Some thoughts inspired by graphene

In viXra log there has been some discussion inspired by Phil's posting about the Nobel prize of physics received by Andre Geim and Konstantin Novoselov for discovering graphene. The discussion had the effect that I clicked "graphene" in Wikipedia to refresh my mental images about graphene.

By looking at Wikipedia article, one realizes that graphene is an extremely interesting from the perspective of theoretical physicist willing to challenge the reductionistic belief that everything above weak length scale is perfectly understood by recent day physics (for a really extreme position bringing in mind the days before quantum mechanics see the article of Sean Carroll and a reaction to it by Johannes Koelman).

Addition: I have made some corrections to the text below afer listening the excellent lecture straightenint out some mis-understandings due to the rather informal style of the Wikipedia article.

Quantum Hall effect and graphene

From Wikipedia one learns that quantum Hall effect (QHE) in graphene corresponds to the multiples N= 4× (2r+1)/2 of minimal transversal conductivity σ_xy. This could be understood as integer quantum Hall effect (QHE) allowing only even integers. Why even integers? This one should understand. This is possible. I learned from a nice lecture about graphene by Eva Andrei here that the formula for N is well-understood. The overall factor g=4 corresponds to the degeneracy of edge states and 1/2 in half odd integer comes from the effective masslessness of electrons at the lowest Landau level meaning that only second chirality for a given momentum is possible. This is so called γ₅ anomaly having analog in particle physics. From the lecture one learns that also FQHE has been observed by Eva Andrei and her group for n=1/3 and there are excellent reasons to expect that it will be found also for other values of n. Also the prospects for graphene super-conductivity are excellent. Therefore the following TGD based explanation of FQHE in terms of quantization of Planck constant is well motivated.

I have considered several variants for the quantization of Planck constant in TGD framework.

The first option postulates the quantization of Planck constant as a first principle and in this case the spectrum of Planck constants would be given by rational numbers: hbar= q× hbar₀ in the most general case but their are arguments favoring rationals for which the quantum phases exp(iq2π) are algebraically simple, say those representable in terms of square root operation alone (rules and compass integers as denominators of q). For q= 1/2 so that Planck constant would be hbar₀/2, one would obtain even integer QHE but this explanation is not needed by the above facts from Eva Andrei's lecture.

There is a slight indication for fractional quantization of Planck constant from hydrino atom of Mills for which the energy levels of hydrogen are claimed to be scaled up by a square of integer. Since energies are proportional to 1/hbar² this would follow from rational quantization of hbar. One can however explain the anomaly also by replacing the Laguerre equation for radial parts of the solutions of Schrödinger equation for hydrogen atom with is q-counterpart. Therefore there is no pressing need to assume fractional values of hbar.

This makes me happy since I have a competing argument reducing the quantization of Planck constant to the basic TGD without introducing it as a separate postulate. This option is of course the one which is more attractive since minimalism is an excellent guideline for a theoretician. This option is highly attractive also from the point of view of biology since integer valuedness means that it is possible to understand evolution in terms of drifting in the space of Planck constants to ever larger Planck constants. This is like difficusion in half-space. For rational values one would have analogy with diffusion along real axis to the directions of both small and large Planck constants and no direction of evolution.

For this option the hierarchy of Planck constants gives a straightforward explanation for FQHE since integer multiple hbar=n× hbar₀ implies that the transversal conductivity σ_xy proportional to alpha proportional to 1/hbar is proportional to 1/n and thus fractionized as multiples of 1/n.

The argument giving quantization of Planck constant as integer multiples of ordinary Planck constant goes as follows.

1. Kähler action is extremely nonlinear and possesses enormous vacuum degeneracy since any space-time surface with CP₂ projection which is Lagrange sub-manifold (maximum dimension 2) is vacuum extremal (Kähler gauge potential is pure gauge).

   The U(1) gauge symmetry realized as symplectic transformations of CP₂ is not gauge symmetry but spin glass degeneracy and not present for non-vacuum extremals. TGD Universe would be 4-D spin glass and thus possess extremely rich structure of ground states. The failure of classical non-determinism for vacuum solutions would make possible to generalized quantum classical correspondence so that one would have space-time correlates also for quantum jump sequences and thus symbolic representations at space-time level for contents of consciousness (quantum jumps as moment of consciousness). Preferred extremal property guarantees both holography and generalized Bohr orbit property for space-time surfaces.

