Nobel Prize in Physics to Andre Geim and Konstantin Novoselov for Graphene

The 2010 Nobel Prize in physics has been awarded to Andre Geim and Konstantin Novoselov for the discovery of graphene. Both laureates work at Manchester University making it the second Nobel Prize this year to be awarded to work in the UK. The winners themselves are from Russia.

Graphene is a material one atom thick made of carbon atoms. Graphite which is often used in pencils or as a lubricant is actually just layers of graphene. separating the graphene and studying its properties started with the idea of using sticky tape to peel the layers off. It sounds like a simple idea but you can be sure that it was not an easy process otherwise other people would have done it first.

It turns out that graphene has extraordinary properties for conducting electricity and heat and is very string for its feeble thickness. Recently the world record for rotational speed of objects was taken by a flake of graphene that was spun using light to a million revolutions per second. Any other material would have broken apart but graphene has the potential to go even faster before it breaks.

You may be wondering why a discovery of a new type of molecular substance wins the physics prize instead of the chemistry prize. Me too. Perhaps it makes up for the fact that some recent chemistry prizes would have been better suited to the medicine Nobel.

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23 Responses to Nobel Prize in Physics to Andre Geim and Konstantin Novoselov for Graphene

Graphene is interesting stuff. The statistics of electrons becomes anyonic, since there is one too few dimensions for the spinor field. Also the mass of the electron becomes extremely small, so the motion is relativistic. Graphene is also analogous to M2-branes with quantum hall effects in relativistic settings.

Condensed matter physics produces surprise after surprise and still reductionistic colleagues continue to believe that everything above weak scale is understood in principle and even living matter is just something very complex. Fractality has been here for decades but reductionism seems to be too hard an opponent for it. There are also exceptions: I think that it was Anderson who said that there is no theory of condensed matter.

As far as I understand it is effective mass which vanishes. And the counterpart of light-velocity emerging from dispersion relations is Fermi velocity and much lower than light velocity. Fascinating in any case.

Graphene exhibits quantum Hall effect physics. This has analogues with M2-branes and dyon black holes. The F_{ab}F^{ab} in the spacetime action of an AdS spacetime with a black hole exhibits field quantization similar to the quantum Hall effect.

There are certain structures which have a universality to them. Isospin systems fit this bill, fluid dynamics (Navier Stokes equation) from electron flows to renormalization theory, and now quantum Hall effects.

There are statements flying about how graphene is harder than steel. This means it is stronger than a 2-dimensional 1-atom thickness of steel. These hexagonal structures of carbon, whether in tubes or sheets, will under sufficient sheering unzip, similar to a run in a nylon stocking. So graphene structures will be embedded in some matrix, either interacting with silicon electronics, or in a glass or transparent material.

I would have guessed on Zeilinger with his superposition of matter molecules, but this is almost as interesting.
Maybe the energy gap is an anyonic phase :) The perturbative zone.

From Wikipedia I learned that Quantum Hall effect seems to correspond to minimal fractional quantum hall effect with half odd integers (2n+1)/2. If I remember correctly, n=2 is somewhat problematic in the theory of fractional quantum Hall effect.

In TGD framework the Hierarchy of Planck constants gives a straightforward explanation for quantum Hall effect since integer multiple hbar=nhbar_0 implies that sigma_xy proportional to alpha proportional to 1/hbar is proportional to 1/n and thus fractionized. Graphene would correspond to the minimal 2-sheeted covering and to hbar= 2hbar_0 in TGD framework.

One ends up with the quantization of Planck constant from basic TGD as follows (it could be postulated also independently as I indeed did). Kaehler action is extremely nonlinear and possesses enormous vacuum degeneracy since any space-time surface with CP_2 projection which is Lagrange sub-manifold (maximum dimension 2) is vacuum extremal (Kaehler gauge potential is pure gauge).

The U(1) gauge symmetry realized as symplectic transformations of CP_2 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 correspondences 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.

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 and physics as geometry of the world of classical worlds 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.

Different roots for time derivatives in the formulas for canonical momentum densities correspond to same values of canonical momentum densities and therefore also conserved currents and also to the same value of action if the weak form of electric-magnetic duality is accepted. Therefore it is convenient to introduce n-sheeted covering of imbedding space as a convenient tool to describe the situation. hbar is the effective value of Planck constant at the sheets of covering.

Fractionization means simply division of Kahler 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 theoreticians and takes care that perturbative approach works!

Still a comment about graphene. 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 could be deduced from first principles. This skeptic thought led to the realization that here might be the natural physical interpretation of formally massless Kahler Dirac equation in space-time interior.

