Sunday, August 19, 2018

Large scale fluctuations in metagalactic ionizing background for redshift six

I learned about a very interesting result related to early cosmology and challenging the standard cosmology. The result is described in popular article " Early opaque universe linked to galaxy scarcity" (see this). The original article " Evidence for Large-scale Fluctuations in the Metagalactic Ionizing Background Near Redshift Six" of Becker et al is published in Astrophysical Journal (see this).

The abstract of the article is following.

" The observed scatter in intergalactic Lyα opacity at z ≤ 6 requires large-scale fluctuations in the neutral fraction of the intergalactic medium (IGM) after the expected end of reionization. Post-reionization models that explain this scatter invoke fluctuations in either the ionizing ultraviolet background (UVB) or IGM temperature. These models make very different predictions, however, for the relationship between Lyα opacity and local density. Here, we test these models using Lyα-emitting galaxies (LAEs) to trace the density field surrounding the longest and most opaque known Lyα trough at z < 6. Using deep Subaru Hyper Suprime-Cam narrowband imaging, we find a highly significant deficit of z ≈ 5.7 LAEs within 20 h-1 Mpc of the trough. The results are consistent with a model in which the scatter in Lyα opacity near z ∼ 6 is driven by large-scale UVB fluctuations, and disfavor a scenario in which the scatter is primarily driven by variations in IGM temperature. UVB fluctuations at this epoch present a boundary condition for reionization models, and may help shed light on the nature of the ionizing sources. "

The basic conclusion is that the opaque regions of the early Universe about 12.5 billion years ago (redshift z∼ 6) correspond to to small number of galaxies. This is in contrast to standard model expectations. Opacity is due to the absorption of radiation by atoms and the UV radiation generated by galaxies ionizes atoms and makes Universe transparent. In standard cosmology the radiation would arrive from rather large region. The formation of galaxies is estimated to have begun .5 Gy years after Big Bang but there is evidence for galaxies already for .2 Gy after Big Bang (see this). Since the region studied corresponds to a temporal distance about 12.5 Gly and the age of the Universe is around 13.7 Gy, UV radiation from a region of size about 1 Gly should have reached the intergalactic regions and have caused the ionization.

Second conclusion is that there are large fluctuations in the opacity. What is suggested is that either the intensity of the UV radiation or that the density of intergalactic gas fluctuates. The fluctuations in the intensity of UV radiation could be understood if the radiation from the galaxies propagates only to finite distance in early times. Why this should be the case is difficult to understand in standard cosmology.

Could TGD provide the explanation.

  1. In TGD framework galaxies would have born as cosmic strings thickened to flux tubes. This causes reduction of the string tension as energy per unit length. The liberated dark energy and matter transformed to ordinary matter and radiation. Space-time emerges as thickened magnetic flux tubes. Galaxies would correspond to knots of cosmic strings and stars to their sub-knots.

  2. If the UV light emerging from the galaxies did not get far away from galaxies, the ionization of the intergalactic gas did not occur and these regions became opaque if distance to nearest galaxies was below critical value.

  3. Why the UV radiation at that time would have been unable to leave some region surrounding galaxies? The notion of many-sheeted space-time suggests a solution. Simplest space-time sheets are 2-sheeted structure if one does not allow space-time to have boundaries. The members of the pair with boundary are glued to together along their common boundary. The radiation would have left this surface only partially. Partial reflection should occur as the radiation along first member of pair is reflected as a reflected signal propagating along second member. This model could explain the large fluctuations in the opacity as fluctuations in the density of galaxies.

  4. Cosmic expansion occurring in TGD framework in jerk-wise manner as rapid phase transitions would have expanded the galactic space-time sheets and in the recent Universe this confinement of UV radiation would not occur and intergalactic space would be homogenously ionized and transparent.

The echo phenomenon could be completely general characteristic of the many-sheeted space-time.
  1. The popular article "Evidence in several Gamma Ray Bursts of events where time appears to repeat backwards" (see this) tells about the article " Smoke and Mirrors: Signal-to-Noise and Time-Reversed Structures in Gamma-Ray Burst Pulse Light Curve" of Hakkila et al (see this). The study of gamma ray bursts (GRBs) occurring in the very early Universe with distance of few billion light years (smaller than for opacity measurements by an order of magnitude) has shown that the GRB pulses have complex structures suggesting that the radiation is reflected partially back at some distance and then back in core region.The duration of these pulses varies from 1 ms to 200 s. Could also this phenomenon be caused by the finite size of the space-time sheets assignable to the object creating GRBs?

