Could second generation of weak bosons explain the reduction of proton charge radius?

The discovery by Pohl et al (2010) was that the charge radius of proton deduced from the muonic version of hydrogen atom - is .842 fm and about 4 per cent smaller than .875 fm than the charge radius deduced from hydrogen atom is in complete conflict with the cherished belief that atomic physics belongs to the museum of science (for details see the Wikipedia article). The title of the article Quantum electrodynamics-a chink in the armour? of the article published in Nature expresses well the possible implications, which might actually go well extend beyond QED.

Quite recently (2016) new more precise data has emerged from Pohl et al (see this). Now the reduction of charge radius of muonic variant of deuterium is measured. The charge radius is reduced from 2.1424 fm to 2.1256 fm and the reduction is .012 fm, which is about .8 per cent (see this). The charge radius of proton deduced from it is reported to be consistent with the charge radius deduced from deuterium. The anomaly seems therefore to be real. Deuterium data provide a further challenge for various models.

The finding is a problem of QED or to the standard view about what proton is. Lamb shift is the effect distinguishing between the states hydrogen atom having otherwise the same energy but different angular momentum. The effect is due to the quantum fluctuations of the electromagnetic field. The energy shift factorizes to a product of two expressions. The first one describes the effect of these zero point fluctuations on the position of electron or muon and the second one characterizes the average of nuclear charge density as "seen" by electron or muon. The latter one should be same as in the case of ordinary hydrogen atom but it is not. Does this mean that the presence of muon reduces the charge radius of proton as determined from muon wave function? This of course looks implausible since the radius of proton is so small. Note that the compression of the muon's wave function has the same effect.

Before continuing it is good to recall that QED and quantum field theories in general have difficulties with the description of bound states: something which has not received too much attention. For instance, van der Waals force at molecular scales is a problem. A possible TGD based explanation and a possible solution of difficulties proposed for two decades ago is that for bound states the two charged particles (say nucleus and electron or two atoms) correspond to two 3-D surfaces glued by flux tubes rather than being idealized to points of Minkowski space. This would make the non-relativistic description based on Schrödinger amplitude natural and replace the description based on Bethe-Salpeter equation having horrible mathematical properties.

The basic idea of the original model of the anomaly (see this) is that muon has some probability to end up to the magnetic flux tubes assignable to proton. In this state it would not contribute to the ordinary Schrödinger amplitude. The effect of this would be reduction of |Ψ|² near origin and apparent reduction of the charge radius of proton. The weakness of the model is that it cannot make quantitative prediction for the size of the effect. Even the sign is questionable. Only S-wave binding energy is affected considerably but does the binding energy really increase by the interaction of muon with the quarks at magnetic flux tubes? Is the average of the charge density seen by muon in S wave state larger, in other words does it spend more time
near proton or do the quarks spend more time at the flux tubes?

In the following a new model for the anomaly will be discussed.

1. The model is inspired by data about breaking of universality of weak interactions in neutral B decays possibly manifesting itself also in the anomaly in the magnetic moment of muon. Also the different values of
   the charge radius deduced from hydrogen atom and muonium could reflect the breaking of universality. In the original model the breaking of universality is only effective.
   
   
2. TGD indeed predicts a dynamical U(3) gauge symmetry whose 8+1 gauge bosons correspond to pairs of fermion and anti-fermion at opposite throats of wormhole contact. Throats are characterized by genus g=0,1,2, so that bosons are superpositions of states labelled by (g₁,g₂). Fermions correspond to single wormhole throat carrying fermion number and behave as U(3) triplet labelled by g.
   

   The charged gauge bosons with different genera for wormhole throats are expected to be very massive. The 3 neutral gauge bosons with same genus at both throats are superpositions of states (g,g) are expected to be lighter. Their charge matrices are orthogonal and necessarily break the universality of electroweak interactions. For the lowest boson family - ordinary gauge bosons - the charge matrix is proportional to unit matrix. The exchange of
   second generation bosons Z⁰₁ and γ₁ would give rise to Yukawa potential increasing the binding energies of S-wave states. Therefore Lamb shift defined as difference between energies of S and P waves is increased and the charge radius deduced from Lamb shift becomes smaller.
   
   

3. The model thus predicts a correct sign for the effect but the size of the effect from naive estimate assuming only γ₂ and α₂=α₁== α for M=2.9 TeV is almost by an order of magnitude too small. The values of the gauge couplings α₂ and α_Z,2 are free parameters as also the mixing angles between states (g,g). The effect is also proportional to the ratio (m_μ/M(boson)². It turns out that the inclusion of Z⁰₁ contribution and assumption α₁ and α_Z,1 are near color coupling strength α_s gives a correct prediction.
   
   

Motivations for the breaking of electroweak universality

The anomaly of charge radius could be explained also as breaking of the universality of weak interactions. Also other anomalies challenging the universality exists. The decays of neutral B-meson to lepton pairs should be same apart from corrections coming from different lepton masses by universality but this does not seem to be the case (see this). There is also anomaly in muon's magnetic moment discussed briefly here. This leads to ask whether the breaking of universality could be due to the failure of universality of electroweak interactions.

The proposal for the explanation of the muon's anomalous magnetic moment and anomaly in the decays of B-meson is inspired by a recent very special di-electron event and involves higher generations of weak bosons predicted by TGD leading to a breaking of lepton universality. Both Tommaso Dorigo (see this) and Lubos Motl (see this) tell about a spectacular 2.9 TeV di-electron event not observed in previous LHC runs. Single event of this kind is of course most probably just a fluctuation but human mind is such that it tries to see something deeper in it - even if practically all trials of this kind are chasing of mirages.

Since the decay is leptonic, the typical question is whether the dreamed for state could be an exotic Z boson. This is also the reaction in TGD framework. The first question to ask is whether weak bosons assignable to Mersenne prime M₈₉ have scaled up copies assignable to Gaussian Mersenne M₇₉. The scaling factor for mass would be 2^(89-79)/2= 32. When applied to Z mass equal to about .09 TeV one obtains 2.88 TeV, not far from 2.9 TeV. Eureka!? Looks like a direct scaled up version of Z!? W should have similar variant around 2.6 TeV.

TGD indeed predicts exotic weak bosons and also gluons.

1. TGD based explanation of family replication phenomenon in terms of genus-generation correspondence forces to ask whether gauge bosons identifiable as pairs of fermion and antifermion at opposite throats of wormhole contact could have bosonic counterpart for family replication. Dynamical SU(3) assignable to three lowest fermion generations labelled by the genus of partonic 2-surface (wormhole throat) means that fermions are combinatorially SU(3) triplets. Could 2.9 TeV state - if it would exist - correspond to this kind of state in the tensor product of triplet and antitriplet? The mass of the state should depend besides p-adic mass scale also on the structure of SU(3) state so that the mass would be different. This difference should be very small.
   
   
2. Dynamical SU(3) could be broken so that wormhole contacts with different genera for the throats would be more massive than those with the same genera. This would give SU(3) singlet and two neutral states, which are analogs of η' and η and π⁰ in Gell-Mann's quark model. The masses of the analogs of η and π⁰ and the the analog of η', which I have identified as standard weak boson would have different masses. But how large is the mass difference?
   
   
3. These 3 states are expected top have identical mass for the same p-adic mass scale, if the mass comes mostly from the analog of hadronic string tension assignable to magnetic flux tube. connecting the two wormhole contacts associates with any elementary particle in TGD framework (this is forced by the condition that the flux tube carrying monopole flux is closed and makes a very flattened square shaped structure with the long sides of the square at different space-time sheets). p-Adic thermodynamics would give a very small contribution genus dependent contribution to mass if p-adic temperature is T=1/2 as one must assume for gauge bosons (T=1 for fermions). Hence 2.95 TeV state could indeed correspond to this kind of state.
   
   

The sign of the effect is predicted correctly and the order of magnitude come out correctly

Could the exchange of massive M_G,79 photon and Z⁰ give rise to additional electromagnetic interaction inducing the breaking of Universality? The first observation is that the binding energy of S-wave state increases but there is practically no change in the energy of P wave state. Hence the effective charge radius r_p as deduced from the parameterization of binding energy different terms of proton charge radius indeed decreases.

Also the order of magnitude for the effect must come out correctly.

1. The additional contribution in the effective Coulomb potential is Yukawa potential. In S-wave state this would give a contribution to the binding energy in a good approximation given by the expectation value of the Yukawa potential, which can be parameterized as
   

   V(r)= g² e^-Mr/r ,&g² = 4π kα .
   

   The expectation differs from zero significantly only in S-wave state characterized by principal quantum number n. Since the exponent function goes exponentially to zero in the p-adic length scale associated with 2.9 TeV mass, which is roughly by a factor 32 times shorter than intermediate boson mass scale, hydrogen atom wave function is constant in excellent approximation in the effective integration volume. This gives for the energy shift
   

   Δ E= g²| Ψ(0)|² × I ,
   

   Ψ(0) ² =[2²/n²]×(1/a₀³) ,
   

   a₀= 1/(mα) ,
   

   I= ∫ (e^-Mr/r) r²drdΩ =4π/3M².
   

   For the energy shift and its ratio to ground state energy
   

   E_n= α²/2n²× m
   

   one obtains the expression
   

   Δ E_n= 64π² α/n² α³ (m/M)² × m ,
   

   Δ E_n/E_n= (2⁷/3) π²α² k²(m/M)² .
   

   For k=1 and M=2.9 one has Δ E_n/E_n ≈ 3× 10^-11 for muon.
   

Consider next Lamb shift.

1. Lamb shift as difference of energies between S and P wave states (see this) is approximately given by
   

   Δ_n (Lamb)/E_n= 13α³/2n .
   

   For n=2 this gives Δ₂ (Lamb)/E₂= 4.9× 10^-7.
   