Dirac equation appears in three forms in TGD.

a) 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 coder of what Penrose calls U-matrix.

b) 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 Yangian symmetry discovered in this approach indeed generalizes in very straightforward manner and leads also the realization that TGD allows also formulation in terms of CP_3 x p is quantized and 1/pslahs defines propagator in lines of generalized Feynman diagrams.

c) There is Kahler Dirac equation in the interior of space-time. The gamma matrices are replaced with modified gamma matrices defined by the contractions of canonical momentum currents Pi^alpha_k = partial L/partial_alpha h^k with imbedding space gammas gamma_k.

Could Kahler 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? The resulting dispersion relation for the square of the operator assuming that induced like metric, Kahler field, etc. are 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 Kahler Dirac gamma matrices are combinations of M^4 and CP_2 gammas so that modified Dirac mixes different M^4 chiralities (basic signal for massivation).

Still one comment about graphene. I have represented it in my blog for years ago and also in some of the books about TGD.

The question is whether the reduction of light-velocity to Fermi velocity could be interpreted
as actual reduction of light-velocity at space-time surface. This could be possible.

TGD allows infinite family of warped imbeddings of M^4 to M^4xCP_2. They are analogous to the 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.

An example of this kind of imbedding is obtained as graph of a map from M^4 to the geodesic circle of CP_2 with angle coordinate Phi linear in M^4 time coordinate:

Phi= omega*t.

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

v =sqrt(g_tt)c= sqrt(1-R^2omega^2)c

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.

In previous posting I talked about Kahler Dirac equation as a possible first principle counterpart for the phenomenological Dirac equation used in the modeling of graphene. Could it be that strongly warped space-time space-time surfaces obtained as deformations of warped imbeddings of flat Minkowski geometries (vacuum extremals) could provide a model for graphene?

I noticed an error in my claim for QHE in graphene. The correct formula involves factor of so that one would have even integer QHE. See
for the discussion at my blog.

The previous comment about the interpretation of the Kaehler Dirac equation
inspired a vision about how the AdS/CFT correspondence could be realized in TGD framework in 4-D context.

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 argument about Kaehler Dirac equation is on the right track, this effective metric should have applications in condensed matter theory. What comes in mind is the expression for frequency dependent viscocity expressible in terms of zero frequency Fourier component for the commutator of components of quantal energy momentum tensor at two points (see this). The energy metric is quadratic in the components of canonical momentum densities. This is vaguely analogous to a reverse Fourier transform of four-momentum dependent viscocity so that it becomes a space-time dependent quantity. This motivates the question whether the local transport coefficients characterizing condensed matter might be expressible in terms of the “energy” metric.

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 Kaehler 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. I am however afraid that the idea that space-time might be four-dimensional is regarded as too childish and old-fashioned to be taken seriously by any well-informed theoretician. Branes and landscape are much more interesting and allow really large amplitude hand waving;-).

In any case, 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?

Wow, there is a lot which you write here. I will confess that a lot of this is not entirely clear to me. The CP^2 ~ U(2)/(U(1)xU(1)) seems to have some reference to standard model. At least one could have a weak interaction model as U(2) ~ CP^2xU(1)xU(1) or an standard EW model as U(2)xU(1) = CP^2xU(1)^3 or something. Also in a general setting for U(n) this might be a cohomology or classifying space for general bundles.

A lot of this with hierarchies of ħ constants and the like I have trouble understanding.

I thought I would indicate something about the structure of this type of theory and its connection with 2-dimensional systems like graphene. Much of this stems from well understood physics.

The k = -1 curvature manifold in two dimensions, the Poincare disk, half-plane or hypersphere, describes by the Gauss-Codazzi a wave motion governed by the sine-Gordon equation

∂_{tt}φ – ∂_{xx}φ = sin(φ)

which is a fascinating equation. This is usually written as

∂_{uv}φ = sin(φ),

for u = (x + t)/2, v = (x – t)/2. This describes the motion of a particle with the line element

ds^2 = du^2 + dv^2 + 2cos(φ)dudv

which is the AdS_2 spacetime for the hyperbolic replacement cos(φ) – -> cosh(φ) with the sinh-Gordon equation ∂_{uv}φ = sinh(φ).

Exact quantum scattering matrix for this sine-Gordon equation was discovered by Alexander Zamolodchikov, which is S-dual to the Thirring model. This is a theory of fermions in two dimensions with the Lagrangian,

L = ψ-bar(γ^a∂_a – m)ψ – g(ψ-bar γ^aψ)(ψγ_aψ),

which is a fermionic theory of bosonization — similar to superconductivity. Zamolodchikov solved this theory and removed the UV divergence with the Bethe hypothesis. The solution is S-dual to the sine-Gordon equation. This points to very deep relationships with respect to gravitation. Gravitation has as its underlying quantum theory a fermionic quantum system with bosonization (a quantum critical point), where gravitation itself is not really quantized.

