About Langlands program

The following represents an off-topic comment to Not-Even-Wrong which I never even considered to post. Peter Woit has a nice posting about Langlands program about which I wrote a chapter for year or so ago with emphasis on what I call number theoretic braids, on the identification of Galois group Gal of algebraic numbers as a permutation group S_∞ for infinite number of objects, and on the connection with hyper-finite factors of type II₁ emerging via the fact that the group algebra of S_∞ is HFF of type II₁.

This background perhaps explains why I had several fleeting impressions of "At some primitive level I might understand this!" while reading Peter's summary. The question about the physical interpretation of the constructs of Geometric Langlands was raised but Peter made with an admirable clarity clear that any comments relating to fundamental physics by non-names would be deleted.

1. Some ideas related to number theoretic Langlands program and their counterparts in TGD framework

The first nice idea of the number theoretic Langlands program is that rationals can be formally regarded as "rational functions" in a discrete space of primes and that the extensions of rationals can be regarded as covering spaces of this "function space" characterized by Galois groups.

In TGD framework this analogy appears in the reverse direction. Infinite rationals and their algebraic extensions (in particular infinite primes) forming an infinite hierarchy can be constructed by an iterated second quantization of an arithmetic quantum field theory as quantum states of a generalized arithmetic QFT. The reverse of analogy means that infinite rationals can be mapped to rational functions.

Langlands program has local and global aspects which in TGD framework would physically correspond to various p-adic physics and real physics (p-adic physics would relate to cognition and intentionality). Note however that p-adic space-time sheets have literally infinite size in the real sense (rational points are common to real and p-adic space-time sheets). Local aspect means that to each prime one can assign a local function field and it corresponds to a p-adic number field Q_p. The elements of Q_p are analogs of formal Laurent series which in general do not converge in real sense. The local Langlands conjecture gives a correspondence between the representations of Gal(Q_p) into a complex Lie group G and complex representations of the corresponding algebraic group ^LG(Q_p) with Q_p coefficients. The global aspect is about Gal(Q) and Langlands relates the representations of Gal(Q) in group G and the dual group ^LG (A_Q), where A_Q denotes adeles.

2. Geometric Langlands program

Witten and others are developing geometric Langlands program which is a generalization of Langlands program from number fields to holomorphic function fields defined at 2-dimensional Riemann surfaces. Now duality corresponds to electric-magnetic duality originally conjectured by Olive and Montonen that one can assign to a gauge theory dual gauge theory formulated in terms of magnetic charges regarded as gauge charges of the dual group ^LG instead of electric charges in G. It would be interesting to relate G and its dual to the groups related to the inclusions of HFFs.

Conformal field theory is a central element of the approach. In particular, disks with punctures appear as a basic notion. Some kind of 4-D theory (twisted 4-D SYM) giving rise to Chern-Simons theory giving rise to 2-D conformal field theory is conjectured to provide representations for groups and their duals.

3. Quantum theory according to TGD

Before comparison with TGD I want to emphasize that quantum theory according to TGD differs from standard quantum theory in many respects.

1. TGD can be formulated using several philosophies about what quantum physics is: quantum physics as Kähler geometry and spinor structure for the world of classical worlds; quantum physics as almost topological QFT; quantum physics as generalized number theory with associativity defining the fundamental dynamical principle; quantum physics reduced to the notion of measurement resolution formulated in terms of inclusions for HFFs of type II₁ (or possibly more general algebras).

2. Kähler geometry of the world of classical worlds provide a geometrization of quantum theory. Kähler function as a functional of light-like 3-surface is defined as Kähler action for the preferred extremal. Fermionic oscillator operators are building blocks of gamma matrices of this space and define super-generators of super-canonical algebra.

3. Zero energy ontology implies radical deviation from the standard QFT framework. S-matrix is generalized to M-matrix defined as the complex square root of density matrix with unitary S-matrix appearing as analog of phase factor. The idea is that quantum theory is square root of statistical physics. M-matrix defines time-like entanglement coefficients between positive and negative energy parts of the zero energy state having interpretation as initial and final states of particle reaction.

4. Measurement resolution is realized in terms of inclusions of HFFs of type II₁ leading to a highly unique identification of M-matrix in terms of Connes tensor product. The non-uniqueness corresponds to statistical physics degrees of freedom. M-matrix has Hermitian operators of the included algebra defining measurement resolution as symmetries. These symmetries should include super-conformal symmetries.

5. Hierarchy of Planck constants realized in terms of book like structure of the imbedding space and introduction of p-adic copies of imbedding space glued to real imbedding space along common rational and algebraic points mean radical generalization of the standard quantum theory framework. Hierarchy of Planck constants brings in dark matter as a hierarchy of macroscopically quantum coherent phases and p-adic physics gives rise to the correlates of intentionality and cognition.

4. Resemblance with quantum TGD identified as almost topological conformal QFT

Geometric Langlands program brings strongly to my uneducated mind TGD as an almost topological QFT with generalized conformal symmetries (light-like 3-surfaces are metrically 2-dimensional). 4-D twisted SYM would be replaced with Kähler action plus holography in the restricted sense that 4-D physics provides a (non-faithful) classical representation of the fundamental 3-D lightlike partonic quantum physics in terms of the preferred extrema of Kähler action having identification as generalized Bohr orbits. This is necessary for quantum measurement theory to make sense at the fundamental level.

Punctures, whose interpretation remains unclear in the Langlands program, associate themselves naturally with the intersections of the initial and final partonic 2-surface of particle reaction with the generalized number theoretic braids (strands can now fuse) defined by the orbits of the minima of Higgs expectation identified as a generalized eigenvalue of the modified Dirac operator (certain holomorphic function on partonic 2-surface by the properties of modified Dirac). Partons correspond to quantum states created by applying fermionic fields at the points of the number theoretic braid so that one has a concrete physical interpretation. TGD as a generalized number theory vision of course relies on the idea that fundamental physics provides a representation for number theory understood in some very general sense.

Physically punctures correspond to the lowest step in a dimensional hierarchy. Light-like 3-surface decomposes to cells bounded by 2-D surfaces such that each 3- region is independent dynamical unit. One has effectively discretized 3-D physics. The resulting 2-D surfaces (partons) obey conformal field theory separately and a collection of 1-D curves serves as causal determinants for them. Number theoretic universality in turn forces to select Higgs minima as a subset of points common to real and p-adic partonic 2-surfaces.

What is interesting that the number theoretical braids emerge in the TGD based proposal for the formulation of the number theoretic Langlands program based on Gal(Q)= S_∞ identification and relates also naturally to the conformal field approach appearing in the geometric Langlands. Could number theoretic braids allow to unify number theoretic and geometric Langlands as the unification of number fields and rational function fields provided by the notion of infinite prime suggests? This I cannot of course answer. These are just ideas inspired by the physics of TGD. It would be flattering if some real mathematician would consider these ideas seriously. I dare to believe that since TGD seems to be a working physical theory it could help also to discover "working" mathematics.

The work of Kanarev and Mizuno about cold fusion in electrolysis

The article of Kanarev and Mizuno [1] reports findings supporting the occurrence of cold fusion in NaOH and KOH hydrolysis. The situation is different from standard cold fusion where heavy water D₂O is used instead of H₂O.

1. One can understand the cold fusion reactions reported by Mizuno as nuclear reactions in which part of what I call dark proton string having negatively charged color bonds (essentially a zoomed up variant of ordinary nucleus with large Planck constant) suffers a phase transition to ordinary matter and experiences ordinary strong interactions with the nuclei at the cathode. In the simplest model the final state would contain only ordinary nuclear matter.

2. Negatively charged color bonds could correspond to pairs of quark and antiquark or to pairs of color octet electron and antineutrino having mass of order 1 MeV. Also quantum superpositions of quark and lepton pairs can be considered. Note that TGD predicts that leptons can have colored excitations and production of neutral leptopions formed from them explains the anomalous production of electron-positron pairs associated with heavy ion collisions near Coulomb wall.

3. The so called H_1.5O anomaly of [2] can be understood if 1/4 of protons of water forms dark lithium nuclei or heavier nuclei formed as sequences of these just as ordinary nuclei are constructed as sequences of ⁴He and lighter nuclei in nuclear string model. The results force to consider the possibility that nuclear isotopes unstable as ordinary matter can be stable dark matter. In the formation of these sequence the negative electronic charge of hydrogen atoms goes naturally to the color bonds. The basic interaction would generate charge quark pair (or a pair of color octet electron and antineutrino or a quantum superposition of quark and lepton pair) plus color octet neutrino. By lepton number conservation each electron pair would give rise to a color singlet particle formed by two color octet neutrinos and defining the analog of leptobaryon. Di-neutrino would leave the system unless unless it has large enough mass. Neutrino mass scale .1 eV gives for the Compton time scale the estimate .1 attoseconds which would suggest that di-neutrinos do not leak out. Recall that attosecond is the time scale in which H_1.5O behavior prevails.

4. The data of Mizuno requires that the protonic strings have net charge of three units and by em stability have neutral color bonds at ends and negatively charged bonds in between. Dark variants of Li isotopes would be in question. The so called lithium problem of cosmology (the observed abundance of lithium is by a factor 2.5 lower than predicted by standard cosmology [3]) can be resolved if lithium nuclei transform partially to dark lithium nuclei.

