What's new inTowards MMatrixNote: Newest contributions are at the top! 
Year 2018 
TGD View about Coupling Constant Evolution
Atyiah has recently proposed besides a proof of Riemann Hypothesis also an argument claiming to derive the value of the structure constant (see this). The mathematically elegant arguments of Atyiah involve a lot of refined mathematics including notions of Todd exponential and hyperfinite factors of type II (HFFs) assignable naturally to quaternions. The idea that 1/α could result by coupling constant evolution from π looks however rather weird for a physicist. What makes this interesting from TGD point of view is that in TGD framework coupling constant evolution can be interpreted in terms of inclusions of HFFs with included factor defining measurement resolution. An alternative interpretation is in terms of hierarchy of extensions of rationals with coupling parameters determined by quantum criticality as algebraic numbers in the extension. In the following I will explain what I understood about Atyiah's approach. My critics includes the arguments represented also in the blogs of Lubos Motl (see this) and Sean Carroll (see this). I will also relate Atyiah's approach to TGD view about coupling evolution. The hasty reader can skip this part although for me it served as an inspiration forcing to think more precisely TGD vision. There are two TGD based formulations of scattering amplitudes.

How could Planck length be actually equal to much larger CP_{2} radius?!
The following argument stating that Planck length l_{P} equals to CP_{2} radius R: l_{P}=R and Newton's constant can be identified G= R^{2}/ℏ_{eff}. This idea looking nonsensical at first glance was inspired by an FB discussion with Stephen Paul King. First some background.
To get some perspective, consider first the phase transition replacing hbar and more generally hbar_{eff,i} with hbar_{eff,f}=h_{gr} .
See the chapter Some Questions Related to the Twistor Lift of TGD or the article About the physical interpretation of the velocity parameter in the formula for the gravitational Planck constant. 
Are spacetime surfaces minimal surfaces everywhere except at 2D interaction vertices?The action S determining spacetime surfaces as preferred extremals follows from twistor lift and equals to the sum of volume term Vol and Kähler action S_{K}. The field equation is a geometric generalization of d'Alembert (Laplace) equation in Minkowskian (Eucidian) regions of spacetime surface coupled with induced Kähler form analogous to Maxwell field. Generalization of equations of motion for particle by replacing it with 3D surface is in question and the orbit of particle defines a region of spacetime surface.
See the chapter The Recent View about Twistorialization in TGD Framework or the article with the same title. 
Three dualities at the level of field equationsThe basic field equations of TGD allow several dualities. There are 3 of them at the level of basic field equations (and several other dualities such as M^{8}M^{4}× CP_{2} duality).

New insights about quantum criticality for twistor lift inspired by analogy with ordinary criticalityQuantum criticality (QC) is one of the basic ideas of TGD. Zero energy ontology (ZEO) is second key notion and leads to a theory of consciousness as a formulation of quantum measurement theory solving the basic paradox of standard quantum measurement theory, which is usualy tried to avoid by introducing some "interpretation". ZEO allows to see quantum theory could be seen as "square root" of thermodynamics. It occurred to me that it would be interesting to apply this vision in the case of quantum criticality to perhaps gain additional insights about its meaning. We have a picture about criticality in the framework of thermodynamics: what would be the analogy in ZEO based interpretation of Quantum TGD? Could it help to understand more clearly the somewhat poorly understood views about the notion of self, which as a quantum physical counterpart of observer becomes in ZEO a key concept of fundamental physics? The basic ingredients involved are discrete coupling constant evolution, zero energy ontology (ZEO) implying that quantum theory is analogous to "square root" of thermodynamics, self as generalized Zeno effect as counterpart of observer made part of the quantum physical system, M^{8}M^{4}× CP_{2} duality, and quantum criticality. A further idea is that vacuum functional is analogous to a thermodynamical partition function as exponent of energy E= TSPV. The correspondence rules are simple. The mixture of phases with different 3volumes per particle in a critical region of thermodynamical system is replaced with a superposition of spacetime surfaces of different 4volumes assignable to causal diamonds (CDs) with different sizes. Energy E is replaced with action S for preferred extremals defining Kähler function in the "world of classical worlds" (WCW). S is sum of Kähler action and 4volume term, and these terms correspond to entropy and volume in the generalization E= TSPV → S. P resp. T corresponds to the inverse of Kähler coupling strength α_{K} resp. cosmological constant Λ. Both have discrete spectrum of values determined by number theoretically determined discrete coupling constant evolution. Number theoretical constraints force the analog of microcanonical ensemble so that S as the analog of E is constant for all 4surfaces appearing in the quantum superposition. This implies quantization rules for Kähler action and volume, which are very strong since α_{K} is complex. This kind of quantum critical zero energy state is created in unitary evolution created in single step in the process defining self as a generalized Zeno effect. This unitary process implying time delocalization is followed by a weak measurement reducing the state to a fixed CD so that the clock time idenfified as the distance between its tips is welldefined. The condition that the action is same for all spacetime surfaces in the superposition poses strong quantization conditions between the value of Kähler action (Kähler coupling strength is complex) and volume term proportional to cosmological