2. As a consequence, the correspondence between canonical momentum densities and time derivatives of the imbedding space coordinates is 1-to -many: 1-to-infinite for vacuum extremals. This spoils all hopes about canonical quantization and path integral approach and led within 6 years or so to the realization that quantum physics as geometry of the world of classical worlds vision generalizing Einstein's geometrization program is the only way out of the situation. Much later -during last summer- I realized that this 1-to-many correspondence could allow to understand the quantization of Planck constant as a consequence of quantum TGD rather than as independent postulate.

3. Different roots for the values of time derivatives in the extremely non-linear formulas for canonical momentum densities correspond to same values of canonical momentum densities and therefore also conserved currents and of Kähler action if the weak form of electric-magnetic duality is accepted reducing Kähler action to Chern-Simons term. It is convenient to introduce n-sheeted covering of imbedding space as a convenient tool to describe the situation. hbar= n× hbar₀ is the effective value of Planck constant at the sheets of covering.

4. Fractionization means simply division of Kähler action and various conserved charges between the n sheets. In this manner the amount of charge at given sheet is reduced by a factor 1/n and perturbation theory applies. One could say that the space-time sheet is unstable against this kind of splitting and in zero energy ontology the space-time sheets split at the boundaries of the causal diamond (intersection of future and past directed light-cones) to n sheets of the covering. One particular consequence is fractional quantum Hall effect. A very pleasant news for theoretician is that Mother Nature loves her theoreticing children and takes care that perturbative approach works!

Kähler Dirac equation and graphene: a useful mis-understanding

As I looked at Wikipedia article, I found that Dirac equation is applied by treating electron as a massless particle and by replacing light velocity with Fermi velocity. I must say, that I find it very difficult to believe that this description could be deduced from first principles. This skeptic thought led to the realization that here might be the natural physical interpretation of formally massless Kähler Dirac equation in space-time interior.

Addition: Here again Eva's lecture clarified a lot. The spinors in question are not genuine Dirac spinors. There are two sub-lattices in graphene such that the wave functions of electron are localized to either of them. This is conveniently described in terms of spinors: the value of spin corresponds to a localization to either sub-lattice. Condensed matter physics uses rather informal Wikipedia terminology! "Schrödinger spinor" mentioned in the lecture would help enormously the random Wikipedia visitor. To avoid possible confusions let us stress that the linear dispersion relation has absolutely nothing to do with the dispersion relation of real electron in relativistic theory and and reflects only the dependence of electrons non-relativistic energy on momentum. Also spin is only a formal concept in this context.

This irritatingly informal use of the notion of spinor caused very useful mis-understanding since it forced to ask whether these strange spinors describing effectively massless electrons could have a first principle counterpart in TGD. They do not and there is not need for this that but one ends up with a proposal for the physical interpretation of the Kähler Dirac equation for the induced spinor fields in the interior of space-time surface.

To begin with, Dirac equation appears in three forms in TGD.

1. The Dirac equation in world of classical worlds codes for the super Virasoro conditions for the super Kac-Moody and similar representations formed by the states of wormhole contacts forming the counterpart of string like objects (throats correspond to the ends of the string. This Dirac generalizes the Dirac of 8-D imbedding space by bringing in vibrational degrees of freedom. This Dirac equation should gives as its solutions zero energy states and corresponding M-matrices generalizing S-matrix and their collection defining the unitary U-matrix whose natural application appears in consciousness theory as a coder of what Penrose calls U-process.

2. There is generalized eigenvalue equation for Chern-Simons Dirac operator at light-like wormhole throats. The generalized eigenvalue is pslash. The interpretation of pseudo-momentum p has been a problem but twistor Grassmannian approach suggests strongly that it can be interpreted as the counterpart of equally mysterious region momentum appearing in momentum twistor Grassmannian approach to N=4 SYM. The pseudo-/region momentum p is quantized (this does not spoil the basics of Grasssmannian residues integral approach) and 1/pslahs defines propagator in lines of generalized Feynman diagrams. The Yangian symmetry discovered generalizes in a very straightforward manner and leads alsoto the realization that TGD could allow also a twistorial formulation in terms of product CP₃ ×CP₃ of two twistor spaces. General arguments lead to a proposal for explicit form for the solutions of field equation represented identified as holomorphic 6-surfaces in this space subject to additional partial different equations for homogenenous functions of projective twistor coordinates suggesting strongly the quantal interpretation as analogs of partial waves. Therefore quantum-classical correspondence would be realize in beatiful manner.

