Machian Principle and TGD

I have never considered Machian Principle very seriously but with inspiration coming from Kea's posting I will try to do it now!

1. Non-conserved gravitational four-momentum and conserved inertial momentum at 4-D space-time level

Consider first the situation at the level of classical theory identifiable in terms of classical dynamics for space-time surfaces.

1. In TGD framework one must distinguish between non-conserved gravitational four-momentum and conserved inertial four-momentum identified as conserved Poincare four-momentum at the level of 4-D space-time dynamics and associated with the preferred extremals of Kähler action defining the analogs of Bohr orbits (no path integral over all possible space-time surfaces but functional integral over light-like partonic 3-surfaces). A collection of conserved vector currents rather than tensor results and this resolves the problems due to ill-definedness of four-momentum in General Relativity which served as the primary motivation for the identification of space-times as 4-surfaces of H=M⁴×CP₂.

2. Non-conserved gravitational four-momentum densities can be identified as a linear combination of Einstein tensor and metric tensor (cosmological constant) by contracting them with the Killing vectors of M⁴ translations. Collection of, in general non-conserved, 4-currents result but gravitational four-momentum is well-defined quite generally unlike in General Relativity. Only for the asymptotic stationary cosmologies corresponding to extremals of the curvature scalar plus constant for the induced metric gravitational four-momentum is conserved.

2. Inertial four-momentum as the average of gravitational four-momentum

The first question is how non-conserved gravitational and conserved inertial four-momentum relate to each other. Certainly Equivalence Principle in a strong form cannot hold true.

1. In zero energy ontology the total quantum numbers of states vanish and positive and negative energy parts of states have interpretation as initial and final states of particle reaction at elementary particle level where geometro-temporal distance between them is short (TGD inspired theory of consciousness forces to distinguish between geometric time and subjective time). Positive energy ontology emerges as an effective ontology at observational level when the temporal distance between positive and negative energy parts of the state is long as compared to the time scale of conscious observer. The recent understanding about bosons as wormhole contacts between space-time sheets with positive and negative time orientation suggests that the two space-time sheets in question correspond to positive and negative energy parts of the state. This brings in mind the picture of Connes about Higgs mechanism involving two copies of Minkowski space.

2. The intuitive idea is that the conserved inertial four-momentum assignable to the positive energy part of the state is the average of the non-conserved gravitational four momentum and depends on the p-adic length scale characterizing the pair of space-time sheets connecting positive and negative energy states. The average is over a p-adic time scale characterizing the temporal span of the space-time sheet. This average is coded by the classical dynamics for the preferred extremal of Kähler action defining the generalized Bohr orbit.

3. Non-conserved gravitational four-momentum and conserved inertial momentum at parton level

A deeper level description of the situation is achieved at parton level. For light-like partonic 3-surfaces the dynamics is defined by almost topological QFT defined by Chern-Simons action for the induced Kähler form. The extrema have 2-D CP₂ projection. Light-likeness implies the replacement of "topological" with "almost topological" by bringing in the notions of metric and four-momentum.

1. The world of classical worlds (WCW) decomposes into a union of sub-WCW:s associated with preferred points of imbedding space H= M⁴_+/-× CP₂. The selection of preferred point of H means means a selection of tip of future/past directed light-cone in the case of M⁴_+/- and selection of U(2) subgroup of SU(3) in the case of CP₂. There is a further selection fixing rest system and angular momentum quantization axis (preferred plane in M⁴ defining non-physical polarizations for massless bosons) and quantization axis of color isospin and hyper-charge. That configuration space geometry reflects these choices conforms with quantum-classical correspondence requiring that everything quantal has a geometric correlate.

2. At the level of S-matrix the preferred points of H defining past/future directed light-cones correspond to the arguments of n-point function. In the construction of S-matrix one integrates over the tips of the light-cones parameterizing sub-WCW:s consisting of partonic 3-surfaces residing inside these light-cones (×CP₂). Hence a full Poincare invariance results meaning the emergence of conserved four-momentum identifiable as inertial four-momentum assignable to the preferred extremals of Kähler action defining Bohr orbits. These light-cones give rise to Russian doll cosmology with cosmologies within cosmologies such that elementary particles formally correspond to the lowest level in the hierarchy.

3. Parton dynamics is associated with a given future/past light-cone. At parton level one has Lorentz invariance and only the mass squared is conserved for the partonic time evolution dictated by random light-likeness. There is a very delicate point involved here. Partonic four-momentum is non-vanishing only if CP₂ Kähler gauge potential has also M⁴+/- component which is pure gauge. Mass squared is conserved (Lorentz invariance) if this component is in the direction of proper time coordinate a of the light-cone and if its magnitude is constant. From the point of view of spinor structure M⁴_+/- and CP₂ are not totally decoupled. This does not break gauge invariance since Kähler gauge potential does not give rise to U(1) gauge degeneracy but only to 4-D spin glass degeneracy.

4. The natural identification of the conserved classical partonic four-momentum is as the non-conserved gravitational four-momentum defined for a space-time sheet characterized by a p-adic time scale. In accordance with zero energy ontology, a length scale dependent notion is in question. At single parton level Equivalence Principle would state that the conserved gravitational mass is equal to inertial mass but would not require equivalence of four-momenta.

4. Inertial four-momentum as average of partonic four-momentum and p-adic thermodynamics

1. The natural hypothesis is that inertial four-momentum at partonic level is the temporal average of non-conserved gravitational four-momentum. This implies particle massivation in general since the motion of light-like parton is in general random zitterbewegung so that only mass squared is conserved. The average is defined always in some time scale identifiable as the p-adic time scale defining the mass scale via Uncertainty Principle. There is actually hierarchy of p-adic time scales coming as powers of p. Inertial mass vanishes only if the motion is non-random in the time scale considered and this never occurs exactly for even photon and graviton.

2. The quantitative formulation of the averaging relies on p-adic thermodynamics for the integer valued conformal weight characterizing the particle. By number theoretic universality this description must be equivalent to real thermodynamics with quantized temperature. Quantization of the mass scale is purely number theoretical: p-adic thermodynamics based on standard Boltzman weight e^L₀/T does not make sense since exp(x) has always unit p-adic norm so that partition sum does not converge. One can however replace this Boltzman weight with p^L₀/T_p, which exists for T_p=1/n, n=1,2,..., if L₀ is a generator of conformal scaling having non-negative integer spectrum. This predicts a discrete spectrum of p-adic mass scales and real thermodynamics is obtained by reversing the sign of exponent. Assuming a reasonable cutoff on conformal weight (only two lowest terms give non-negligible contributions to thermal average) and a prescription for the mapping of p-adic mass squared to its real counterpart the two descriptions are equivalent. Note that mass squared is the average of conformal weight rather than the average of four-momentum squared so that Lorentz invariance is not lost. Note also that in the construction of S-matrix four-momenta emerge only via the Fourier transform of n-point function and do not appear at fundamental vertices.

3. Also the coupling to Higgs gives a contribution to the mass. Higgs corresponds to a wormhole contact with wormhole throats carrying fermion and antifermion quantum numbers as do all gauge bosons. Higgs expectation should have space-time correlate appearing in the modified Dirac operator. A good candidate is p-adic thermal average for the generalized eigenvalue of the modified Dirac operator vanishing for the zero modes. Thermal mass squared as opposed to Higgs contribution would correspond to the average of integer valued conformal weight. For bosons (in particular Higgs boson!) it is simply the sum of expectations for the two wormhole throats.

4. Both contributions are basically thermal which raises the question whether the interpretation in terms of coherent state of Higgs field (and essentially quantal notion) is really appropriate unless also thermal states can be regarded as genuine quantum states. The matrix characterizing time-like entanglement for the zero energy quantum state can be also thermal S-matrix with respect to the incoming and outgoing partons (hyper-finite factors of type III allow the analog of thermal QFT at the level of quantum states). This allows also a first principle description of p-adic thermodynamics.

5. Various interpretations of Machian Principle

TGD allows several interpretations of Machian Principle and leads also to a generalization of the Principle.

1. Machian Principle is true in the sense that the notion of completely free particle is non-sensible. Free CP₂ type extremal (having random light-like curve as M⁴projection) is a pure vacuum extremal and only its topological condensation creates a wormhole throat (two of them) in the case of fermion (boson). Topological condensation to space-time sheet(s) generates all quantum numbers, not only mass. Both thermal massivation and massivation via the generation of coherent state of Higgs type wormhole contacts are due to topological condensation.

