Saturday, January 23, 2010

Verlinde's thermal origin of gravitation from TGD view point of view

Eric Verlinde has posted an interesting eprint titled On the Origin of Gravity and the Laws of Newton to Lubos has commented the article here and also here. What Linde heuristically derives is Newton's F=ma and gravitational force F= GMm/R2 from thermodynamical considerations plus something else which I try to clarify (at least to myself!) in the following.

1. Verlinde's argument for F=ma

The idea is to deduce Newton's F=ma and gravitational force from thermodynamics by assuming that space-time emerges in some sense. There are however various assumptions involved which more or less impy that both special and general relativity has been feeded in besides quantum theory and thermodynamics.

  1. Time translation invariance is required in order to have the notions of conserved energy and thermodynamics. This assumption requires not only require time but also symmetry with respect to time translations. This is quite a powerful assumption and time translation symmetry not hold true in General Relativity- this was actually the basic motivation for quantum TGD.

  2. Holography is assumed. Information stored on surfaces, or screens and discretization is assumed. Again this means in practice the assumption of space-time since otherwise the notion of holography does not make sense. One could of course say that one considers the situation in the already emerged region of space-time but this idea does not look very convincing to me.

    Comment: In TGD framework holography is an essential piece of theory: light-like 3-surfaces code for the physics and space-time sheets are analogous to Bohr orbits fixed by the light-like 3-surfaces defining the generalized Feynman diagrams.

  3. The first law of thermodynamics in the form

    dE= TdS-Fdx

    Here F denotes generalized force and x some coordinate variable. In usual thermodynamics pressure P would appear in the role of F and volume V in the role of x. Also chemical potential and particle number form a similar pair. If energy is conserved for the motion one has

    Fdx= TdS.

    This equation is basic thermodynamics and is used to deduce Newton's equations.

After this some quantum tricks -a rather standard game with Uncertainty Principle and quantization when nothing concrete is available- are needed to obtain F=ma which as such does not involve hbar nor Boltzmann constant kB. What is needed are thermal expression for acceleration and force and identifying these one obtains F=ma.

  1. Δ S= 2π kB states that entropy is quantized with a unit of 2π appearing as a unit. log(2) would be more natural unit if bit is the unit of information.

  2. The identification Δ x =hbar/mc involves Uncertainty principle for momentum and position. The presence of light velocity c in the formula means that Minkowski space and Special Relativity creeps in. At this stage I would not speak about emergence of space-time anymore.

    This gives T= FΔ x/Δ S= F×hbar/[2π×mc×kB]

    F has been exressed in terms of thermal parameters and mass.

  3. Next one feeds in something from General Relativity to obtain expression for acceleration in terms of thermal parameters. Unruh effect means that in an accelerted motion system measures temperate proportional to acceleration :

    kBT= hbar a/2π .

    This quantum effect is known as Unruh effect. This temperature is extremely low for accelerations encountered in everyday life - something like 10-16 K for free fall near Earth's surface.

    Using this expression for T in previous equation one obtains the desired F=ma, which would thus have a thermodynamical interpretation.

    At this stage I have even less motivations for talking about emergence of space-time. Essentially the basic conceptual framework of Special and General Relativities, of wave mechanics and of thermodynamics are introduced by the formulas containing the basic parameters involved.

2. Verlinde's argument for F= GMm/R2

The next challenge is to derive gravitational force from thermodynamic consideration. Now holography with a very specially chosen screen is needed.

Comment: In TGD framework light-like 3-surfaces (or equivalently their space-like duals) represent the holographic screens and in principle there is a slicing of space-time surface by equivalent screens. Also Verlinde introduces a slicing of space-time surfaces by holographic screens identified as surfaces for which gravitational potential is constant. Also I have considered this kind of identification.

  1. The number of bits for the information represented on the holographic screen is assumed to be proportional to area.

    N =A/Ghbar.

    This means bringing in blackhole thermodynamics and general relativity since the notion of area requires geometry.

    Comment: In TGD framework the counterpart for the finite number of bits is finite measurement resolution meaning that the 2-dimensional partonic surface is effectively replaced with a set of points carrying fermion or antifermion number or possibly purely bosonic symmetry generator. The orbits of these points define braid giving a connection with topological QFTs for knots, links and braids and also with topological quantum computation.

  2. It is assumed that A=4π R2, where R is the distance between the masses. This means a very special choice of the holographic screen.

    Comment: In TGD framework the counterpart of the area would be the symplectic area of partonic 2-surfaces. This is invariant under symplectic transformations of light-cone boundary. These "partonic" 2-surfaces can have macroscopic size and the counterpart for blackhole horizon is one example of this kind of surface. Anyonic phases are second example of a phase assigned with a macroscopic partonic 2-surface.

  3. Special relativity is brought in via the bomb formula


    One introduces also other expression for the rest energy. Thermodynamics gives for non-relativistic thermal energy the expression

    E= 1/2N kBT.

    This thermal energy is identified with the rest mass. This identification looks to me completely ad hoc and I think that kind of holographic duality is assumed to justify it. The interpretation is that the points/bits on the holographic screen behave as particles in thermodynamical equilibrium and represent the mass inside the spherical screen. What are these particles on the screen? Do they correspond to gravitational flux?

    Comment: In TGD framework p-adic thermodynamics replaces Higgs mechanism and identify particle's mass squared as thermal conformal weight. In this sense inertia has thermal origin in TGD framework. Gravitational flux is mediated by flux tubes with gigantic value of gravitational Planck constant and the intersections of the flux tubes with sphere could be TGD counterparts for the points of the screen in TGD. These 2-D intersections of flux tubes should be in thermal equilibrium at Unruh temperature. The light-like 3-surfaces indeed contain the particles so that the matter at this surface represents the system. Since all light-like 3-surfaces in the slicing are equivalent means that one can choose the reresentation of the system rather freely .

  4. Eliminating the rest energy E from these two formulas one obtains NT= 2mc2 and using the expression for N in terms of area identified as that of a sphere with radius equal to the distance R between the two masses, one obtains the standard form for gravitational force.

It is difficult to say whether the outcome is something genuinely new or just something resulting unavoidably by feeding in basic formulas

from general thermodynamics, special relativity, and general relativity and using holography principle in highly questionable and ad hoc manner.

3. In TGD quantum classical correspondence predicts that thermodynamics has space-time correlates

From TGD point of view entropic gravity is a misconception. On basis of quantum classical correspondence - the basic guiding principle of quantum TGD - one expects that all quantal notions have space-time correlates. If thermodynamics is a genuine part of quantum theory, also temperature and entropy should have the space-time correlates and the analog of Verlinde's formula could exist. Even more, the generalization of this formula is expected to make sense for all interactions.

Zero energy ontology makes thermodynamics an integral part of quantum theory.