   

2. The parameterization for the Lamb shift reads as
   

   Δ E(r_p) =a - br_p² +cr_p³
   = 209.968(5) - 5.2248 × r²_p + 0.0347 × r³_p meV ,
   

   where the charge radius r_p=.8750 is expressed in femtometers and energy in meVs.
   
   

3. The reduction of r_p by 3.3 per cent allows to estimate the reduction of Lamb shift (attractive additional potential reduces it). The relative change of the Lamb shift is
   

   x=[Δ E(r_p))-Δ E(r_p(exp))]/Δ E(r_p)
   

   = [- 5.2248 × (r²_p- r²_p(exp)) + 0.0347 × ( r³_p-r³_p(exp))]/[209.968(5) - 5.2248 × r²_p + 0.0347 × r³_p(th)] .
   

   The estimate gives x= 1.2× 10^-3.
   
   

This value can be compared with the prediction. For n=2 ratio of Δ E_n/Δ E_n(Lamb) is

x=Δ E_n/Δ E_n (Lamb)= k² × [2⁹π²/3×13α] × (m/M)² .

For M=2.9 TeV the numerical estimate gives x≈ (1/3)×k² × 10^-4. The value of x deduced from experimental data is x≈ 1.2× 10^-3. There is discrepancy of one order of magnitude. For k≈ 5 a correct order of magnitude is obtained. There are thus good hopes that the model works.

The contribution of Z⁰₁ exchange is neglected in the above estimate. Is it present and can it explain the discrepancy?

1. In the case of deuterium the weak isospins of proton and deuterium are opposite so that their contributions to the Z⁰₁ vector potential cancel. If Z⁰₁ contribution for proton can be neglected, one has Δ r_p=Δ r_d.
   

   One however has Δ r_p≈ 2.75 Δ r_d. Hence Z⁰₁ contribution to Δ r_p should satisfy Δ r_p(Z⁰₁)≈ 1.75×Δ r_p(γ₁). This requires α_Z,1>α₁, which is true also for the ordinary gauge bosons. The weak isospins of electron and proton are opposite so that the atom is weak isospin singlet in Abelian sense, and one has I³_pI³_μ= -1/4 and attractive interaction. The condition relating r_p and r_Z suggests
   

   α_Z,1/α₁≈ 286=4+13 .
   

   In standard model one has α_Z/α= 1/[sin²(θ_W)cos²(θ_W)] =5.6 for sin²(θ_W)=.23 . One has upper bound α_Z,1/α₁ ≥ 4 saturated for sin²(θ_W,1) =1/2. Weinberg angle can be expressed as
   

   sin²(θ_W,1)= (1/2)[1 - (1-4( α₁/α_Z,1)^1/2] .
   

   α_Z,1/α₁≈ 28/6 gives sin²(θ_W,1) = (1/2)[1 -(1/7)^1/2] ≈ .31.
   

   The contribution to the axial part of the potential depending on spin need not cancel and could give a spin dependent contribution for both proton and deuteron.
   
   

2. If the scale of α₁ and α_Z,1 is that of α_s and if
   the factor 2.75 emerges in the proposed manner, one has k²≈ 2.75× 10= 27.5 rather near to the rough estimate k²≈ 27 from data for proton.
   

   Note however than there are mixing angles involved corresponding to the diagonal hermitian family charge matrix Q= (a,b,c) satisfying a²+b²+c²=1 and the condition a+b+c=0 expressing the orthogonality with the electromagnetic charge matrix (1,1,1)/3^1/2 expressing electroweak universality for ordinary electroweak bosons. For instance, one could have (a,b,c)= (0,1,-1)/2^1/2 for the second generation and (a,b,c)= (2,-1,-1)/6^1/2 for the third generation. In this case the above estimate would would be scaled down: α₁→ 2α₁/3≈ 1/20.5.
   
   

To sum up, the proposed model is successful at quantitative level allowing to understand the different changes for charge radius for proton and deuteron and estimate the values of electroweak couplings of the second generation of weak bosons apart from the uncertainty due to the family charge matrix. Muon's magnetic moment anomaly and decays of neutral B allow to test the model and perhaps fix the remaining two mixing angles.

See the article Could second generation of weak bosons explain the reduction of proton charge radius?

For background see the chapters New Physics Predicted by TGD: Part I and New Physics Predicted by TGD: Part II.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

What about actual realization of Lorentz invariant synchronization?

I wrote one day ago about synchronization of clocks and found that the clocks distributed at the hyperboloids of light-cone assignable to CD can in principle be synchronized in Lorentz invariant manner (see this). But what about actual Lorentz invariant synchronization of the clocks? Could TGD say something non-trivial about this problem? I received an interesting link relating to this (see this). The proposed theory deals with fundamental uncertainty of clock time due to quantum-gravitational effects. There are of course several uncertainties involved since quantum theory of gravity does not exist (officially) yet!

1. Operationalistic definition of time is adopted in the spirit with the empiristic tradition. Einstein was also empirist and talked about networks of synchronized clocks. Nowadays particle physicists do not talk much about them. Symmetry based thinking dominates and Special Relativity is taken as a postulate about symmetries.
   
   
2. In quantum gravity situation becomes even rather complex. If quantization attempt tries to realize quantum states as superpositions of 3-geometries one loses time totally. If GRT space-time is taken to be small deformation of Minkowski space one has path integral and classical solutions of Einstein's equation define the background.
   

   The difficult problem is the identification of Minkowski coordinates unless one regards GRT as QFT in Minkowski space. In astrophysical scales QFT picture one must consider solutions of Einstein's equations representing astrophysical objects. For the basic solutions of Einstein's equations the identification of Minkowski coordinates is obvious but in general case such as many-particle system this is not anymore so. This is a serious obstacle in the interpretation of the classical limit of GRT and its application to planetary systems.
   
   

What about the situation in TGD? Particle physicist inside me trusts symmetry based thinking and has been somewhat reluctant to fill space-time with clocks but I am ready to start the job if necessarily! Since I am lazy I of course hope that Nature might have done this already and the following argument suggests that this might be the case!

1. Quantum states can be regarded as superpositions of space-time surfaces inside causal diamond of imbedding space H= M⁴× CP₂ in quantum TGD. This raises the question how one can define universal time coordinate for them. Some kind of absolute time seems to be necessary.
   
   
2. In TGD the introduction of zero energy ontology (ZEO) and causal diamonds (CDs) as perceptive fields of conscious entities certainly brings in something new, which might help. CD is the intersection of future and past directed light-cones analogous to a big bang followed by big crunch. This is however only analogy since CD represents only perceptive field not the entire Universe.
   

   The imbeddability of space-time as to CD× CP₂ ⊂ H= M⁴× CP₂ allows the proper time coordinate a² =t²-r² near either CD bouneary as a universal time coordinate, "cosmic time". At a= constant hyperboloids Lorentz invariant synchronisation is possible. The coordinate a is kind of absolute time near a given boundary of CD representing the perceptive field of a particular conscious observer and serves as a common time for all space-time surfaces in the superposition. Newton would not have been so wrong after all.
   

   Also adelic vision involving number theoretic arguments selects a as a unique time coordinate. In p-adic sectors of adele number theoretic universality (NTU) forces discretization since the coordinates of hyperboloid consist of hyperbolic angle and ordinary angles. p-Adicallhy one cannot realize either angles nor their hyperbolic counterparts. This demands discretization in terms of roots of unity (phases) and roots of e (exponents of hyperbolic angles) inducing finite-D extension of p-adic number fields in accordance with finiteness of cognition. a as Lorentz
   invariant would be genuine p-adic coordinate which can in principle be continuous in p-adic sense. Measurement resolution however discretizes also a.
   

   This discretization leads to tesselations of a=constant hyperboloid having interpretation in terms of cognitive representation in the intersection of real and various p-adic variants of space-time surface with points having coordinates in the extension of rationals involved. There are two choices for a. The correct choice
   corresponds to the passive boundary of CD unaffected in state function reductions.
   
   

3. Clearly, the vision about space-time as 4-surface of H and NTU show their predictive power. Even more, adelic physics itself might solve the problem of Lorentz invariant synchronization in terms of a clock network assignable to the nodes of tesselation!
   

   Suppose that tesselation defines a clock network. What synchronization could mean? Certainly strong correlations between the nodes of the network Could the correlation be due to maximal quantum entanglement (maximal at least in p-adic sense) so that the network of clocks would behave like a single quantum clock? Bose-Einstein condensate of clocks as one might say? Could quantum entanglement in astrophysical scales predicted by TGD via h_gr= h_eff=n× h hypothesis help to establish synchronized clock networks even in astrophysical scales? Could Nature guarantee Lorentz invariant synchronization automatically?
   

   What would be needed would be not only 3-D lattice but also oscillatory behaviour in time. This is more or less time crystal (see this and this)! Time crystal like states have been observed but they require feed of energy in contrast to what Wilzek proposed. In TGD Universe this would be due to the need to generate large h_eff/h=n phases since the energy of states with n increases with n. In biological systems this requires metabolic energy feed. Can one imageine even cosmic 4-D lattice for which there would be the analog of metabolic energy feed?
   

   I have already a model for tensor networks and also here a appears naturally (see this). Tensor networks would correspond at imbedding space level to tesselations of hyperboloid t²-r²=a² analogous to 3-D lattices but with recession velocity taking the role of quantized position for the point of lattice. They would induce tesselations of space-time surface: space-time surface would go through the points of the tesselation (having also CP₂ counterpart). The number of these tesselations is huge. Clocks would be at the nodes of these lattice like structures. Maximal entanglement would be key feature of this network. This would make the clocks at the nodes one big cosmic clock.
   

   If astrophysical objects serving as clocks tend to be at the nodes of tesselation, quantization of cosmic redshifts is predicted! What is fascinating is that there is evidence for this: for TGD based model for this see (see this and this)! Maybe dark matter fraction of Universe might have taken care of the Lorentz invariant synchronization so that we need not worry about that!
   

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Is Lorentz invariant synchronization of clocks possible?