2. Machian Principle has also interpretation in terms of p-adic physics. Most points of p-adic space-time sheets have infinite distance from the tip light-cone in the real sense. The discrete algebraic intersection of the p-adic space-time sheet with the real space-time sheet gives rise to effective p-adicity of the topology of the real space-time sheet if the number of these points is large enough. Hence p-adic thermodynamics with given p also assigned to the partonic 3-surface by the modified Dirac operator makes sense. The continuity and smoothness of the dynamics corresponds to the p-adic fractality and long range correlations for the real dynamics and allows to apply p-adic thermodynamics in the real context. p-Adic variant of Machian Principle says that p-adic dynamics of cognition and intentionality in literally infinite scale in the real sense dictates the values of masses among other things.

3. A further interpretation of Machian Principle is in terms of number theoretic Brahman=Atman identity or equivalently, Algebraic Holography. This principle states that the number theoretic structure of the space-time point is so rich due to the presence of infinite hierarchy of real units obtained as ratios of infinite integers that single space-time point can represent the entire world of classical worlds. This could be generalized also to a criterion for a good mathematics: only those mathematical structures which are representable in the set of real units associated with the coordinates of single space-time point are really fundamental.

For more details see the end of the chapter The Relationship Between TGD and GRT of "Classical Physics in Many-Sheeted Space-Time".

P.S. Anyone can some day wake up to the realization of having a scientific Archnemesis. Even the God of Old Testament can forgive but not your Archnemesis. The curse of Archnemesis follows you even from grave and to grave. What you can do is to cross your fingers and pray the help of mightier gods. For more about the topic here.

Could also gauge bosons correspond to wormhole contacts?

The developments in the formulation of quantum TGD which have taken place during the period 2005-2007 (see this , this , and this ) suggest dramatic simplifications of the general picture about elementary particle spectrum. p-Adic mass calculations (see this , this , this , and this ) leave a lot of freedom concerning the detailed identification of elementary particles. The basic open question is whether the theory is free at parton level as suggested by the recent view about the construction of S-matrix and by the almost topological QFT property of quantum TGD at parton level (see this and this ). Or more concretely: do partonic 2-surfaces carry only free many-fermion states or can they carry also bound states of fermions and anti-fermions identifiable as bosons?

What is known that Higgs boson corresponds naturally to a wormhole contact (see this ). The wormhole contact connects two space-time sheets with induced metric having Minkowski signature. Wormhole contact itself has an Euclidian metric signature so that there are two wormhole throats which are light-like 3-surfaces and would carry fermion and anti-fermion number in the case of Higgs. Irrespective of the identification of the remaining elementary particles MEs (massless extremals, topological light rays) would serve as space-time correlates for elementary bosons. Higgs type wormhole contacts would connect MEs to the larger space-time sheet and the coherent state of neutral Higgs would generate gauge boson mass and could contribute also to fermion mass.

The basic question is whether this identification applies also to gauge bosons (certainly not to graviton). this identification would imply quite a dramatic simplification since the theory would be free at single parton level and the only stable parton states would be fermions and anti-fermions. As will be found this identification allows to understand the dramatic difference between graviton and other gauge bosons and the weakness of gravitational coupling, gives a connection with the string picture of gravitons, and predicts that stringy states are directly relevant for nuclear and condensed matter physics as has been proposed already earlier (see this , this , and this ).

1. Option I: Only Higgs as a wormhole contact

The only possibility considered hitherto has been that elementary bosons correspond to partonic 2-surfaces carrying fermion-anti-fermion pair such that either fermion or anti-fermion has a non-physical polarization. For this option CP₂ type extremals condensed on MEs and travelling with light velocity would serve as a model for both fermions and bosons. MEs are not absolutely necessary for this option. The couplings of fermions and gauge bosons to Higgs would be very similar topologically. Consider now the counter arguments.

1. This option fails if the theory at partonic level is free field theory so that anti-fermions and elementary bosons cannot be identified as bound states of fermion and anti-fermion with either of them having non-physical polarization.

2. Mathematically oriented mind could also argue that the asymmetry between Higgs and elementary gauge bosons is not plausible whereas asymmetry between fermions and gauge bosons is. Mathematician could continue by arguing that if wormhole contacts with net quantum numbers of Higgs boson are possible, also those with gauge boson quantum numbers are unavoidable.

3. Physics oriented thinker could argue that since gauge bosons do not exhibit family replication phenomenon (having topological explanation in TGD framework) there must be a profound difference between fermions and bosons.

2. Option II: All elementary bosons as wormhole contacts

The hypothesis that quantum TGD reduces to a free field theory at parton level is consistent with the almost topological QFT character of the theory at this level. Hence there are good motivations for studying explicitly the consequences of this hypothesis.

2.1 Elementary bosons must correspond to wormhole contacts if the theory is free at parton level

Also gauge bosons could correspond to wormhole contacts connecting MEs (see this ) to larger space-time sheet and propagating with light velocity. For this option there would be no need to assume the presence of non-physical fermion or anti-fermion polarization since fermion and anti-fermion would reside at different wormhole throats. Only the definition of what it is to be non-physical would be different on the light-like 3-surfaces defining the throats.

The difference would naturally relate to the different time orientations of wormhole throats and make itself manifest via the definition of light-like operator o=x^k γ_k appearing in the generalized eigenvalue equation for the modified Dirac operator (see this and this ). For the first throat o^k would correspond to a light-like tangent vector t^k of the partonic 3-surface and for the second throat to its M⁴ dual t_d^k in a preferred rest system in M⁴ (implied by the basic construction of quantum TGD). What is nice that this picture non-asks the question whether t^k or t_d^k should appear in the modified Dirac operator.

Rather satisfactorily, MEs (massless extremals, topological light rays) would be necessary for the propagation of wormhole contacts so that they would naturally emerge as classical correlates of bosons. The simplest model for fermions would be as CP₂ type extremals topologically condensed on MEs and for bosons as pieces of CP₂ type extremals connecting ME to the larger space-time sheet. For fermions topological condensation is possible to either space-time sheet.

2.2 Phase conjugate states and matter-antimatter asymmetry

By fermion number conservation fermion-boson and boson-boson couplings must involve the fusion of partonic 3-surfaces along their ends identified as wormhole throats. Bosonic couplings would differ from fermionic couplings only in that the process would be 2→ 4 rather than 1→ 3 at the level of throats.

The decay of boson to an ordinary fermion pair with fermion and anti-fermion at the same space-time sheet would take place via the basic vertex at which the 2-dimensional ends of light-like 3-surfaces are identified. The sign of the boson energy would tell whether boson is ordinary boson or its phase conjugate (say phase conjugate photon of laser light) and also dictate the sign of the time orientation of fermion and anti-fermion resulting in the decay.

Also a candidate for a new kind interaction vertex emerges. The splitting of bosonic wormhole contact would generate fermion and time-reversed anti-fermion having interpretation as a phase conjugate fermion. this process cannot correspond to a decay of boson to ordinary fermion pair. The splitting process could generate matter-antimatter asymmetry in the sense that fermionic antimatter would consist dominantly of negative energy anti-fermions at space-time sheets having negative time orientation (see this and this ).

This vertex would define the fundamental interaction between matter and phase conjugate matter. Phase conjugate photons are in a key role in TGD based quantum model of living matter. this involves a model for memory as communications in time reversed direction, mechanism of intentional action involving signalling to geometric past, and mechanism of remote metabolism involving sending of negative energy photons to the energy reservoir (see this ). The splitting of wormhole contacts has been considered as a candidate for a mechanism realizing Boolean cognition in terms of "cognitive neutrino pairs" resulting in the splitting of wormhole contacts with net quantum numbers of Z⁰ boson (see this ).

3. Graviton and other stringy states

Fermion and anti-fermion can give rise to only single unit of spin since it is impossible to assign angular momentum with the relative motion of wormhole throats. Hence the identification of graviton as single wormhole contact is not possible. The only conclusion is that graviton must be a superposition of fermion-anti-fermion pairs and boson-anti-boson pairs with coefficients determined by the coupling of the parton to graviton. Graviton-graviton pairs might emerge in higher orders. Fermion and anti-fermion would reside at the same space-time sheet and would have a non-vanishing relative angular momentum. Also bosons could have non-vanishing relative angular momentum and Higgs bosons must indeed possess it.

Gravitons are stable if the throats of wormhole contacts carry non-vanishing gauge fluxes so that the throats of wormhole contacts are connected by flux tubes carrying the gauge flux. The mechanism producing gravitons would the splitting of partonic 2-surfaces via the basic vertex. A connection with string picture emerges with the counterpart of string identified as the flux tube connecting the wormhole throats. Gravitational constant would relate directly to the value of the string tension.