  1. In zero energy ontology quantum states become zero energy states consisting of pairs of the positive and negative energy states with opposite conserved quantum numbers and interpreted in the usual ontology as physical events. These states are located at opposite light-like boundaries of causal diamond (CD) defined as the intersection of future and past directed light-cones. There is a fractal hierarchy of them. M-matrix generalizing S-matrix defines time-like entanglement coefficients between positive and negative energy states. M-matrix is essentially a "complex" square root of density matrix expressible as positive square root of diagonalized density matrix and unitary S-matrix. Thermodynamics reduces to quantum physics and should have correlate at the level of space-time geometry. The failure of the classical determinism in standard sense of the word makes this possible in quantum TGD (special properties of Kähler action (Maxwell action for induced Kahler form of CP2) due to its vacuum degeneracy analogous to gauge degeneracy). Zero energy ontology allows also to speak about coherent states of bosons, say of Cooper pairs of fermions- without problems with conservation laws and the undeniable existence of these states supports zero energy ontology.

  2. Quantum classical correspondence is very strong requirement. For instance, it requires also that electrons traveling via several routes in double slit experiment have classical correlates. They have. The light-like 3-surfaces describing electrons can branch and the induced spinor fields at them "branch" also and interfere again. Same branching occurs also for photons so that electrodynamics has hydrodynamical aspect too emphasize in recent empirical report about knotted light beams. This picture explains the findings of Afshar challenging the Copenhagen interpretation.

    These diagrams could be seen as generalizations of stringy diagrams but do not describe particle decays in TGD framework. In TGD framework stringy diagrams are replaced with a direct generalization of Feynman diagrams in which the ends of 3-D lightlike lines meet along 2-D partonic surfaces at their ends. The mathematical description of vertices becomes much simpler since the 2-D manifolds describing vertices are not singular unlike the 1-D manifolds associated with string diagrams ("eyeglass" in fusion of closed strings).

  3. If entropy has a space-time correlate then also first and second law should have such and Verlinde's argument that gravitational force attraction follows from first law assuming energy correlation might identify this correlate. This of course applies only to the classical gravitation. Also other classical forces should allow analogous interpretation as space-time correlates for something quantal.

4. The simplest identification of thermodynamical correlates in TGD framework

The first questions that pop up are following. Inertial mass emerges from p-adic thermodynamics as thermal conformal weight. Could the first law for p-adic thermodynamics, which allows to calculate particle masses in terms of thermal conformal weights, allow to deduce also other classical forces? One could think that by adding to the Hamiltonian defining partition function chemical potential terms characterizing charge conservation it might be possible to obtain also other forces.

In fact, the situation might be much simpler. The basic structure of quantum TGD allows a very natural thermodynamical interpretation.

  1. The basic structure of quantum TGD suggests a thermodynamic interpretation. The basic observation is that the vacuum functional identified as the exponent of Kähler function is analogous to a square root of partition function and Kähler coupling strength is analogous to critical temperature. Kähler function identified as Kähler action for a preferred extremal appears in the role of Hamiltonian. Preferred extremal property realizes holography identifying space-time surface as analog of Bohr orbit. One can interpret the exponent of Kähler function as the density of states in the world of classical worlds so that Kähler function would be analogous to entropy density. Ensemble entropy is average of Kähler function involving functional integral over the world of classical worlds. This exponent is the counterpart for the quantity Ω appearing in Verlinde's basic formula.

  2. The addition of a measurement interaction term to the modified Dirac action gives rise to a coupling to conserved charges. Vacuum functional is identified as Dirac determinant and this addition is visible as an addition of an interaction term to Kähler function. The interaction gives rise to forces coupling to various charges at classical level for quantum states with fixed quantum numbers for positive energy part of the state. These terms are analogous to chemical potential terms in thermodynamics fixing the average values of various charges or particle numbers. In ordinary non-relativistic thermodynamics energy is in a special role. In the recent case there is a complete quantum number democracy very natural in a framework with coordinate invariance and with four-momentum assigned with the isometries of the 8-D imbedding space. In Verlinde's formula there is exponential factor exp(-E/T- Fx) analogous to the measurement interaction term. In TGD however conserved charges multiplied by chemical potentials defining generalized forces appear in the exponent.

  3. This gives an analog of thermodynamics in the world of classical worlds (WCW) for fixed values of quantum numbers of the positive energy part of state. For zero energy states one however has also additional thermodynamics- or rather its square root. This thermodynamics is for the conserved quantum numbers whose averages are fixed. For general zero energy states one has sum over state pairs labelled by momenta and various other quantum numbers labelling the positive energy part of the state. The coefficients of the conserved quantities of the measurement interaction term linear in conserved quantum numbers define the analogs of temperature and various chemical potentials. The field equations defined by Kähler function and chemical potential terms have thermodynamical interpretation and give coupling to conserved charges and also to their thermal averages.

    What is important is that temperature and various chemical potentials assigned to positive and negative energy parts of the state allow a complete geometrization in a general coordinate invariant manner and allow explicit expressions in terms of functions expressible in terms of the induced geometry.

  4. The explicit expressions must be deduced from Dirac determinant defining exponent of Kähler function plus measurement interaction term, in which the conserved isometry charges of Cartan algebra (necessarily!) appearing in the exponent are contracted with the analogs of chemical potentials. One make two rather detailed educated guesses for the chemical potentials. For the modified Dirac action the measurement interaction term is 4-dimensional and essentially unique. For the Kähler action one can imagine two candidates for the measurement interaction term. For the first option the term is 4-dimensional and for the second one 3-dimensional.

5. Some details related to the measurement interaction term

As noticed, one can imagine two options for the measurement interaction term defining the chemical potentials in terms of the space-time geometry.

  1. For both options the M4 part of the interaction term is proportional to n(M4)G/R and CP2 part to a dimensionless constant n(CP2), and the condition that there is no dependence of hbar excludes the dependence on the dimensionless constant Ghbar/R2.

  2. One can consider two different forms of the measurement interaction part in Kähler function. For the first option the conserved Kähler current replaces fermion current in the modified Dirac action and defines a 4-dimensional interaction term highly analogous to that assigned with the modified Dirac action. The source term induced to the field equations corresponds to the variation of

    [(G/R)× n(M4)pq,A gAB(M4)jA,α +n(CP2)Qq,A gABJA,α(CP2)] Jα .

    Here Jα is Kähler current.

  3. For the second option the measurement interaction term in Kähler action is sum over contractions of quantum Cartan charges with corresponding classical Noether charges giving the sum of the term

    (G/R)× n(M4)pq,A pcl,A +n(CP2)Qq,A Qcl,A

    from both ends of the space-time sheet. For a general space-time sheet the classical charges are different at its ends so that the variation gives non-trivial boundary conditions equating the normal (time-like) component of the canonical momentum current with the contraction of the variation of classical Noether charges contracted with quantum charges. By the extremal property the measurement interaction terms at the ends of the space-time sheet cancel each other so that the effect on Kähler function is only via the boundary conditions in accordance with zero energy ontology. For this option the thermodynamics for conserved charges is visible at space-time level only via the appearence of the average quantal charges and universal chemical potentials.