I participated an FB discussion with several anti-Einsteinians. As a referee I have expressed my opinion about numerous articles claiming that Einstein's special or general relativity contains a fatal error not noticed by any-one before. I have tried to tell that colleagues are extremely eager to find a mistake in the work of colleague (unless they can silence the colleague) so that logical errors can be safely excluded. If something goes wrong it is at the level of basic postulates. In vain.

Once I had a long email discussion with a professor of logic who claimed to have found logical mistake in the deduction of time dilation formula. It was easy to find that he thought in terms of Newtonian space-time and this was of course in conflict with relativistic view. The logical error was his, not Einstein's. I tried to tell this. In vain again.

At this time I was demanded to explain why the 2 page article of Stephen Crothers (see this). This article was a good example of own logical error projected to that of Einstein. The author assumed besides the basic formulas for Lorentz transformation also synchronization of clocks so that they show the same time everywhere (about how this is achieved see this).

Even more: Crothers assumes that Einstein assumed that this synchronization is Lorentz invariant. Lorentz invariant synchronization of clocks is not however possible for the linear time coordinate of Minkowski space as also Crothers demonstrates. Einstein was wrong! Or was he? No!: Einstein of course did not assume Lorentz invariant synchronization!

The assumption that the synchronization of clock network is invariant under Lorentz transformations is of course in conflict with SR. In Lorentz boosted system the clocks are not in synchrony. This expresses just Einstein's basic idea about the relativity of simultaneity. Basic message of Einstein is misunderstood! The Newtonian notion of absolute time again!

The basic predictions of SR - time dilation and Lorentz contraction - do not depend on the model of synchronization of clocks. Time dilation and Lorentz contraction follow from basic geometry of Minkowskian space-time extremely easily.

Draw system K and K' moving with constant velocity with respect to K. The t' and x' axis of K' have angle smaller than π/2 and are in first quadrant.

1. Assume first that K corresponds to the rest system of particle. You see that the projection of segment=(0,t') t'-axis to t-axis is shorter than the segment (0,t'): time dilation.
   
   
2. Take K to be the system of stationary observer. Project the segment L=(0,x') to segment on x axis. It is shorter than L: Lorentz contraction.
   
   

There is therefore no need to build synchronized networks of clocks to deduce time dilation and Lorentz contraction. They follow from Minkowskian geometry.

This however raises a question. Is it possible to find a system in which synchronization is possible in Lorentz invariant manner? The quantity a²=t²-x² defines proper time coordinate a along time like geodesics as Lorentz invariant time coordinate of light-one. a = constant hyper-surfaces are now hyperboloids. If you have a synchronized network of clocks, its Lorentz boost is also synchronized. General coordinate invariance of course allows this choice of time coordinate.

For Robertson-Walker cosmologies with sub-critical mass time coordinate a is Lorenz invariant so that one can have Lorentz invariant synchronization of clocks. General Coordinate Invariance allows infinitely many choices of time coordinate and the condition of Lorentz invariant synchronization fixes the time coordinate to cosmic time (or its function to be precise). To my opinion this is rather intesting fact.

What about TGD? In TGD space-time is 4-D surface in H=M⁴×CP₂. a²= t²-r² defines Lorentz invariant time coordinate a in future light-cone M⁴₊ ⊂ M⁴ which can be used as time-coordinate also for space-time surfaces.

Robertson-Walker cosmologies can be imbedded as 4-surfaces to H=M⁴×CP₂. The empty cosmology would be just the lightcone M⁴₊ imbedded in H by putting CP₂ coordinates constant. If CP₂ coordinates depend on M⁴₊ proper time a, one obtains more general expanding RW cosmologies. One can have also sub-critical and critical cosmologies for which Lorentz transformations are not isometries of a= constant section. Also in this case clocks are synchronized in Lorentz invariant manner. The duration of these cosmologies is finite: the mass density diverges after finite time.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Are viruses fragments of topological quantum computer code?

I was listening a highly interesting talk about viruses in Helsinki by Dr. Matti Jalasvuori, a molecular biologist working in the University of Jyväskylä as a researcher (see this). He has published a book about viruses in finnish titled "Virus. Elämän synnyttäjä, kuoleman kylväjä, ajatusten tartuttaja" (see this).

I learned an extremely interesting new-to-me fact about viruses. They might be far from a mere nuisance, In TGD Universe they could be quantum memes, short pieces of a code of quantum computer code, wandering around and attaching to the existing quantum computer code represented by DNA! Replication of viruses would be replication of memes. If the infected organism survives the virus attack by taming the virus and making it part of its non-coding DNA, it will gain more strength! If my computer survives the updating of the operating system, it works better!

Some basic facts

Viruses are very small, few nanometers is the size scale. Virus contains short pieces of RNA or DNA coding for the virus, in particular the protein shell around it, which virus must have in the "non-living" state outside the host cell to which it can penerate. Inside its host this shell melts and virus attaches to DNA and is able to to replicate. The copies of virus leave the host cell to search for their own host cells.

Usually viruses are regarded as a nuisance. But a new more holistic vision is evolving about viruses and their actual role. Viruses have been present perhaps even before the cell was present in its recent form, they might have been crucial for the emergence of life as we know it and would be also now. The system would consist of various kinds of cells, not necesary those of single organism. The contain several kinds of DNA and RNA: cell nucleus and mitochondria contain their own genomes; there are circular plasmids, and also viruses.

There is a continual exchange of information between cells including viruses as form form of information exchange. In this framework virus represents a meme represented by its DNA ,which does not code for protein shell. This meme wants to replicate and must use the genetic machinery to achieve this. But does virus do this to only replicate and produce more nuisance?

The organism manages to survive the virus attack if it is able to transform the virus so that it cannot replicate. One manner to achieve this would be transformation of the DNA portion due to the attached virus DNA (possible reverse transcribed from the RNA of virus) to a non-coding DNA often referred to as "junk" DNA. Non-coding DNA includes both intragenic regions - introns - and intergenic regions containing for instance promoters and enhancers crucial for the control of gene expression as proteins (see this). Introns are portions of genes, whose contribution to mRNA is sliced away in translation to proteins. The decomposition to introns and translated regions is dynamical, which gives rise to a rich spectrum of different translations of the gene.

In fact, most of non-coding DNA might be due to viruses! The portion of non-coding DNA increases for speciei at higher evolutionary level. For our species it is estimated to be 98 percent! Most of our genome is "junk" as many biologists still would put it. But can this really be the case? On might think that immune system would have invented some mechanism to prevent the infection of DNA by junk DNA? The size of the trash bin cannot be a measure for evolutionary level! It is also known that virus infections force the organism to change and in some cases to become a better surviver. Viruses would drive evolution.

One can speculate that during the very early period in evolution there were only viruses and proto-cells. There is no need for them to be coded by genes. Self-organization can produce cell membrane like structures: soap films represent an example. The DNA fragments could survive inside these proto-cells but according to simulations done by the Jyäskylä group in which Matti Jalasvuori is working, eventually the evolution would lead to the emergence of parasitic DNA strands, which would soon begin to dominate and kill the protocell.

Viruses might solve the problem. Viruses would attract DNA fragments and replicate with themto build a protein wall around the fragment containg also a piece of DNA of proto-cell. Viruses would leave the proto cell before its death and find another protocell. Gradually genome would be formed as viruses would steal pieces of DNA fragments from protocells. One step in the later evolution could be the elimination of the part of virus coding for the protein shell and the use of the rest as protein coding DNA. For eukariotes the transformation to non-coding DNA including intronic and intergenic DNA becomes possible.

Viruses as pieces of quantum computer code?

Computational thinking would suggest that viruses might make possible the emergence of new biological program modules allowing to use existing program modules coding for proteins more effectively. The different slicings of mRNA dropping some pieces away would correspond to different manners to transform DNA sequences to proteins. But what about intragenic portions of DNA: are they just junk?

Could the non-coding DNA and viruses have a much deeper purpose of existence than mere replication? In TGD Universe this kind of purpose is easy to imagine if the system formed by DNA - say intragenic portions of DNA - and nuclear membrane (or cell membrane) system serves as a topological quantum computer. DNA codons would be connected to lipids of the lipid layer of cell nucleus by magnetic flux tubes carrying dark charged particles. These connections could be also to cell membrane and even to cell membranes of other cells.

The braiding of the flux tubes would define the space-time realization of a quantum computer program. This would represent a new expression of DNA and would explain why so small differences between our DNA and that of our cousins give rise to so huge differences. What is important that genetic code would not be terribly important: it is braiding that matters now. The realization as quantum computer programs would give rise to cultural evolution, the realization as proteins to biological evolution. There would be a transition from the level of genes to that of memes.

Viruses would correspond to pieces of quantum computer code - memes. They would be wandering between cells and infecting them to get fused to the DNA. If DNA is able to transform them to introns it gets the code. Otherwise it dies. Infection is the necessary price for achieving meme replication. Living cells could be seen quantum computer programs updating them continually. Sounds somehow familiar!

See the chapters DNA as topological quantum computer, Three new physics realizations of the genetic code and the role of dark matter in bio-systems, and More Precise TGD Based View about Quantum Biology and Prebiotic Evolution of "Genes and Memes".

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

About double slit experiments of Dean Radin

Dean Radin and his team have done a very interesting experiment (see this and this) testing the idea that observer induces state function reduction.

Experiment

The experiment is a modified double slit experiment. In double slit experiment a laser beam arrives to the screen via two slits and interference pattern is generated as if photons would behave like waves localized at screen. If one adds detectors at the slits, either detector fires and detects the passing-by by photon, and interference pattern disappears with optimal detection efficiency.

The idea is to add a subject person (S) at distance of two meters. S imagines of measuring that electron passes through either slit. One can say that S intends to add a "detector" to either slit or both of them so that a state function reduction selecting either slit occurs. This experiment differs from experiments in which S tries to affect the ratio of frequencies of 0:s and 1:s in random series of bits: S does not try to force the electrons to pass by either slit. There is a feedback represented as sound/yellow light whose height/intensity coded for the amount of the reduction of the height of the peak. There are two kinds of participants: meditators and those who have no experience in meditation.