The TGD view about coupling constant evolution (see this ) predicts G proportional to L_p² , where L_p is p-adic length scale, and that physical graviton corresponds to p=M₁₂₇=2¹²⁷ -1. Thus graviton would have geometric size of order Compton length of electron which is something totally new from the point of view of usual Planck length scale dogmatism. In principle an entire p-adic hierarchy of gravitational forces is possible with increasing value of G.

The explanation for the small value of the gravitational coupling strength serves as a test for the proposed picture. The exchange of ordinary gauge boson involves the exchange of single CP₂ type extremal giving the exponent of Kähler action compensated by state normalization. In the case of graviton exchange two wormhole contacts are exchanged and this gives second power for the exponent of Kähler action which is not compensated. It would be this additional exponent that would give rise to the huge reduction of gravitational coupling strength from the naive estimate G ≈ L_p².

Gravitons are obviously not the only stringy states. For instance, one obtains spin 1 states when the ends of string correspond to gauge boson and Higgs. Also non-vanishing electro-weak and color quantum numbers are possible and stringy states couple to elementary partons via standard couplings in this case. TGD based model for nuclei as nuclear strings having length of order L(127) (see this ) suggests that the strings with light M₁₂₇quark and anti-quark at their ends identifiable as companions of the ordinary graviton are responsible for the strong nuclear force instead of exchanges of ordinary mesons or color van der Waals forces.

Also the TGD based model of high T_c super-conductivity involves stringy states connecting the space-time sheets associated with the electrons of the exotic Cooper pair (see this and this ). Thus stringy states would play a key role in nuclear and condensed matter physics, which means a profound departure from stringy wisdom, and breakdown of the standard reductionistic picture.

4. Spectrum of non-stringy states

The 1-throat character of fermions is consistent with the generation-genus correspondence. The 2-throat character of bosons predicts that bosons are characterized by the genera (g₁,g₂) of the wormhole throats. Note that the interpretation of fundamental fermions as wormhole contacts with second throat identified as a Fock vacuum is excluded.

The general bosonic wave-function would be expressible as a matrix M_g1,g2 and ordinary gauge bosons would correspond to a diagonal matrix M_g1,g2 δ_g1,g2 as required by the absence of neutral flavor changing currents (say gluons transforming quark genera to each other). 8 new gauge bosons are predicted if one allows all 3× 3 matrices with complex entries orthonormalized with respect to trace meaning additional dynamical SU(3) symmetry. Ordinary gauge bosons would be SU(3) singlets in this sense. The existing bounds on flavor changing neutral currents give bounds on the masses of the boson octet. The 2-throat character of bosons should relate to the low value T=1/n<< 1.

If one forgets the complications due to the stringy states (including graviton), the spectrum of elementary fermions and bosons is amazingly simple and almost reduces to the spectrum of standard model. In the fermionic sector one would have fermions of standard model. By simple counting leptonic wormhole throat could carry 2³ =8 states corresponding to 2 polarization states, 2 charge states, and sign of lepton number giving 8+8=16 states altogether. Taking into account phase conjugates gives 16+16=32 states.

In the non-stringy boson sector one would have bound states of fermions and phase conjugate fermions. Since only two polarization states are allowed for massless states, one obtains (2+1)×(3+1)=12 states plus phase conjugates giving 12+12=24 states. The addition of color singlet states for quarks gives 48 gauge bosons with vanishing fermion number and color quantum numbers. Besides 12 electro-weak bosons and their 12 phase conjugates there are 12 exotic bosons and their 12 phase conjugates. For the exotic bosons the couplings to quarks and leptons are determined by the orthogonality of the coupling matrices of ordinary and boson states. For exotic counterparts of Wbosons and Higgs the sign of the coupling to quarks is opposite. For photon and Z⁰ also the relative magnitudes of the couplings to quarks must change. Altogether this makes 48+16+16=80 states. Gluons would result as color octet states. Family replication would extend each elementary boson state into SU(3)octet and singlet and elementary fermion states into SU(3)triplets.

5. Higgs mechanism

Consider next the generation of mass as a vacuum expectation value of Higgs when also gauge bosons correspond to wormhole contacts. The presence of Higgs condensate should make the simple rectilinear ME curved so that the average propagation of fields would occur with a velocity less than light velocity. Field equations allow MEs of this kind as solutions (see this ).

The finite range of interaction characterized by the gauge boson mass should correlate with the finite range for the free propagation of wormhole contacts representing bosons along corresponding ME. The finite range would result from the emission of Higgs like wormhole contacts from gauge boson like wormhole contact leading to the generation of coherent states of neutral Higgs particles. The emission would also induce non-rectilinearity of ME as a correlate for the recoil in the emission of Higgs.

For more details see the end of the chapter Hyperfinite Factors and Construction of S-matrix of "Towards S-matrix" or the chapters Construction of Elementary Particle Vacuum Functionals and Massless states and Particle Massivation of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy".

Now and Then

Now and then it is not possible to avoid the situation when the head is completely empty of ideas, questions, inspirations, and all that which makes you a happy thinker. This week has been such a period of time. Luckily, blog discussions have helped to live through this period and the best manner to get a new idea is to explain old idea. Thank you for Kea and Mahndisa (Resources in Category Theory, Braid Theory, Coxeter Dynkin Diagrams and Spider Diagrams with Tesselations for good measure) and Carl Brannen for patience.

This morning there was a nice piece of text by Connes about universal time evolution for factors of type III and the idea about emergent time in Noncommutative Geometry. There was already discussion in Kea's blog about this.

Personally I am not sure about the emergence of time. But on the other hand, I have proposed that imbedding space and the "world of classical worlds" could indeed emerge from von Neumann algebra framework... I should be able to decide. Be as it may, I find the idea about the universality of time evolution really fascinating. Some questions.

1. Universal time evolution is fixed apart from inner automorphisms. How to achieve completely unique dynamics? Could inner automorphisms be identified as universal local gauge symmetries? In the case of HFFs of type II₁ one can also ask whether outer automorphisms could be seen as analogs of global gauge transformations.

2. Could the unique time evolution of HFF of type III operator algebra induce such an evolution in HFF of type II₁ as one restricts the consideration to physical states and to quantized values of time parameter? From "Noncommutative Geometry" I remember the representation of HFF of type III as a crossed product of Z and HFF of type II₁ with Z represented as shifts in tensor factor (I hope I understood correctly!). Could the reduction of Z to N, analogous to restriction of oscillator operators to creation operators, imply reduction to a HFF of type II₁?

3. The idea about emergence of time is fascinating but leaves a lot of room for models. A weaker form of this idea would be a universal dynamics at single particle level but assuming that quantum dynamics has space-time correlates. My own proposal identifies light-like 3-surfaces as basic dynamical objects and non-commutative dynamics would result at the level state space from the finite measurement resolution characterized by inclusion. Also quantum geometry could be a manner to describe finite measurement resolution.

   Universal geometric time could correspond to the light-like coordinate and the universal time evolution would be that associated with the light-like orbits of partons appearing as lines in the generalized Feynman diagrams defined by light-like 3-surfaces and would give rise to universal propagators as one integrates over the end points of internal lines. This would be single particle time evolution and would not give vertices: they would correspond to isomorphisms between tensor products of incoming and outgoing HFFs of type II₁. If the above ideas make sense, the dynamics provided by the generalized eigen modes of what I call modified Dirac operator should boil down to a universal dynamics.

Planar algebras and generalized Feynman diagrams

There has been an interesting discussion in Kea's blog about planar algebras and related concepts and I decided to add here the posting that I sent also there. You can find information about issues related to planar algebras in Kea's blog. I found also an article about planar algebras in Wikipedia.

What occurred to me is that planar algebras might have interpretation in terms of planar projections of generalized Feynman diagrams (these structures are metrically 2-D by presence of one light-like direction so that 2-D representation is especially natural).

1. Planar algebra very briefly

First a brief definition of planar algebra.

1. One starts from planar k-tangles obtained by putting disks inside a big disk. Inner disks are empty. Big disk contains 2k braid strands starting from its boundary and returning back or ending to the boundaries of small empty disks in the interior containing also even number of incoming lines. It is possible to have also loops. Disk boundaries and braid strands connecting them are different objects. A black-white coloring of the disjoint regions of k-tangle is assumed and there are two possible options. Equivalence of planar tangles under diffeomorphisms is assumed.

2. One can define a product of k-tangles by identifying k-tangle along its outer boundary with some inner disk of another k-tangle. Obviously the product is not unique when the number of inner disks is larger than 1. In the product one deletes the inner disk boundary but if one interprets this disk as a vertex-parton, it would be better to keep the boundary.