constant. The outcome is that after sufficiently large number of steps no spacetime surfaces satisfying the conditions can be found, and the first reduction to the opposite boundary of CD must occur  self dies. This is the classical counterpart for the fact that eventually all state function reduction leaving the members of state pairs at the passive boundary of CD invariant are made and the first reduction to the opposite boundary remains the only option. The generation of magnetic flux tubes provides a manner to satisfy the constancy conditions for the action so that the existing phenomenology as well as TGD counterpart of cyclic cosmology as reincarnations of cosmic self follows as a prediction. This picture generalizes to the twistor lift of TGD and cosmology provides an interesting application. One ends up with a precise model for the padic coupling constant evolution of the cosmological constant Λ explaining the positive sign and smallness of Λ in long length scales as a cancellation effect for M^{4} and CP_{2} parts of the Kähler action for the sphere of twistor bundle in dimensional reduction, a prediction for the radius of the sphere of M^{4} twistor bundle as Compton length associated with Planck mass (2π times Planck length), and a prediction for the padic coupling constant evolution for Λ and coupling strength of M^{4} part of Kähler action giving also insights to the CP breaking and matter antimatter asymmetry. The observed two values of Λ could correspond to two different padic length scales differing by a factor of 2^{1/2}. See the chapter The Recent View about Twistorialization in TGD Framework or the article with the same title or the shorter article New insights about quantum criticality for twistor lift inspired by analogy with ordinary criticality. 
Complex 8momenta are necessary for the realization of massless manyparticle states implying unitary without loopsI have proposed a realization of unitarity in twistor approach without loops and with discrete coupling constant evolution dictated by number theory (see this). The proposal relies crucially on the identification quantum numbers in M^{8} picture as likelight quaternionic 8momenta and the assumption that also manyparticle states are massles. The 8momenta are also complex already at classical level with corresponding imaginary unit i commuting with octonionic imaginary units I_{k} of M^{8}. The essential assumption was that the 8momenta of also manyparticle states are lightlike. It is easy to see that this cannot make sense if single particle states have lightlike 8momenta unless they are also parallel. For a moment I thought that complexification of single particle 8momenta might help but it did not. Next came the realization that BCFW construction actually gives analogs of zero energy states having complex lightlike momenta. The single particle momenta are not however lightlike anymore. In TGD these states can be assigned with the interior regions of causal diamonds and have interpretation as resonances/bound states with complex momenta. The following tries to articular this more precisely.

Connection between quaternionicity and causalityThe notion of quaternionicity is a central element of M^{8}H duality. At the level of momentum space it means that 8momenta , which by M^{8}Hduality correspond to 4momenta at level of M^{4} and color quantum numbers at the level of CP_{2}  are quaternionic. Quaternionicity means that the time component of 8momentum, which is parallel to real octonion unit, is nonvanishing. The 8momentum itself must be timelike, in fact lightlike. In this case one can always regard the momentum as momentum in some quaternionic subspace. Causality requires a fixed sign for the time component of the momentum. It must be however noticed that 8momentum can be complex: also the 4momentum can be complex at the level of M× CP_{2} already classically. A possible interpretation is in terms of decay width as part of momentum as it indeed is in phenomenological description of unstable particles. Remark: At spacetime level either the tangent space or normal space of spacetime surface in M^{8} is quaternionic (equivalently associative) in the regions having interpretation as external particles arriving inside causal diamond (CD). Inside CD this assumption is not made. The two options correspond to spacetime regions with Minkowskian and Euclidian signatures of the induced metric. Could one require that the quaternionic momenta form a linear space with respect to octonionic sum? This is the case if the energy  that is the timelike part parallel to the real octonionic unit  has a fixed sign. The sum of the momenta is quaternionic in this case since the sum of lightlike momenta is in general timelike and in special case lightlike. If momenta with opposite signs of energy are allowed, the sum can become spacelike and the sum of momenta is coquaternionic. This result is technically completely trivial as such but has a deep physical meaning. Quaternionicity at the level of 8momenta implies standard view about causality: only timelike or at most lightlike momenta and fixed sign of timecomponent of momentum. Remark: The twistorial construction of Smatrix in TGD framework based on generalization of twistors leads to a proposal allowing to have unitary Smatrix with vanishing loop corrections and number theoretically determined discrete coupling constant evolution. Also the problems caused by nonplanar diagrams disappear and one can have particles, which are massive in M^{4} sense. The proposal boils down to the condition that the 8momenta of manyparticle states are lightlike (in complex sense). One has however a superposition over states with different directions of the projection of lightlike 8momentum to E^{4} in M^{8}= M^{4}× E^{4}). At the level of CP_{2} one has massive state but in color representation for which color spin and hypercharge vanish but color Casimir operator can have value of the order of the mass squared for the state. This prediction sharply distinguishes TGD from QCD. See the chapter The Recent View about Twistorialization in TGD Framework or the article with the same title. 