  4. The vanishing of Kähler gauge current resp. classical Noether charges for the first resp. second option would suggest an interpretation in terms of infinite temperature limit. The fact that momenta and color charges are in completely symmetric position suggests however the vanishing of chemical potentials. One can in fact fix the value of the temperature to say T= R/G without loss of information and code thermodynamics in terms of the chemical potentials alone.

    The vanishing of the measurement interaction term occurs for the vacuum extremals. For CP2 type vacuum extemals with Euclidian signature of the induced metric interpretation in terms of vanishing chemical potentials is more natural. For vacuum extremals with Minkowskian signature of the induced metric fluctuations and consequently classical non-determinism are maximal so that the interpretation in terms of high temperature phase cannot be excluded. One must however notice that CP2 projection for vacuum extremals is 2-dimensional whereas high temperature limit would suggest 4-D projection so that the interpretation in terms of vanishing chemical potentials is more natural also now.

To sum up, TGD suggests two thermodynamical interpretations. p-Adic thermodynamics gives inertial mass squared as thermal conformal weight and also the basic formulation of quantum TGD allows thermodynamical interpretation. The thermodynamical structure of quantum TGD has of course been guiding principle for two decades. In particular, quantum criticality as the counterpart of thermal criticality has been extremely useful guide line and led to a breakthrough in the understanding of the modified Dirac equation during the last year. Also p-adic thermodynamics has been in the scene for more than 15 years and makes TGD a theory able to make precise quantitative predictions.

Some conclusions drawn from Verlinde's argument is that gravitation is entropic interaction, that gravitons do not exist, and that string models and theories introducing higher-dimensional space-time are a failure. TGD view is different. Only a generalization of string model allowing to realize space-time as surface is needed and this requires fixed 8-D imbedding space. Gravitons also exist and only classical gravitation as well as other classical interactions code for thermodynamical information by quantum classical correspondence. In any case, it is encouraging that also colleagues might be finally beginning to get on the right track although the path from Verlinde's arguments to quantum TGD as it is now will be desperately long and tortuous if colleagues continually refuse to receive the helping hand.

For more details see the brief pdf file or the chapter Does the Modified Dirac Equation Define the Fundamental Action Principle? of "Quantum TGD as Infinite-dimensional Spinor Geometry".

Monday, January 18, 2010

Twenty four questions

Lubos Motl provided his own answer to Sean Carroll's 24 questions. Lubos answered these questions as a super string fanatic. In the following I will do the same as a TGD fanatic;-).

1. What breaks electroweak symmetry?

Lubos gives the text book answer: the electroweak symmetry is broken by the Higgs field's vacuum expectation value. TGD allows Higgs but reduces the description of the symmetry breaking to much deeper level. CP2 geometry breaks the electroweak symmetry: for instance, color partial waves for different weak isospin states of imbedding space spinors have hugely different masses. The point is that electroweak gauge group is the holonomy group of spinor connection and not a symmetry group unlike color group, which acts as isometries.

For physical states are massless before p-adic thermal massivation due to the compensation of conformal weights of various operators. The most plausible option is that both the non-half integer part of vacuum conformal weight for particle and Higgs expectation are expressible in terms of the same parameter which corresponds to a generalized eigenvalue of the modified Dirac operator. Higgs expectation-massivation relation is transformed from causation to correlation.

2. What is the ultraviolet extrapolation of the Standard Model?

As Lubos violently explains that "UV extrapolation" in the above statement should be replaced with "UV completion". I would replace it with "the unified theory of fundamental interactions". Lubos of course answers as a proponent of string theory. The problem is that there is practically infinite number of completions so that the predictivity is lost.

TGD geometrizes the symmetries of the standard model and reduces them to the symmetries of classical number fields. Also octonionic infinite primes, one of the most exotic notions inspired by TGD, code standard model symmetries. The most general formulation of the World of Classical Worlds is as the space of hyper-quaternionic of co-hyper-quaternionic subalgebras of the local hyper-octonionic Clifford algebra of M8 or equivalent M4× CP2.

The answers by both Lubos and me involve also supersymmetry but in different sense. In TGD framework the oscillator operators of the induced spinor fields define the analog of the space-time SUSY so that the algebra of second quantization is replaced with N=∞ SUSY. This requires a modification of SUSY formalism but N=1 SUSY associated with the right handed coveriantly constant neutrinos emerges as preferred sub-SUSY and counterpart of N=1 SUSY. The construction of infinite primes involves also supersymmetry.

3. Why is there a large hierarchy between the Planck scale, the weak scale, and the vacuum energy?

These are the two most famous hierarchy problems of current physics as Lubos notices. In TGD framework Planck scale is replaced with CP2 length scale, which is roughly by a factor 104 longer than Planck length scale. Instead of Planck length it might be more appropriate to talk about gravitational constant which follows as a prediction in TGD framework.

p-Adic length scale hierarchy is needed to understand the hierarchy of mass scales. The inverse of the mass squared scale comes as primes which are very near to octaves of a fundamental scale. Powers of two near Mersenne primes or Gaussian Mersennes are favored and this predicts a scaled up copy of hadron physics, which should become visible at LHC. Quite generally, unlimited number of scaled versions of standard model physics are in principle possible.

The vacuum energy density is the basic problem of super string approach. How desperate the situation is is clear from the fact that rhetoric tricks such as anthropic principle are considered seriously. Empirical findings- for some reason neglected by colleagues - suggests that cosmological constant depends on time. In TGD framework the cosmological constant is predicted to depend on the p-adic length scale of the space-time sheet and behaves roughly like 1/a2, where a is cosmic time identified as light-cone property time. Actually the time parameter a is replaced by a corresponding p-adic length scale. The recent value is predicted correctly under natural assumptions.

What dark energy is is a second question. TGD suggests the identification as a matter at space-time sheets mediating gravitational interaction having gigantic values of Planck constant implying extremely long Compton lengths for elementary particles. This guarantees that the energy density is constant in excellent approximation. If gravitational space-time sheets correspond to dark magnetic flux tubes- expanded cosmic strings- the mysterious negative pressure can be identified classically in terms of magnetic tension. If one takes seriously the correlation of the intelligence of conscious entities with the value o Planck constant, these gravitational space-time sheets can be God like entities.

4. How do strongly-interacting degrees of freedom resolve into weakly-interacting ones?

Lubos regards this question as strange and expresses this using colorful rhetoric. Maybe Carroll refers to QCD and hadronization. M8-M4× CP2 duality relates low energy and higher energy hadron physics to each other in TGD framework and corresponds group theoretically to SU(3)-SO(4) duality, where SO(4) is the well-known strong isospin symmetry of low energy hadron physics. Or maybe Carroll talks about the technical problem of calculating the behavior of strongly interacting systems. Nature might have solved the latter problem by a phase transition increasing Planck constant so that perturbation theory based on larger value of Planck constant works. The particle spectrum however changes and system becomes anyonic in general.