This fermion theory, which might be the underlying quantum theory of gravitation (gravitation might not need to be quantized directly beyond a few loop level) is the graded portion of the anyonic field on the two dimensional surface. This is described by a Chern-Simons Lagrangian, which in a more general setting of the 3×3 Jordan algebra describes associators with 3 octonions or E_8 groups. This is the possible connection between graphene and these M-theoretic foundations.

The above fermionic Lagrangian has a bosonization according to Bogoliubov functions. This is a bosonization similar to superfluidity or superconductivity. The interpretation with respect to the emergence of gravity is the onset of decoherent quantum fields in curved spacetime. The emergence of spacetime might then be a phase transition, where the parameter of “disorder” is the scale of quantum fluctuations. This plays the role of temperature if the iHt/ħ is wick rotated i – -> 1 and equated to the Boltzmann term E/kT so the Euclidean time t = ħ/kT serves as the “β” term. So this means the set of quantum fluctuation of the Ferm-Dirac field have a critical point or “attractor,” similar to the condition for a Fermi surface, where there is a bosonization.

Graphene exhibits structure similar to this, and suggests a sort of universality to this sort of physics. In other settings this is also apparent with the quantum phase transition in heavy metal http://arxiv.org/abs/0904.1993 and http://arxiv.org/abs/1003.1728.

a fermionic theory of bosonization (new to me) — similar to superconductivity. But this superconductivity is at high temp. It is quite different from low temp superconductivity. I found an explanation on this high temp phenomen some days ago. Try to find the link.

“This is described by a Chern-Simons Lagrangian, which in a more general setting of the 3×3 Jordan algebra describes associators with 3 octonions or E_8 groups. This is the possible connection between graphene and these M-theoretic foundations.”
This is where Kea comes in?

BTW anyonic phase could be the reaction center, depending on bosons mainly. Matti coupled this to antimatter and dark matter if I remember right. Means a center for the light cone?

the tricky part. You need some way to turn electrons from fermions into bosons if you’re going to invoke BEC physics to explain superconductivity. the tricky part. You need some way to turn electrons from fermions into bosons if you’re going to invoke BEC physics to explain superconductivity.
But at high temp? – Whoever finally figures it out is pretty much guaranteed a Nobel Prize very shortly thereafter.

http://www.technologyreview.com/blog/arxiv/24769/ noticed that p-doped diamond has two of these characteristics but superconducts only at 4K.
However, they calculate that p-doped graphane fits the bill exactly and should superconduct in the old-fashioned BCS way at 90K.

http://www.nature.com/nphys/journal/v6/n6/full/nphys1656.html Evidence for a Lifshitz transition in electron-doped iron arsenic superconductors at the onset of superconductivity. Superconductivity and magnetism are competing states in this system: when petal-like hole pockets are present, superconductivity is fully suppressed, whereas in their absence the two states can coexist.

“This volume provides a detailed account of bosonization. This important
technique represents one of the most powerful nonperturbative approaches to many-body systems currently available.”

The AdS/CFT correspondence is an equivalency between the isometries of the AdS boundary and the conformal symmetries of a quantum field theory. This occurs in the large N limit, or the many body case with many degrees of freedom. Bosonization is a case of a phase change in a fermionic system, such as BCS Cooper pairing of electrons. So there exists two phases of a system here. One is a soliton dynamics, which for AdS_2 is a sine-Gordon wave dynamics, and a nonperturbative QFT under bosonization. There is I think a third phase which is a perturbative field theory in some broken symmetry phase. The broken symmetry phase I think is gravitation, which might for some n-loop correction exhibit a broken symmetry.

This is a part of why I think gravitation is only partially quantized, in a sense semi-classical, where for some loop order it breaks down. I think that something else underlies quantum gravity. Sakharov basically argued for something similar to this back in the late 1960s, where he argued that the fabric of spacetime had some underlying quantal system of pre-geometry or “atoms.”

Ah, Sakharov again. I have not had time to make that ready.

The third phase is anyonic, fractional angular momentum, kinematics of 3+1D, charge-flux-tubes composites. geometrical QFT in 3+1D, quasiparticles (solitons) in FQHE and quantum tunnelling, lying behind quantum biology… look in the book. Solitons are bosonic always?

I look for a way antimatter can be bosonic and anyonic, behind the characters of Life :) Antimatter is annihilated into photons (bosonic) but that do not sounds the way it should be? I don’t know.
If carbon, as a boson, could interfere with the antimatter?