5. Biologically important ions K⁺, Cl^-, Ca⁺⁺ appear in cathode in plasma electrolysis and would be produced in cold nuclear reactions of dark Li nuclei of water and Na⁺. This suggests that cold nuclear reactions occur also in living cell and produce metabolic energy. There exists evidence for nuclear transmutations in living matter [4]. In particular, Kervran claims that it is very difficult to understand where the Ca in egg shells comes from. Cell membrane would provide the extremely strong electric field perhaps creating the plasma needed for cold nuclear reactions somewhat like in plasma electrolysis.

6. The model is consistent with the model for cold fusion of deuterium nuclei [5]. In this case nuclear reaction would however occur on the "dark side". The absence of He from reaction products can be understood if the D nuclei in Pd target are transformed by weak interactions between D and Pd nuclei to their neutral counterparts analogous to di-neutrons. Neutral color bond could transform to negatively charged one by the exchange of W⁺ boson of a scaled version of weak interactions with the range of interaction given by atomic length scale. Also exchange of charge ρ meson of scaled down variant of QCD could affect the same thing. This interaction might be at work also for ordinary nuclei in condensed matter and ordinary nuclei could contain protons and negatively charged color bonds neutrons. The difference in mass would be very small since the quarks have mass of order MeV.

The model leads also to a new understanding of ordinary [6] and plasma electrolysis of water [7], and allows to identify hydrogen bond as dark OH bond.

1. The model for plasma hydrolysis relies on the observation of Kanarev that the energy of OH bonds in water is reduce from about 8 eV to a value around .5 eV which corresponds to the fundamental metabolic energy quantum resulting in dropping of proton from atomic k=137 space-time sheet and also to a typical energy of hydrogen bond. This suggests the possibility that hydrogen bond is actually a dark OH bond. From 1/hbar-proportionality of perturbative contribution of Coulomb energy for bond one obtains that dark bond energy scales as 1/hbar so that dark OH bond could be in question. In Kanarev's plasma electrolysis the temperature is between .5-1 eV and thermal radiation could induce producing 2H₂+O₂ by the splitting of the dark OH bonds. One could have hbar=2⁴×hbar₀. Also in the ordinary electrolysis the OH bond energy is reduced by a factor of order 2 which suggest that in this case one has hbar=2×hbar₀.

2. The transformation of OH bonds to their dark counterparts requires energy and this energy would come from dark nuclear reactions. The liberated (dark) photons could kick protons from (dark) atomic space-time sheets to smaller space-time sheets and remote metabolism would provide the energy for the transformation of OH bond. The existence of dark hydrogen bonds with energies differing by integer scaling is predicted and powers of 2 are favored. It is known that at least two hydrogen bonds for which energies differ by factor 2 exist in ice [8].

3. In plasma electrolysis the increase of the input voltage implies a mysterious reduction of the electron current with the simultaneous increase of the size of the plasma region near the cathode. The electronic charge must go somewhere and the natural place are negative color bonds connecting dark protons to dark lithium isotopes. The energy liberated in cold nuclear reactions would create plasma by ionizing hydrogen atoms which in turn would generate more dark protons fused to dark lithium isotopes and increase the rate of energy production by dark nuclear reactions. This means a positive feedback loop analogous to that occurring in ordinary nuclear reactions.

The model explains also the burning of salt water discovered by Kanzius [9] as a special case of plasma electrolysis since the mechanism does not necessitate the presence of either anode, cathode, or electron current.

1. The temperature of the flame is estimated to be 1500 C. The temperature in water could be considerably higher and 1500 C defines a very conservative estimate. Hydrolysis would be preceded by the transformation of HO bonds to hydrogen bonds and dark nuclear reactions would provide the energy. Again positive feedback loop should be created. Dark radio wave photons would transform to microwave photons and together with nuclear energy production would keep the water at the temperature corresponding to the energy of.017 eV (for conservative estimate T=.17 eV in water) so that dark OH bonds would break down thermally.

2. For T=1500 C the energy of dark OH bond (hydrogen bond) would be very low, around .04 eV for hbar=180×hbar₀ and nominal value 8 eV OH bond energy (this is not far from the energy assignable to the membrane resting potential) from the condition that dark radio wave frequency 13.65 MHz corresponds to the microwave frequency needed to heat water by the rotational excitation of water molecules.

3. Visible light would result as dark protons drop from k=165 space-time sheet to any larger space-time sheet or from k=164 to k=165 space-time sheet (2 eV radiation). 2 eV photons would explain the yellow color in the flame (not red as I have claimed earlier). The red light present in Kanarev's experiment can be also understood since there is entire series E(n)= E_∞× (1-2^-n) of energies corresponding to transitions to space-time sheets with increasing p-adic length scale. For k=165 n<6 corresponds to red or infrared light and n>5 to yellow light.

4. There is no detectable or perceivable effect on hand by the radio wave radiation. The explanation would be that dark hydrogen bonds in cellular water correspond to a different values of Planck constant. One should of course check whether the effect is really absent.

For more details see the chapter Nuclear String Hypothesis.

References

[1] Cold fusion by plasma electrolysis of water, Ph. M. Kanarev and T. Mizuno (2002),
http://www.guns.connect.fi/innoplaza/energy/story/Kanarev/coldfusion/.

[2] M. Chaplin (2005), Water Structure and Behavior,
http://www.lsbu.ac.uk/water/index.html.
For 41 anomalies see http://www.lsbu.ac.uk/water/anmlies.html.
For the icosahedral clustering see http://www.lsbu.ac.uk/water/clusters.html.
J. K. Borchardt(2003), The chemical formula H₂O - a misnomer, The Alchemist 8 Aug (2003).
R. A. Cowley (2004), Neutron-scattering experiments and quantum entanglement, Physica B 350 (2004) 243-245.
R. Moreh, R. C. Block, Y. Danon, and M. Neumann (2005), Search for anomalous scattering of keV neutrons from H₂O-D₂O mixtures, Phys. Rev. Lett. 94, 185301.

[3] C. Charbonnel and F. Primas (2005), The lithium content of the Galactic Halo stars.
See also Lithium.

[4]C. L. Kervran (1972), Biological transmutations, and their applications in chemistry, physics, biology, ecology, medicine, nutrition, agriculture, geology, Swan House Publishing Co.
P. Tompkins and C. Bird (1973), The secret life of plants, Harper and Row, New York.

[5] Cold fusion is back at the American Chemical Society.
See also Cold fusion - hot news again .

[6] Electrolysis of water.

[7] P. Kanarev (2002), Water is New Source of Energy, Krasnodar.

[8] J-C. Li and D.K. Ross (1993), Evidence of Two Kinds of Hydrogen Bonds in Ices. J-C. Li and D.K. Ross, Nature, 365, 327-329.

[9] Burning salt water.

The Lie-algebra of symmetries of M-matrix forms an infinite-dimensional Jordan algebra

The notion of Jordan algebra was born as a mathematization of algebra of observables. Also Lie algebra can be seen as this kind of mathematization of the notion of observable. The linear combinations of Hermitian operators with real coefficients are Hermitian and define an algebra under the product A*B= (AB+BA)/2, which is commutative and non-associative but satisfies the weaker associativity condition (xy)(xx)= x(y(xx)).

There exists four infinite families of Jordan algebras plus one exceptional Jordan algebra. The finite-dimensional real, complex, and quaternionic matrix algebras with product defined as above are Jordan algebras. Also the Euclidian gamma matrix algebra defined by Euclidian inner product and with real coefficients is Jordan algebra and known as so called spin factor: now the commutativity is not put in by hand. The exceptional Jordan algebra consists of a real linear space of Hermitian 3× 3 matrices with octonionic coefficients and with symmetrized product.

1. The notion of finite measurement resolution leads to infinite-dimensional Jordan algebra in TGD framework

Hermitian operators of N subset M, where N and M are hyperfinite factors of type II₁ and N specifies the measurement resolution, act as maximal symmetries of M-matrix so that finite measurement resolution corresponds to an infinite-D symmetry group and Jordan algebra corresponds now to operators whose action has no detectable physical effect rather than algebra of observables. A so called Hermitian Jordan algebra is in question. Of course, also Lie-algebra commutator i(AB-BA) defines a Hermitian operator in N.

The maximal symmetries of M-matrix mean that the Hermitian generators of the algebra define a generalization of finite-dimensional Jordan algebra. The condition that all Hermitian operators involved are finite-dimensional brings in mind the definition of the permutation group S_∞ as consisting of finite permutations only and also the definition of infinite-dimensional Clifford algebra. Thus the natural interpretation of the algebra in question would be as maximal possible dynamical gauge symmetry implied by the finite measurement resolution. The active symmetries would be analogous to global gauge transformations and act non-trivially on all tensor factors in tensor product representation as a tensor product of 2× 2 Clifford algebras.

Quaternionic Jordan algebra is natural in TGD framework since 2× 2 Clifford algebra reduces to complexified quaternions and contains as sub-algebras real and complex Jordan algebras. Also Clifford algebra of world of classical worlds is a generalized Jordan algebra.

2. Octonions and TGD

There are intriguing hints that octonions might be important for TGD.