5. Is there a pattern/explanation behind the family structure and parameters of the Standard Model?

I can only echo Lubos: of course there is. In super string models the large number of explanations tells that the real explanation is lacking. In TGD framework fermion families correspond to various genera for partonic 2-surfaces (genus tells the number of handles attached to sphere to get the 2-dimensional topology). There is an infinite number of genera but the 3 lowest genera are mathematically very special (hyper-ellipticity as a universal property), which makes them excellent candidates for light fermion families. The successful predictions for masses using p-adic thermodynamics and relying strongly on the genus dependent contribution from conformal moduli supports the explanation.

Bosons correspond to wormhole contacts and are labeled by pairs of general implying a dynamical SU(3) symmetry with ordinary bosons identified as SU(3) singlets. SU(3) octet bosons perhaps making themselves visible at LHC are predicted and serve as a killer test.

The symmetries of standard model reduce to the geometry of CP2 having a purely number theoretical interpretation in terms of the hyper-octonionic structure. Number theory fixes through associativity condition the dynamics of space-surfaces completely (hyper-quaternionicity or its co-property in appropriate sense).

6. What is the phenomenology of the dark sector?

Lubos sees the dark matter as something relatively uninteresting. Just some exotic weakly acting particles. How incredibly blind a theorist accepting 11-D space-time and landscape having absolutely no empirical support can be when it comes to actual experimental facts!

In TGD framework dark matter means a revolution in the world view. Its description relies on the hierarchy of Planck constants requiring a generalization of the 8-D imbedding space M4 × CP2 to a book like structure with pages partially characterized by the value of Planck constant. The most fascinating implications are in biology. Also the implications for our view about the nature of consciousness and our position in World Order are profound.

7. What symmetries appear in useful descriptions of nature?

As Lubos says, one must be careful what types of symmetries we are talking about. As Lubos says "Only global unbroken symmetries are "really objective" features of the reality. It's very likely that we have found the full list and it includes the CPT-symmetry, Poincare symmetry (including Lorentz, translational, and rotational symmetries), and the U(1) from the conservation of the electric charge. By adding color symmetry and separate baryuon and lepton conservation one obtains the symmetries of quantum TGD: this prediction follows from number theoretical vision alone.

Lubos mentions dualities relating descriptions based on different symmetries. In TGD M8-M42 duality manifests as the dual descriptions of hadrons using low energy hadron phenomenology (SO(4))and parton picture at high energies (color SU(3)).

There are good reasons to believe that TGD Universe is able to emulate almost any gauge theory for which gauge group is simply laced Lie group and stringy system (Mc-Kay correspondence, inclusions of hyper-finite factors and the book like structure of generalized imbedding space). These symmetries would be however engineered rather than fundamental symmetries.

8. Are there surprises at low masses/energies?

Lubos believes that there are no surprises without realizing that we ourselves are the most surprising surprise. Eye is not able to see itself without a mirror. The fact is that standard physics cannot say anything really interesting about life and consciousness. p-Adic physics, hierarchy of Planck constants, zero energy ontology,.... ; I believe that all this is necessary if one really wants to understand living matter.

9. How does the observable universe evolve?

Lubos believes in standard cosmology described by General Relativity as such. TGD predicts quantum version of standard cosmology. Smooth cosmological evolution is replaced by a sequence of rapid expansion periods serving as space-time correlates for quantum jumps increasing Planck constant for appropriate space-time sheets. This applies in all length scales and one especially fascinating application is to the evolution of Earth itself. Expanding Earth hypothesis finds a physical justification and one ends up to an amazingly simple and predictive vision about pre-Cambrian and Cambrian periods: this includes both meteorology, geology, and biology.

Zero energy ontology strongly suggests that the proper quantum description is in terms of the moduli space for causal diamonds (CDs identified as intersections of future and past light-cones). The entire future light-cone labeling the "upper" tips of CD and analogous to Robertson-Walker cosmology is replaced with a discrete set of points. In particular, the values of cosmic time come as octaves of basic scale for a given value of Planck constant. The spectrum of planck constants means that all rational multiples of CP2 time scale are in principle possible. Cosmic evolution as endless re-creation of the Universe- can be seen as the emergence of CDs with larger and larger size.

10. How does gravity work on macroscopic scales?

General Relativity is part of the description but zero energy ontology and hierarchy of Planck constants bring in new elements. The gigantic values of gravitational Planck constant make possible astroscopic quantum coherence for the dark matter at magnetic flux tubes mediating gravitational interaction and explain dark energy. Quantum classical correspondence suggests that the exchanges of virtual particles has classical description allowed by Einstein's tensor. In the case of planetary system a possible manifestation is the observation of Grusenick that a Michelson interferometer rotating horizontal plane produces constant interference pattern but in a vertical plane the interference pattern varies during rotation. If real this find is revolutionary. It might also directly relate also to the finding that the measured values of gravitational constant varies within 1 per cent. There has been no reaction from academic circles.

The assumption that gravitation in long length scales has been understood more or less completely is the basic dogma of string theorists. This despite the fact that the list of anomalies and intriguing regularities is really long. It is much more rewarding to impress colleagues with long and complex calculations than using the professional lifetime to a risky attempt to solve a real problem.

11. What is the topology and geometry of space-time and dynamical degrees of freedom on small scales?

In TGD framework "on small scales" can be dropped from the question. Many-sheeted space-time, hierarchy of Planck constants, p-adic space-time sheets serving as correlates of cognition and intentionality, zero energy ontology... All this means a dramatic generalization of the view about space-time in all length scales and a profoundly new way to interpret what we observe. If TGD is correct we really "see" the dark matter in biology and we really "see" p-adic physics via its interaction giving rise to effective p-adic topology of real space-time sheets leading to to extremely successful predictions for elementary particle masses.

Quantum group enthusiasts believe that space-time time becomes non-commutative in Planckian length scales. Some theoreticians believe that some kind of Planckian discreteness emerges. In TGD framework quantum groups emerge as a natural part of description in terms of a finite measurement resolution and in all length scales. Discretization appears as a space-time correlate for a finite measurement resolution but not as an actual discreteness. The finite resolution of cognition and sensory perception implies also an apparent discreteness. Also the hierarchy of infinite primes suggests description in terms of hierarchy of discrete structures.

At fundamental level everything is however continuous- in real or in p-adic sense in accordance with the generalization of number concept involving both fusion of real and p-adic number fields to a larger super structure and providing single space-time point with infinitely rich number theoretic anatomy. The talk about infinite primes (infinite only in real sense) sounds very unpractical but to my great surprise infinite primes lead to highly detailed predictions for the spectrum of states and quantum numbers.

12. How does quantum gravity work in the real world?

Lubos restates the basic belief of string theorists that Einstein's equations follow at long length scale QFT limit of super string models. In TGD framework Einstein's equation hold true too at this limit but quantal aspects are also present. The hierarchy of Planck constants -in particular gigantic values of the gravitational Planck constant at dark magnetic flux tubes mediating gravitational interaction- are essential for the gravitational physics of dark matter.