“Everyone was looking at these materials as ordered and homogeneous,” said Bianconi. That is not the case — but neither, he found, was the position of oxygen atoms truly random. Instead, they assumed complex geometries, possessing a fractal form: A small part of the pattern resembles a larger part, which in turn resembles a larger part, and so on.

“Such fractals are ubiquitous elsewhere in nature,” wrote Leiden University theoretical physicist Jan Zaanen in an accompanying commentary, but “it comes as a complete surprise that crystal defects can accomplish this feat.”

Geometries, dirty fractals, remember Matti talked of a vibration a little out of phase for the soliton in the nerve pulse :) A fractal nerve pulse that is a ‘surfer dude’ :)

The connection to cuprates does indicate this is a physical analogue to high Tc physics. The paper links I give discuss some of this physics and the relationship between AdS_n physics and quantum critical phase transitions in these metals.

According to Zaanen, the closest mathematical description of superconductive behavior comes from something called “Anti de Sitter space / Conformal Field Theory correspondence,” a subset of string theory that attempts to describe the physics of black holes.

That’s a dramatic connection. But as Zaanen wrote, “This fractal defect structure is astonishing, and there is nothing in the textbooks even hinting at an explanation.”

(Link: wired.com above) Now I shall stop, so i don’t flood this blog :)

A little correction to CP_2: it is just complex projective space or Grassmannian G(1,3) (appearing in twistor approach to N=4 SYM) consisting of complex 1 planes or equivalently of dual 2 planes in C^3.

SU(3)/U(2) is second representation. The idea is to represent a given complex line by frame. This frame is not unique. You can do U(1) transformation which acts on line but does not change it as a whole. You can also act on normal space by SU(2) without affective the line. Frames are parametrize by SU(3) and you form equivalence classes by dividing with SU(2)xU(1).

The physical motivation for CP_2 is that its isometry group is SU(3) and holonomy group after constructing a respectable spinor structure by coupling spinors to Kahler gauge potential has structure of electroweak gauge group. One obtains just the correct selection rules,parity breaking, quarks and leptons correspond to different chiralities of M^4xCP_2 spinors coupling to n=1 and n=3 multiples of Kahler gauge potential. Actually Hawking and others did the construction long time ago but did not realize that one obtains electroweak structure.

Sine-Gordon appeared first time in my the model for nerve pulses as solitons of Josephson junction assumed to be defined by cell membrane. Phi corresponds to the phase differences of super-conducting order parameter over the membrane. Few years ago I modified the model. The ground state would be a propagating soliton sequence: imagine a sequence of gravitational penduli (for which Sine-Gordon is a continuum idealization) oscillating a little bit out of phase so that wave results. Nerve pulse results when you kick one of gravitational penduli so that perturbation starts to propagate.

I would not pass in exam about dualities. Thirring-Sine-Gordon: good to learn. The bridge to Thirring model to space-time requires something and I find it difficult to believe that two additional dimensions would just pop up from nowhere. I stubbornly believe that the 4-D dynamics should effectively reduce to almost 2-D dynamics giving holography in strong form. The direction for dimensions would not be 2–>4 but 4–> a little bit more than 2.

My proposal is that space-time surface are analogous to Bohr orbit: 4-D(!) general coordinate invariance indeed implies this in world of classical worlds picture where the WCW metric must assign a 4-D space-time surface to given 3-surface for 4-D coordinate transformations to act on.

A further reduction to effective 2-dimensionality comes from extremum property for Chern-Simons, which for light-like 3-surfaces is the only possibility since metric based action principles are excluded by singularity of the metric.

Weak form of electric-magnetic duality saves from complete topologization, which would be a catastrophe. This duality gives a constraint between Kahler electric and magnetic fields at 3-surface and reduces quantization of Kahler electric charge to that of Kahler magnetic charge. This constraint depends on induced metric because electric field involves metric. Very delicate.

CP^2 is under a Euclideanization i –> 1 RP^4 ~ G(3,1) which has the bundle structure O(3)/O(2)xO(1). The complex universal bundle has the structure SU(2)/U(1)xU(1).

[…] here from all around the world, but scientists like Andre Geim and Konstantin Novoselov who won the Nobel prize for work done in the UK will no longer be welcome here. Young scientists are very mobile. […]

Condensed matter physics produces surprise after surprise and still reductionistic colleagues continue to believe that everything above weak scale is understood in principle and even living matter is just something very complex. Fractality has been here for decades but reductionism seems to be too hard an opponent for it. There are also exceptions: I think that it was Anderson who said that there is no theory of condensed matter.