1. U(1), SU(2), and SU(3) are the factors of standard model gauge group and also the natural symmetries of minimal Jordan algebras relying on complex numbers, quaternions, and octonions. These symmetry groups relate also naturally to the geometry of CP₂.

2. 8-D Clifford algebra allows also octonionic representation.

3. The idea that one could make HFF of type II₁ a genuine local algebra analogous to gauge algebra can be realized only if the coordinate is non-associative since otherwise the coordinate can be represented as a tensor factor represented by a matrix algebra. Octonionic coordinate means an exception and would make 8-D imbedding space unique in that it would allow local version of HFF of type II₁.

4. These observations partially motivate a nebulous concept that I have christened HO-H duality (see this) - admittedly a rather speculative idea -stating that TGD can be formulated alternatively using hyper-octonions (subspace of complexified octonions with Minkowskian signature of metric) as the imbedding space and assuming that the dynamics is determined by the condition that space-time surfaces are hyper-quaternionic or co-hyper-quaternionic (and thus associative or co-associative). Associativity condition would determine the dynamics.

3. What about the octonionic Jordan algebra?

The question is therefore whether also 3×3 octonionic Jordan algebra might have some role in TGD framework.

1. Suppose for a moment that the above interpretation for the Hermitian operators as elements of a sub-factor N defining the measurement resolution generalizes also to the case of octonionic state space and operators represented as octonionic matrices. Also the direct sums of octonion valued matrices belonging to the octonionic Jordan algebra define a Jordan algebra and included algebras would now correspond to direct sums for copies of this Jordan algebra. One could perhaps say that the gauge symmetries associated with octonionic N would reduce to the power SU(3)_oⁿ= SU(3)_o× SU(3)_o×... of the octonionic SU(3) acting on the fundamental triplet representation.

2. Triplet character is obviously problematic and one way out could be projectivization leading to the octonionic counterpart of CP₂. Octonionic scalings should not affect the physical state so that physical states as octonionic rays would correspond to octonionic CP_n. It is not however possible to realize the linear superposition of quantum states in CP_n. The octonionic (quaternionic) counterpart of CP₂ would be 2× 8-dimensional and U(2)_o would act as a matrix multiplication in this space. Realizing associativity (commutativity) condition for 2× 8 spinors defined by octonionic CP₂ by replacing octonions with quaternions (complex numbers) would give 2× 4-dimensional (2× 2-dimensional) space.

3. The first question is whether CP₂ as a factor of imbedding space could somehow relate to the octonionic Jordan algebra. Could one think that this factor relates to the configuration space degrees of freedom assignable to CP₂ rather than Clifford algebra degrees of freedom? That color does not define spin like quantum numbers in TGD would conform with this. Note that the partial waves associated S² associated with light-cone boundary would correspond naturally to SU(2) and quaternionic algebra.

4. Second question is whether the HFF of type II₁ could result from its possibly existing octonionic generalization by these two steps and whether the reduction of the octonionic symmetries to complex situation would give SU(3)× SU(3)... reducing to U(2)× U(2)× .... The Lie-algebra of symmetries of M-matrix forms a Jordan algebra.

For a background see the chapter Construction of Quantum Theory: S-matrix of "Towards S-matrix".

Comments about E8 theory of Garrett Lisi

I have been a week in travel and during this time there has been a lot of fuss about the E₈ theory proposed by Garrett Lisi in physics blogs such as Not-Even-Wrong and Reference Frame, in media, and even New Scientist wrote about the topic. I have been also asked to explain whether there is some connection between Lisi's theory and TGD.

1. Objections against Lisi’s theory

The basic claim of Lisi is that one can understand the particle spectrum of standard model in terms of the adjoint representation of a noncompact version of E₈ group.

There are several objections against E₈ gauge theory interpretation of Lisi.

1. Statistics does not allow to put fermions and bosons in the same gauge multiplet. Also the identification of graviton as a part of a gauge multiplet seems very strange if not wrong since there are no roots corresponding to a spin 2 two state.

2. Gauge couplings come out wrong for fermions and one must replace YM action with an ad hoc action.

3. Poincare invariance is a problem. There is no clear relationship with the space-time geometry so that the interpretation of spin as E₈ quantum numbers is not really justified.

4. Finite-dimensional representations of non-compact E₈ are non-unitary. Non-compact gauge groups are however not possible since one would need unitary infinite-dimensional representations which would change the physical interpretation completely. Note that also Lorentz group has only infinite-D unitary representations and only the extension to Poincare group allows to have fields transforming according to finite-D representations.

5. The prediction of three fermion families is nice but one can question the whole idea of putting particles with mass scales differing by a factor of order 10¹² (top and neutrinos) into same multiplet. For some reason colleagues stubbornly continue to see fundamental gauge symmetries where there seems to be no such symmetry. Accepting the existence of a hierarchy of mass scales seems to be impossible for a theoretical physicistin main main stream although fractals have been here for decades.

6. Also some exotic particles not present in standard model are predicted: these carry weak hyper charge and color (6-plet representation) and are arranged in three families.

2. Three attempts to save Lisi’s theory

To my opinion, the shortcomings of E₈ theory as a gauge theory are fatal but the possibility to put gauge bosons and fermions of the standard model to E₈ multiplets is intriguing and motivatse the question whether the model could be somehow saved by replacing gauge theory with a theory based on extended fundamental objects possessing conformal invariance.

1. In TGD framework H-HO duality allows to consider Super-Kac Moody algebra with rank 8 with Cartan algebra assigned with the quantized coordinates of partonic 2-surface in 8-D Minkowski space M⁸ (identifiable as hyper-octonions HO). The standard construction for the representations of simply laced Kac-Moody algebras allows quite a number of possibilities concerning the choice of Kac-Moody algebra and the non-compact E₈ would be the maximal choice.

2. The first attempt to rescue the situation would be the identification of the weird spin 1/2 bosons in terms of supersymmetry involving addition of righthanded neutrino to the state giving it spin 1. This options does not seem to work.

3. The construction of representations of non-simply laced Kac-Moody algebras (performed by Goddard and Olive at eighties) leads naturally to the introduction of fermionic fields for algebras of type B, C, and F: I do not know whether the construction has been made for G₂. E₆, E₇, and E₈ are however simply laced Lie groups with single root length 2 so that one does not obtain fermions in this manner.

4. The third resuscitation attempt is based on fractional statistics. Since the partonic 2-surfaces are 2-dimensional and because one has a hierarchy of Planck constants, one can have also fractional statistics. Spin 1/2 gauge bosons could perhaps be interpreted as anyonic gauge bosons meaning that particle exchange as permutation is replaced with braiding homotopy. If so, E₈ would not describe standard model particles and the possibility of states transforming according to its representations would reflect the ability of TGD to emulate any gauge or Kac-Moody symmetry.

   The standard construction for simply laced Kac-Moody algebras might be generalized considerably to allow also more general algebras and fractionization of spin and other quantum numbers would suggest fractionization of roots. In stringy picture the symmetry group would be reduced considerably since longitudinal degrees of freedom (time and one spatial direction) are unphysical. This would suggest a symmetry breaking to SO(1,1)× E₆ representations with ground states created by tachyonic Lie allebra generators and carrying mass squared –2 in suitable units. In TGD framework the tachyonic conformal weight can be compensated by super-canonical conformal weight so that massless states getting their masses via Higgs mechanism and p-adic thermodynamics would be obtained.

3. Could supersymmetry rescue the situation?

E₈ is unique among Lie algebras in that its adjoint rather than fundamental representation has the smallest dimension. One can decompose the 240 roots of E₈ to 112 roots for which two components of SO(7,1) root vector are +/- 1 and to 128 vectors for which all components are +/- 1/2 such that the sum of components is even. The latter roots Lisi assigns to fermionic states. This is not consistent with spin and statistics although SO(3,1) spin is half-integer in M⁸ picture.

The first idea which comes in mind is that these states correspond to super-partners of the ordinary fermions. In TGD framework they might be obtained by just adding covariantly constant right-handed neutrino or antineutrino state to a given particle state. The simplest option is that fermionic super-partners are complex scalar fields and sbosons are spin 1/2 fermions. It however seems that the super-conformal symmetries associated with the right-handed neutrino are strictly local in the sense that global super-generators vanish. This would mean that super-conformal super-symmetries change the color and angular momentum quantum numbers of states. This is a pity if indeed true since super-symmetry could be broken by different p-adic mass scale for super partners so that no explicit breaking would be needed.

4. Could Kac Moody variant of E₈ make sense in TGD?

One can leave gauge theory framework and consider stringy picture and its generalization in TGD framework obtained by replacing string orbits with 3-D light-like surfaces allowing a generalization of conformal symmetries.

H-HO duality is one of the speculative aspects of TGD. The duality states that one can either regard imbedding space as H=M⁴×CP₂ or as 8-D Minkowski space M⁸ identifiable as the space HO of hyper-octonions which is a subspace of complexified octonions. Spontaneous compactification for M⁸ described as a phenomenon occurring at the level of Kac-Moody algebra would relate HO-picture to H-picture which is definitely the fundamental picture. For instance, standard model symmetries have purely number theoretic meaning in the resulting picture.