There are also several delicate effects such as Allais effect suggesting that the ultraconservative view of Lubos is wrong. With all respect, the builders of quantum gravity theories should really consider returning to the roots and also a serious consideration of experimental data. Otherwise they continue to produce useless formalism without any connection with the observed reality.

13. Why was the early universe hot, dense, and very smooth but not perfectly smooth?

The standard answer echoed by Lubos is in terms of inflationary cosmology. In TGD framework very early cosmology is cosmic string dominated. Space-time sheets appear later (at certain proper time distance from light-cone boundary). Inflationary cosmology is replaced with a sequence of expansion periods during which the cosmology is quantum critical at appropriate space-time sheets. No scales are present and 3-space is flat. The critical cosmology, which is unique apart from a parameter telling its duration describes the situation. This is extremely powerful prediction following from the imbeddability to M4× CP2 alone. Quantum criticality implies the universality of the dynamics during expansion periods.

Big Bang is replaced by a "silent whisper amplified to a Bang" since the energy density of cosmic strings behaves as 1/a2, where a denotes the proper time of light-cone. The moduli space of CDs suggests a cartesian product of M4×CP2 labeling the lower tips of CDs with its discrete version labeling the upper tips of CD. One must ask whether a CD corresponds to a counterpart of Big Bang followed eventually by a Big Crush.

14. What is beyond the observable universe?

"What is beyond the universe observable to us" would be a more precise formulation. The hierarchies of Planck constants and p-adic length scales, the hierarchy of conscious entities in which we correspond to one particular relatively low lying level, the hierarchy of infinite primes mathematically similar to an infinite hierarchy of second quantizations, the infinitely complex structure of single space-time point realizing algebraic holography,.... I find myself standing at the shore of an infinitely vast sea. The fundamental symmetries are the basic elementary particle quantum numbers are universal. This by the simple requirement that the geometry of the world of classical worlds exists mathematically and has number theoretic interpretation.

15. Why is there a low-entropy boundary condition in the past but not the future?

The form of the question reflects the erratic identification of the experienced time appearing in second law with the geometric time appearing as one space-time coordinate. After these 32 years this identification looks to me incredibly stupid but is made by most of colleagues despite the that the fact that these times are completely different. Irreversibility contra reversibility, only the recent moment and past contra entire eternity, etc... Here only consciousness theory could help but the patient stubbornly refuses to receive the medication.

Lubos however intuitively realizes that future and past are not in symmetric position in second law but is unable to ask what this means. He really believes that Boltzmann equations are all that is needed and never consider the possibility that these wonderful equations might make sense only under certain conditions.

In TGD framework the geometric correlate for the arrow of subjective time which by definition is always the same (consciousness as sequence of quantum jumps with past identified as quantum jumps that have already occurred and contribute to conscious experience) can in principle have both directions. Phase conjugate laser beams provide a basic example about the situation in which second law applies in "wrong" direction of geometric time. Also self assembly for biological molecules can be interpreted in this manner. Hierarchy of Planck constants implies that for given CD Boltzmann's equations make sense only for smaller CDs inside it. In living matter the Boltzmannian description fails.

In TGD framework the concept of low entropy boundary condition does not make sense. The subjective evolution applies the evolution of entire CD of cosmological size quantum jump by quantum jump. Boltzmann's equation apply only in scales considerably shorter than cosmological time. What is clear that one can speak only initial condition rather than boundary condition.

It is however not clear whether one can speak about the evolution of entropy as a function of cosmic time if identified as a coordinate of the imbedding space. Quantum classical correspondence might allow also the mapping of subjective time evolution to a geometric time evolution with respect to cosmic time. The low entropy of very early universe could correspond to that assignable to cosmic strings. The energy density of cosmic strings goes down as 1/a2 and entropy density as 1/a so that for a given comoving volume the entropy approaches to zero. The structure of moduli space of CDs suggests that positive of the upper tip of CD relative to the lower one defines a discretized cosmic time and the space-time correlate for entropy corresponds to the growth of entropy of CD as a function of this time in an ensemble of CDs. The asummetry between tips could be seen as a correlate for the arrow of time.

Carroll's idea about boundary conditions in future might make sense in the following sense. In zero energy ontology one has pairs of positive and negative energy sense and there is large temptation to think that there are two choices for the tip which corresponds to the discrete version of future light-cone.

16. Why aren't we fluctuations in de Sitter space?

If I have understood correctly the emotional rhetoric of Lubos, the idea of Carroll seems to be that intelligent life is just a random fluctuation rather than a long lasting evolution. For some reason he locates this fluctuation in de Sitter space. In the standard physics framework this view is however more or less unavoidable. The colleagues should really use some of their time to learn what we understand and what we do not understand about consciousness and brain to realize that the physics as they understand really fails to describe the physics of life.

Also Lubos is so fixated in his materialistic and reductionistic dogmas that he is unable to propose anything constructive. For instance, he does not ask how this undeniable evolution is possible in the framework of standard physics.

In TGD framework the hierarchy of Planck constants meaning a hierarchy of macroscopic quantum phases and hierarchy of time scales of memory and intentional action leads to a coherent overall view about what life is. Zero energy ontology provides a concrete realization how volition is realized in accordance with the laws of physics and makes possible a continual re-creation of the Universe.

17. How do we compare probabilities for different classes of observers?

I do not repeat the violent reaction of Lubos to this question. I am myself not at all sure whether I can catch the meaning of this question. Maybe I could interpret in terms of finite measurement resolution. Different measurement resolutions give rise to different M-matrices and probabilities and the comparison would require rules allowing to compare these probabilities. This comparison requires relationship between M-matrices at quantum level: probabilities are not enough. Renormalization group evolution as function of measurement resolution could provide the answer to ho compare the probabilities.

18. What rules govern the evolution of complex structures?

The text book answer of Lubos is "The detailed evolution of all complex structures is governed by the microscopic laws that govern the elementary building blocks, applied to a large number of ingredients".

The TGD inspired answer is based on the acceptance of fractal hierarchies: reductionistic dogma is replaced with fractality. The laws at various levels are essentially similar but every level brings something new: Mandelbrot set does not look exactly the same in the new zoom. It is not possible to reduce the behavior at higher levels that at the lowest level.

The hierarchy of infinite primes characterizes this idea number theoretically and -as there are reasons to believe- also physically. The construction of hyper-octonionic infinite primes is structurally similar to a second quantization of an arithmetic quantum field theory with states labeled by primes (rational, quaternionic, or octonionic). There is infinite hierarchy of second quantization with many particle states of the previous level becoming single particle states of the new level. At each level one has infinite primes analogous to free many particle states plus primes analogous to bound states.

