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This Concept Map, created with IHMC CmapTools, has information related to: Quantum Hall effect and TGD, QUANTUM HALL EFFECT AND TGD 2. Composite model has at least following problems. a) The flux quanta must be assumed to behave like fer- mions. What gives them the fermionic statistics and may- be also fermion number? b) How both the number of flux quanta per electron and the number of electrons per flux quantum can be larger than one? c) How to understand charge fractionization. How general phenomenon the fractioniza- tion is?, QUANTUM HALL EFFECT AND TGD 1. Quantum Hall effect (QHE). a) Hall effect occurs in 2-D systems. Current flowing along voltage V in magnetic field B or- thogonal to the direction of incoming current and experiences Lorentz force. current develops a transversal component proportional to the vol- tage V. Classically the transverse conductivity is σ_xy= ne/B, n 2-D electron density. b) In integer QHE this happens at quantum level the transversal conductivity is quantized as σ_xy = νe^2h, ν an integer. This can be under- stood from standard quantum mechanics just by solving Schrödinger equation in external magne- tic field. n is is equals to n= ν eB/h guaranteeing that the number of electrons is ν-multiple of the magnetic flux using elementary flux quantum as unit. c) For fractional QHE (FQHE) the ν is however fractional. The most plausible explanation is provided by composite electron moded reducing the effect to IQHE but replacing electron with a composite of electron - anyon - with even number of magnetic flux quanta extracted from the external magnetic field and forming a bound state with electron, which now responds to the external magnetic field. d) Composite model fails to explain the fractions ν=5/2 and 7/2. The conjecture is that Cooper pair like bound stats are formed from anyons. Anyons obey fractional braid statistics made possible by 2-dimensionality of the system. One can generalize abelian braid statistics to non-abelian one so that rotating through 2π corresponds to multiplication with matrix rat- her than phase factor. It has been proposed that the exotic states obey to non-abelian braid statistics. e) The charge of the anyon is fractionized., QUANTUM HALL EFFECT AND TGD 3. e) What gives for the flux tube fermio- nic statistics? One possibility is based on the fact that a magnetic flux tube carrying Kähler magnetic flux equal to Kähler electric flux at its end is dyon with minimal magnetic charge and odd electric charge. By a well-known argu- ment dyons obey fermionic statistics. The objection is that physical fermions are obtained by adding "ur-fermions" at dyonic wormhole throats. Does this mean that they are bosons after all! f) A solution of the problem consistent with basic TGD could be that the dyo- nic flux tube assignable to elementary particle defines only a classical space- time correlate for fermion. g) An alternative explanation would be that the flux tube contains two parts located at parallel space-time sheets connected by wormhole contacts at the ends of the flux tubes so that a closed flux tube results (no genuine magnetic mono- poles in TGD). One would have two flux tube portions of this kind and statistics would be bosonic. The appearance of monopole fluxes as pairs in FQHE would conform with this picture about electron. h) Another - less attractive - possibility is that covariantly constant right handed neutrino assignable to the flux tube gi- ves rise to fermion number without con- tributing to four-momentum. The first geometric explanation would only define space-time correlate for spin 1/2. Note that this is consistent with the assumption that neutrino pairs neutralizes the weak isospin of electron. For larger h_eff the fractionization of weak isospin would be required., QUANTUM HALL EFFECT AND TGD 3. Could TGD based model provide deeper justification for the com- posite model? a) Composite model as starting point. Flux quanta now realized as magnetic flux tubes. An interesting possibility is that flux quanta correspond to mono- pole fluxes for which the transversal section of the flux tube is closed 2- surface rather than disk or annulus. b) Interesting possibility is that the underlying 2-D system corresponds to a partonic 2-surface of macroscopic size at which electrons and accompa- nying flux tubes are attached. This kind of surfaces are proposed to appe- ar even in astrophysical scales in TGD Universe and carry dark matter. This would give first a principle justification for braid statistics. c) Could the braid statistics has justifi- cation in terms of hierachy of Planck constants? The proposal is that h_eff= n×h corresponds to a formation of ef- fective n-sheeted covering of imbed- ding space. M^4 (CP_2) could be cove- red n_1 (n_2) times. This means that ordinary rotations and color rotations in Cartan algebra induce only a phase correspond to 2pi/n_1 (2pi/n_2). This would bring in various kinds of fractio- nizations. d) A de-localization of em charge to n sheets implies 1/n fractionization. e) Non-abelian braid statistics is pos- sible only for nɰ and n=4would be minimal value of n.