3. One assigns to the planar k-tangle a vector space V_k and a linear map from the tensor product of spaces V_ki associated with the inner disks to V_k such that this map is consistent with the decomposition k-tangles. Under certain additional conditions the resulting algebra gives rise to an algebra characterizing multi-step inclusion of HFFs of type II₁.

4. It is possible to bring in additional structure and in TGD framework it seems necessary to assign to each line of tangle an arrow telling whether it corresponds to a strand of a braid associated with positive or negative energy parton. One can also wonder whether disks could be replaced with closed 2-D surfaces characterized by genus if braids are defined on partonic surfaces of genus g. In this case there is no topological distinction between big disk and small disks. One can also ask why not allow the strands to get linked (as suggested by the interpretation as planar projecitons of generalized Feynman diagrams) in which case one would not have a planar tangle anymore.

2. General arguments favoring the assignment of a planar algebra to a generalized Feynman diagram

There are some general arguments in favor of the assignment of planar algebra to generalized Feynman diagrams.

1. Planar diagrams describe sequences of inclusions of HFF:s and assign to them a multi-parameter algebra corresponding indices of inclusions. They describe also Connes tensor powers in the simplest situation corresponding to Jones inclusion sequence. Suppose that also general Connes tensor product has a description in terms of planar diagrams. This might be trivial.

2. Generalized vertices identified geometrically as partonic 2-surfaces indeed contain Connes tensor products. The smallest sub-factor N would play the role of complex numbers meaning that due to a finite measurement resolution one can speak only about N-rays of state space and the situation becomes effectively finite-dimensional but non-commutative.

3. The product of planar diagrams could be seen as a projection of 3-D Feynman diagram to plane or to one of the partonic vertices. It would contain a set of 2-D partonic 2-surfaces. Some of them would correspond vertices and the rest to partonic 2-surfaces at future and past directed light-cones corresponding to the incoming and outgoing particles.

4. The question is how to distinguish between vertex-partons and incoming and outgoing partons. If one does not delete the disk boundary of inner disk in the product, the fact that lines arrive at it from both sides could distinguish it as a vertex-parton whereas outgoing partons would correspond to empty disks. The direction of the arrows associated with the lines of planar diagram would allow to distinguish between positive and negative energy partons (note however line returning back).

5. One could worry about preferred role of the big disk identifiable as incoming or outgoing parton but this role is only apparent since by compactifying to say S² the big disk exterior becomes an interior of a small disk.

3. A more detailed view

The basic fact about planar algebras is that in the product of planar diagrams one glues two disks with identical boundary data together. One should understand the counterpart of this in more detail.

1. The boundaries of disks would correspond to 1-D closed space-like stringy curves at partonic 2-surfaces along which fermionic anti-commutators vanish.

2. The lines connecting the boundaries of disks to each other would correspond to the strands of number theoretic braids and thus to braidy time evolutions. The intersection points of lines with disk boundaries would correspond to the intersection points of strands of number theoretic braids meeting at the generalized vertex.

   [Number theoretic braid belongs to an algebraic intersection of a real parton 3-surface and its p-adic counterpart obeying same algebraic equations: of course, in time direction algebraicity allows only a sequence of snapshots about braid evolution].

3. Planar diagrams contain lines, which begin and return to the same disk boundary. Also "vacuum bubbles" are possible. Braid strands would disappear or appear in pairwise manner since they correspond to zeros of a polynomial and can transform from complex to real and vice versa under rather stringent algebraic conditions.

4. Planar diagrams contain also lines connecting any pair of disk boundaries. Stringy decay of partonic 2-surfaces with some strands of braid taken by the first and some strands by the second parton might bring in the lines connecting boundaries of any given pair of disks (if really possible!).

5. There is also something to worry about. The number of lines associated with disks is even in the case of k-tangles. In TGD framework incoming and outgoing tangles could have odd number of strands whereas partonic vertices would contain even number of k-tangles from fermion number conservation. One can wonder whether the replacement of boson lines with fermion lines could imply naturally the notion of half-k-tangle or whether one could assign half-k-tangles to the spinors of the configuration space ("world of classical worlds") whereas corresponding Clifford algebra defining HFF of type II₁ would correspond to k-tangles.

For the recent TGD view about generalized Feynman graphics see the chapter Hyperfinite Factors and Construction of S-matrix of "Towards S-matrix".

TGD Universe from the condition that all possible statistics are possible

By simple physical arguments H=M⁴×CP₂ is the unique choice for the imbedding space in TGD Universe. This choice follows also from the number theoretic vision. M⁴ has interpretation as hyper-quaternions and CP₂ as space of space of quaternionic planes at a point of (hyper)-octonion space. Space-time surfaces can be seen as hyper-quaternionic sub-manifolds of hyper-octonionic space M⁸ and correspond to sub-manifolds of H. The basic conjecture is that hyper-quaternionicity corresponds to the property of being a certain preferred extremal of so called Kähler action with corresponding Kähler action defining Kähler function for the "world of classical worlds". The corresponding isometry group SO(3,1)× SU(3) and vielbein group SL(2,C)× U(2)_ew have therefore a distinguished position both in physics and quantum TGD.

A new observation induced by a little discussion with Kea is that also the general pattern for inclusions of hyperfinite factors of type II₁ selects these groups as something very special: the condition that all possible statistics are realized is guaranteed by the choice M⁴× CP₂.

1. Inclusions are partially characterized by quantum phase q=exp(i2π/n). n>2 for the quantum counterparts of the fundamental representation of SU(2) means that braid statistics for Jones inclusions cannot give the usual fermionic statistics. That Fermi statistics cannot "emerge" conforms with the role of infinite-D Clifford algebra as a canonical representation of HFF of type II₁. SO(3,1) as isometries of H gives Z₂ statistics via the action on spinors of M⁴ and U(2) holonomies for CP₂ realize Z₂ statistics in CP₂ degrees of freedom.

2. n>3 for more general inclusions [I learned this from the thesis "Hecke algebras of type A_n and subfactors" of Hans Wenzl (Inv. Math. 92, 349-383 (1988)] in turn excludes Z₃ statistics as braid statistics in the general case. SU(3) as isometries induces a non-trivial Z₃ action on quark spinors but trivial action at the imbedding space level so that Z₃ statistics becomes possible in quark sector quite generally.

For more details about general inclusions see the appendix of the chapter Was von Neumann Right After All? of "Towards S-matrix".

Inverting reductionism upside down

In the recent New Scientists there is an article about string nets. For a more technical description the article Photons and electrons as emergent phenomena of Michael Levin and Xiao-Gang Wen is recommended. I found myself resonating with several ideas appearing in this work but also enjoy of disagreeing.

1. Inverting reductionism upside down

1. What makes me happy is that the work challenges the basic dogmas of reductionism and locality which have led to the recent blind alley in the official theoretical physics. One might say that the standard reductionistic view is turned upside down: local structures such as gauge bosons and fermions emerge from non-local ones. Condensed matter physics is taken as starting point in attempt to build a model for fundamental physics. It is really enjoyable to see that genuine thinking is still taking occasionally place in theoretical physics.

2. The basic inspiration comes from topological quantum computation. The crucial element of quantum computation is quantum entanglement which in the case of topological quantum computation is robust against perturbations. This implies now non-locality at fundamental level. Universal (topological) quantum computers are able to mimic any dynamics at the level of discrete approximation. The point of view taken by the authors is that that there is no fundamental level involving elementary gauge bosons and fermions. Particle spectrum and dynamics of gauge theories could result as this kind of mimicry. Hence the breakdown of reductionism in the sense that everything emerges from the dance of quarks and leptons or some even smaller structures.

2. String nets as fundamental structures

Consider now a more concrete view about the model.

1. One might think that braid like structures would have been introduced as fundamental objects. This was not the case, and probably because they are not enough for the desired inverse reduction. Instead, strings are taken as the basic objects. Strings can form string nets by fusing together along their ends and string network becomes the fundamental notion giving rise to elementary particles as its excitations. There are different type of strings and only those fusion vertices are favored for which the interaction energy is small. Besides this Hamiltonian contains kinetic energy and string tension term. If string tension is very high one obtains large number of small string nets. If it is small, large string nets emerge.

2. These models predict excitations having interpretation in terms of gauge bosons and fermions. From string model point of view this is perhaps not surprising. For instance, transversal excitations of string would naturally give rise to bosonic excitations and ends of string would behave like fermions.