In his famous More is Different published in journal Science, Anderson stated why condensed matter is not just applied particle physics, and emphasized the arrogance of the particle physicists. It is just funny to read to particle physicists and string theories saying nonsense about other disciplines (e.g. thermodynamics, chemistry, biology) as if they know of what they were talking…

Congratulations to Andre Geim and Konstantin Novoselov!

Huping

Graphene is interesting stuff. The statistics of electrons becomes anyonic, since there is one too few dimensions for the spinor field. Also the mass of the electron becomes extremely small, so the motion is relativistic. Graphene is also analogous to M2-branes with quantum hall effects in relativistic settings.

Truly amazing. I wonder what this material’s conductivity is when supercooled. Imagine a transparent tablet computer that’s nearly indestructible.

Condensed matter physics produces surprise after surprise and still reductionistic colleagues continue to believe that everything above weak scale is understood in principle and even living matter is just something very complex. Fractality has been here for decades but reductionism seems to be too hard an opponent for it. There are also exceptions: I think that it was Anderson who said that there is no theory of condensed matter.

As far as I understand it is effective mass which vanishes. And the counterpart of light-velocity emerging from dispersion relations is Fermi velocity and much lower than light velocity. Fascinating in any case.

Graphene exhibits quantum Hall effect physics. This has analogues with M2-branes and dyon black holes. The F_{ab}F^{ab} in the spacetime action of an AdS spacetime with a black hole exhibits field quantization similar to the quantum Hall effect.

There are certain structures which have a universality to them. Isospin systems fit this bill, fluid dynamics (Navier Stokes equation) from electron flows to renormalization theory, and now quantum Hall effects.

There are statements flying about how graphene is harder than steel. This means it is stronger than a 2-dimensional 1-atom thickness of steel. These hexagonal structures of carbon, whether in tubes or sheets, will under sufficient sheering unzip, similar to a run in a nylon stocking. So graphene structures will be embedded in some matrix, either interacting with silicon electronics, or in a glass or transparent material.

I would have guessed on Zeilinger with his superposition of matter molecules, but this is almost as interesting.

Maybe the energy gap is an anyonic phase :) The perturbative zone.

From Wikipedia I learned that Quantum Hall effect seems to correspond to minimal fractional quantum hall effect with half odd integers (2n+1)/2. If I remember correctly, n=2 is somewhat problematic in the theory of fractional quantum Hall effect.

In TGD framework the Hierarchy of Planck constants gives a straightforward explanation for quantum Hall effect since integer multiple hbar=nhbar_0 implies that sigma_xy proportional to alpha proportional to 1/hbar is proportional to 1/n and thus fractionized. Graphene would correspond to the minimal 2-sheeted covering and to hbar= 2hbar_0 in TGD framework.

One ends up with the quantization of Planck constant from basic TGD as follows (it could be postulated also independently as I indeed did). Kaehler action is extremely nonlinear and possesses enormous vacuum degeneracy since any space-time surface with CP_2 projection which is Lagrange sub-manifold (maximum dimension 2) is vacuum extremal (Kaehler gauge potential is pure gauge).

The U(1) gauge symmetry realized as symplectic transformations of CP_2 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 correspondences 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.

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 and physics as geometry of the world of classical worlds 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.

Different roots for time derivatives in the formulas for canonical momentum densities correspond to same values of canonical momentum densities and therefore also conserved currents and also to the same value of action if the weak form of electric-magnetic duality is accepted. Therefore it is convenient to introduce n-sheeted covering of imbedding space as a convenient tool to describe the situation. hbar is the effective value of Planck constant at the sheets of covering.

Fractionization means simply division of Kahler 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 theoreticians and takes care that perturbative approach works!

Still a comment about graphene. 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 could be deduced from first principles. This skeptic thought led to the realization that here might be the natural physical interpretation of formally massless Kahler Dirac equation in space-time interior.

Dirac equation appears in three forms in TGD.

a) 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 coder of what Penrose calls U-matrix.

b) 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 Yangian symmetry discovered in this approach indeed generalizes in very straightforward manner and leads also the realization that TGD allows also formulation in terms of CP_3 x p is quantized and 1/pslahs defines propagator in lines of generalized Feynman diagrams.

c) There is Kahler Dirac equation in the interior of space-time. The gamma matrices are replaced with modified gamma matrices defined by the contractions of canonical momentum currents Pi^alpha_k = partial L/partial_alpha h^k with imbedding space gammas gamma_k.

Could Kahler 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? The resulting dispersion relation for the square of the operator assuming that induced like metric, Kahler field, etc. are 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 Kahler Dirac gamma matrices are combinations of M^4 and CP_2 gammas so that modified Dirac mixes different M^4 chiralities (basic signal for massivation).

Still one comment about graphene. I have represented it in my blog for years ago and also in some of the books about TGD.