3. There is Kähler Dirac equation in the interior of space-time. In this equation the gamma matrices are replaced with modified gamma matrices defined by the contractions of canonical momentum currents T^&alpha_k = ∂ L/∂_α h^k with imbedding space gamma matrices γ_k. This replacement is required by internal consistency and by super-conformal symmetries.

Could Kähler Dirac equation provide a first principle justification for the light-hearted use of effective mass and the analog of Dirac equation in condensed manner physics? This would conform with the holographic philosophy. Partonic 2-surfaces with tangent space data and their light-like orbits would give hologram like representation of physics and the interior of space-time the 4-D representation of physics. Holography would have in the recent situation interpretation also as quantum classical correspondence between representations of physics in terms of quantized spinor fields at the light-like 3-surfaces on one hand and in terms of classical fields on the other hand.

The resulting dispersion relation for the square of the Kähler-Dirac operator assuming that induced like metric, Kähler field, etc. are very slowly varying contains quadratic and linear terms in momentum components plus a term corresponding to magnetic moment coupling. In general massive dispersion relation is obtained as is also clear from the fact that Kähler Dirac gamma matrices are combinations of M⁴ and CP₂ gammas so that modified Dirac mixes different M⁴ chiralities (basic signal for massivation). If one takes into account the dependence of the induced geometric quantities on space-time point dispersion relations become non-local. Let us however add again that this dispersion relation has nothing to do with the dispersion relation for Schrödinger spinors in graphene.

Does energy metric provided the gravitational dual for condensed matter systems?

The modified gamma matrices define an effective metric via their anticommutators which are quadratic in components of energy momentum tensor (canonical momentum densities). This effective metric vanishes for vacuum extremals. Note that the use of modified gamma matrices guarantees among other things internal consistency and super-conformal symmetries of the theory. The physical interpretation has remained obscure hitherto although corresponding effective metric for Chern-Simons Dirac action has now a clear physical interpretation.

If the above argument is on the right track, this effective metric should have applications in condensed matter theory. In fact, energy metric has a natural interpretation in terms of effective light velocities which depend on direction of propagation. One can diagonalize the energy metric g_e^αβ (contravariant form results from the anticommutators) and one can denote its eigenvalues by (v₀,v_i) in the case that the signature of the effective metric is (1,-1,-1,-1). The 3-vector v_i/v₀ has interpretation as components of effective light velocity in various directions as becomes clear by thinking the d'Alember equation for the energy metric. This velocity field could be interpreted as that of hydrodynamic flow. The study of the extremals of Kauml;hler action shows that if this flow is actually Beltrami flow so that the flow parameter associated with the flow lines extends to global coordinate, Kähler action reduces to a 3-D Chern-Simons action and one obtains effective topological QFT. The conserved fermion current

Ψbar&Gamma_e^αΨ

has interpretation as incompressible hydrodynamical flow.

This would give also a nice analogy with AdS/CFT correspondence allowing to describe various kinds of physical systems in terms of higher-dimensional gravitation and black holes are introduced quite routinely to describe condensed matter systems: probably also graphene has already fallen in some 10-D black hole or even many of them.

In TGD framework one would have an analogous situation but with 10-D space-time replaced with the interior of 4-D space-time and the boundary of AdS representing Minkowski space with the light-like 3-surfaces carrying matter. The effective gravitation would correspond to the "energy metric". One can associate with it curvature tensor, Ricci tensor and Einstein tensor using standard formulas and identify effective energy momentum tensor associated as Einstein tensor with effective Newton's constant appearing as constant of proportionality. Note however that the besides ordinary metric and "energy" metric one would have also the induced classical gauge fields having purely geometric interpretation and action would be Kähler action. This 4-D holography would provide a precise, dramatically simpler, and also a very concrete dual description. This cannot be said about model of graphene based on the introduction of 10-dimensional black holes, branes, and strings chosen in more or less ad hoc manner.