3. Some criticism

I do not try to pretend of being objective and I confess that my criticism reflects strongly my TGD based belief system also profoundly influenced by the beautiful ideas of topological quantum computation.

1. In order to pass as a unified theory the model should be able to provide fundamental dynamics based on some simple principle. The string net Hamiltonian can be however tailored to reproduce any gauge theory. This smells first like a reincarnation of string landscape. At the second thought there is nothing wrong with this kind of flexibility but I would take it as a reflection of the fact that universal topological quantum computer is able to mimic any system when discretization is allowed. To me the correct question would be "What is the fundamental dynamics allowing topological quantum computation as a fundamental process?".

   Perhaps it is un-necessary to tell the reader that the belief that standard model based physics really allows universal topological quantum computation is only a belief. Personally I regard this belief as wrong. The point is that much more than a mere construction of a discrete Hamiltonian characterizing the quantum computation is needed. Fundamental physics must allow the entire process culminating to a conscious experience about the result of the computation. The basic challenge is the identification of the principles of the fundamental dynamics allowing any topological computation. One cannot circumvent the fact before there can be a topological computer there must also be intention to build it and theory must describe also this.

2. Could string nets (or something more general) then be fundamental non-local structures? My answer based on my personal belief system is a qubit somewhere between yes and no.

   Most of the qubit consists of "No". I do not believe that strings of any kind are really the fundamental objects. In TGD framework strings are replaced by 3-dimensional light-like 3-surfaces as basic objects and they define generalized Feynman diagrams. The interpretation is as orbits of 2-dimensional partons (in honor of Feynman's deep intuition) which can have arbitrarily large size. Classical space-time dynamics emerges in a well-defined sense as a classical correlate for the quantum physics of these light-like 3-surfaces. Quantum physics is not anymore local at space-time level since 3-surface behaves as a single coherent whole (very important for the proper understanding of living systems). Interestingly, quantum physics however remains local and formally completely classical (apart from quantum jump) at the level of the "world of classical worlds".

   Qubit contains also some "Yes".

   • The dynamics of TGD involves definitely stringy aspects, in particular generalized super-conformal symmetries and the stringy character of fermionic anti-commutation relations.

   • Also TGD Universe predicts the existence of string like objects and of fractal string networks consisting of magnetic flux quanta. This network plays a key role in cosmology: for instance, dark energy corresponds to magnetic energy. In TGD inspired nuclear physics nuclei are identified as highly tangled nuclear strings.

3. I would have been happy to see braids, which served as starting point, as fundamental objects because they represent really deep mathematics.

4. Braid statistics is an essentially 2-dimensional phenomenon. Hence I would have expected that 2-dimensional surfaces would have been introduced explicitly as fundamental objects.

5. In the model of Levin and Wen one could see bosons (strings) and fermions (string ends) as dual manners to describe dynamics and this kind of eliminative reductionism makes me very skeptic. In TGD framework the counterpart of string is interior dynamics of space-time and corresponds to classical physics as exact part of quantum theory necessary for quantum measurement theory whereas bosons and fermions as counterparts of string ends correspond to quantum states of light-like partonic 3-surfaces.

6. Effective 2-dimensionality is absolutely essential for the braid statistics and is of course only an assumption subject to criticism. Standard model skeptic might argue that standard model quantum physics does not allow to achieve effective 2-dimensionality in such a good an approximation as to guarantee the effective topologization of the dynamics. Standard model skeptic might be right. It is of course experimental fact that anyons are there but their existence might only demonstrate that the belief system of standard model skeptic is in need of updating.

7. The idea of giving up fermions as fundamental dynamical objects looks to me questionable. For instance, for Jones inclusions for which braid statistics emerges naturally only q=exp(iπ/n), n≥3 is allowed as quantum phase and one does allow fermionic braid statistics. The second reason for my skepticism is more personal. In TGD framework the Clifford algebra of the world of classical worlds has interpretation in terms of fermionic oscillator operators so that a beautiful geometrization of fermion number results. This algebra is identifiable as fundamental hyper-finite factor of type II₁ responsible for the beauty of TGD and also lurking behind the magic of braids. I am ready to describe bosons as fermion-antifermion bound states but refuse to continue further.

4. What if one requires that Universe is topological quantum computer?

Topological quantum computation has strongly inspired also the development of quantum TGD. This inspires me to pose the key question differently. Suppose that Universe is topological quantum computer. What can one conclude about the fundamental dynamics? You can of course guess the outcome and for me this is still one exercise to deduce TGD Universe from some simple basic assumptions in the hope (I am really incurable optimist!) that the message could finally permeate through the magnificently effective cognitive immune (or its it insulation-?) system of main stream theoretical physicist.

1. The first requirement is that, as far as fermionic quantum dynamics is considered, the fundamental objects are effectively 2-dimensional and carry braids as fundamental objects. This makes braid statistics a genuine statistics and allow topological quantum computation at the fundamental level. In particular, topological braid dynamics would be genuinely topological and not only approximately so.

   Conformal symmetries are natural in 2-dimensional context and lead naturally to braid statistics. Requiring the generalization of super-conformal invariance as a fundamental symmetry leads to the identification of fundamental dynamical objects as light-like 3-surfaces identifiable as orbits of partonic 2-surfaces containing braids. Chern-Simons type action emerges as the only possible action principle and gives rise to almost (light-likeness!) topological QFT.

   A generalization of topological quantum computation emerges since parton replication accompanied by braid replication is involved and has interpretation in terms of copying of information. Particle exchanges in generalized Feynman diagrams have interpretation in terms of communication whereas incoming and outgoing lines could be interpreted as involving topological quantum computations.

2. The condition that braids emerge as fundamental structures is very strong and it is very difficult to imagine how they could emerge naturally in the standard mathematical framework of physics. In the physics based on the fusion of real physics and various p-adic physics interpreted in terms of cognition and intentionality, the fundamental braids emerge naturally as subsets of the rational (more generally algebraic) intersection of real parton and its p-adic counterpart obeying same algebraic equations. It seems that we have got a lot of TGD already. Bringing in the condition that standard model quantum numbers appear at fundamental level or accepting the vision about physics as a generalized number theory gives the rest of TGD (or all of it).

3. The quantum dynamics of TQC Universe need not rely on standard quantum theory. Indeed, new quantum physics based on hyper-finite factors of type II₁ with brand new quantum measurement theory with measurement resolution as a basic concept implying automatically non-commutative physics is involved. This physics emerges from both the Clifford algebra of the world of classical worlds and from the braid models of topological quantum computation. TGD suggests also strongly the quantization of Planck constant so that quantum entanglement in arbitrarily long length scales becomes possible and one can understand dark matter as macroscopic quantum phases responsible also for the very special properties of living matter.

Still about Higgs candidate

I have already discussed the slight indications for Higgs at 160 GeV in two postings (see this and this). In the original posting I mentioned briefly the possibility that the exotic pion of the 512-fold scaled up fractal copy of ordinary hadron physics might be responsible for the excess but by a stupid calculational error the mass estimate of the exotic pion came almost twice too large whereas correct estimate is 152 GeV not too far from 160 GeV. I attach below a short piece of the text from the first posting.

M₈₉ hadron physics might be required in TGD framework by the requirement of perturbative unitarity. Thus the mesons of M₈₉ hadron physics might be involved. By a very naive scaling by factor 2^(107-89)/2=2⁹ the mass of the pion of M₈₉ physics would be about 70 GeV. This estimate is not reliable since color spin-spin splittings distinguishing between pion and ρ mass do not scale naively. For M₈₉ mesons this splitting should be very small since color magnetic moments are very small (for calculation of color-magnetic splittings see this). The mass of ordinary pion in absence of splitting would be around 297 MeV and 512-fold scaling gives M(π₈₉)= 152 GeV which is not too far from 160 GeV. Could the decays of this exotic pion give rise to the excess of fermion pairs? (Note that the mass was erratically estimated to be 250 GeV in the original posting). This interpretation might also allow to understand why b-pair and t-pair excesses are not consistent. Monochromatic photon pairs with photon energy around 76 GeV would be the probably easily testable experimental signature of this option.

If this interpretation is correct, a whole new hadron physics might have been already seen in laboratory! Few of us can irritate their colleagues to the verge of madness by boasting with a discovery of a brand new physics and remaining still unemployed;-)! Be as it may, LHC will tell the truth.

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

About the construction of vertices

The understanding about the construction of S-matrix has increased considerably and I can now attack seriously the challenge of writing down the generalized Feynman rules.

1. Generalized Feynman diagrams

Let us first summarize the general picture.