The results of the experiment are thoroughly discussed in the Youtube lecture or Radin (see this). To my opinion the results are amazing. In one experiment it was found that the height of the peak of the Fourier transform of the intensity distribution of the diffraction pattern is reduced. In second experiment the depth of bottom of the through of distribution was reduced. As if the intention would induce with some probability to perform the measurement selecting the photon path. The effect was small but appeared systematically for a group consisting of meditators. For persons without experience in meditation the effect averaged out also in this case it was present in the beginning of the experiment when subject person were not bored by the repetitive character of the experiment. The longer attention span of meditators could partially explain this.

Even more amazing finding was that in a variant of the experiment realized in internet the results were also positive although the persons intending to induce the experiment.

Arguments of different skeptic

The standard argument of skeptic is that statistics is poor, that the experiment is even fraud, etc... One can however consider more refined and more imaginative objections. Let us make a digress from the usual behavior of skeptic and assume that the effect was real.

If the meditators could induce the measurement by intention, one expects that also the experimenter could have done it. To how high degree the outcome was due to the experimenters and how much due to the meditators? Experimenter also had the theoretical expectation that meditators are better in inducing the slit detection. Could the wish that the theory is correct have caused subconscious intention about performing the detection in the case of meditators or not doing it in the case of non-meditating subject persons?

In the case of net experiment situation becomes even more problematic. One can imagine that also in this case the intention of experimenter could induce the detection - at least if experimenter is near to the system. Should experimenters have spent the period of experiments in Mars or at least in a distant holiday resort! Experimenters studying remote mental interactions are usually not rich people and presumably they did not do this.

The experimenter effect is well-known in parapsychology. Some experimenters are extremely successful. Could one think that they have strong intentional powers? Ironically, this would demonstrate the reality of paranormal effects of this kind but in a manner that can never convince the skeptics. There is evidence for this kind of effect in the testing of new medicines. Good results are obtained when the testers are enthusiastic and dream of a positive result. When they do same tests after some years, the results are worse.

TGD based model

The challenge is to understand how the S imagining a measurement telling that photon went through either slit could realize this intention. What does the detection mean and what it demands?

1. The measurement should involve a state function reduction selecting between the slits entangled with observer. In principle it is enough to have an interaction of photon in either slit localing the path of the photon to that slit. It is enough that photon interacts with charged particles in either slit with some probability. This measurement is of course not optimal since the interference diagram is only partially changed. Only some fraction of these measurements take place and produce single slit pattern so that the observer pattern is a weighted average of double slit and single slit patterns. In principle one can estimate the probability for single slit pattern from the data.
   
   
2. Quantum classical correspondence requires that in order that the intention to detect could be realized, one must have a physical connection from the S to both slits or at least either of them. Also charged particles assignable to the connection should be involved to make scattering of photon possible. Also entanglement entangling detector fires/does not fire with corresponding states of some other system, say the S would be needed.

How could one realize these connections in TGD?

1. In TGD framework the magnetic flux tubes serve as correlates of entanglement and directed attention. To direct attention to a system means to connected with it by flux tubes. Flux tubes carry dark charged particles essential for TGD view about quantum biology.
   
   
2. Every system has U-shaped flux tubes emanating from it and acting as kind of tentacles scanning the environment. As a U-shaped flux tube from system A encounters another similar flux tube from system B, a reconnection takes place if the quantized fluxes are same. The outcome is a pair of flux tubes connection A and B. The flux tube pair can carry Cooper pairs with members of the pair at the flux tubes. The photons from laser could scatter from the charged particles.
   
   
3. The dark particles the flux tube are dark with h_eff/h=n satisfying an additional condition implying that n is proportional to the mass of the charged particle in turn implying that cyclotron energies E_c= hbar_eff eB/m are universal and assumed to correspond energies in the range of visible and UV.
   

   In order that photon scatters from the charged particles it must have the same value of h_eff as the particles at magnetic flux tubes emanating from the S. Some fraction of laser photons could satisfy this condition. Note that if perturbative quantum theory applies, the classical predictions are same as lowest order quantum predictions so that h_eff makes it visible only in higher orders assuming that perturbation theory works when h_eff/h=n holds true. Unfortunately, it is not possible to estimate the probability that photon enters to the flux tube. Note that the probability depends also on the density of the flux tubes.
   
   

The effect is reported in net experiments for which distances can be long and there is no visual contact. Can one understand this?

1. If there quantum entanglement between A and B already exists one can increase the distance without spoiling the entanglement. But how to achieve the entanglement if n the systems are at large distance from beginning?
   
   
2. The length of the magnetic flux tubes is not a problem. The size scale for the layers of magnetic flux tube corresponding to EEG frequency 7.8 Hz is circumference of Earth. The condition that the size of the flux tube is at least of the order of the cyclotron wavelength λ for cyclotron photons at the flux tube implies that length of the flux tube of of the order of the size scale of Earth for EEG frequencies.
   

   In fact, our MBs could have much larger layers if biological rhythms have cyclotron frequencies as counterparts. The size scales could be of order light-life-time or even longer. This changes totally the view about the role of length scales in biology and consciousness. There is some evidence that galactic day defines the natural rhythm for precognitive phenomena: precognitive phenomena tend to occur at galactic midday. Galactic cyclotron frequencies (the galactic magnetic field is of order nT) could correspond to bio-rhythms up to 12 hours.
   
   

In net experiment the problem is how to generate the connection to a correct target. The same problem is encountered in the attempts to explain the claimed results of remote viewing experiments. Could the density of flux tubes of personal magnetic body (MB) be so high that the connection is generated with high enough probability. S receives data through the web. Could this help to build the desired connection.

1. Skeptic would explain the reported positive result in web experiments by saying that the results were actually induced by the intention of the experimenter who was near to the system. This might of course be the case.
   
   
2. The first possibility is that an entanglement is generated between the camera monitoring the system and slits involving flux tubes. The communication of the image from the camera to computer builds another flux tube bridge. The radiation reflected in satellite to the computer at Earth involves propagation along flux tubes. At the receiver ends similar bridges are build. There is therefore a flux tube connection with the computer of used by S, who generates the last piece of the connection. This kind of flux tube connection would be between all communicating systems. Also the experiments would belong to this entanglement network.
   
   
3. MB has layers with size scale of order Earth size. Could it be able to meet the challenge by using the information coming from web. Could the U-shaped flux tubes be so dense as to be able to build a contact with the experimental arrangement with high enough probability? If they are to represent Maxwellian magnetic field in good approximation, they should be dense. What is important that these flux tubes correspond do different space-time sheets for distinct observers: this is actually the basic distinction between the field concepts of Maxwell and TGD.
   

   Could it be that the feedback from S at her computer via the net to the computer at the other end generates quantum correlated events and this correlation has as correlates magnetic flux tubes connecting the distant systems.
   
   

4. The hyper-imaginative option is that S can delegate the problem with collective consciousness assignable to the magnetosphere of Earth and having all the engineering knowledge that Earth has! Could we be neurons of a gigantic brain of Mother Gaia, which would help S to realize their intention. Can single neuron realize its intention on a distant neuron in brain in the similar manner? Could some kind of resonance mechanism be involved? Could MB detect correlations between distant events and generate flux tube connection and entanglement between these places? Could brain do the same for neutrons?
   

See the article About the double slit experiment of Dean Radin or the chapter TGD inspired view about remote mental interactions and paranormal of "TGD based view about living matter and remote mental interactions".

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

About unitarity of twistor amplitudes

The first question is what one means with S-matrix in ZEO. I have considered several proposals for the counterparts of S-matrix. In the original U-matrix, M-matrix and S-matrix were introduced but it seems that U-matrix is not needed.

1. The first question is whether the unitary matrix is between zero energy states or whether it characterizes zero energy states themselves as time-like entanglement coefficients between positive and negative energy parts of zero energy states associated with the ends of CD. One can argue that the first option is not sensible since positive and negative energy parts of zero energy states are strongly correlated rather than forming a tensor product: the S-matrix would in fact characterize this correlation partially.
   

   The latter option is simpler and is natural in the proposed identification of conscious entity - self - as a generalized Zeno effect, that is as a sequence of repeated state function reductions at either boundary of CD shifting also the boundary of CD farther away from the second boundary so that the temporal distance between the tips of CD increases. Each shift of this kind is a step in which superposition of states with different distances of upper boundary from lower boundary results followed by a localization fixing the active boundary and inducing unitary transformation for the states at the original boundary.
   
   

2. The proposal is that the the proper object of study for given CD is M-matrix. M-matrix is a product for a hermitian square root of diagonalized density matrix ρ with positive elements and unitary S-matrix S : M= ρ^1/2S. Density matrix ρ could be interpreted in this approach as a non-trivial Hilbert space metric. Unitarity conditions are replaced with the conditions MM^†= ρ and M^†M=ρ. For the single step in the sequence of reductions at active boundary of CD one has M→ MS (Δ T) so that one has S→ SS(Δ T). S(Δ T) depends on the time interval Δ T measured as the increase in the proper time distance between the tips of CD assignable to the step.
   
   

What does unitarity mean in the twistorial approach?

1. In accordance with the idea that scattering diagrams is a representation for a computation, suppose that the deformations of space-time surfaces defining a given topological diagram as a maximum of the exponent of Kähler function, are the basic objects. They would define different quantum phases of a larger quantum theory regarded as a square root of thermodynamics in ZEO and analogous to those appearing also in QFTs. Unitarity would hold true for each phase separately.
   

   The topological diagrams would not play the role of Feynman diagrams in unitarity conditions although their vertices would be analogous to those appearing in Feynman diagrams. This would reduce the unitarity conditions to those for fermionic states at partonic 2-surfaces at the ends of CDs, actually at the ends of fermionic lines assigned to the boundaries of string world sheets.
   