The question is whether the non-compact E₈ could be replaced with the corresponding Kac Moody algebra and act as a stringy symmetry. Note that this would be by no means anything new. The Kac-Moody analogs of E₁₀ and E₁₁ algebras appear in M-theory speculations. Very little is known about these algebras. Already E_n, n>8 is infinite-dimensional as an analog of Lie algebra. The following argument shows that E₈ representations do not work in TGD context unless one allows anyonic statistics.

1. In TGD framework space-time dimension is D=8. The speculative hypothesis of HO-H duality (see this) inspired by string model dualities states that the descriptions based on the two choices of imbedding space are dual. One can start from 8-D Cartan algebra defined by quantized M⁸ coordinates regarded as fields at string orbit just as in string model. A natural constraint is that the symmetries act as isometries or holonomies of the effectively compactified M⁸. The article Octonions of John Baez discusses exceptional Lie groups and shows that compact form of E₈ appears as isometry group of 16-dimensional octo-octonionic projective plane E₈/(Spin(16)Z₂): the analog of CP₂ for complexified octonions. There is no 8-D space allowing E₈ as an isometry group. Only SO(1,7) can be realized as the maximal Lorentz group with 8-D translational invariance.

2. In HO picture some Kac Moody algebra with rank 8 acting on quantized M⁸ coordinates defining stringy fields is natural. The charged generators of this algebra are constructible using the standard recipe involving operators creating coherent states and their conjugates obtained as operator counterparts of plane waves with momenta replaced by roots of the simply laced algebra in question and by normal ordering.

3. Poincare group has 4-D maximal Cartan algebra and this means that only 4 Euclidian dimensions remain. Lorentz generators can be constructed in standard manner in terms of Kac-Moody generators as Noether currents.

4. The natural Kac-Moody counterpart for spontaneous compactification to CP₂ would be that these dimensions give rise to the generators of electroweak gauge group identifiable as a product of isometry and holonomy groups of CP₂ in the dual H-picture based on M⁴×CP₂. Note that in this picture electroweak symmetries would act geometrically in E⁴ whereas in CP₂ picture they would act only as holonomies.

Could one weaken the assumption that Kac-Moody generators act as symmetries and that spin-statistics relation would be satisfied?

1. The hierarchy of Planck constants relying on the generalization of the notion of imbedding space breaks Poincare symmetry to Lorentz symmetry for a given sector of the world of classical worlds for which one considers light-like 3-surfaces inside future and past directed light cones. Translational invariance is obtained from the wave function for the position of the tip of the light cone in M⁴. In this kind of situation one could consider even E₈ symmetry as a dynamical symmetry.

2. The hierarchy of Planck constants involves a hierarchy of groups and fractional statistics at the partonic 2-surface with rotations interpreted as braiding homotopies. The fractionization of spin allows anyonic statistics and could allow bosons with anyonic half-odd integer spin. Also more general fractional spins are possible so that one can consider also more general algebras than Kac-Moody algebras by allowing roots to have more general values. Quantum versions of Kac-Moody algebras would be in question. This picture would be consistent with the view that TGD can emulate any gauge algebra with 8-D Cartan algebra and Kac-Moody algebra dynamically. This vision was originally inspired by the study of the inclusions of hyper-finite factors of type II₁. Even higher dimensional Kac-Moody algebras are predicted to be possible.

3. It must be emphasized that these considerations relate in TGD framework to Super-Kac Moody algebra only. The so called super-canonical algebra is the second quitessential part of the story. In particular, color is not spinlike quantum number for quarks and quark color corresponds to color partial waves in the world of classical worlds or more concretely, to the rotational degrees of freedom in CP₂ analogous to ordinary rotational degrees of freedom of rigid body. Arbitrarily high color partial waves are possible and also leptons can move in triality zero color partial waves and there is a considerable experimental evidence for color octet excitations of electron and muon but put under the rug.

5. Can one interpret three fermion families in terms of E₈ in TGD framework?

The prediction of three fermion generations by E₈ picture must be taken very seriously. In TGD three fermion generations correspond to three lowest genera g=0,1,2 (handle number) for which all 2-surfaces have Z₂ as global conformal symmetry (hyper-ellipticity). One can assign to the three genera a dynamical SU(3) symmetry. They are related by SU(3) triality, which brings in mind the triality symmetry acting on fermion generations in E₈ model. SU(3) octet and singlet bosons correspond to pairs of light-like 3-surfaces defining the throats of a wormhole contact and since their genera can be different one has color singlet and octet bosons. Singlet corresponds to ordinary bosons. Color octet bosons must be heavy since they define neutral currents between fermion families.

The three E₈ anyonic boson families cannot represent family replication since these symmetries are not local conformal symmetries: it obviously does not make sense to assign a handle number to a given point of partonic 2-surface! Also bosonic octet would be missing in E₈ picture.

One could of course say that in E₈ picture based on fractional statistics, anyonic gauge bosons can mimic the dynamical symmetry associated with the family replication. This is in spirit with the idea that TGD Universe is able to emulate practically any gauge - or Kac-Moody symmetry and that TGD Universe is busily mimicking also itself.

To sum up, the rank 8 Kac-Moody algebra - emerging naturally if one takes HO-H duality seriously - corresponds very naturally to Kac-Moody representations in terms of free stringy fields for Poincare-, color-, and electro-weak symmetries. One can however consider the possibility of anyonic symmetries and the emergence of non-compact version of E₈ as a dynamical symmetry, and TGD suggests much more general dynamical symmetries if TGD Universe is able to act as the physics analog of the Universal Turing machine.

For more details see the chapter TGD as a Generalized Number Theory II: Quaternions, Octonions, and their Hyper Counterparts of "TGD as Generalized Number Theory".

References

[1] G. Lisi (2007), An exceptionally simple theory of everything,

[2] Z. Merali (1007), Is mathematical pattern the theory of everything?, New Scientist issue 2630.

[3] E₈ .

[4] J. Baez (2002), The Octonions.

Cosmic rays above GKZ bound from distant galactic nuclei

Lubos tells about the announcement of Pierre Auger Collaboration relating to ultrahigh energy cosmic rays. I glue below a popular summary of the findings.

Scientists of the Pierre Auger Collaboration announced today (8 Nov. 2007) that active galactic nuclei are the most likely candidate for the source of the highest-energy cosmic rays that hit Earth. Using the Pierre Auger Observatory in Argentina, the largest cosmic-ray observatory in the world, a team of scientists from 17 countries found that the sources of the highest-energy particles are not distributed uniformly across the sky. Instead, the Auger results link the origins of these mysterious particles to the locations of nearby galaxies that have active nuclei in their centers. The results appear in the Nov. 9 issue of the journal Science.

Active Galactic Nuclei (AGN) are thought to be powered by supermassive black holes that are devouring large amounts of matter. They have long been considered sites where high-energy particle production might take place. They swallow gas, dust and other matter from their host galaxies and spew out particles and energy. While most galaxies have black holes at their center, only a fraction of all galaxies have an AGN. The exact mechanism of how AGNs can accelerate particles to energies 100 million times higher than the most powerful particle accelerator on Earth is still a mystery.

About million cosmic ray events have been recorded and 80 of them correspond to particles with energy above the so called GKZ bound, which is .54 × 10¹¹ GeV. Electromagnetically interacting particles with these energies from distant galaxies should not be able to reach Earth. This would be due to the scattering from the photons of the microwave background. About 20 particles of this kind however comes from the direction of distant active galactic nuclei and the probability that this is an accident is about 1 per cent. Particles having only strong interactions would be in question. The problem is that this kind of particles are not predicted by the standard model (gluons are confined).

1. What does TGD say about the finding?

TGD provides an explanation for the new kind of particles.

1. The original TGD based model for the galactic nucleus is as a highly tangled cosmic string (in TGD sense of course, see this). Much later it became clear that also TGD based model for black-hole is as this kind of string like object near Hagedorn temperature (see this and this). Ultrahigh energy particles could result as decay products of a decaying split cosmic string as an extremely energetic galactic jet. Kind of cosmic fire cracker would be in question. Originally I proposed this decay as an explanation for the gamma ray bursts. It seems that gamma ray bursts however come from thickened cosmic strings having weaker magnetic field and much lower energy density (see this).

2. TGD predicts particles having only strong interactions (see this). I have christened these particles super-canonical quanta. These particles correspond to the vibrational degrees of freedom of partonic 2-surface and are not visible at the quantum field theory limit for which partonic 2-surfaces become points.

2. What super-canonical quanta are?

Super-canonical quanta are created by the elements of super-canonical algebra, which creates quantum states besides the super Kac-Moody algebra present also in super string model. Both algebras relate closely to the conformal invariance of light-like 3-surfaces.

1. The elements of super-canonical algebra are in one-one correspondence with the Hamiltonians generating symplectic transformations of δM⁴₊× CP₂. Note that the 3-D light-cone boundary is metrically 2-dimensional and possesses degenerate symplectic and Kähler structures so that one can indeed speak about symplectic (canonical) transformations.