Could functional equation and Riemann hypothesis generalize?Number theoretical considerations lead to the modification of zeta function by replacing the powers n^{s} with= exp(log(n)s) with powers exp(Log(n)s), where rational valued number theoretic logarithm Log(n) is defined as sum_{p}p p/π(p) corresponding to the decomposion of n to a product of powers of prime. For large primes Log(p) equals in good approximation to log(p). The point of the replacement is that Log(n) carriers number theoretical information so that the definition is very natural. This number theoretical zeta will denoted with Ζ to distinguish it from ordinary zeta function denoted by ζ. It is interesting to list the elementary properties of the Ζ before trying to see whether functional equation for ζ and Riemann hypothesis generalize.
For background see the chapter The Recent View about Twistorialization in TGD Framework or the article with the same title. 
Considerations related to coupling constant evolution and Riemann zetaI have made several number theoretic peculations related to the possible role of zeros of Riemann zeta in coupling constant evolution. The basic problem is that it is not even known whether the zeros of zeta are rationals, algebraic numbers or genuine transcendentals or belong to all these categories. Also the question whether number theoretic analogs of ζ defined for padic number fields could make sense in some sense is interesting. 1. Is number theoretic analog of ζ possible using Log(p) instead of log(p)? The definition of Log(n) based on factorization Log(n)==∑_{p}k_{p}Log(p) allows to define the number theoretic version of Riemann Zeta ζ(s)=∑ n^{s} via the replacement n^{s}=exp(log(n)s)→ exp(Log(n)s).
2. Could the values of 1/α_{K} be given as zeros of ζ or of modified ζ I have discussed the possibility that the zeros s=1/2+iy of Riemann zeta at critical line correspond to the values of complex valued Kähler coupling strength α_{K}: s=i/α_{K} (see this). The assumption that p^{iy} is root of unity for some combinations of p and y [log(p)y =(r/s)2π] was made. This does not allow s to be complex rational. If the exponent of Kähler action disappears from the scattering amplitudes as M^{8}H duality requires, one could assume that s has rational values but also algebraic values are allowed.
Consider now the argument suggesting that the roots of zeta cannot be complex rationals. On basis of numerical evidence Dyson (see this) has conjectured that the Fourier transform for the characteristic function for the critical zeros of zeta consists of multiples of logarithms log(p) of primes so that one could regard zeros as onedimensional quasicrystal. This hypothesis makes sense if the zeros of zeta decompose into disjoint sets such that each set corresponds to its own prime (and its powers) and one has p^{iy}= U_{m/n}=exp(i2π m/n) (see the appendix of this). This hypothesis is also motivated by number theoretical universality (see this).
For background see the chapter The Recent View about Twistorialization in TGD Framework or the article with the same title. 
General ideas about coupling constant evolutionThe discrete coupling constant evolution would be associated with the scale hierarchy for CDs and the hierarchy of extensions of rationals.
In both cases one expects approximate logarithmic dependence and the challenge is to define "number theoretic logarithm" as a rational number valued function making thus sense also for padic number fields as required by the number theoretical universality. Consider first the coupling constant as a function of the length scale l_{CD}(n)/l_{CD}(1)=n.
For background see the chapter The Recent View about Twistorialization in TGD Framework or the article with the same title. 
Summary about twistorialization in TGD frameworkSince the contribution means in welldefined sense a breakthrough in the understanding of TGD counterparts of scattering amplitudes, it is useful to summarize the basic results deduced above as a polished answer to a Facebook question. There are two diagrammatics: Feynman diagrammatics and twistor diagrammatics.

The Recent View about Twistorialization in TGD FrameworkThe recent view about the twistorialization in TGD framework is discussed.