   

2. The unitarity conditions be interpreted stating the orthonormality of the basis of zero energy states assignable with given topological diagram. Since 3-surfaces as points of WCW appearing as argument of WCW spinor field are pairs consisting of 3-surfaces at the opposite boundaries of CD, unitarity condition would state the orthonormality of modes of WCW spinor field. If might be even that no mathematically well-defined inner product assignable to either boundary of CD exists since it does not conform with the view provided by WCW geometry. Perhaps this approach might help in identifying the correct form of S-matrix.
   
   
3. If only tree diagrams constructed using 4-fermion twistorial vertex are allowed, the unitarity relations would be analogous to those obtained using only tree diagrams. They should express the discontinuity for T in S=1+iT along unitary cut as Disc(T)= TT^†. T and T^† would be T-matrix and its time reversal.
   
   
4. The correlation between the structure of the fermionic scattering diagram and topological scattering diagrams poses very strong restrictions on allowed scattering reactions for given topological scattering diagram. One can of course have many-fermion states at partonic 2-surfaces and this would allow arbitrarily high fermion numbers but physical intuition suggests that for given partonic 2-surface (throat of wormhole contact) the fermion number is only 0, 1, or perhaps 2 in the case of supersymmetry possibly generated by right-handed neutrino.
   

   The number of fundamental fermions both in initial and final states would be finite for this option. In quantum field theory with only masive particles the total energy in the final state poses upper bound on the number of particles in the final state. When massless particles are allowed there is no upper bound. Now the complexity of partonic 2-surface poses an upper bound on fermions.
   

   This would dramatically simplify the unitarity conditions but might also make impossible to satisfy them. The finite number of conditions would be in spirit with the general philosophy behind the notion of hyper-finite factor. The larger the number of fundamental fermions associated with the state, the higher the complexity of the topological diagram. This would conform with the idea about QCC. One can make non-trivial conclusions about the total energy at which the phase transitions changing the topology of space-time surface defined by a topological diagram must take place.
   
   

See the article About twistor lift of TGD.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Kerr effect, breaking of T symmetry, and Kähler form of M4

I encountered in Facebook (thanks to Ulla) a link to a very interesting article Here is the abstract.

We prove an instance of the Reciprocity Theorem that demonstrates that Kerr rotation, also known as the magneto-optical Kerr effect, may only arise in materials that break microscopic time reversal symmetry. This argument applies in the linear response regime, and only fails for nonlinear effects. Recent measurements with a modified Sagnac Interferometer have found finite Kerr rotation in a variety of superconductors. The Sagnac Interferometer is a probe for nonreciprocity, so it must be that time reversal symmetry is broken in these materials.

I had to learn some basic condensed matter physics. Magneto-optic Kerr effect occurs when a circularly polarized plane wave - often with normal incidence - reflects from a sample with planar boundary. In magneto-optic Kerr effect there are many options depending on the relative directions of the reflection plane (incidence is not normal in the general case so that one can talk about reflection plane) and magnetization. Also the incoming polarization can be linear or circular. Reflected circular polarized beams suffers a phase change in the reflection: as if they would spend some time at the surface before reflecting. Linearly polarized light reflects as elliptically polarized light.

Kerr angle θ_K is defined as 1/2 of the difference of the phase angle increments caused by reflection for oppositely circularly polarized plane wave beams. As the name tells, magneto-optic Kerr effect is often associated with magnetic materials.

Kerr effect has been however observed also for high Tc superconductors and this has raised controversy. As a layman in these issues I can naively wonder whether the controversy is created by the expectation that there are no magnetic fields inside the super-conductor. Anti-ferromagnetism is however important for high Tc superconductivity. In TGD based model for high Tc superconductors the supracurrents would flow along pairs of flux tubes with the members of S=0 (S=1) Cooper pairs at parallel flux tubes carrying magnetic fields with opposite (parallel) magnetic fluxes. Therefore magneto-optic Kerr effect could be in question after all.

The author claims to have proven that Kerr effect in general requires breaking of microscopic time reversal symmetry. Time reversal symmetry breaking (TRSB) caused by the presence of magnetic field and in the case of unconventional superconductors is explained nicely here. Magnetic field is required. Magnetic field is generated by a rotating current and by right-hand rule time reversal changes the direction of the current and also of magnetic field. For spin 1 Cooper pairs the analog of magnetization is generated, and this leads to T breaking.

This result is very interesting from the point of TGD. The reason is that twistorial lift of TGD requires that imbedding space M⁴× CP₂ has Kähler structure in generalized sense. M⁴ has the analog of Kähler form, call it J(M⁴). J(M⁴) is assumed to be self-dual and covariantly constant as also CP₂ Kähler form, and contributes to the Abelian electroweak U(1) gauge field (electroweak hypercharge) and therefore also to electromagnetic field.

J(M⁴) implies breaking of Lorentz invariance since it defines decomposition M⁴= M²× E² Implying preferred rest frame and preferred spatial direction identifiable as direction of spin quantization axis. In zero energy ontology (ZEO) one has moduli space of causal diamonds (CDs) and therefore also moduli space of Kähler forms and the breaking of Lorentz invariance cancels. Note that a similar Kähler form is conjectured in quantum group inspired non-commutative quantum field theories and the problem is the breaking of Lorentz invariance.

What is interesting that the action of P,CP, and T on Kähler form transforms it from self-dual to anti-self-dual form and vice versa. If J(M⁴) is self-dual as also J(CP₂), all these 3 discrete symmetries are broken in arbitrarily long length scales. On basis of tensor property of J(M⁴) one expects P: (J(M²),J(E²)→ (J(M²),-J(E²) and T: (J(M²),J(E²)→ (-J(M²),J(E²). Under C one has (J(M²),J(E²)→ (-J(M²),-J(E²). This gives CPT: (J(M²),J(E²)→ (J(M²),J(E²) as expected.

One can imagine several consequences at the level of fundamental physics.

1. One implication is a first principle explanation for the mysterious CP violation and matter antimatter asymmetry not predicted by standard model (see the recent blog post).
   
   
2. A new kind of parity breaking is predicted. This breaking is separate from electroweak parity breaking and perhaps closely related to the chiral selection in living matter.
   
   
3. The breaking of T might in turn relate to Kerr effect if the argument of authors is correct. It could occur in high Tc superconductors in macroscopic scales. Also large h_eff/h=n scaling up quantum scales in high Tc superconductors could be involved as with the breaking of chiral symmetry in living matter. Strontium ruthenate for which Cooper pairs are in S=1 state is is indeed found to exhibit TRSB (for references and explanation see this).
   

   In TGD based model of high Tc superconductivity the members of the Cooper pair are at parallel magnetic flux tubes with the same spin direction of magnetic field. The magnetic fields and thus the direction of spin component in this direction changes under T causing TRSB. The breaking of T for S=1 Cooper pairs is not spontaneous but would occur at the level of physics laws: the time reversed system finds itself experiences in the original self-dual J(M⁴)) rather than in (-J(M²),J(E²)) demanded by T symmetry.
   
   

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Key ideas related to the twistor lift of TGD

The generalization of twistor approach from M⁴ to H=M⁴× CP₂ involves the replacement of twistor space of M⁴ with that of H. M⁸-H duality allows also an alternative approach in which one constructs twistor space of octonionic M⁸. Note that M⁴,E⁴, S⁴, and CP₂ are the unique 4-D spaces allowing twistor space with Kähler structure. This makes TGD essentially unique.

Ordinary twistor approach has two problems.

1. It applies only if the particles are massless. In TGD particles are massless in 8-D sense but the projection of 8-momentum to given M⁴ is in general massive in 4-D sense. This solves the problem. Note that the 4-D M⁴ momenta can be light-like for a suitable choice of M⁴⊂ H. There exist even a choice of M² for which this is the case. For given M² the choices of quaternionic M⁴ are parametrized by CP₂.
   
   
2. The twistor approach has second problem: it works nicely in signature (2,2) rather (1,3) for Minkowski space. For instance, twistor Fourier transform cannot be defined as an ordinary integral. The very nice results by Nima Arkani-Hamed et al about positive Grassmannian follow only in the signature (2,2).
   

   One can always find M²⊂ M⁸ in which the 8-momentum lies and is therefore light-like in 2-D sense. Furthermore, the light-like 8-momenta and thus 2-momenta are prediced already at classical level to be complex. M² as subspace of momentum space M⁸ effectively extends to its complex version with signature (2,2)!
   

   At classical space-time level the presence of preferred M² reflects itself in the properties of massless extremals with M⁴= M²× E² decomposition such that light-like momentum is in M² and polarization in E².
   

   4-D conformal invariance is restricted to its 2-D variant in M². Twistor space of M⁴ reduces to that of M². This is SO(2,2)/SO(2,1)=RP₃. This is 3-D RP₃, the real variant of twistor space CP₃. Complexification of light-like momenta replaces RP₃ with CP₃.
   
   

Light-like M⁸-momenta are in question but they are not arbitrary.

1. They must lie in some quaternionic plane containing fixed M², which corresponds to the plane spanned by real octonion unit and some imaginary unit. . This condition is analogous to the condition that the space-time surfaces as preferred extremals in M⁸ have quaternionic tangent planes.
   
   
2. In particular, the wave functions can be expressed as products of plane waves in M², wave functions in the plane of transverse momenta in E²⊂ M⁴, where M⁴ is quaternionic plane containing M² and wave function in the space for the choices of M⁴, which is CP₂. One obtains exactly the same result in M⁴× CP₂ if delocalization in transversal E² momenta taking place of quarks inside hadrons takes place.
   Transversal wave function can also concentrate on single momentum value.
   

   It should be noticed that quaternionicity forces number theoretical spontaneous compactification. It would be very clumsy to realize the condition that allowed 8-momenta are qiuaternionic. Instead going to M⁴× CP₂, "spontaneously compactifying", description everyting becomes easy.
   