2. This algebra is the analog of Kac-Moody algebra with finite-dimensional Lie group replaced with the infinite-dimensional group of symplectic transformations (see this). This should give an idea about how gigantic a symmetry is in question. This is as it should be since these symmetries act as the largest possible symmetry group for the Kähler geometry of the world of classical worlds (WCW) consisting of light-like 3-surfaces in 8-D imbedding space for given values of zero modes (labelling the spaces in the union of infinite-dimensional symmetric spaces). This implies that for the given values of zero modes all points of WCW are metrically equivalent: a generalization of the perfect cosmological principle making theory calculable and guaranteing that WCW metric exists mathematically. Super-canonical generators correspond to gamma matrices of WCW and have the quantum numbers of right handed neutrino (no electro-weak interactions). Note that a geometrization of fermionic statistics is achieved.

3. The Hamiltonians and super-Hamiltonians have only color and angular momentum quantum numbers and no electro-weak quantum numbers so that electro-weak interactions are absent. Super-canonical quanta however interact strongly.

3. Also hadrons contain super-canonical quanta

One can say that TGD based model for hadron is at space-time level kind of combination of QCD and old fashioned string model forgotten when QCD came in fashion and then transformed to the highly unsuccessful but equally fashionable theory of everything.

1. At quantum level the energy corresponding to string tension explaining about 70 per cent of proton mass corresponds to super-canonical quanta (see this). Supercanonical quanta allow to understand hadron masses with a precision better than 1 per cent.

2. Super-canonical degrees of freedom allow also to solve spin puzzle of the proton: the average quark spin would be zero since same net angular momentum of hadron can be obtained by coupling quarks of opposite spin with angular momentum eigen states with different projection to the direction of quantization axis.

3. If one considers proton without valence quarks and gluons, one obtains a boson with mass very nearly equal to that of proton (for proton super-canonical binding energy compensates quark masses with high precision). These kind of pseudo protons might be created in high energy collisions when the space-time sheets carrying valence quarks and super-canonical space-time sheet separate from each other. Super-canonical quanta might be produced in accelerators in this manner and there is actually experimental support for this from Hera (see this).

4. The exotic particles could correspond to some p-adic copy of hadron physics predicted by TGD and have very large mass smaller however than the energy. Mersenne primes M_n= 2ⁿ-1 define excellent candidates for these copies. Ordinary hadrons correspond to M₁₀₇. The protons of M₃₁ hadron physics would have the mass of proton scaled up by a factor 2^(107-31)/2=2³⁸≈ 2.6×10¹¹. Energy should be above 2.6 × 10¹¹ GeV and is above .54 × 10¹¹ GeV for the particles above the GKZ limit. Even super-canonical quanta associated with proton of this kind could be in question. Note that CP₂ mass corresponds roughly to about 10¹⁴ proton masses.

5. Ideal blackholes would be very long highly tangled string like objects, scaled up hadrons, containing only super-canonical quanta. Hence it would not be surprising if they would emit super-canonical quanta. The transformation of supernovas to neutron stars and possibly blackholes would involve the fusion of hadronic strings to longer strings and eventual annihilation and evaporation of the ordinary matter so that only super-canonical matter would remain eventually. A wide variety of intermediate states with different values of string tension would be possible and the ultimate blackhole would correspond to highly tangled cosmic string. Dark matter would be in question in the sense that Planck constant could be very large.

Quantum version of Expanding Earth theory

TGD predicts that cosmic expansion at the level of individual astrophysical systems does not take place continuously as in classical gravitation but through discrete quantum phase transitions increasing gravitational Planck constant and thus various quantum length and time scales. The reason would be that stationary quantum states for dark matter in astrophysical length scales cannot expand. One would have the analog of atomic physics in cosmic scales. Increases of hbar by a power of two are favored in these transitions but also other scalings are possible.

This has quite far reaching implications.

1. These periods have a highly unique description in terms of a critical cosmology for the expanding space-time sheet. The expansion is accelerating. The accelerating cosmic expansion can be assigned to this kind of phase transition in some length scale (TGD Universe is fractal). There is no need to introduce cosmological constant and dark energy would be actually dark matter.

2. The recently observed void which has same size of about 10⁸ light years as large voids having galaxies near their boundaries but having an age which is much higher than that of the large voids, would represent one example of jerk-wise expansion.

3. This picture applies also to solar system and planets might be perhaps seen as having once been parts of a more or less connected system, the primordial Sun. The Bohr orbits for inner and outer planets correspond to gravitational Planck constant which is 5 times larger for outer planets. This suggests that the space-time sheet of outer planets has suffered a phase transition increasing the size scale by a factor of 5. Earth can be regarded either as n=1 orbit for Planck constant associated with outer planets or n= 5 orbit for inner planetary system. This might have something to do with the very special position of Earth in planetary system. One could even consider the possibility that both orbits are present as dark matter structures. The phase transition would also explain why n=1 and n=2 Bohr orbits are absent and one only n=3,4, and 5 are present.

4. Also planets should have experienced this kind of phase transitions increasing the radius: the increase by a factor two would be the simplest situation.

The obvious question - that I did not ask - is whether this kind of phase transition might have occurred for Earth and led from a completely granite covered Earth -Pangeia without seas- to the recent Earth. Neither it did not occur to me to check whether there is any support for a rapid expansion of Earth during some period of its history.

Situation changed when my son Paavo visited me last Saturday and told me about a Youtube video by Neal Adams, an American comic book and commercial artist who has also produced animations for geologists. We looked the amazing video a couple of times and I looked it again yesterday. The video is very impressive (no wonder!) but in the lack of references skeptic probably cannot avoid the feeling that Neal Adams might use his highly developed animation skills to cheat you. I found also a polemic article of Adams but again the references were lacking. Perhaps the reason of polemic tone was that the concrete animation models make the expanding Earth hypothesis very convincing but geologists dare not consider seriously arguments by a layman without a formal academic background.

1. The claims of Adams

The basic claims of Adams were following.

1. The radius of Earth has increased during last 185 million years (dinosaurs appeared for about 230 million years ago) by about factor 2. If this is assumed all continents have formed at that time a single super-continent, Pangeia, filling the entire Earth surface rather than only 1/4 of it since the total area would have grown by a factor of 4. The basic argument was that it is very difficult to imagine Earth with 1/4 of surface containing granite and 3/4 covered by basalt. If the initial situation was covering by mere granite -as would look natural- it is very difficult for a believer in thermodynamics to imagine how the granite would have gathered to a single connected continent.

2. Adams claims that Earth has grown by keeping its density constant, rather than expanded, so that the mass of Earth has grown linearly with radius. Gravitational acceleration would have thus doubled and could provide a partial explanation for the disappearance of dinosaurs: it is difficult to cope in evolving environment when you get slower all the time.

3. Most of the sea floor is very young and the areas covered by the youngest basalt are the largest ones. This Adams interprets this by saying that the expansion of Earth is accelerating. The alternative interpretation is that the flow rate of the magma slows down as it recedes from the ridge where it erupts. The upper bound of 185 million years for the age of sea floor requires that the expansion period - if it is already over - lasted about 185 million years after which the flow increasing the area of the sea floor transformed to a convective flow with subduction so that the area is not increasing anymore.

4. The fact that the continents fit together -not only at the Atlantic side - but also at the Pacific side gives strong support for the idea that the entire planet was once covered by the super-continent. After the emergence of subduction theory this evidence as been dismissed: sounds very odd to me. It seems that geologists are doing "Wegeners" again.

5. I am not sure whether Adams mentions this objection. Subduction only occurs on the other side of the subduction zone so that the other side should show evidence of being much older in the case that oceanic subduction zones are in question. This is definitely not the case. This is explained in plate tectonics as a change of the subduction direction. My explanation would be that by the symmetry of the situation both oceanic plates bend down so that this would represent new type of boundary not assumed in the tectonic plate theory.

6. As a master visualizer Adams notices that Africa and South-America do not actually fit together in absence of expansion unless one assumes that these continents have suffered a deformation. Continents are not easily deformable stuff. The assumption of expansion implies a perfect fit of all continents without deformation.

Knowing that the devil is in the details, I must admit that some of these arguments look rather convincing to me and what I learned from Wikipedia articles supports this picture.

2. The critic of Adams of the subduction mechanism

The prevailing tectonic plate theory has been compared to the Copernican revolution in geology. The theory explains the young age of the seafloor in terms of the decomposition of the litosphere to tectonic plates and the convective flow of magma to which oceanic tectonic plates participate. The magma emerges from the crests of the mid ocean ridges representing a boundary of two plates and leads to the expansion of sea floor. The variations of the polarity of Earth's magnetic field coded in sea floor provide a strong support for the hypothesis that magma emerges from the crests.

The flow back to would take place at so called oceanic trenches near continents which represent the deepest parts of ocean. This process is known as subduction. In subduction oceanic tectonic plate bends and penetrates below the continental tectonic plate, the material in the oceanic plate gets denser and sinks into the magma. In this manner the oceanic tectonic plate suffers a metamorphosis returning back to the magma: everything which comes from Earth's interior returns back. Subduction mechanism explains elegantly formation of mountains (orogeny), earth quake zones, and associated zones of volcanic activity.

Adams is very polemic about the notion of subduction, in particular about the assumption that it generates steady convective cycle. The basic objections of Adams against subduction are following.