1. Feynman diagrams are replaced with their higher dimensional variants with lines replaced with lightlike 3-surfaces identifiable as partonic orbits and with vertices replaced with partonic 2-surfaces along which lines meet. Lightlike 3-surfaces corresponding of maxima of Kähler function define generalized Feynman diagrams. There is no summation over the diagrams and each reaction corresponds to single minimal diagram. Quantum dynamics is 2-dimensional in the sense that vertices are defined by partonic 2-surfaces and 3-dimensional in the sense that different maxima of Kähler function defining points of spin glass energy landscape give rise to additional degeneracy essentially due to the presence of light-like direction.

2. S-matrix reduces to a unitary S-matrix depending parametrically on points of M⁴ defining arguments of N-point function in QFT approach. The momentum representation of S-matrix is obtained by taking a Fourier transform of this and is also unitary.

3. S-matrix is a generalization of braiding S-matrix in the sense that one assigns to the incoming/outgoing and internal lines a unitary braiding matrix. To the vertices, where braids replicate, one assigns a unitary isomorphism between tensor product of hyper-finite II₁ factors (HFFs) associated with incoming resp. outgoing lines. A crucial element in the construction is that these tensor products are themselves HFFs of type II₁.

4. Since also bosons are fermion-antifermion states located at partonic 2-surfaces, the construction of vertices reduces basically to that in the fermionic Fock space associated with the vertex and the space of small deformations of the generalized Feynman diagram around the maximum of Kähler function. The discrete set of points defining number theoretic strand define the basic unitary S-matrix and these points carry various quantum numbers. The natural assumption is that one can use at the vertex same fermionic basis for all incoming and outgoing lines and that unitary braiding S-matrix associated with lines induces a unitary transformation of basis. Its presence in internal lines gives rise to propagators as one integrates over the positions for tips of future and past lightcones containing at their light-like boundaries incoming and outgoing partons.

One can proceed by making simple guesses about the unitary isomorphism associated with the vertex.

1. The simplest guess would be that vertices involve only simple Fock space inner product. This would be like old fashioned quark model in which the quarks of incoming hadrons are re-arranged to from outgoing hadrons without pair creation or gluon emission. This trial does not work since it would not allow bosons which can be regarded as fermion-antifermion pair with either of them having non-physical helicity. This observation however serves as a valuable guideline.

2. An alternative guess is based on the observation that partonic 2-surface with punctures defined by number theoretical braids is analogous to closed bosonic string emitting particles. This would suggest that unitary S-matrix could be assigned with some conformal field theory or possibly string model. At least for non-specialist in conformal field theories this approach looks too abstract.

2. Vertices from free field theory defined by the modified Dirac operator

Something more concrete is required and to proceed one can try to apply the mathematical constraints from the basic definition of TGD.

1. The vertices should come out naturally from the modified Dirac action which contains the classical coupling of the gauge potentials (induced spinor connection) to fermions. Hence the modified Dirac action defining the analog of free field theory should appear as a basic building block in the definition of the inner product. Perturbation theory with respect to the induced gauge potential would conform with standard QFT but does not make sense. There is simply no decomposition of the modified Dirac operator D to "free" part and interaction term.

2. The vacuum expectation for the exponent of the modified Dirac action gives vacuum functional identified as exponent of Kähler function. When one sandwiches the exponent of Dirac action between many-fermion states, one obtains an inner product analogous to that in free field theory Feynman rules. How however the states are are not annihilated by D but are its generalized eigenstates with eigenvalues λ depending on p-adic prime by an overall scaling factor log(p) responsible for the coupling constant evolution. The generalized eigenvalue equation reads as DΨ= λ t^kΓ_kΨ, where t^k is lightlike vector t^k defining the tangent vector of partonic 3-surface or its M⁴ dual fixed once rest system and quantization axis of angular momentum has been fixed (it is not yet quite clear which option is correct). The notion of generalized eigenmode allows also to define Dirac determinant without giving up the separate conservation of H-chiralities (B and L). The generalized eigenstates are analogs of solutions of massless wave equation in the sense that the square of D annihilates them. Between states created by a monomial of fermionic oscillator operators the inner product reduces to a product of propagators.

3. A strict correspondence with free field theory would require that the incoming and outgoing states correspond to zero modes with λ=0 whereas internal lines as off mass shell states would correspond to non-vanishing eigenvalues λ . This assumption is however un-necessary since the four-momentum dependence comes only through the Fourier transform and one can regard all generalized eigenmodes as counterparts of massless modes. The restriction might be also inconsistent with unitarity.

4. For generalized eigenstates of D the modified Dirac propagator 1/D reduces to o^kΓ_k/λ. o^k is the light-like M⁴ dual of the lightlike vector t^k and λ is the generalized eigenvalue of D proportional to log(p). The propagator can be non-vanishing between vacuum and a boson consisting of fermion with physical helicity and antifermion with non-physical helicity so that non-trivial boson emission vertices are possible. At first it would seem that the inverse of the generalized eigenvalue λ contributes to the p-adic coupling constant evolution an overall 1/log(p) proportionality factor. However, since the inner product of un-normalized "bare" boson states (just fermion pair) is proportional to 1/log(p), the normalization of bosonic states cancels this factor so that algebraic number results. Thus fermionic contributions to the vertices are extremely simple since only the matrix o^kΓ_k remains. The conclusion made already earlier is that the p-adic coupling constant evolution must be due to the time evolution along parton lines dictated by the modified Dirac operator.

5. The fermionic contribution to the vertex says nothings about gauge couplings. All gauge coupling strengths must be proportional to the RG invariant Kähler coupling strength α_K, which can emerge only from the functional integral over small fluctuations around maximum of Kähler function K when the operator inverse of the covariant configuration space Kähler metric defining propagator is contracted between bosonic vector fields generating Kac-Moody and super-canonical symmetries in terms of which the deformation of the partonic 3-surface can be expressed. Obviously the configuration space spinor fields representing bosonic states must vanish at the maximum of K: otherwise coupling strength is of order unity. Geometrically this means that the maxima of Kähler function correspond to fixed points of these isometries.

3. Number theoretical constraints

The condition that S-matrix elements are algebraic numbers is an additional powerful guideline.

1. The most straightforward manner to guarantee that S-matrix elements are algebraic numbers is that vertex factors and propagators are separately algebraic numbers. log(p)-factors are obviously problematic number theoretically but normalization of the Fock space inner products cancels these factors. Thus coupling constant evolution can come only from the unitary time evolution with respect to the light-like coordinate of propagator lines dictated by the modified Dirac operator. Fermionic oscillator operators suffer a non-trivial unitary transformation depending on the p-adic prime p since (expressing it schematically) e^iHt is replaced by p^iHt.

2. The fundamental number theoretic conjecture is that the numbers p^s_n, where s_n=1/2+iy_n correspond to non-trivial zeros of Riemann zeta (or of more general zetas possibly involved), are algebraic numbers. If this is the case, then also the products and sums with rational coefficients involving finite number of nontrivial zeros of zeta are algebraic numbers and define a commutative algebra. The effect of the unitary time evolution operator should be expressible as an element of this algebra. Also larger algebraic extensions can be considered.

3. A simplified picture is provided by the dynamics of free number theoretic Hamiltonian for which eigenstates are labelled by primes and energy eigenvalues are given by E_p= log(p). Time evolution gives rise to phase factors exp(iE_pt)=p^it which are algebraic numbers in given extension of rationals for some quantized values of light-like coordinate t. If the conjectures about zeros of zeta hold true this is achieved if t is a linear combination of imaginary parts of zeros of zeta with integer coefficients: t= ∑_n k(n) y_n.

For more details see the chapter Hyper-Finite Factors and Construction of S-Matrix of the book "Towards S-matrix".

P.S. The following saying of the week from Tommaso Dorigo's blog somehow resonates with my inner feelings.

I make a living as a lawyer, and I spend a lot of time in the world of physics. I have encountered a lot of sleazy scumbags in the world of lawyers, but none of them are as bad as the bad guys in physics, and, although the bad guys in law do a lot of damage, I really think that the bad guys in physics do more damage to human civilization.

Non-commutative geometry: warmly recommended

I love to visit the good old This Week's Finds . Down-to-earth explanations of mathematical concepts and ideas, good humor, and no putting down mentality. What more one could hope for.

This Week's Finds is not anymore the only one of its kind. A new highly interesting blog has appeared in the blogsphere at the boundary layer between mathematics and physics: Noncommutative Geometry. I warmly recommend it to myself and theoretical physicists who might still dare to dream about the day when M theory is not anymore the only possible theory of even more than everything and that the proponents of competing theories are not anymore regarded as mindless crackpots and human waste.