   

3. What is amusing that the geometric twistor space M⁴× S² of M⁴ having bundle projections to M⁴ and ordinary twistor spaces is nothing but the space of choices of causal diamonds with preferred M² and fixed rest frame (time axis connecting the tips). M⁴ point fixes the tip of causal diamond (CD) and S² the spatial direction fixing M² plane. In case of CP₂ the point of twistor space fixes point of CP₂ as analog for tip of CD: the complex CP₂ coordinates have origin at this point. The point of twistor sphere of SU(3)/U(1)× U(1) codes for the selection of quantization axis for hypercharge Y and isospin I₃. The corresponding subgroup U(1)× U(1) affects only the phases of the preferred complex coordinates transforming linearly under SU(2)× U(1).
   

   At the level of momentum space M⁴ twistor codes for the momentum and helicity of particle. For CP₂ it codes for the selection of M⁴⊂ M⁸ and for em charge as analog of helicity. Now one has actually wave function for the selections of CP₂ point labelled by the color numbers of the particle.
   
   

Number theoretical vision inspires the idea that scattering ampitudes define representations for algebraic computations leading from initial set of algebraic objects to to final set of objects. If so, the amplitudes should not depend on how the computation is done and there should exist a minimal computation possibly represented by a tree diagram. There would be no summation over the equivalent diagrams: one can choose any-one of them and the best choice is the simplest one.

To develop this idea one must understand what scattering diagrams are. The scattering diagrams involve two kinds of lines.

1. There are topological "lines" corresponding to light-like orbits of partonic 2-surfaces playing the role of lines of Feynman diagrams. The topological diagram formed by these lines gives boundary conditions for 4-surface: at these light-like partonic orbits Euclidian space-time region changes to Minkowskian one. Vertices correspond to 2-surfaces at which these 3-D lines meet just like line in the case of Feynman diagrams.
   
   
2. There are also fermion lines assignable to fundamental fermions serving as building bricks of elementary particles. They correspond to the boundaries of string world sheets at the orbits of partonic 2-surfaces. Fundamental fermion-fermion scattering takes place via classical interactions at partonic 2-surfaces: there is no 4-vertex in the usual sense (this would lead to non-renormalizable theory).
   

   The conjecture is that he 4-vertex is described by twistor amplitude fixed apart from over all scaling factor. Fermion lines are along parton orbits. Boson lines correspond to pairs of fermion and antifermion at the same parton orbit.
   

   As a matter fact, the situation is more complex for elementary particles since they correspond to pairs of wormhole contacts connected by monopole magnetic tubes and wormhole contacts has two wormhole throats - partonic 2-surfaces.
   
   

For the idea about diagrams as representations of computations to make sense, there should exist moves which allow to glide the 4-fermion vertex and associated flux tubes along the topological line of scattering diagrams in the vicinity of the second end of the loop. Second move should allow to snip away the loop. Is this possible? The possibility to find M² for which momentum is light-like is central in the argument claiming that this is indeed possible.

The basic problem is that the kinematics for 4-fermion vertices need not be consistent with the gliding of vertex past another one so that this move is not possible.

1. Clearly, one must assume something. If all momenta along at vertices along fermion line are in same M² then they parallel as light-like M²-momenta. Kinematical conditions allow the gliding of two vertices of this kind past each other as is easy to show. The scattering would mean only redistribution of parallel light-like momenta in this particular M².
   

   This kind of scattering would be more general than the scattering in integrable quantum field theories in M²: in this case the scattering would not affect the momenta but would induce phase shifts: particles would spend some time in the vertex before continuing. What is crucial for having non-trivial scatterings, is that in the general frame M²⊂ M⁴ ⊂ M⁸ the momenta would be massive and also different.
   
   

2. The condition would be that all four-fermion vertices along given fermion line correspond to the same preferred M². M²:s can differ only for fermionic sub-diagrams which do not have common vertices.
   

   Note however that tree diagrams for which lines can have different M²s can give rise to non-trivial scattering. One can take tree diagram and assign to the internal lines networks with same M²s as the internal line has. It is quite possible that for general graphs allowing different M²s in internal lines and loops, the reduction to tree graph is not possible.
   

   At least this idea could define precisely what the equivalence of diagrams, if vertices in which M²:s can be different are allowed. One can of course argue, that there is not deep reason for not allowing more general loopy graphs in which the incoming lines can have arbitrary M²:s.
   

One implication is that the BCFW recursion formula allowing to generate loop diagrams from those with lower number of loops must be trivial in TGD - this of course only if one accepts that BCFW formula makes sense in TGD. This requires that the entangled removal appearing as second term in the right hand side of BCFW formula and adding loop gives zero. One can develop and argument for why this must be the case in TGD framework. Also the second term corresponding to removal of BCFW bridge should give zero so that allowed diagrams cannot have BCFW bridges.

In TGD Universe allowed diagrams would represent closed objects in what one might call BCFW homology. The operation appearing at the right hand side of BCFW recursion formula is indeed boundary operation, whose square by definition gives zero.

For background see the article Some questions related to the twistor lift of TGD.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Issues related to the precise formulation of twistor lift of TGD

During last two weeks I have worked hardly to deduce the implications of some observations relating to the twistor lift of Kähler action. Some of these observations were very encouraging but some observations were a cold shower forcing a thorough criticism of the first view about the details of the twistor lift of TGD.

New formulation of Kähler action

The first observation was that the correct formulation of 6-D Kähler action in the framework of adelic physics implies that the classical physics of TGD does not depend on the overall scaling of Kähler action.

1. Kähler form has dimension length squared. Kähler form projected to the space-time surface defines Mawell field, which should be however dimensionless. I had assumed that one can just divide Kähler form by CP₂ radius squared to achieve this. The skeptic realizes immediately that this parameter is free coupling parameter albeit CP₂ radius is good guess for it. The correct formulation of the action principle must keep Kähler form dimensional and divides Kähler action by a dimensional parameter with dimension 4: this is new coupling contant type parameter besides α_K. The classical field equations do not depend at all on this scaling parameter. The exponent of action defining vacuum functional however depends on it.
   
   
2. What is so nice that all couplings disappear from classical field equations in the new formulation, and number theoretical universality (NTU) is automatically achieved. In particular, the preferred extremals need not be minimal surface extremals of Kähler action to achieve this as in the original proposal for the twstor lift. It is enough that they are so asymptotically - near the boundaries of CDs, where they behave like free particles. In the interior they couple to Kähler force. This also nicely conforms with the physical idea that they are 4-D generalizations for orbits of particles in induced Kähler field.
   
   
3. I also realized that the exchange of conserved quantities between Euclidian and Minkowskian space-time regions is not possible for the original version of twistor lift. This does not sound physical: quantal interactions should have classical correlates. The reason for the catastrophe is simple. Metric determinant appearing in action integral is identified as g₄^1/2. In Minkowskian regions it is purely imaginary but real in Euclidian regions. Boundary conditions lead to decoupling of Minkowskian and Euclidian regions.
   

   This forced to return to an old nagging question whether one should use a) g₄^1/2 (imaginary in Minkowskian regions) or b)|g₄^1/2| in the action. For real α_K the option a) is unavoidable and the need to have exponent of imaginary action in Minkowskian regions indeed motivated option a).
   

   For complex α_K forced by other considerations the situation however changes - something that I had not noticed. Complex α_K allows |g₄^1/2|. The study of so called CP₂ extremals assuming that 1/α_K= s, s=1/2+iy zero of Riemann zeta shows that NTU is realized in the sense that the exponent of action exists in some extension of rationals, provided that the imaginary part of zero of zeta satisfies y= qπ, q rational, implying that the exponent of y is root of unity. This possibility has been considered already earlier. This is highly non-trivial hypothesis about zeros of zeta.
   

4. Option b) allows transfer of conserved quantities between Minkowskian and Euclidian regions as required. Option a) also predicts separate conservation of Noether charges for Kähler action and volume term. This can make sense only asymptotically. Therefore only Option b) remains under serious consideration. In the new picture the interaction region in particle physics experiences corresponds to the region, where there is coupling between volume and Kähler terms: extrenal particles correpond to minimal surface extremals of Kähler action and all known extremals indeed are such.
   

Realizing NTU

The independence of the classical physics on the scale of the action in the new formulation inspires a detailed discussion of the number theoretic vision.

1. Quantum Classical Correspondence (QCC) breaks the invariance with respect to the scalings via fermionic anti-commutation relations and NTU can fix the spectrum of values of the over-all scaling parameter of the action. Fermionic anticommutation relations introduce the constraint removing the projective invariance.
   
   
2. One ends up to a condition guaranteeing NTU of the action exponentiale xp(S). One must have S= q₁+iq₂π , q_i rational. This guarantees that exp(S) is in some extension
   of rationals and therefore number theoretically universal. S itself is however not number theoretically universal.
   The overall scaling parameter for action contrained by fermionic anticommutations must have a value allowing to satisfy the condition.
   

3. The vision about scattering amplitude as a representation of computation however suggests the action exponential disappears from twistorial scattering amplitudes altogether as it does in quantum field theories. This would require that one defines scattering amplitude - actually zero energy state - by allowing functional integral
   only around single maximum of action. Whether this makes sense is not obvious but ZEO might allow it. I have not yet discussed seriously the constraints from unitary - or its generalization to ZEO, and these constraints might force sum
   over several maxima.
   

   This looks at first a catastrophe but the scattering amplitudes depend on the preferred extremal in implicit manner. For instance, the h_eff/h= n depends on extremal. Also quantum classical correspondence (QCC) realized as boundary conditions stating that the classical Noether charges are equal to the eigenvalues of fermionic charges in Cartan algebra bring in the dependence of scattering amplitudes on preferred extremal. Furhermore, the maxima of Kähler function could correspond to the points of WCW for which WCW coordinates are in the extension of rationals: if the exponent of action is such a coordinate this could be the case.
   