This raises questions. Does this give a general dual gravitational description of dissipative effects in terms of the "energy" metric and induced gauge fields? Does one obtain the counterparts of black holes? Do the general theorems of general relativity about the irreversible evolution leading to black holes generalize to describe analogous fate of condensed matter systems caused by dissipation? Can one describe non-equilibrium thermodynamics and self-organization in this manner?

One might argue that the incompressible Beltrami flow defined by the dynamics of the preferred extremals is dissipationless and viscosity must therefore vanish locally. The failure of complete non-determinism of Kähler action however means generation of entropy since the knowledge about the state decreases gradually. This in turn should have a phenomenological local description in terms of viscosity which characterizes the transfer of energy to shorter scales and eventually to radiation. The deeper description should be non-local and basically topological and might lead to quantization rules. For instance, one can imagine the quantization of the ratio η/s of the viscosity to entropy density as multiples of a basic unit defined by its lower bound (note that this would be analogous to Quantum Hall effect). For the first M-theory inspired derivation of the lower bound of η/s see this. The lower bound for η/s is satisfied in good approximation by what should have been QCD plasma but found to be something different (RHIC and the first evidence for new physics from LHC: I have discussed TGD based understanding of these anomalies in previous posting).

An encouraring sign comes from the observation that for so called massless extremals representing classically arbitrarily shaped pulses of radiation propagating without dissipation and dispersion along single direction the canonical momentum currents are light-like. The effective contravariant metric vanishes identically so that fermions cannot propate in the interior of massless extremals! This is of course the case also for vacuum extremals. Massless extremals are purely bosonic and represent bosonic radiation. Many-sheeted space-time decomposes into matter containing regions and radiation containing regions. Note that when wormhole contact (particle) is glued to a massless extremal, it is deformed so that CP₂ projection becomes 4-D guaranteing that the weak form of electric magnetic duality can be satisfied. Therefore massless extremals can be seen as asymptotic regions. Perhaps one could say that dissipation corresponds to a decoherence process creating space-time sheets consisting of matter and radiation. Those containing matter might be even seen as analogs blackholes as far as energy metric is considered.

Could warped imbeddings relate to graphene?

An interesting question is whether the reduction of light-velocity to Fermi velocity could be interpreted as an actual reduction of light-velocity at space-time surface. I have discussed this possibility for some years in some blog posting and the argument is also buried in some chapter of some of the seven books about TGD. The proposed interpretation of energy metric in terms of hydrodynamic velocities does not allow this interpretation. Rather, the velocity in question should be assigned to the ordinary radiation.

TGD allows infinite family of warped imbeddings of M⁴ to M⁴xCP₂. They are analogous to different imbeddings of flat plane to 3-D space. In real world the warped imbeddings of 2-D flat space are obtained spontaneously when you have a thin plane of metal or just a sheet of paper: it gets spontaneously warped. The resulting induced geometry is flat as long as no stretching occurs.

A very simple example of this kind of imbedding is obtained as graph of a map from M⁴ to the geodesic circle S¹ of CP₂ with angle coordinate Φ linear in M⁴ time coordinate t:

Φ= &omega× t.

What is interesting is that although their is no gravitation in the standard sense, the light velocity is in this simple situation reduced to

v =(g_tt)^1/2c= (1-R²ω²)^1/2c

in the sense that it takes time T=L/v to move from point A to B along light-like geodesic of warped space-time surface whereas along non-warped space-time surface the time would be only T= L/c. The reason is of course that the imbedding space distance travelled is longer due to the warping. One particular effect is anomalous time dilation which could be much larger than the usual special relativistic and general relativistic time dilations.

Suppose that Kähler Dirac equation and Kähler action itself can be used as a possible first principle counterpart for the phenomenological Dirac equation and Maxwell's equations in the modeling of condensed matter systems. This is kind of description might make sense for so called slow photons with very slow group velocity. These surfaces could provide a holographic description for the reduction of the light-velocity also in di-electrics caused by interactions between particles described in terms of light-like 3-surfaces.

Strongly warped space-time surfaces obtained as deformations of warped imbeddings of flat Minkowski geometries (vacuum extremals) do not seem to provide a natural model for graphene. The basic objection is that electrons are in question and this light velocity is associated with genuinely massless particles. As already proposed, one could however assign effective light-velocity also to the "energy" metric. This velocity could be assigned to electrons in condensed matter.