Number theory as generalized physics

Amusingly, and rather satisfyingly, the basic theme during last days has been number theory as generalized quantum physics whereas my own basic theme (at much lower level of sophistication) has been physics as a generalized number theory.

One of very interesting articles briefly discussed in the blog is Quantum Statistical Mechanics and Class Field Theory.

1. A physical model allowing to construct the generators of the maximal Abelian extension of rationals and identify the action of corresponding Galois group (call it Gal) on these generators, is proposed. Someone might build an association with Hilbert's twelth problem.
2. The idea is to assign a C^* algebra to so called Hecke algebra appearing in Langlands program. The quantal time evolution and thermodynamics of this simple number theoretical quantum system is essentially unique since it corresponds to a state in a hyperfinite factor of type III.
3. What is amazing is that the numerical values of states of this system at the infinite temperature limit code for the algebraic numbers generating the maximal Abelian extension of rationals. This kind of condition is fundamental in TGD program of physics as generalized number theory having universality with respect to number field. Furthermore, Gal acts as symmetries of the system.
4. Partition function is nothing but Riemann Zeta as a function of temperature. Rieman Zeta and more general zetas have become an essential element of quantum TGD although I am very far from rigorous formulations yet.
5. There are two phases: low temperature phase corresponding to 1/T> 1 and high temperature phase corresponding to critical strip 1/T≤ 1. At T=1 partition function diverges: a signature of phase transition. At low temperature phase the action of Gal is non-trivial and thermal expectations of observables transforming non-trivially under Gal are non-vanishing. At high temperature phase the action of Gal is trivial meaning that thermal expectations of observables transforming non-trivially under Gal vanish. There is complete analogy with non-trivial action of gauge group on Higgs vacuum expectation value below critical temperature.

6. In TGD framework somewhat different breaking of gauge symmetry occurs for the Galois group of closure or rationals identified as infinite symmetric group whose group algebra is..., you guessed correctly: hyper-finite factor of type II₁! The finite Galois groups to which symmetry breaking occurs correspond are analogous to unbroken subgroups of gauge groups and directly related to Jones inclusions.

The article inspires interesting questions which only an innocent novice blessed with deep ignorance about basics can articulate.

1. At the critical line at which partition function is real. Could one consider quantum field theory at a finite temperature 1/T= Re(s)=1/2 and finite time interval t=Im[s]?

2. The partition function vanishes for the values of t corresponding to non-trivial zeros of zeta. What could this mean physically? Could it be that along critical line symmetry is broken in step wise manner. At first zero the thermal expectations for some non-singlets under some subgroup H of Gal become non-vanishing. At the next zero H is extended. In accordance with some earlier TGD inspired speculations this process would gradually lead to the entire Abelian Galois group and emergence of all phases generating maximal abelian extension of rationals. This suggests also a possible connection with Jones inclusions coded by phases exp(iπ/n) and cyclic group Z_n.

3. The reckless number theoretic speculations inspired by TGD inspire further questions. Could it be that for zeros the numbers p^iy appearing in the product decomposition of zeta are algebraic numbers in maximal Abelian extension? Could also zeros be algebraic numbers as the most stringent speculations suggest?

Physics as generalized number theory

I find this article highly interesting for many reasons.

1. In TGD framework finite subgroups of Galois group of Galois group of algebraic closure of rationals play a fundamental role as group of symmetries of number theoretic braids (see earlier postings).

2. Number theoretic universality requires that everything reduces to algebraic numbers belonging to the algebraic extension of rationals or p-adics corresponding to the level of predicted number theoretic hierarchy. For instance, the maxima of Kähler function coding for the geometry of the world of classical worlds would be algebraic number (also Neper number e and its roots could be allowed in p-adic context if extensions of p-adics are required to be finite-dimensional). S-matrix elements would be algebraic numbers and so on. The model discussed realizes this dream for thermodynamical states at infinite temperature limit.

3. The notion of number theoretic braid defined as a subset of the intersection of real parton and its p-adic counterpart would consist of algebraic points so that the construction of S-matrix would reduce to discrete number theory.

4. It seems that number theoretical QFT at finite temperature based on hyperfinite factors of type III could be associated with the strands of number theoretic braids representing incoming partons propagating in generalized Feynman diagrams generalizing the braid diagrams. This would bring in p-adic thermodynamics at the fundamental level. Coupling constant evolution would be at the level of free states and no summation over loops would be needed.

Some ideas relating to physics, number theory, and biology

Before closing I want to repeat myself by recalling some associated ideas suggesting deep connections between number theory, quantum computation, and biology.

1. Vertices are direct generalizations of those for Feynman diagrams with point-like vertex being replaced with 2-dimensional partonic surface. They are totally different from string diagrams.

2. At the vertices number theoretic braids replicate and also DNA replication might involve this process as a deeper level process. Isomorphisms between tensor products of HFFs of type II₁ associated with incoming resp. outgoing lines would define vertices. Everything would be unitary and definitely non-trivial.

3. I ended up with braids from TGD inspired model for topological quantum computation using braids. Universe could be topological quantum computer at all levels of fractal hierarhcy. Copying of information represented by number theoretic braids would occur at vertices, its communication would takes place at propagator lines, and quantum computation would be performed at incoming lines.

To conclude, I think that Noncommutative Geometry might provide an excellent opportunity for a theoretical physicists to enjoy simple explanations of ideas which are difficult to extract from formal mathematical papers.

More about Higgs candidate

As I told in previous positng, there have been cautious claims (see New Scientist article, the postings in the blog of Tommaso Dorigo, and the postings of John Conway in Cosmic Variance) about the possible detection of first Higgs events.

According to simple argument of John Conway based on branching ratios of Z⁰ and standard model Higgs to τ-τbar and b-bbar, Z⁰→ τ-τbar excess predicts that the ratio of Higgs events to Z⁰ events for Z⁰→ b-bbar is related by a scaling factor

[B(H→ b-bbar)/B(H→ τ-τbar)]:[B(Z⁰→ b-bbar)/B(Z⁰→ τ-τbar)] ≈ 10/5.6=1.8

to that in Z⁰→ τ-τbar case. The prediction seems to be too high which raises doubts against the identification of the excecss in terms of Higgs.

In a shamelessly optimistic mood and forgetting that mere statistical fluctuations might be in question, one might ask whether the inconsistency of τ-τbar and b-bbar excesses could be understood in TGD framework.

1. The couplings of Higgs to fermions need not scale as mass in TGD framework. Rather, the simplest guess is that the Yukawa couplings scale like p-adic mass scale m(k)=1/L(k), where L(k) is the p-adic length scale of fermion. Fermionic masses can be written as m(F)= x(F)/L(k), where the numerical factor x(F)>1 depends on electro-weak quantum numbers and is different for quarks and leptons. If the leading contribution to the fermion mass comes from p-adic thermodynamics, Yukawa couplings in TGD framework can be written as h(F)= ε(F) m(F)/x(F), ε<< 1. The parameter ε should be same for all quarks resp. leptons but need not be same for leptons and quarks so that that one can write ε (quark)= ε_Q and ε (lepton)= ε_L. This is obviously an important feature distinguishing between Higgs decays in TGD and standard model.

2. The dominating contribution to the mass highest generation fermion which in absence of topological mixing correspond to genus g=2 partonic 2-surface comes from the modular degrees of freedom and is same for quarks and leptons and does not depend on electro-weak quantum numbers at all (p-adic length scale is what matters only). Topological mixing inducing CKM mixing affects x(F) and tends to reduce x(τ), x(b), and x(t).

3. In TGD framework the details of the dynamics leading to the final states involving Z⁰ bosons and Higgs bosons are different since one expects that it fermion-Higgs vertices suppressed to the degree that weak-boson-Higgs vertices could dominate in the production of Higgs. Since these details should not be relevant for the experimental determination of Z⁰→ τ-τbar and Z⁰→ b-bbar distributions, then the above argument can be modified in a straightforward manner by looking how the branching ratio R(b-bbar)/R(τ-τbar) is affected by the modification of Yukawa couplings for b and τ. What happens is following:

   B(H→ b-bbar)/B(H→ τ-τbar)= m_b²/m_τ² → B(H→ b-bbar)/B(H→ τ-τbar)×X ,

   X=(ε²(q)/ε² (L))× (x_τ²/x_b²).

   Generalizing the simple argument of Conway one therefore has

   (H/Z)⁰(b-bbar)= 1.8 × (ε²_Q/ε²_L )×(x_τ²/x_b²)× (H/Z)⁰(τ-τbar).