   One could see the situation in two manners. The standard view in which preferred extremals are maxima of Kähler function, whose exponentials however disappear from the scattering amplitudes, and the number theoretic view in which maxima correspond to WCW points in the intersection of real and various p-adic WCWs defining cognitive representation at the level of WCW similar to that provided by the discretization at the level of space-time surface. Maybe there is a maximization of cognitive information (classical correlate for NMP): say in the sense that the number of points in the intersection of real and p-adic space-time surfaces is maximal for the preferred extremals.
   

   This kind of connection would mean deep connection between cognition and sensory perception, p-adic physics and real physics, and geometric and number theoretic views about physics.
   

Trouble with cosmological constant

Also an unpleasant observation about cosmological constant forces to challenge the original view about twistor lift.

1. The original vision for the p-adic evolution of cosmological constant assumed that α_K(M⁴) and α_K(CP₂) are different for the twistor lift. This is definitely somewhat ad hoc choice but in principle possible. If one assumes that the Kähler form has also M⁴ part J(M⁴) this option becomes very artificial. In fact, the assumption
   that the twistor space M⁴× S² associated with M⁴ allows Kähler structure, J(M⁴) must be non-vanishing and is completely fixed. It is now clear that
   J(M⁴) allows to understand both CP breaking and parity breaking (in particular chiral selection in living matter). The introduction of moduli space for CDs means also introduction of moduli space for the choices of J(M⁴), which is nothing but the twistor space T(M⁴)!
   
   
2. One indeed finds in the more geometric formulation of 6-D Kähler action that single value of α_K is the only natural choice. The nice outcome guaranteeing NTU is that the preferred extremals do not depend on the coupling parameters at all. In the original version one had to assume that extremals of Kähler action are also minimal surfaces to guarantee this.
   
   
3. One however loses the original proposal for the p-adic length scale evolution of cosmological constant explaining why it is so small in cosmological scale. The solution to the problem would be that the entire 6-D action decomposing to 4-D Kähler action and volume term is identified in terms of cosmological constant. The cancellation of Kähler electric contribution and remaining contributions would explain why the cosmological constant is so small in cosmological scales and also allows to understand p-adic coupling constant evolution of cosmological constant.
   

   One important implication is that there are two kind string like objects. Those for which string tension is very large and which are analogous to the strings of super-string theories and those for which string tension is small due to the cancellation of Kähler action and volume term. These strings appear in all scales and they also mediate gravitational interaction. Also hadronic strings are this kind of strings as also elementary particles as string like objects. In this framework one additional reason for the superstring tragedy becomes manifest: they predict only the strings giving rise to a gigantic cosmological constant.
   
   

To sum up, it is fair to say that the twistor lift of TGD has now achieved rather stable form. There are also a lot of details to be polished but this requires only hard work and a lot of counter argumentation. What is so fascinating is that the formalism produces now rather precise predictions and new detailed fresh insights to the basic problems of standard model. The problems of cosmological constant and CP breaking represent only two examples in this respect. There is also an explicit proposal for twistor four-fermion amplitudes and one can understand how the QFT picture with central role played by the loops emerges although there are no loops at the fundamental level: when particles are approximated by point like objects, some tree diagrams are contracted to loop diagrams. Consider only exchange between two particle lines replaced with single line in pointlike this approximation.

See the article About twistor lift of TGD.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

What causes CP violation?

CP violation and matter antimatter asymmetry involving it represent white regions in the map provided by recent day physics. Standard model does not predict CP violation necessarily accompanied by the violation of time reflection symmetry T by CPT symmetry assumed to be exact. The violation of T must be distinguished from the emergence of time arrow implies by the randomness associated with state function reduction.

CP violation was originally observed for mesons via the mixing of neutral kaon and antikaon having quark content nsbar and nbars. The lifetimes of kaon and antikaon are different and they transform to each other. CP violation has been also observed for neutral mesons of type nbbar. Now it has been observed also for baryons Λ_b with quark composition u-d-b and its antiparticle (see this). Standard model gives the Feynman graphs describing the mixing in standard model in terms of CKM matrix (see this).

The CKM mixing matrix associated with weak interactions codes for the CP violation. More precisely, the small imaginary part for the determinant of CKM matrix defines the invariant coding for the CP violation. The standard model description of CP violation involves box diagrams in which the coupling to heavy quarks takes place. b quark gives rise to anomalously large CP violation effect also for mesons and this is not quite understood. Possible new heavy fermions in the loops could explain the anomaly.

Quite generally, the origin of CP violation has remained a mystery as also CKM mixing. In TGD framework CKM mixing has topological explanation in terms of genus of partonic 2-surface assignable to quark (sphere, torus or sphere with two handles). Topological mixings of U and D type quarks are different and the difference is not same for quarks and antiquarks. But this explains only CKM mixing, not CP violation.

Classical electric field - not necessary electromagnetic - prevailing inside hadrons could cause CP violation. So called instantons are basic prediction of gauge field theories and could cause strong CP violation since self-dual gauge field is involved with electric and magnetic fields having same strength and direction. That this strong CP violation is not observed is a problem of QCD. There are however proposals that instantons in vacuum could explain the CP violation of hadron physics (see this).

What says TGD? I have considered this here and in the earlier blog posting (see this).

1. M⁴ and CP₂ are unique in allowing twistor space with Kähler structure (in generalized sense for M⁴). If the twistor space T(M⁴)= M⁴× S² having bundle projections to both M⁴ and to the conventional twistor space CP₃, or rather its non-compact version) allows Kähler structure then also M⁴ allow the generalized Kähler structure and the analog symplectic structure.
   

   This boils down to the existence of self-dual and covariantly constant U(1) gauge field J(M⁴) for which electric and magnetic fields E and B are equal and constant and have the same direction. This field is not dynamical like gauge fields but would characterize the geometry of M⁴. J(M⁴) implies violation Lorentz invariance. TGD however leads to a moduli space for causal diamonds (CDs) effectively labelled by different choices of direction for these self-dual Maxwell fields. The common direction of E and B could correspond to that for spin quantization axis. J(M⁴) has nothing to do with instanton field.
   It should be noticed that also the quantum group inspired attempts to build quantum field theories for which space-time geometry is non-commutative introduce the analog of Kähler form in M⁴, and are indeed plagued by the breaking of Lorentz invariance. Here there is no moduli space saving the situation (see this) .
   
   

2. The choice of quantization axis would therefore have a correlate at the level of "world of classical worlds" (WCW). Different choices would correspond to different sectors of WCW. The moduli space for the choices of preferred point of CP₂ and color quantization axis corresponds to the twistor space T(CP₂)= SU(3)/U(1)× U(1) of WCW. One could interpret also the twistor space T(M⁴)= M⁴× S² as the space with given point representing the position of the tip of CD and the direction of the quantization axis of angular momentum. This choice requires a characterization of a unique rest system and the directions of quantization axis and time axes defines plane M² playing a key role in TGD approach to twstorialization(see this) .
   
   
3. The prediction would be CP violation for a given choice of J(M⁴). Usually this violation would be averaged out in the average over the moduli space for the choices of M² but in some situation this would not happen. Why the CP violation does not average out when there is CKM mixing of quarks? Why the parity violation due to the preferred direction is not compensated by C violation meaning that the directions of E and B fields would be exactly opposite for quarks and antiquarks. Could the fact that quarks are not free but inside hadron induce CP violation? Could a more abstract formulation say that the wave function in the moduli space for J(M⁴) (wave function for the choices of spin quantization axis!) is not CP symmetric and this is reflected in the CKM matrix.
   
   
4. An important delicacy is that J(M⁴) can be both self-dual and anti-self-dual depending on whether the magnetic and electric field have same or opposite directions. It will be found that reflection P and CP transform self-dual J(M⁴) to anti-self-dual one. If only self-dual J(M⁴) is allowed, one has both parity breaking and CP violations at the level of WCW.
   
   

Can one understand the emergence of CP violation in TGD framework?

1. Zero energy state is pair of two positive and negative energy parts. Let us assume that positive energy part is fixed - one can call corresponding boundary of CD passive. This state corresponds to the outcome of state function reduction fixing the direction of quantization axes and producing eigenstates of measured observables, for instance spin. Single system at passive boundary is by definition unentangled with the other systems. It can consists of entangled subsystems hadrons are basic example of systems having entanglement in spin degrees of freedom of quarks: only the total spin of hadron is precisely defined.
   

   The states at the active boundary of CD evolve by repeated unitary steps by the action of the analog of S-matrix and are not anymore eigenstates of single particle observables but entangled. There is a sequence of trivial state function reductions at passive boundary inducing sequence of unitary time evolutions to the state at the active boundary of CD and shifting it. This gives rise to self as a generalized Zeno effect.
   

   Classically the time evolution of hadron corresponds to a superposition of space-time surfaces inside CD. The passive ends of the space-time surface or rather, the quantum superposition of them - is fixed. At the active end one has a superposition of 3-surfaces defining classical correlates for quantum states at the active end: this superposition changes in each unitary step during repeated measurements not affecting the passive end. Also time flows, which means that the distance between the tips of CD defining clock-time increases as the active boundary of CD shifts farther away.
   
   

2. The classical field equations for space-time surface follow from an action, which at space-time level is sum of Kähler action and volume term. If Kähler form at space-time surface is induced (projected to space-time surface) from J=J(M⁴)+J(CP₂), the classical time evolution is CP violating. CKM mixing is induced by different topological mixings for U and D type quarks (recall that 3 particle generations correspond to different genera for partonic 2-surfaces: sphere, torus, and sphere with two handles). J(M⁴)+J(CP₂) defines the electroweak U(1) component of electric field so that J(M⁴) contributes to U(1) part of em field and is thus physically observable.
   
   
3. Topological mixing of quarks corresponds to a superposition of time evolutions for the partonic 2-surfaces, which can also change the genus of partonic 2-surface defined as the number of handles attached to 2-sphere. For instance, sphere can transform to torus or torus to a sphere with two handles. This induces mixing of quantum states. For instance, one can say that a spherical partonic 2-surface containing quark would develop to quantum superposition of sphere, torus, and sphere with two handles. The sequence of state function reductions leaving the passive boundary of CD unaffected (generalized Zeno effect) by shifting the active boundary from its position after the first state function reduction to the passive boundary could but need not give rise to a further evolution of CKM matrix.
   