The question is whether the reduction of light-velocity to Fermi velocity could be interpreted

as actual reduction of light-velocity at space-time surface. This could be possible.

TGD allows infinite family of warped imbeddings of M^4 to M^4xCP_2. They are analogous to the 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.

An example of this kind of imbedding is obtained as graph of a map from M^4 to the geodesic circle of CP_2 with angle coordinate Phi linear in M^4 time coordinate:

Phi= omega*t.

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

v =sqrt(g_tt)c= sqrt(1-R^2omega^2)c

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.

In previous posting I talked about Kahler Dirac equation as a possible first principle counterpart for the phenomenological Dirac equation used in the modeling of graphene. Could it be that strongly warped space-time space-time surfaces obtained as deformations of warped imbeddings of flat Minkowski geometries (vacuum extremals) could provide a model for graphene?

I noticed an error in my claim for QHE in graphene. The correct formula involves factor of so that one would have even integer QHE. See

for the discussion at my blog.

The previous comment about the interpretation of the Kaehler Dirac equation

inspired a vision about how the AdS/CFT correspondence could be realized in TGD framework in 4-D context.

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 argument about Kaehler Dirac equation is on the right track, this effective metric should have applications in condensed matter theory. What comes in mind is the expression for frequency dependent viscocity expressible in terms of zero frequency Fourier component for the commutator of components of quantal energy momentum tensor at two points (see this). The energy metric is quadratic in the components of canonical momentum densities. This is vaguely analogous to a reverse Fourier transform of four-momentum dependent viscocity so that it becomes a space-time dependent quantity. This motivates the question whether the local transport coefficients characterizing condensed matter might be expressible in terms of the “energy” metric.

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 Kaehler 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. I am however afraid that the idea that space-time might be four-dimensional is regarded as too childish and old-fashioned to be taken seriously by any well-informed theoretician. Branes and landscape are much more interesting and allow really large amplitude hand waving;-).

In any case, 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?

Matti,

Wow, there is a lot which you write here. I will confess that a lot of this is not entirely clear to me. The CP^2 ~ U(2)/(U(1)xU(1)) seems to have some reference to standard model. At least one could have a weak interaction model as U(2) ~ CP^2xU(1)xU(1) or an standard EW model as U(2)xU(1) = CP^2xU(1)^3 or something. Also in a general setting for U(n) this might be a cohomology or classifying space for general bundles.

A lot of this with hierarchies of ħ constants and the like I have trouble understanding.

I thought I would indicate something about the structure of this type of theory and its connection with 2-dimensional systems like graphene. Much of this stems from well understood physics.

The k = -1 curvature manifold in two dimensions, the Poincare disk, half-plane or hypersphere, describes by the Gauss-Codazzi a wave motion governed by the sine-Gordon equation

∂_{tt}φ – ∂_{xx}φ = sin(φ)

which is a fascinating equation. This is usually written as

∂_{uv}φ = sin(φ),

for u = (x + t)/2, v = (x – t)/2. This describes the motion of a particle with the line element

ds^2 = du^2 + dv^2 + 2cos(φ)dudv

which is the AdS_2 spacetime for the hyperbolic replacement cos(φ) – -> cosh(φ) with the sinh-Gordon equation ∂_{uv}φ = sinh(φ).

Exact quantum scattering matrix for this sine-Gordon equation was discovered by Alexander Zamolodchikov, which is S-dual to the Thirring model. This is a theory of fermions in two dimensions with the Lagrangian,

L = ψ-bar(γ^a∂_a – m)ψ – g(ψ-bar γ^aψ)(ψγ_aψ),

which is a fermionic theory of bosonization — similar to superconductivity. Zamolodchikov solved this theory and removed the UV divergence with the Bethe hypothesis. The solution is S-dual to the sine-Gordon equation. This points to very deep relationships with respect to gravitation. Gravitation has as its underlying quantum theory a fermionic quantum system with bosonization (a quantum critical point), where gravitation itself is not really quantized.

This fermion theory, which might be the underlying quantum theory of gravitation (gravitation might not need to be quantized directly beyond a few loop level) is the graded portion of the anyonic field on the two dimensional surface. This is described by a Chern-Simons Lagrangian, which in a more general setting of the 3×3 Jordan algebra describes associators with 3 octonions or E_8 groups. This is the possible connection between graphene and these M-theoretic foundations.

The above fermionic Lagrangian has a bosonization according to Bogoliubov functions. This is a bosonization similar to superfluidity or superconductivity. The interpretation with respect to the emergence of gravity is the onset of decoherent quantum fields in curved spacetime. The emergence of spacetime might then be a phase transition, where the parameter of “disorder” is the scale of quantum fluctuations. This plays the role of temperature if the iHt/ħ is wick rotated i – -> 1 and equated to the Boltzmann term E/kT so the Euclidean time t = ħ/kT serves as the “β” term. So this means the set of quantum fluctuation of the Ferm-Dirac field have a critical point or “attractor,” similar to the condition for a Fermi surface, where there is a bosonization.