   Since the topological mixing of both charged leptons and quarks of genus 2 with lower genera is predicted to be very small (see this) , x_τ/x_b≈ 1 is expected to hold true. Hence the situation is not improved unless one has ε_Q/ε_L<1 meaning that the coupling of Higgs to the p-adic mass scale would be weaker for quarks than for leptons.

Can one then guess then value of r and perhaps even Yukawa coupling from general arguments?

1. The actual value of r should relate to electro-weak physics at very fundamental level. The ratio r=1/3 of Kähler couplings of quarks and leptons is certainly this kind of number. This would reduce the prediction for (H/Z⁰)(b-bbar) by a factor of 1/9. To my best understanding, this improves the situation considerably (see for yourself).

2. Kähler charge Q_K equals electro-weak U(1) charge Q_U(1). Furthermore, Kähler coupling strength which is RG invariant equals to U(1) coupling strength at the p-adic length scale of electron but not generally (see this). This observation encourages the guess that, apart from a numerical factor of order unity, ε² itself is given by either α_KQ_K² and thus RG invariant or by α_U(1)Q_U(1)². The contribution of Higgs vacuum expectation to fermionic mass would be roughly a fraction 10^-2-10^-3 about fermion mass in consistency with p-adic mass calculations.

Of course, it might turn out that fake Higgs is in question. What is however important is that the deviation of the Yukawa coupling allowed by TGD for Higgs from those predicted by standard model could manifest itself in the ratio of Z_0→ b-bbar and Z⁰→ τ-τbar excesses.

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

Indications for Higgs with mass of 160 GeV

There have been cautious claims (see New Scientist article, the postings in the blog of Tommaso Dorigo, and the postings of John Conway in Cosmic Variance) about the possible detection of first Higgs events.

This inspires more precise considerations of the experimental signatures of TGD counterpart of Higgs. This kind of theorizing is of course speculative and remains on general qualitative level only since no calculational formalism exists and one must assume that gauge field theory provides an approximate description of the situation.

Has Higgs been detected?

The indications for Higgs comes from two sources. In both cases Higgs would have been produced as gluons decay to two b-bbar pairs and virtual b-bbar pair fuses to Higgs, which then decays either to tau-lepton pair or b-quark pair.

John Conway, the leader of CDF team analyzing data from Tevatron, has reported about a slight indication for Higgs with mass m_H=160 GeV as a small excess of events in the large bump produced by the decays of Z⁰ bosons with mass of m_Z≈ 94 GeV to tau-taubar pairs in the blog Cosmic Variance. These events have 2σ significance level meaning that the probability that they are statistical fluctuations is about 2 per cent.

The interpretation suggested by Conway is as Higgs of minimal super-symmetric extension of standard model (MSSM). In MSSM there are two complex Higgs doublets and this predicts three neutral Higgs particles denoted by h, H, and A. If A is light then the rate for the production of Higgs bosons is proportional to the parameter tan(β) define as the ratio of vacuum expectation values of the two doublets. The rate for Higgs production is by a factor tan(β)² higher than in standard model and this has been taken as a justification for the identification as MSSM Higgs (the proposed value is tan(β)≈ 50). If the identification is correct, about recorded 100 Higgs candidates should already exist so that this interpretation can be checked.

Also Tommaso Dorigo, the blogging member of second team analyzing CDF results, has reported at his blog site a slight evidence for an excess of b-bbar pairs in Z⁰→ b-bbar decays at the same mass m_H=160 GeV. The confidence level is around 2 sigma. The excess could result from the decays of Higgs to b-bbar pair associated with b-bbar production.

What forces to take these reports with some seriousness is that the value of m_H is same in both cases. John Conway has however noticed that if both signals correspond to Higgs then it is possible to deduce estimate for the number of excess events in Z⁰→ b-bbar peak from the excess in tau-taubar peak. The predicted excess is considerably larger than the real excess. Therefore a statistical fluke could be in question, or staying in an optimistic mood, there is some new particle there but it is not Higgs.

m_H=160 GeV is not consistent with the standard model estimate by D0 collaboration for the mass of standard model Higgs boson mass based on high precision measurement of electro-weak parameters sin(θ_W), α, α_s , m_t and m_Z depending on log(m_H) via the radiative corrections. The best fit is in the range 96-117 GeV. The upper bound from the same analysis for Higgs mass is 251 GeV with 95 per cent confidence level. The estimate m_t=178.0+/- 4.3 GeV for the mass of top quark is used. The range for the best estimate is not consistent with the lower bound of 114 GeV on m_H coming from the consistency conditions on the renormalization group evolution of the effective potential V(H) for Higgs (see the illustration here). Here one must of course remember that the estimates vary considerably.

TGD picture about Higgs briefly

Since TGD cannot yet be coded to precise Feynman rules, the comparison of TGD to standard model is not possible without some additional assumptions. It is assumed that p-adic coupling constant evolution reduces in a reasonable approximation to the coupling constant evolution predicted by a gauge theory so that one can apply at qualitative level the basic wisdom about the effects of various couplings of Higgs to the coupling constant evolution of the self coupling λ of Higgs giving upper and lower bounds for the Higgs mass. This makes also possible to judge the determinations of Higgs mass from high precision measurements of electro-weak parameters in TGD framework.

In TGD framework the Yukawa coupling of Higgs to fermions can be much weaker than in standard model. This has several implications.

1. The rate for the production of Higgs via channels involving fermions is much lower. This could explain why Higgs has not been observed even if it had mass around 100 GeV.

2. The radiative corrections to electro-weak parameters coming from fermion-Higgs vertices are much smaller than in standard model and cannot be used to deduce Higgs mass from the high precision measurements of electro-weak parameters. Hence one cannot anymore localize Higgs mass to the range 96-117 GeV.

3. In standard model the large Yukawa coupling of Higgs to top, call it h, tends to reduce the quartic self coupling constant λ for Higgs in ultraviolet. The condition that the minimum for Higgs potential is not transformed to a maximum gives a lower bound on the initial value of λ and thus to the value of m_H. In TGD framework the weakness of fermionic couplings implies that there is no lower bound to Higgs mass.

4. The weakness of Yukawa couplings means that self coupling of Higgs tends to increase λ faster than in standard model. Note also that when Yukawa coupling h_t to top is small (h_t²< λ, see arXiv:hep-ph/9409458), its contribution tends to increase the value of β_λ. Thus the upper bound from perturbative unitarity to the scalar coupling λ (and m_H) is reduced. This would force the value of Higgs mass to be even lower than in standard model.

   In TGD framework new physics can however emerge in the length scales corresponding to Mersenne primes M_n=2ⁿ-1. Ordinary QCD corresponds to M₁₀₇ and one cannot exclude even M₈₉ copy of QCD. M₆₁ would define the next candidate. The quarks of M₈₉ QCD would give to the beta function β_λ a negative contribution tending to reduce the value λ so that unitary bound would not be violated. If this new physics is accepted m_H=160 GeV can be considered.

Can one then identify the Higgs candidate with m_H=160 with the TGD variant of standard model Higgs? This is far from clear.

1. Even in standard model the rate for the production of Higgs is low. In TGD the rate for the production of the counterpart of standard model Higgs is reduced since the coupling of quarks to Higgs is expected to be much smaller than in standard model. This might exclude the interpretation as Higgs.

2. The slow rate for the production of Higgs could also allow the presence of Higgs at much lower mass and explain why Higgs has not been detected in the mass range m_H<114>

3. In TGD framework one can consider also other interpretations of the excess events at 160 GeV (taking the findings of both Dorigo's and Conway's group seriously and the fact that they do not seem to be consistent). p-Adically scaled up variants of ordinary quarks which might have something to do with the bumpy nature of top quark mass distribution.

   M₈₉ hadron physics might be required in TGD framework by the requirement of perturbative unitarity. Thus the mesons of M₈₉ hadron physics might be involved. By a very naive scaling by factor 2^(107-89)/2=2⁹ the mass of the pion of M₈₉ physics would be about 70 GeV. This estimate is not reliable since color spin-spin splittings distinguishing between pion and ρ mass do not scale naively. For M₈₉ mesons this splitting should be very small since color magnetic moments are very small. The mass of pion in absence of splitting would be around 297 MeV and 512-fold scaling gives M(π₈₉)≈ 152 GeV which is not too far from 160 GeV. Could the decays of this exotic pion give rise to the excess of fermion pairs? Note that he mass was given erratically in the original posting. This interpretation might also allow to understand why b-pair and t-pair excesses are not consistent.

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