One new element of emergence is association statistics. Permutations and associations are basic stuff of number theory and algebra. Quantum commutativity- invariance of the physical state under permutations in quantum sense leads to Fermi-, Bose- and quantum group statistics in effectively 2-D situation. Quantum associativity requires association statistics with respect to different associations of particles (replacing A(BC) with (AB)C can induce multiplication with +1,-1, or more complex phase).

At space-time level the hierarchy of space-time sheets is the counterpart for this hierarchy. p-Adic length scales define one hierarchy. Also space-time sheets characterized by a large value of Planck constant emerge as systems migrate to the the pages of the Big Book partially characterized increasing value of Planck constant and at which matter is dark relative to the observer with standard value of Planck constant, which corresponds to rational number equal to 1.

There is also a hierarchy of cognitive descriptions of the physical system. The higher the level in the hierarchy, the more abstract the description is and the less details it has. This is like the view of boss of big company as compared to that of a person doing something very concrete job.

p-Adic physics turns upside the reductionistic hierarchy proceeding from short to long scales. What is infinitesimal p-adically is infinitely large in real sense. This p-adic aspects is necessarily if we want to understand intentional systems able to plan their own behavior. p-Adic effectively topology means precise long range correlations and short range chaos which indeed characterizes the behavior of living matter. One can also say that p-adic physics provides the IR completion of physics.

19. Is quantum mechanics correct?

Quantum mechanics is not wrong. It however requires a profound generalization if we want to understand life. Also the gravitational anomalies and unexpected regularities at the level of planetary system suggest a generalization. Planck constant must be replaced with a hierarchy of Planck constants realized in terms of the "Big Book". Positive energy ontology must be replaced with zero energy ontology for which states correspond to physical events in standard positive energy ontology. S-matrix is replaced with its "complex square" root - M-matrix- having interpretation as square root of density matrix and making thermodynamics part of quantum theory. This generalization answers several frustrating questions raised in standard ontology. A further important modification is the introduction of the notion of finite measurement resolution realized in terms of inclusions of hyper-finite factors and having discretization as space-time correlate.

20. What happens when wave functions collapse?

The answer of Lubos is from the few pages of the standard quantum mechanics text book devoted to measurement problem. "A wave function is nothing else than a tool to predict probabilities; it is no real wave. When such an object "collapses", the only thing that it means is that we learned something about the random outcomes of some measurements, so we may eliminate the possibilities that - as we know - can no longer happen. For our further predictions, we only keep the probabilities of the possibilities that can still happen."

This answer brings in "we" but says nothing about what this "we" might be. This "We" remains an outsider to the physical world. Here we encounter the amazing ability of even admittedly intelligent persons to see the problem although it is staring directly at their face.

In TGD framework wave function collapse is involved with quantum jump re-creating the quantum universe. Speaking about space-time correlates this means that entire space-time surface (or rather their quantum superposition) is replaced with a new one. Both geometric past and future are replaced with a new one in quantum jump. There is no conflict with deterministic field equations (in generalized sense in TGD framework) since the non-determinism relates to subjective time identified as a sequence of quantum jumps rather than with geometric time appearing at classical field equations and Schrödinger equation.

Negentropy Maximization Principle stating the reduction of entanglement entropy in quantum jump is maximal implies standard quantum measurement theory. There are fascinating possibilities opened by the fact that for rational and even algebraic entanglement probabilities number theoretic analogs of Shannon entropy make sense and allow negentropic entanglement (emergence of information carrying stable quantum entangled states).

21. How do we go from the quantum Hamiltonian to a quasiclassical configuration space?

A more appropriate question would be "How to go from quantum description to classical description". Hamiltonian formulism relies on on Newtonian time and is given up already in Special Relativity. In General Relativity General Coordinate invariance makes Hamiltonian formalism even more un-natural.

In zero energy ontology the basic mathematical object coding for the predictions of the theory is M-matrix characterizing the physics inside given CD. It decomposes into a product of positive square root of diagonal density matrix and unitary S-matrix. The latter characterizes given CD and need not have any natural representation as an exponentiation of infinitesimal Hermitian operator- the Hamiltonian. This kind of picture is also in conflict with General Coordinate Invariance. In p-adic context unitary evolution becomes highly questionable also for number theoretical reasons. The counterpart of exponential function in p-adic context does not have the properties as it has in real context and the natural unitary operators involve roots of unity typically requiring algebraic extension of p-adic numbers and therefore have no description as unitary time evolutions.

In the formalism without Hamiltonian observables are replaced with algebras of various symmetries. Various super-conformal symmetries make these algebras infinite-dimensional. Modified Dirac equation brings in second quantization which reduces to an infinite-dimensional analog of space-time SUSY algebra.

How classical physics emerges from quantum theory is of course extremely important un-answered question although Lubos claims the opposite. This emergence has two meanings corresponding to geometric time and subjective time.

  1. Consider first geometric time. In TGD framework space-time surface is a preferred extremal of K\"ahler action and analogous to Bohr orbit. Classical physics in the geometric sense becomes an exact part of quantum physics and the geometry of the World of Classical Worlds. This is forced by the General Coordinate Invariance alone. Even more preferred space-time surfaces correspond to maxima of the K\"ahler function identified as value of K\"ahler action for a preferred space-time surface. In mathematically non-existing path integral formalism stationary phase approximation gives something believed to be enough for classical physics in this sense.

  2. Lubos talks also about de-coherence as a mechanism leading to classicality. This notion applies when one speaks about subjective time. When the time scale of observer is long as compared to the time scale of observed events (the CD of observer is much larger than those of observed systems so that quantum statistical determinism applies) decoherence taking place in sub-quantum jumps guarantees that all phase information is lost and quantum mechanical interference effects are masked out. The world looks classical in Boltzmannian sense but only for an observer looking the situation from a longer time scale.

22. Is physics deterministic?

Determinism is not valid in quantum universe as Lubos states. Determinism is valid at the level of field equations. These statements are contradictory unless one realizes that there are two different times. To understand these two times and their relationship one is forced bo make observer a part of the Universe instead of being outsider, that is to develop a quantum theory of consciousness. Amusingly, Lubos admits the non-determinism is a fact but denies that Schrödinger amplitudes which must behave non-deterministically in standard ontology, are real.

23. How many bits are required to describe the universe?

Currently around 10100 says Lubos. For me both the question and its answer are nonsense for the same reason as some other questions above. That people waste their time with this kind of questions shows how desperately physics needs an extension to a theory of consciousness. This is required also by neuroscience and biology. Lubos identifies this number as the entropy of the observed Universe. The notion in principle makes sense but not the identification. In TGD framework the entropy is also dependent on the resolution used. The better the measurement resolution, the larger the number of degrees of freedom, and the larger the entropy.

24. Will elementary physics ultimately be finished?

The answer depends on what one means with "elementary particle" and what one means with "finished"! TGD predicts in principle infinite hierarchy of scaled versions of what we have used to call elementary particle physics corresponding to hierarchies of p-adic length scales and Planck constants. The hierarchy of infinite primes suggests a generalization of elementary particle in which many particle states of given hierarchy level (space-time sheets) become single particle states of the new level (space-time sheets topologically condensed at large space-time sheets). Same Universal mathematical description applies at all levels but always something new emerges. Therefore my answer is realistic "No".

