3rd Non-compact Hopf Maps Ultra-Hyperboloid with signature (p,q)
SO(3,2) Hopf spinor SO(3,2) Hopf spinor generators Incidence Relation Stereographic coordinates
One-particle Mechanics on Hyperboloid SU(1,1) monopole One-particle action (c.f.) constraint SO(3,2) symmetry
LLL projection LLL-limit Fundamental variable Symmetry is Enhanced from SO(3,2) to SU(2,2)! constraint SU(2,2) symmetry
Realization of the fuzzy geometry Then, the hyperboloid also becomes fuzzy. This demonstrates the philosophy of Twistor ! First, the Hopf spinor space becomes fuzzy. The space(-time) non-commutativity comes out from that of the more fundamental space.
Twistor QHE Quantize and rather than ! Analogies • Massless Condition SU(2,2) Enhanced Symmetry • More Fundamental Quantity than Space-Time • Complex conjugation = Derivative Noncommutative Geometry, Holomorphic functions
Summary Table Non-compact 4D QHE Twistor Theory Fundamental Quantity Hopf spinor Twistor Quantized value Monopole charge Helicity Base manifold Hyperboloid Minkowski space Original symmetry Poincare Special limit LLL zero-mass Enhanced symmetry Emergent manifold Noncommutative Geometry Fuzzy Hyperboloid Fuzzy Twistor space
One-particle Problem Landau problem on a ultra-hyperboloid Thermodynamic limit : fixed
Many-body Groudstate Laughlin-Haldane wavefunction The groundstate is invariant under SO(3,2) isometry of , and does not include complex conjugations. On the QH groundstate, particles are distributed uniformly on the basemanifold.
Topological Excitations • Topological excitations are generated by flux penetrations. • The flux has SU(1,1) internal structures. Membrane-like excitations !
Uniqueness Everything is uniquely determined by the geometry of the Hopf map ! (For instance) n-c. 2nd Hopf map Gauge Symmetry Global Symmetry in LLL Base manifold
Extra-Time Physics ? Base manifold 2T Gauge Symmetry • This set-up exactly corresponds to 2T physics developed by I. Bars ! • Sp(2,R) gauge symmetry is required to eliminate the negative norms. • The present model fulfills this requirement from the very beginning ! • There may be some kind of ``duality’’ ?? C.M. Hull (99)
Magic Dimensions of Space-Time ? Compact Hopf maps Non-compact maps 1st 2nd 3rd
Non-compact Hopf Maps Split-algebras Higher D. quantum liquid Membrane-like excitation Non-commutative Geometry Twistor Theory Uniqueness Magic Dimensions Extra-time physics After ALL
Deep mathematical structure exists behind the model ! Entire picture is stillMystery ! END