If the topological mixings are different for U and D type quarks, one obtains CKM mixing. How could the classical time evolution for quarks and for antiquarks as their CP transforms differ? To answer the question one must look how J(M⁴) transforms under C, P, T and CP.

1. J(M⁴)=(J_0z, J_xy= ε J_0z), ε=+/- 1, characterizes hadronic space-time sheet (all space-time sheets in fact). Since J(M⁴) is tensor, P changes only the sign of J_0z giving J(M⁴)→ (-J_0z, J_xy). Since C changes the signs of charges and therefore the signs of fields created by them, one expects J(M⁴)→ -J(M4) under C. CP would give J(M⁴)→ (J_0z, -J_xy) transforming selfdual J(M⁴) to anti-selfdual J(M⁴). If WCW has no anti-self-dual sector, CP is violated at the level of WCW.
   
   
2. If CPT leaves J(M⁴) invariant, one must have J(M⁴) → (J_0z, -J_xy) under T rather than J(M⁴)→ (-J_0z, J_xy). The anti-unitary character of T could correspond for additional change of sign under T. Otherwise CPT should act as J(M⁴)→ -J(M⁴) and only (CPT)² would correspond to unity.
   
   
3. Same considerations apply to J(CP₂) but the difference would be that induced J(M⁴) for space-time surfaces, which are small deformations of M⁴ covariantly constant in good approximation. Also for string world sheets corresponding to small cosmological constant J(M⁴)× J(M⁴)-2≈ 0 holds true in good approximation and induced J(M⁴) at string world sheet is in good approximation covariantly constant. If the string world sheet is just M² characterizing J(M⁴) the condition is exact and was has Kähler electric field induced by J(M⁴) but no corresponding magnetic field. This would make the CP breaking effect large.
   
   

If CP is not violated, particles and their CP transforms correspond to different sectors of WCW with self dual and anti-self dual J(M⁴). If only self-dual sector of WCW is present then CP is violated. Also P is violated at the level of WCW and this parity breaking is different from that associated with weak interactions and could relate to the geometric parity breaking manifesting itself via chiral selection in living matter. Classical time evolutions induce different CKM mixings for quarks and antiquarks reflecting itself in the small imaginary part of the determinant of CKM matrix. CP breaking at the level of WCW could explain also matter-antimatter asymmetry. For instance, antimatter could be dark with different value of h_eff/h=n.

See the articles About twistor lift of TGD and Questions related to the twistor lift of TGD.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

Criticizing the TGD based construction of twistor amplitudes

I have developed a rather detailed vision about twistorial construction of scattering amplitudes of fundamental fermions in TGD framework. These amplitudes serve as building bricks of scattering amplitudes of elementary particles. The construction allows to solve the basic problems of ordinary twistor approach.

Some of the key notions are 8-D light-likeness allowing to get rid of the problems produced by the mass of particles in 4-D sense, M⁸-M⁴× CP₂ duality having nice interpretation in twistor space of $H$, quantum criticality demanding the vanishing of loops associated with functional integral and together with Kähler property implying that functional integral reduces to mere action exponential around given maximum of K\"ahler function, and number theoretical universality (NTU) suggesting that scattering diagrams could be seen as representations of computations reducible to minimal computation represented by tree diagram. One ends up with an explicit representations for the fundamentl 4-fermion scattering amplitude.

The vision is discussed in Questions related to twistor lift TGD. For the necessary background see About twistor lift of TGD. One can however criticize the proposed vision.

What about loops of QFT?

The idea about cancellation of loop corrections in functional integral and moves allowing to transform scattering diagrams represented as networks of partonic orbits meeting at partonic 2-surfaces defining topological vertices is nice.

Loops are however unavoidable in QFT description and their importance is undeniable. Photon-photon cattering is described by a loop diagram in which fermions appear in box like loop. Magnetic moment of muon) involves a triangle loop. A further interesting case is CP violation for mesons involving box-like loop diagrams.

Apart from divergence problems and problems with bound states, QFT works magically well and loops are important. How can one understand QFT loops if there are no fundamental loops? How could QFT emerge from TGD as an approximate description assuming lengths scale cutoff?

The key observation is that QFT basically replaces extended particles by point like particles. Maybe loop diagrams can be "unlooped" by introducing a better resolution revealing the non-point like character of the particles. What looks like loop for a particle line becomes in an improved resolution a tree diagram describing exchange of particle between sub-lines of line of the original diagram. In the optimal resolution one would have the scattering diagrams for fundamental fermions serving as building bricks of elementary particles.

To see the concrete meaning of the "unlooping" in TGD framework, it is necessary to recall the qualitative view about what elementary particles are in TGD framework.

1. The fundamental fermions are assigned to the boundaries of string world sheets at the light-like orbits of partonic 2-surfaces: both fermions and bosons are built from them. The classical scatterings of fundamental fermions at the 2-D partonic 2-surface defining the vertices of topological scattering diagrams give rise to scattering amplitudes at the level of fundamental fermions and twistor lift with 8-D light-likeness suggests essentially unique expressions for the 4-fermion vertex.
   
   
2. Elementary particle is modelled as a pair of wormhole contacts (Euclidian signature of metric) connecting two space-time sheets with throats at the two sheets connected by monopole flux tubes. All elementary particles are hadronlike systems but at recent energies the substructure is not visible. The fundamental fermions at the wormhole throats at given space-time sheet are connected by strings. There are altogether 4 wormhole throats per elementary particle in the simplest model.
   

   Elementary boson corresponds to fundamental fermion and antifermion at opposite wormhole throats with very small size (CP₂ size). Elementary fermion has only single fundamental fermion at either throat. There is ν_Lνbar_R pair or its CP conjugate at the other end of the flux tube to neutralize the weak isospin. The flux tube has length of order Compton length (or elementary particle or of weak boson) gigantic as compared to the size of the wormhole contact.
   
   

3. The vertices of topological diagram involve joining of the stringy diagrams associated with elementary particles at their ends defined by wormhole contacts. Wormhole contacts defining the ends of partonic orbits of say 3 interacting particles meet at the vertex - like lines in Feynman diagram - and fundamental fermion scattering redistributes fundamental fermions between the outgoing partonic orbits.
   
   
4. The important point is that there are 2× 2=4 manners for the wormhole contacts at the ends of two elementary particle flux tubes to join together. This makes a possible a diagrams in which particle described by a string like object is emitted at either end and glued back at the other end of string like object. This is basically tree diagram at the level of wormhole contacts but if one looks it at a resolution reducing string to a point, it becomes a loop diagram.
   
   
5. Improvement of the resolution reveals particles inside particles, which can scatter by tree diagrams. This allows to "unloop" the QFT loops. By increasing resolution new space-time sheets with smaller size emerge and one obtains "unlooped" loops in shorter scales. The space-time sheets are characterized by p-adic length scale and primes near powers of 2 are favored. p-Adic coupling constant evolution corresponds to the gradual "unlooping" by going to shorter and shorter p-adic length scales revealing smaller and smaller space-time sheets.
   
   

The loop diagrams of QFTs could thus be seen as a direct evidence of the fractal many-sheeted space-time and quantum criticality and number theoretical universality (NTU) of TGD Universe. Quantum critical dynamics makes the dynamics universal and this explains the unreasonable success of QFT models as far as length scale dependence of couplings constants is considered. The weak point of QFT models is that they are not able to describe bound states: this indeed requires that the extended structure of particles as 3-surfaces is taken into account.

Can action exponentials really disappear?

The disappearance of the action exponentials from the scattering amplitudes can be criticized. In standard approach the action exponentials associated with extremals determine which configurations are important. In the recent case they should be the 3-surfaces for which Kähler action is maximum and has stationary phase. But what would select them if the action exponentials disappear in scattering amplitudes?

The first thing to notice is that one has functional integral around a maximum of vacuum functional and the disappearance of loops is assumed to follow from quantum criticality. This would produce exponential since Gaussian and metric determinants cancel, and exponentials would cancel for the proposal inspired by the interpretation of diagrams as computations. One could in fact define the functional integral in this manner so that a discretization making possible NTU would result.

Fermionic scattering amplitudes should depend on space-time surface somehow to reveal that space-time dynamics matters. In fact, QCC stating that classical Noether charges for bosonic action are equal to the eigenvalues of quantal charges for fermionic action in Cartan algebra would bring in the dependence of scattering amplitudes on space-time surface via the values of Noether charges. For four-momentum this dependence is obvious. The identification of h_eff/h=n as order of Galois group would mean that the basic unit for discrete charges depends on the extension characterizing the space-time surface.

Also the cognitive representations defined by the set of points for which preferred imbedding space coordinates are in this extension. Could the cognitive representations carry maximum amount of information for maxima? For instance, the number of the points in extension be maximal. Could the maximum configurations correspond to just those points of WCW, which have preferred coordinates in the extension of rationals defining the adele? These 3-surfaces would be in the intersection of reality and p-adicities and would define cognitive representation.

These ideas suggest that the usual quantitative criterion for the importance of configurations could be equivalent with a purely number theoretical criterion. p-Adic physics describing cognition and real physics describing matter would lead to the same result. Maximization for action would correspond to maximization for information.

Irrespective of these arguments, the intuitive feeling is that the exponent of the bosonic action must have physical meaning. It is number theoretically universal if action satisfies S= q₁+iq₂π. This condition could actually be used to fix the dependence of the coupling parameters on the extension of rationals (see this). By allowing sum over several maxima of vacuum functional these exponentials become important. Therefore the above ideas are interesting speculations but should be taken with a big grain of salt.

See the article Questions related to twistor lift TGD and for background the article About twistor lift of TGD.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.