Wednesday, January 13, 2010

How infinite primes could correspond to quantum states and space-time surfaces?

I became conscious of infinite primes for almost 15 years ago. These numbers were the first mathematical fruit of TGD inspired theory of consciousness and define one of the most unpractical looking aspects of quantum TGD.

Their construction is however structurally similar to a repeated second quantization of an arithmetic super-symmetry quantum field theory with states labeled by primes. An attractive identification of the hierarchy is in terms of the many-sheeted space-time. Also the abstraction hierarchy of conscious thought and hierarchy of n:th order logics naturally correspond to this infinite hierarchy. We ourselves are at rather lowest level of this hierarchy. Propositional logic and first order logic at best and usually no logic at all;-)

By generalizing from rational primes to hyper-octonionic primes one has good hopes about a direct connection with physics. The reason is that the automorphism group of octonions respecting a preferred imaginary unit is SU(3)subset G2 and physically corresponds to color group in the formulation of the number theoretical compactification stating equivalence of the formulations of TGD based on the identification of imbedding space with 8-dimensional hyperquaternions M8 and M4× CP2. The components of hyper-octonion behave like two color singlets and triplet and antitriplet. For a given hyper-octonionic prime there exists a discrete subgroup of SU(3) respecting the prime property and generating a set of primes at octonionic 6-sphere. For a given prime one can realize a finite number of color multiplets in this discrete space. The components in the hyper-complex subspace M2 remaining invariant under SU(3) can be identified as components of momentum in this subspace. M2 is needed for massless particles the preferred extremals of Kähler action assign this space to each point of space-time surface as space non-physical polarizations.

There are two kinds of infinite primes differing only by the sign of the "small" part of the infinite prime and for second kind of primes one can consider the action of SU(2) subgroup of SU(3) and corresponding discrete subgroups of SU(2) respecting prime property (note that this suggests a direct connection with the Jones inclusions of hyper-finite factors of type II1!). These representations give rise to two SU(2) multiplets and their orbital excitations identifiable as deformations of the partonic 2-surface. Four components of hyper-octonion remain invariant under SU(2) and have interpretation as momentum in M2 and electroweak charges. Therefore a pair of these primes characterizes standard model quantum numbers of particle if discrete wave functions in the space of primes are allowed. For color singlet particles single prime is enough. At the level of infinite primes one obtains extremely rich structure and it is possible to map the states of quantum TGD to these number theoretical states. Only the genus of partonic 2-surface responsible for family replication phenomenon fails to find an obvious interpretation in this picture.

The completely unexpected by-product is a prediction for the spectrum of quantum states and quantum numbers including masses so that infinite primes and rationals are not so unpractical as one might think! This prediction is really incredible since it applies to the entire hierarchy of second quantizations in which many particle states of previous level become particles of the new level (corresponding physically to space-time sheets condensed to a larger space-time sheet or causal diamonds inside larger causal diamond CD).

In zero energy ontology positive and negative energy states correspond to infinite integers and their inverses respectively and their ratio to a hyper-octonionic unit. The wave functions in this space induced from those for finite hyper-octonionic primes define the quantum states of the sub-Universe defined by given CD and sub-CDs. These phases can be assigned to any point of the 8-dimensional imbedding space M8 interpreted as hyper-octonions so that number theoretic Brahman=Atman identity or algebraic holography is realized! These incredibly beautiful infinite primes are both highly spiritual and highly practical just as a real spiritual person experienced directly Brahman=Atman state is;-).

A fascinating possibility is that even M-matrix- which is nothing but a characterization of zero energy state- could find an elegant formulation as entanglement coefficients associated with the pair of the integer and inverse integer characterizing the positive and negative energy states.

  1. The great vision is that associativity and commutativity conditions fix the number theoretical quantum dynamics completely. Quantum associativity states that the wave functions in the space of infinite primes, integers, and rationals are invariant under associations of finite hyper-octonionic primes (A(BC) and (AB)C are the basic associations), physics requires associativity only apart from a phase factor. The condition of commutativity poses a more familiar condition implying that permutations induce only a phase factor which is +/- 1 for boson and fermion statistics and a more general phase for quantum group statistics for the anyonic phases, which correspond to nonstandard values of Planck constant in TGD framework. These symmetries induce time-like entanglement for zero energy stats and perhaps non-trivial enough M-matrix.

  2. One must also remember that besides the infinite primes defining the counterparts of free Fock states of supersymmetric QFT, also infinite primes analogous to bound states are predicted. The analogy with polynomial primes illustrates what is involved. In the space of polynomials with integer coefficients polynomials of degree one correspond free single particle states and one can form free many particle states as their products. Higher degree polynomials with algebraic roots correspond to bound states being not decomposable to a product of polynomials of first degree in the field of rationals. Could also positive and negative energy parts of zero energy states form a analog of bound state giving rise to highly non-trivial M-matrix?

Also a rigorous interpretation of complexified octonions emerges in zero energy ontology.

  1. The two tips of causal diamond CD define two preferred points of M4. The fixing of quantization axes of color fixes in CP2 also a point at both light-like boundaries of CD. The moduli space for CDs is therefore M4× CP2 × M4++× CP2 and its M8 counterpart is obtained by replacing CP2 with E4 so that a space which correspond locally to complexified octonions is the outcome. p-Adic length scale hypothesis suggests very strongly a quantization of the second factor to a set of hyperboloids with light-cone proper time come as powers of 2. For other values of Planck constant rational multiples of these are obtained. This suggests quantization also for hyperboloids and CP2.

  2. An attractive hypothesis is that infinite-primes determine the discretization as Ga subset SU(2)subset SU(3) and Gb subset SU(3) orbits of the points of hyperboloid and CP2. The interpretation would be in terms of cosmology. The Robertson Walker space-time would be replaced with this discrete space meaning in particular that cosmic time identified as Minkowski proper time is quantized in powers of two. One prediction is quantization of cosmic redshift resulting from quantization of Lorentz boosts and has been indeed observed and extremely difficult to understand in standard cosmology. We would observe infinite primes directly!

I do not bother to type more. Interested reader can read the brief pdf file explaining all this in detail or read the chapter Physics as Generalized Number Theory III: Infinite Primes of "TGD as a Generalized Number Theory".

Saturday, January 09, 2010

Exceptional symmetries in condensed matter system?

Lubos commented an interesting abstract reporting evidence for a realization of the mathematically extremely interesting exceptional Lie group E8 as symmetries of a condensed matter system known as Ising chain consisting of a chain of spins in a strong transversal magnetic field causing magnetization. At criticality for phase transition destroying the magnetization excitations appear and E8 would appear as a symmetry of these excitations. Here is the abstract.