1. There are not enough subduction zones to allow a steady situation. According to Adams, the situation resembles that for a flow in a tube which becomes narrower. In a steady situation the flow should accelerate as it approaches subduction zones rather than slow down. Subduction zones should be surrounded by large areas of sea floor with constant age. Just the opposite is suggested by the fact that the youngest portion of sea-floor near the ridges is largest. The presence of zones at which both ocean plates bend down could improve the situation. Also jamming of the flow could occur so that the thickness of oceanic plate increases with the distance from the eruption ridge. Jamming could increase also the density of the oceanic plate and thus the effectiveness of subduction.

2. There is no clear evidence that subduction has occurred at other planets. The usual defense is that the presence of sea is essential for the subduction mechanism.

3. One can also wonder what is the mechanism that led to the formation of single super continent Pangeia covering 1/4 of Earth's surface. How probable the gathering of all separate continents to form single cluster is? The later events would suggest that just the opposite should have occurred from the beginning.

3. Expanding Earth theories are not new

After I had decided to check the claims of Adams, the first thing that I learned is that Expanding Earth theory, whose existence Adams actually mentions, is by no means new. There are actually many of them.

The general reason why these theories were rejected by the main stream community was the absence of a convincing physical mechanism of expansion or of growth in which the density of Earth remains constant.

1. 1888 Yarkovski postulated some sort of aether absorbed by Earth and transforming to chemical elements (TGD version of aether could be dark matter). 1909 Mantovani postulated thermal expansion but no growth of the Earth's mass.

2. Paul Dirac's idea about changing Planck constant led Pascual Jordan in 1964 to a modification of general relativity predicting slow expansion of planets. The recent measurement of the gravitational constant imply that the upper bound for the relative change of gravitational constant is 10 time too small to produce large enough rate of expansion. Also many other theories have been proposed but they are in general conflict with modern physics.

3. The most modern version of Expanding Earth theory is by Australian geologist Samuel W. Carey. He calculated that in Cambrian period (about 500 million years ago) all continents were stuck together and covered the entire Earth. Deep seas began to evolve then.

4. Summary of TGD based theory of Expanding Earth

TGD based model differs from the tectonic plate model but allows subduction which cannot imply considerable back flow of magma. Let us sum up the basic assumptions and implications.

1. The expansion is due to a quantum phase transition increasing the value of gravitational Planck constant and forced by the cosmic expansion in the average sense.

2. Tectonic plates do not participate to the expansion and therefore new plate must be formed and the flow of magma from the crests of mid ocean ridges is needed. The decomposition of a single plate covering the entire planet to plates to create the mid ocean ridges is necessary for the generation of new tectonic plate. The decomposition into tectonic plates is thus prediction rather than assumption.

3. The expansion forced the decomposition of Pangeia super-continent covering entire Earth for about 530 million years ago to split into tectonic plates which began to recede as new non-expanding tectonic plate was generated at the ridges creating expanding sea floor. The initiation of the phase transition generated formation of deep seas.

4. The eruption of plasma from the crests of ocean ridges generated oceanic tectonic plates which did not participate to the expansion by density reduction but by growing in size. This led to a reduction of density in the interior of the Earth roughly by a factor 1/8. From the upper bound for the age of the seafloor one can conclude that the period lasted for about 185 million years after which it transformed to convective flow in which the material returned back to the Earth interior. Subduction at continent-ocean floor boundaries and downwards double bending of tectonic plates at the boundaries between two ocean floors were the mechanisms. Thus tectonic plate theory would be more or less the correct description for the recent situation.

5. One can consider the possibility that the subducted tectonic plate does not transform to magma but is fused to the tectonic layer below continent so that it grows to an iceberg like structure. This need not lead to a loss of the successful predictions of plate tectonics explaining the generation of mountains, earthquake zones, zones of volcanic activity, etc...

6. From the video of Adams it becomes clear that the tectonic flow is East-West asymmetric in the sense that the western side is more irregular at large distances from the ocean ridge at the western side. If the magma rotates with a slightly lower velocity than the surface of Earth (like liquid in a rotating vessel), the erupting magma would rotate slightly slower than the tectonic plate and asymmetry would be generated.

7. If the planet has not experienced a phase transition increasing the value of Planck constant, there is no need for the decomposition to tectonic plates and one can understand why there is no clear evidence for tectonic plates and subduction in other planets. The conductive flow of magma could occur below this plate and remain invisible.

The biological implications might provide a possibility to test the hypothesis.

1. Great steps of progress in biological evolution are associated with catastrophic geological events generating new evolutionary pressures forcing new solutions to cope in the new situation. Cambrian explosion indeed occurred about 530 years ago (the book Wonderful Life of Stephen Gould explains this revolution in detail) and led to the emergence of multicellular creatures, and generated huge number of new life forms living in seas. Later most of them suffered extinction: large number of phylae and groups emerged which are not present nowadays.

   Thus Cambrian explosion is completely exceptional as compared to all other dramatic events in the evolution in the sense that it created something totally new rather than only making more complex something which already existed. Gould also emphasizes the failure to identify any great change in the environment as a fundamental puzzle of Cambrian explosion. Cambrian explosion is also regarded in many quantum theories of consciousness (including TGD) as a revolution in the evolution of consciousness: for instance, micro-tubuli emerged at this time. The periods of expansion might be necessary for the emergence of multicellular life forms on planets and the fact that they unavoidably occur sooner or later suggests that also life develops unavoidably.

2. TGD predicts a decrease of the surface gravity by a factor 1/4 during this period. The reduction of the surface gravity would have naturally led to the emergence of dinosaurs 230 million years ago as a response coming 45 million years after the accelerated expansion ceased. Other reasons led then to the decline and eventual catastrophic disappearance of the dinosaurs. The reduction of gravity might have had some gradually increasing effects on the shape of organisms also at microscopic level and manifest itself in the evolution of genome during expansion period.

3. A possibly testable prediction following from angular momentum conservation (ωR²= constant) is that the duration of day has increased gradually and was four times shorter during the pre-Cambrian era. For instance, genetically coded bio-clocks of simple organisms during the expansion period could have followed the increase of the length of day with certain lag or failed to follow it completely. The simplest known circadian clock is that of the prokaryotic cyanobacteria. Recent research has demonstrated that the circadian clock of Synechococcus elongatus can be reconstituted in vitro with just the three proteins of their central oscillator. This clock has been shown to sustain a 22 hour rhythm over several days upon the addition of ATP: the rhythm is indeed faster than the circadian rhythm. For humans the average innate circadian rhythm is however 24 hours 11 minutes and thus conforms with the fact that human genome has evolved much later than the expansion ceased.

4. Addition: My son told that scientists have found a fossil of a sea scorpion with size of 2.5 meters which has lived for about 10 million years for 400 million years ago in Germany (see also the article in Biology Letters). The finding would conform nicely with the much smaller value of surface gravity at that time. Also the emergence of trees could be understood in terms of a gradual growth of the maximum plant size as the surface gravity was reduced. The fact that the oldest known tree fossil is 385 million years old conforms with this picture.

5. Did intra-terrestrial life burst to the surface of Earth during Cambrian expansion?

Intra-terrestrial hypothesis is one of the craziest TGD inspired ideas about the evolution of life and it is quite possible that in its strongest form the hypothesis is unrealistic. One can however try to find what one obtains from the combination of the IT hypothesis with the idea of pre-Cambrian granite Earth. Could the harsh pre-Cambrian conditions have allowed only intra-terrestrial multicellular life? Could the Cambrian explosion correspond to the moment of birth for this life in the very concrete sense that the magma flow brought it into the day-light?

1. Gould emphasizes the mysterious fact that very many life forms of Cambrian explosion looked like final products of a long evolutionary process. Could the eruption of magma from the Earth interior have induced a burst of intra-terrestrial life forms to the Earth's surface? This might make sense: the life forms living at the bottom of sea do not need direct solar light so that they could have had intra-terrestrial origin. It is quite possible that Earth's mantle contained low temperature water pockets, where the complex life forms might have evolved in an environment shielded from meteoric bombardment and UV radiation.

2. Sea water is salty (for why this is the case see this). It is often claimed that the average salt concentration inside cell is that of the primordial sea: I do not know whether this claim can be really justified. If the claim is true, the cellular salt concentration should reflect the salt concentration of the water inside the pockets. The water inside water pockets could have been salty due to the diffusion of the salt from ground but need not have been same as that for the ocean water (higher than for cell interior and for obvious reasons). Indeed, the water in the underground reservoirs in arid regions such as Sahara is salty, which is the reason for why agriculture is absent in these regions. Note also that the cells of marine invertebrates are osmoconformers able to cope with the changing salinity of the environment so that the Cambrian revolutionaries could have survived the change in the salt concentration of environment.

3. What applies to Earth should apply also to other similar planets and Mars is very similar to Earth. The radius is .533 times that for Earth so that after quantum leap doubling the radius and thus Schumann frequency scale (7.8 Hz would be the lowest Schumann frequency) would be essentially same as for Earth now. Mass is .131 times that for Earth so that surface gravity would be .532 of that for Earth now and would be reduced to .131 meaning quite big dinosaurs! We have learned that Mars probably contains large water reservoirs in it's interior and that there is an un-identified source of methane gas usually assigned with the presence of life. Could it be that Mother Mars is pregnant and just waiting for the great quantum leap when it starts to expand and gives rise to a birth of multicellular life forms. Or expressing freely how Bible describes the moment of birth: in the beginning there was only darkness and water and then God said: Let the light come!