Graphene exhibits structure similar to this, and suggests a sort of universality to this sort of physics. In other settings this is also apparent with the quantum phase transition in heavy metal http://arxiv.org/abs/0904.1993 and http://arxiv.org/abs/1003.1728.

a fermionic theory of bosonization (new to me) — similar to superconductivity. But this superconductivity is at high temp. It is quite different from low temp superconductivity. I found an explanation on this high temp phenomen some days ago. Try to find the link.

“This is described by a Chern-Simons Lagrangian, which in a more general setting of the 3×3 Jordan algebra describes associators with 3 octonions or E_8 groups. This is the possible connection between graphene and these M-theoretic foundations.”

This is where Kea comes in?

BTW anyonic phase could be the reaction center, depending on bosons mainly. Matti coupled this to antimatter and dark matter if I remember right. Means a center for the light cone?

The links to high temp superconduction

http://www.physorg.com/news198326815.html

http://scienceblogs.com/principles/2010/08/how_do_superconductors_work.php

the tricky part. You need some way to turn electrons from fermions into bosons if you’re going to invoke BEC physics to explain superconductivity. the tricky part. You need some way to turn electrons from fermions into bosons if you’re going to invoke BEC physics to explain superconductivity.

But at high temp? – Whoever finally figures it out is pretty much guaranteed a Nobel Prize very shortly thereafter.

http://www.technologyreview.com/blog/arxiv/24769/ noticed that p-doped diamond has two of these characteristics but superconducts only at 4K.

However, they calculate that p-doped graphane fits the bill exactly and should superconduct in the old-fashioned BCS way at 90K.

http://www.nature.com/nphys/journal/v6/n9/full/nphys1759.html Iron and magnets: superconductivity in layered iron-based materials, with transition temperatures climbing as high as 55 K

http://www.nature.com/nphys/journal/v6/n6/full/nphys1656.html Evidence for a Lifshitz transition in electron-doped iron arsenic superconductors at the onset of superconductivity. Superconductivity and magnetism are competing states in this system: when petal-like hole pockets are present, superconductivity is fully suppressed, whereas in their absence the two states can coexist.

http://arxiv.org/abs/1007.2736

When I tried to get information about this I stumbled on a google book

E Wilczek, ed., Fractional Statistics and Anyon Superconductivity 1990

Looks interesting, also with thought of Keas figures. He is still a Nobelist.

Also this http://arxiv.org/PS_cache/cond-mat/pdf/9909/9909069v1.pdf

a bit old, from 1998, but the beginning looked good

“This volume provides a detailed account of bosonization. This important

technique represents one of the most powerful nonperturbative approaches to many-body systems currently available.”

The AdS/CFT correspondence is an equivalency between the isometries of the AdS boundary and the conformal symmetries of a quantum field theory. This occurs in the large N limit, or the many body case with many degrees of freedom. Bosonization is a case of a phase change in a fermionic system, such as BCS Cooper pairing of electrons. So there exists two phases of a system here. One is a soliton dynamics, which for AdS_2 is a sine-Gordon wave dynamics, and a nonperturbative QFT under bosonization. There is I think a third phase which is a perturbative field theory in some broken symmetry phase. The broken symmetry phase I think is gravitation, which might for some n-loop correction exhibit a broken symmetry.

This is a part of why I think gravitation is only partially quantized, in a sense semi-classical, where for some loop order it breaks down. I think that something else underlies quantum gravity. Sakharov basically argued for something similar to this back in the late 1960s, where he argued that the fabric of spacetime had some underlying quantal system of pre-geometry or “atoms.”

Ah, Sakharov again. I have not had time to make that ready.

The third phase is anyonic, fractional angular momentum, kinematics of 3+1D, charge-flux-tubes composites. geometrical QFT in 3+1D, quasiparticles (solitons) in FQHE and quantum tunnelling, lying behind quantum biology… look in the book. Solitons are bosonic always?

I look for a way antimatter can be bosonic and anyonic, behind the characters of Life :) Antimatter is annihilated into photons (bosonic) but that do not sounds the way it should be? I don’t know.

If carbon, as a boson, could interfere with the antimatter?

“The benefit of fractal dirt.” By Jan Zaanen. Nature, Vol. 466 No. 7308, August 12, 2010.

Bianconi. Nature, Vol. 466 No. 7308, August 12, 2010.