Quantum phase transitions take place between distinct phases of matter at zero temperature. Near the transition point, exotic quantum symmetries can emerge that govern the excitation spectrum of the system. A symmetry described by the E8 Lie group with a spectrum of eight particles was long predicted to appear near the critical point of an Ising chain. We realize this system experimentally by using strong transverse magnetic fields to tune the quasi–one-dimensional Ising ferromagnet CoNb2O6 (cobalt niobate) through its critical point. Spin excitations are observed to change character from pairs of kinks in the ordered phase to spin-flips in the paramagnetic phase. Just below the critical field, the spin dynamics shows a fine structure with two sharp modes at low energies, in a ratio that approaches the golden mean predicted for the first two meson particles of the E8 spectrum. Our results demonstrate the power of symmetry to describe complex quantum behaviors.

The relation of the results to string theory and TGD

Lubos gives a nice summary of E8, which I recommend. Unfortunately Lubos takes a completely non-critical attitude accepting the experimental evidence as a proof and also creates the impression that this as a victory of super string model. The emergent dynamical E8 symmetry is actually predicted by conformal field theory approach to 1-D critical systems alone and has nothing to with the fundamental E8×E8 symmetry of heterotic strings as Lubos actually admits. E8 symmetry is predicted to be possible by conformal symmetry characterizing 2-dimensional criticality and the Kac-Moody representation is obtained once one has 8 complex scalar fields describing excitations of a conformally invariant system. The associated Kac-Moody symmetry predicts also a presence of a large number of other excitations created by the Kac-Moody generators obtained as normal ordered exponentials of complex scalar fields and their presence in the spectrum should be shown.

Of course, also string models as well as TGD are characterized by conformal symmetry. In TGD conformal symmetries have interpretation as a 3-D generalization of 2-D conformal symmetries acting at light-like boundaries of light-cone of M4 and also at light-like 3-surfaces of H=M4×CP2 (because of their metric 2-dimensionality). Also string theories apply the exponentiation trick so that the 10-D target space of superstring models could be a purely formal construct in which case the notion of spontaneous compactification, which has led to the landscape catastrophe, would not make sense physically. In TGD framework compactication is replaced by number theoretical compactification, which is not a dynamical process but a duality stating the equivalence of formulations of quantum TGD based on the possibility to interpret 8-D imbedding space either as M8 or H=M4×CP2 (M8-H duality).

Could E8 emerge in TGD?

E8 is interesting also from the TGD point of view. Of course, to say anything detailed about the finding in TGD framework would require hard work and in the following I can make just speculative general remarks.

  1. The rank of E8 group is 8, which means that the Cartan algebra of E8 spanned by maximum number of commuting algebra elements has dimension 8. The eigenvalues of the Cartan algebra generators define the 8 quantum numbers of a physical state belonging to a representation of E8.

    In TGD framework the quantum numbers of particle correspond to Cartan algebra of the product of Poincare group color group SU(3) and electroweak group SU(2)×U(1). The dimension of the corresponding Cartan algebra is also 8 corresponding to 4 components of four-momentum, 2 color quantum numbers and 2 electroweak quantum numbers.

    In conformal field theories Lie groups are extended to Kac-Moody algebras. One can construct rank 8 Kac-Moody algebras by starting from 8 complex scalar fields which could be interpreted in terms of coordinates of 8-D Minkowski space. One would obtain both the complex form of E8 and the current algebra defined by symmetries of TGD (and of standard model).

  2. Hyper-finite factors of type II1 (HFFs) are a particular class of von Neumann algebras, which is very interesting from the point of view of quantum theories and the mathematics of quantum groups relates to them very closely. The spinors of world of classical worlds (the 4-surfaces in 8-D imbedding space) define a canonical representative for HFF. The inclusions of HFFs known as Jones inclusions are in one-one correspondence with finite discrete subgroups of SO(3) and these in turn are in one-one correspondence with simply laced Lie groups containing also E8. E6,E7 and E8 correspond to tedrahedon, octahedron, and dodecahedron, which are 3-D polygons. For other subgroups the minimal orbit is 2-D polygon. The conjecture is roughly that these Lie groups appear as dynamical symmetries of quantum TGD so that TGD Universe is like a universal computer able to emulate any other computer. Now the emulation is emulation of any gauge theory and also string model type system. These symmetries would not be fundamental but achieved by engineering.

  3. Also the hierarchy of Planck constants realized in terms of the book like structure of the 8-D imbedding space could involve the mathematics of Jones inclusions. The pages of the big book are singular coverings and factor spaces of both CP2 and what I call causal diamond (CD). CD is the intersection of future and past directed light-cones of 4-D Minkowski space M4. At least cyclic subgroups Zn are involved. Also Zn with reflection added and perhaps all finite discrete subgroups of the rotation group as symmetries permuting the copies of M8 or CP2 of the covering or permuting the identified points of the singular factor space.

    E8 gauge symmetry could emerge as a dynamical symmetry at corresponding pages. Even E8×E8 of heterotic strings models could appear. The two E8:s would be associated with M4 and CP2: maybe TGD Universe is able to emulate also E8×E8 and heterotic super string model. In the case of E8 the symmetries of dodecahedron would identify equivalent points of M4 for singular factor space option.

    These symmetries would be engineering symmetries requiring quantum criticality. The system should be very near to the back of the big book so that the 3-surface describing the physical system can leak to the other pages of the book. The E8 symmetry would appear only at the other side of criticality (E8 page) and would correspond to a non-standard value of Planck constant. The change of the value of Planck constant would stabilize the phase unstable for the standard value of Planck constant. The claimed condensed matter E8 symmetry is indeed assigned with quantum criticality rather than thermal criticality. Maybe the space-time sheets serving as correlates for the magnetic excitations of the system reside at the E8 page and correspond to dark matter in TGD framework.

  4. The fundamental representation of E8 is identical with its adjoint representation and obtained by combining the rotation generators of SO(16) acting as rotations of points of 16-D Euclidian space E16 and the spinors of the same space to form a Lie-algebra in which E8 acts. The question whether TGD could allow to identify some natural 16-D space inspires some reckless numerology.

    The definition of singular covering and factor spaces means a choice of two points of M4 in case of CD so that the moduli space for CDs is M4×M4+, where M4+ is 8-D light-cone: p-adic length scale hypothesis is obtained if M4+ reduces to a union of hyperboloids for which proper time is quantized as powers of two. A possible interpretation is in terms of quantum cosmology with quantization of cosmological time. This procedure fixes quantization axes and means fixing of preferred time-like direction and spatial direction at either tip of CD (rest system and quantization axes of spin).

    In the case of CP2 the selection of quantization axes should fix of point of CP2 and a direction of geodesic line at that point. Therefore this part of the moduli space is CP2×E4. Altogether the moduli space labeling CD×CP2 with fixed quantization axes and thus sectors of the world of classical worlds is 16-D space M4×M4+ ×CP2×E4. Could the tangent space of this space provide a natural realization of the generators of the complex form of E8?