To sum up, TGD would provide only the long sought mechanism of expansion and a possible connection with the biological evolution. It would be indeed fascinating if Planck constant changing quantum phase transitions in planetary scale would have profoundly affected the biosphere.

For more details see the chapter Quantum Astrophysics of "Classical Physics in Many-Sheeted Spacetime".

Does Higgs boson appear with two p-adic mass scales?

The p-adic mass scale of quarks is in TGD Universe dynamical and several mass scales appear already in low energy hadron mass formulas. Also neutrinos seem to correspond to several mass scales and the large variation of electron's effective mass in condensed matter might be also partially due to the variation of p-adic mass scale. The values of Higgs mass deduced from high precision electro-weak observables converges to two values differing by order of magnitude (see this and this) and this raises the question whether also Higgs mass scale could vary and depend on experimental situation.

1. Higgs mass in standard model

In standard model Higgs and W boson masses are given by

m_H²= 2v²λ=μ²λ³,

m_W²= g²v²/4= [e²/8sin²(θ_W)] μ²λ² .

This gives

λ= [π/2α_emsin²(θ_W)] (m_H/m_W)² .

In standard model one cannot predict the value of m_H.

2. Higgs mass in TGD

In TGD framework one can try to understand Higgs mass from p-adic thermodynamics as resulting via the same mechanism as fermion masses so that the value of the parameter λ would follow as a prediction.

One must assume that p-adic temperature equals to T_p=1. The natural assumption is that Higgs can be regarded as superposition of pairs of fermion and anti-fermion at opposite throats of wormhole contact. With these assumptions the thermal expectation of the Higgs conformal weight is just the sum of contributions from both throats and two times the average of the conformal weight over that for quarks and leptons:

s_H= 2× <s> = 2× [∑_q s_q +∑_L s_L]/(N_q+N_L)

= 2∑_g=0² s_mod(g)/3+ (s_L+s_νL+ s_U+s_D)/2

= 26+5+4+5+8/2= 37 .

A couple of comments about the formula are in order.

1. The first term - two times the average of the genus dependent modular contribution to the conformal weight - equals to 26, and comes from modular degrees of freedom and does not depend on the charge of fermion.

2. The contribution of p-adic thermodynamics for super-conformal generators gives same contribution for all fermion families and depends on the em charge of fermion. The values of thermal conformal weights deduced earlier have been used. Note that only the value s_νL=4 (also s_νL=5 could be considered) is possible if one requires that the conformal weight is integer. If the standard form of the canonical identification mapping p-adics to reals is used, this must be the case since otherwise real mass would be super-heavy.

3. What p-adic mass scale Higgs corresponds?

The first guess would be that the p-adic length scale associated with Higgs boson is M₈₉. Second option is p≈ 2^k, k=97 (restricting k to be prime). If one allows k to be non-prime (these values of k are also realized) one can consider also k=91=7×13. By scaling from the expression for the electron mass, one obtains the estimates

m_H(89)≈ (37/5)^1/2× 2²¹m_e≈ 727.3 GeV , m_H(91)≈ (37/5)^1/2× 2¹⁸m_e≈ 363.5 GeV, m_H(97)≈ (37/5)^1/2× 2¹⁵m_e≈ 45.5 GeV.

A couple of comments are in order.

1. From the article of Giudice one learns that the latest estimates for Higgs mass give two widely different values, namely m_H= 31³³_-19 GeV and m_H=420⁴²⁰_-190 GeV. Since the p-adic mass scale of both neutrinos and quarks and possibly even electron can vary in TGD framework, one cannot avoid the question whether - depending on experimental situation- Higgs could appear in two different mass scales corresponding to k=91 and 97.

2. The low value of m_H(97) might be consistent with experimental facts since the couplings of fermions to Higgs can in TGD framework be weaker than in standard model because Higgs expectation does not contribute to fermion masses.

4. Unitary bound and Higgs mass

The value of λ is given in the three cases by

λ(89)≈ 4.41 , λ(91)≈ 1.10, λ(97)= .2757.

Unitarity would thus favor k=97 and k=91 also favored by the high precision data and k=91 is just at the unitarity bound λ=1) (here I am perhaps naive!). A possible interpretation is that for M₈₉ Higgs mass forces λ to break unitarity bound and that this corresponds to the emergence of M₈₉ copy of hadron physics.

For more details see the chapter p-Adic Particle Massivation: Elementary Particle Masses of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy".

Can one assign a continuous Schrödinger time evolution to light-like 3-surfaces?

Alain Connes wrote very interesting comments about factors of various types using as an example Schrödinger equation for various kinds of foliations of space-time to time=constant slices. If this kind of foliation does not exist, one cannot speak about time evolution of Schrödinger equation at all. Depending on the character of the foliation one can have factor of type I, II, or III. For instance, torus with slicing dx= ady in flat coordinates, gives a factor of type I for rational values of a and factor of type II for irrational values of a.

1. 3-D foliations and type III factors

Connes mentioned 3-D foliations V which give rise to type III factors. Foliation property requires a slicing of V by a one-form v to which slices are orthogonal (this requires metric).

1. The foliation property requires that v multiplied by suitable scalar is gradient. This gives the integrability conditions dv= w\wedge v, w=-dψ/ψ =-dlog(ψ). Something proportional to log(ψ) can be taken as a third coordinate varying along flow lines of v: the flow defines a continuous sequence of maps of 2-dimensional slice to itself.

2. If the so called Godbillon-Vey invariant defined as the integral of dw\wedge w over V is non-vanishing, factor of type III is obtained using Schrödinger amplitudes for which the flow lines of foliation define the time evolution. The operators of the algebra in question are transversal operators acting on Schrödinger amplitudes at each slice. Essentially Schrödinger equation in 3-D space-time would be in question with factor of type III resulting from the exotic choice of the time coordinate defining the slicing.

2. What happens in case of light-like 3-surfaces?

In TGD light-like 3-surfaces are natural candidates for V and it is interesting to look what happens in this case. Light-likeness is of course a disturbing complication since orthogonality condition and thus contravariant metric is involved with the definition of the slicing. Light-likeness is not however involved with the basic conditions.

1. The one-form v defined by the induced Kähler gauge potential A defining also a braiding is a unique identification for v. If foliation exists, the braiding flow defines a continuous sequence of maps of partonic 2-surface to itself.

2. Physically this means the possibility of a super-conducting phase with order parameter satisfying covariant constancy equation Dψ=(d/dt -ieA)ψ=0. This would describe a supra current flowing along flow lines of A.

3. If the integrability fails to be true, one cannot assign Schrödinger time evolution with the flow lines of v. One might perhaps say that 3-surface behaves like single quantum event not allowing slicing into a continuous Schrödinger time evolution.

4. In TGD Schrödinger amplitudes are replaced by second quantized induced spinor fields. Hence one does not face the problem whether it makes sense to speak about Schrödinger time evolution of complex order parameter along the flow lines of a foliation or not. Also the fact that the "time evolution" for the modified Dirac operator corresponds to single position dependent generalized eigenvalue identified as Higgs expectation same for all transversal modes (essentially zⁿ labelled by conformal weight) is crucial since it saves from the problems caused by the possible non-existence of Schrödinger evolution.

It is not at all clear whether the integrability condition can be satisfied at all in TGD framework for non-vacuum extremals. This indeed seems to be the case and this is due to a very important delicacy related to the construction of quantum TGD as an almost topological QFT.

1. The construction of quantum TGD at parton level using light-like 3-surfaces as basic objects forces the introduction of Lorentz invariant component of Kähler gauge potential A_a= constant, where a=(t²-r²)^1/2 denotes light-cone proper time. The value of this component could depend on the sector of the generalized imbedding space partially characterized by the value of the Planck constant. The modification does not affect the Kähler form but has the highly non-trivial implication that Chern-Simons action is non-vanishing even when the CP₂ projection of the light-like 3-surface is 2-dimensional. D=2 holds true for the extremals of Chern-Simons action.

2. Non-vanishing A_a is necessary in order to modify the topological QFT defined by Chern-Simons action to an almost topological QFT. What is of utmost importance is that the Noether currents associated with the four-momentum are non-trivial and non-conserved whereas four-momentum squared is conserved and non-vanishing. The breaking of Poincare invariance does not however take place at the level of the world of classical worlds since the configuration space is a union of sub-configuration spaces for which a choice of preferred future light-cone has been made.

3. Since the integrability conditions for A are not gauge invariant, the non-vanishing value of A_a implies that integrability conditions fail already for D=2 as is easy to see by taking two X³ coordinates to be the coordinates of geodesic sphere of CP₂ and the remaining coordinate a light-like coordinate. The light-like 3-surfaces associated with all non-vacuum extremals would behave like quantum events rather than continuous evolutions of Schrödinger equation. This is in spirit of the zero energy ontology in which the ends of the space-time sheet carry positive and negative energy states defining physical state as zero energy state. It conforms also with the notion of time-like entanglement defined by Connes tensor product, which can be reduced only partially in quantum measurements. The failure of the integrability condition means that the flow lines of A define typically helical structures which means a non-trivial braiding. This brings strongly in mind the helical structures of living matter.