Read More http://www.wired.com/wiredscience/2010/08/superconductor-fractals/

“Everyone was looking at these materials as ordered and homogeneous,” said Bianconi. That is not the case — but neither, he found, was the position of oxygen atoms truly random. Instead, they assumed complex geometries, possessing a fractal form: A small part of the pattern resembles a larger part, which in turn resembles a larger part, and so on.

“Such fractals are ubiquitous elsewhere in nature,” wrote Leiden University theoretical physicist Jan Zaanen in an accompanying commentary, but “it comes as a complete surprise that crystal defects can accomplish this feat.”

Geometries, dirty fractals, remember Matti talked of a vibration a little out of phase for the soliton in the nerve pulse :) A fractal nerve pulse that is a ‘surfer dude’ :)

The connection to cuprates does indicate this is a physical analogue to high Tc physics. The paper links I give discuss some of this physics and the relationship between AdS_n physics and quantum critical phase transitions in these metals.

According to Zaanen, the closest mathematical description of superconductive behavior comes from something called “Anti de Sitter space / Conformal Field Theory correspondence,” a subset of string theory that attempts to describe the physics of black holes.

That’s a dramatic connection. But as Zaanen wrote, “This fractal defect structure is astonishing, and there is nothing in the textbooks even hinting at an explanation.”

(Link: wired.com above) Now I shall stop, so i don’t flood this blog :)

Lawrence,

A little correction to CP_2: it is just complex projective space or Grassmannian G(1,3) (appearing in twistor approach to N=4 SYM) consisting of complex 1 planes or equivalently of dual 2 planes in C^3.

SU(3)/U(2) is second representation. The idea is to represent a given complex line by frame. This frame is not unique. You can do U(1) transformation which acts on line but does not change it as a whole. You can also act on normal space by SU(2) without affective the line. Frames are parametrize by SU(3) and you form equivalence classes by dividing with SU(2)xU(1).

The physical motivation for CP_2 is that its isometry group is SU(3) and holonomy group after constructing a respectable spinor structure by coupling spinors to Kahler gauge potential has structure of electroweak gauge group. One obtains just the correct selection rules,parity breaking, quarks and leptons correspond to different chiralities of M^4xCP_2 spinors coupling to n=1 and n=3 multiples of Kahler gauge potential. Actually Hawking and others did the construction long time ago but did not realize that one obtains electroweak structure.

Sine-Gordon appeared first time in my the model for nerve pulses as solitons of Josephson junction assumed to be defined by cell membrane. Phi corresponds to the phase differences of super-conducting order parameter over the membrane. Few years ago I modified the model. The ground state would be a propagating soliton sequence: imagine a sequence of gravitational penduli (for which Sine-Gordon is a continuum idealization) oscillating a little bit out of phase so that wave results. Nerve pulse results when you kick one of gravitational penduli so that perturbation starts to propagate.

I would not pass in exam about dualities. Thirring-Sine-Gordon: good to learn. The bridge to Thirring model to space-time requires something and I find it difficult to believe that two additional dimensions would just pop up from nowhere. I stubbornly believe that the 4-D dynamics should effectively reduce to almost 2-D dynamics giving holography in strong form. The direction for dimensions would not be 2–>4 but 4–> a little bit more than 2.

My proposal is that space-time surface are analogous to Bohr orbit: 4-D(!) general coordinate invariance indeed implies this in world of classical worlds picture where the WCW metric must assign a 4-D space-time surface to given 3-surface for 4-D coordinate transformations to act on.

A further reduction to effective 2-dimensionality comes from extremum property for Chern-Simons, which for light-like 3-surfaces is the only possibility since metric based action principles are excluded by singularity of the metric.

Weak form of electric-magnetic duality saves from complete topologization, which would be a catastrophe. This duality gives a constraint between Kahler electric and magnetic fields at 3-surface and reduces quantization of Kahler electric charge to that of Kahler magnetic charge. This constraint depends on induced metric because electric field involves metric. Very delicate.

Matti,

CP^2 is under a Euclideanization i –> 1 RP^4 ~ G(3,1) which has the bundle structure O(3)/O(2)xO(1). The complex universal bundle has the structure SU(2)/U(1)xU(1).

[…] here from all around the world, but scientists like Andre Geim and Konstantin Novoselov who won the Nobel prize for work done in the UK will no longer be welcome here. Young scientists are very mobile. […]

Matti Pitkänen Said:

In his famous

More is Differentpublished in journal Science, Anderson stated why condensed matter is not just applied particle physics, and emphasized the . It is just funny to read to particle physicists and string theories saying nonsense about other disciplines (e.g. thermodynamics, chemistry, biology) as if they know of what they were talking…