3. Extremals of Kähler action

Some comments relating to the interpretation of the classification of the extremals of Kähler action by the dimension of their CP projection are in order. In the chapter Basic Extremals of the Kähler Action classical field equations of TGD are studied. It was found that the extremals can be classified according to the dimension D of the CP₂ projection of space-time sheet in the case that A_a=0 holds true.

1. For D=2 integrability conditions for the vector potential can be satisfied for A_a=0 so that one has generalized Beltrami flow and one can speak about Schrödinger time evolution associated with the flow lines of vector potential defined by covariant constancy condition Dψ=0 makes sense. Kähler current is vanishing or lightlike. This phase is analogous to a super-conductor or a ferromagnetic phase. For nonvanishing A_a the Beltrami flow property is lost but the analogy with ferromagnetism makes sense still.

2. For D=3 foliations are lost even for A_a=0. The phase is dominated by helical structures as also D=2 phase with non-vanishing A_a. This phase is analogous to spin glass phase around phase transition point from ferromagnetic to non-magnetized phase and expected to be important in living matter systems.

3. D=4 is analogous to a chaotic phase with vanishing Kähler current and to a phase without magnetization. The interpretation in terms of non-quantum coherent "dead" matter is suggestive.

An interesting question is whether the ordinary 8-D imbedding space which defines one sector of the generalized imbedding space could correspond to A_a=0 phase. If so, then all states for this sector would be vacua with respect to M⁴ quantum numbers. M⁴-trivial zero energy states in this sector could be transformed to non-trivial zero energy states by a leakage to other sectors.

Mandelbrot and Julia sets from fundamental physics?

Kea told at her blog about the Serpentine Gallery of mathematical formulas of twenty first century.

My favourite formula is very simple looking: z→z²+c of Mandelbrot. Mandelbrot set is defined as the set of values of c for which the iteration starting from z=0 does not lead to infinity. To me this is the equation of the last century which is richest of hidden information and beauty. The definition of Mandelbrot set involves two complex spaces. Julia set J(f), where f is holomorphic function in the most general case, looks very similar looking fractal, and its definition involves no parameters. If f is polynomial, Julia set is the boundary of the set of points whose orbits remain bounded under iteration.

One has some kind of mystic dejavu feeling as one looks at these sets: I must have seen them before! By just looking at this fractal you feel that in some profound sense it really contains Universes within Universes. This might make sense in some sense as will be found!

No wonder that any theoretician would be extremely happy if she or he could identify a generalization of Mandelbrot set or Julia set as something physical and concrete. I realized that in the case of TGD there might be some hopes for this.

1. Does partonic 2-surface as the fundamental 2-D space and light-like 3-surface the fundamental iteration map?

The boundary of the Mandelbrot set represents a critical set: the points of the complex plane (parameter c) for which the iteration leads and does not lead to infinity. Obviously also the Julia set is a critical set. Could quantum criticality of TGD Universe be able to produce these miracles from physics?

1. The fundamental 2-D space would be the partonic 2-surface X² at δM⁴₊× CP₂. Here δ M⁴_+/- refers to the boundary of future/past directed lightcone M⁴_+/-. These light-cones form a causal diamond containing the light-like partonic 3-surfaces. X² is mapped by the light-like orbit X³_l of X² to the final state partonic 2-surface Y² at δ M⁴_-× CP₂. This map followed by return to the lower end of the diamond should define the fundamental iteration step. If one could identify Julia set as a subset of X² - the fundamental conformal dynamical object in TGD having arbitrarily large size - one could indeed say that Julia sets contain universes within universes!

2. The iteration map would be defined by a braiding. Kähler magnetic flow lines or flow lines of Kähler potential define two candidates for the braiding at the light-like 3-surface. Only the latter can be considered for the partonic 3-surfaces whereas the first one could be important in the interior of space-time surfaces. Internal consistency requires that the braiding by Kähler gauge potential is homotopic with the braiding associated with the minima of Higgs potential. The ends of braid correspond to X² and Y² at the intersections of future and past directed lightcones forming a causal diamond. The iteration map could be defined by the braiding taking point z at the lower end of diamond to a point point z₁ at the upper end of diamond. Then you would look what happens to z₁ interpreted as a point at the lower end of the diamond. And so on...

3. With a proper choice of complex coordinate for X² the iteration map should be a holomorphic (or perhaps even rational or polynomial) map. Conformal invariance indeed encourages to think that this might be the case. The complex coordinate defined by the generalized eigenvalue of D is very natural candidate in this respect. Also the complex coordinates of the two geodesic spheres S²_II subset CP₂ and S² subset δ M⁴₊ are reasonable candidates.

2. A candidate for the TGD counterpart of Julia set

The identification of the candidate for the Julia set is easier since it is a subset of the space, where the iteration takes place rather than a subset of the parameter space.

1. The number theoretic braid corresponds to the decomposition of the S²_II projection of X² to separate regions having interpretation as domains inside which conformal field theory applies. The conformal fields associated witgh different regions are independent dynamically so that X² remains effectively 2-D in discretized sense. This decomposition can be interpreted as a decomposition of 2-D landscape to (complex) mountains. At the peaks of the mountains the complex eigenvalue of the modified Dirac operator D vanishes: this corresponds physically to the vanishing of Higgs and to quantum criticality.

2. What is the counterpart for the non-convergence of the iteration? Each mountain is separated from the neighbouring mountains by a saddle point curve of Higgs modulus containing one or more Higgs minima. The escape of the iteration to infinity would naturally correspond to a situation in which the outcome of repeated iteration z→z₁ does not belong at the original mountain. Thus one expects each mountain to decompose into regions separated by boundaries at which iteration ceases to lead to the original valley. One can also identify boundaries of regions in which iteration leads out in N steps and in this manner make the might-exist counterpart of Julia set colored.

3. A candidate for the TGD counterpart of Mandelbrot set

Concering the identification of the candidate for the TGD counterpart of Mandelbrot set the basic challenge is to identify the physical counterpart of the complex parameter c.

1. The only identification of the complex parameter c that comes in mind and does not mean a modification of X² is the choice of S²_II defining the quantization axes in color degrees of freedom and parametrized by some complex coordinate c. One should show that different choices of S²_II define a holomorphic iteration map. The counterpart of Mandelbrot set would exist at S²_II.

2. The point at which Higgs vanishes (peak of the complex mountain) is the obvious choice for the initial point z=0 of the iteration map. This favors complex "Higgs" as the coordinate in which iteration is represented by a holomorphic function.

3. Again the open question is whether one really obtains holomorphic/rational/polynomial map with a proper choice of complex coordinate for X² and for the space of geodesic spheres.

Wegener, Susskind, and me too

Lubos Motl writes nice birthday postings (see for instance, the one about Galois). The latest one was about Wegener. Lubos finished his posting with the following comment.

Well, it is probably obvious where I am going. The convoluted properties of the particle spectrum we observe may also be a result of some historical evolution, as eternal inflation combined with the landscape may suggest. But it doesn't have to be so.

Structurally thinking outsider (in a hypothesical world, where outsiders interested on meanderings of a non-name like me might exist) might be wondering why landscape makes me so sick. After all, there are many qualitative similarities.

1. Speaking about evolution at the level of particle mass spectrum. I am indeed talking about particle spectrum as an outcome of evolutionary pressures selecting p-adic length scales corresponding to primes near powers of two. In zero energy ontology measurement resolution comes in powers of 2 very naturally so that it would seem that coupling constant evolution as an analog for a transition to chaos by period doubling would automatically select the preferred p-adic primes. Also for dark matter hierarchy the especially preferred values of Planck constant come as powers of two: you can add to the kettle also the products of different Fermat primes. As matter fact, I am saying that the entire Universe is a gigantic topologically computing living organism at all levels of various hierarchies.

2. Speaking about landscape. I am predicting an infinite hierarchy of dark matters with different values of Planck constant and assume the 8-D imbedding space to have infinite number of branches intersecting along M² or S² subset CP₂ at quantum criticality and M²×S² at maximal quantum criticality: looks rather brany. Any gauge group can appear as a dynamically engineered gauge group (Hermitian operators in included algebra define gauge symmetry algebra) and characterize measurement resolution via inclusions of HFFs with M-matrix identified in terms of a modification of Connes tensor product.

   My only defense against landscape accusations is that standard model symmetries are in a completely different role than those induced by a finite measurement resolution and are symmetries of the 8-D imbedding space selected uniquely by the mathematical existence of the geometry of world of classical worlds. The symmetries related to measurement resolution are engineered.

3. Speaking about eternal inflation. I am saying cosmic evotion is locally a sequence of non-expanding periods with quantum phase transitions increasing the value of the gravitational Planck constant between them and giving rise to accelerated periods of expansion. These period correspond to critical cosmologies with flat three-spaces so that inflation is replaced with quantum criticality with very similar predictions. Eternal inflation + landscape is replaced with quantum criticality + dark matter hierarchy.

So: who he is the Wegener? Is it Lenny Susskind or could it be...? I did not say anything! I didn't! Luckily I heard just in time the authoritative sounding "crackpot" inside me!