  2. There is also evidence for blackhole echoes, which could represent example of a similar phenomenon. Sabine Hossenfelder (see this) tells about the new evidence for blackhole echoes in the fusion of blackholes for GW170817 event observed by LIGO reported by Niayesh Afshordi, Professor of astrophysics at Perimeter Institute in the article " Echoes from the Abyss: A highly spinning black hole remnant for the binary neutron star merger GW170817" (see this). The earlier 2.5 sigma evidence has grown into 4.2 sigma evidence. 5 sigma is regarded as a criterion for discovery. For TGD based comments see this.

See the article Some new strange effects associated with galaxies or the chapter TGD and Astrophysics of "Physics in many-sheeted space-time".

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

Articles and other material related to TGD.

Friday, August 17, 2018

Conformal cyclic cosmology of Penrose and zero energy ontology based cosmology

Penrose has proposed an interesting cyclic cosmology (see this, , and this) in which two subsequent cosmologies are glued along conformal boundary together. The metric of the next cosmology is related to that of previous by conformal scaling factor, which approaches zero at the 3-D conformal boundary. The physical origin of this kind of distance scaling is difficult to understand. The prediction is the existence of concentric circles of cosmic size interpretable as kind of memories about previous cosmic cycles.

In TGD framework zero energy ontology (ZEO) inspired theory of consciousness suggest an analogous sequence of cosmologies. Now the cycles would correspond to life cycles of cosmic size serving as a conscious entity having causal diamond (CD) as imbedding space correlate. The arrow of geometric time is defined as the time direction to which the temporal distance between the ends of CD increases in sequence of state function reductions leaving passive boundary of CD unaffected and having interpretation as weak measurements. The arrow of time changes "big" state function reductions changing the roles of the boundaries of CD and meaning the death and re-incarnation of self with opposite arrow of time. Penrose's gluing procedure would be replaced with "big" state function reduction in TGD framework. This proposal is discussed in some detail and the possibility that also now concentric low variance circles in CMB could carry memories about the previous life cycles of cosmos. This picture applies to all levels in the hierarchy of cosmologies (hierarchy of selves) giving rise to a kind of Russian doll cosmology.

See the article Conformal cyclic cosmology of Penrose and zero energy ontology based cosmology or the chapter TGD based cosmology of "Physics in many-sheeted space-time".

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

Articles and other material related to TGD.

Wednesday, August 15, 2018

Unexpected support for the nuclear string model

Nuclear string model (see this) replaces in TGD framework the shell model. Completely unexpected support for nuclear string model emerged from a research published by CLAS Collaboration in Nature (see this). The popular article "Protons May Have Outsize Influence on Properties of Neutron Stars" refers to possible implications for the understanding of neutron stars but my view is that the implications might dramatically modify the prevailing view about nuclei themselves. The abstract of popular article reads as (see this).

"A study conducted by an international consortium called the CLAS Collaboration, made up of 182 members from 42 institutions in 9 countries, has confirmed that increasing the number of neutrons as compared to protons in the atom’s nucleus also increases the average momentum of its protons. The result, reported in the journal Nature, has implications for the dynamics of neutron stars."

The finding is that protons tend to pair with neutrons. If the number of neutrons increases, the probability for the pairing increases too. The binding energy of the pair is liberated as kinetic energy of the pair - rather than becoming kinetic energy of proton as the popular text inaccurately states.

Pairing does not fit with shell model in which proton and neutron shells correlate very weakly. The weakness of proton-neutron correlations in nuclear shell model looks somewhat paradoxical in this sense since - as text books tell to us - it is just the attractive strong interaction between neutron and proton, which gives rise to the nuclear binding.

In TGD based view about nucleus protons and neutrons are connected to nuclear strings with short color flux tubes connecting nucleons so that one obtains what I call nuclear string (see this). These color flux tubes would bind nucleons rather than nuclear force in the conventional sense.

What can one say about correlations between nucleons in nuclear string model? If the nuclear string has low string tension, one expects that nucleons far away from each other are weakly correlated but neighboring nuclei correlate strongly by the presence of the color flux tube connecting them.

Minimization of repulsive Coulomb energy would favor protons with neutrons as nearest neighbors so that pairing would be favored. For instance, one could have n-n-n... near the ends of the nuclear string and -p-n-p-n-... in the middle region and strong correlations and higher kinetic energy. Even more neutrons could be between protons if the nucleus is neutron rich. This could also relate to neutron halo and the fact that the number of neutrons tends to be larger than that of protons. Optimistic could see the experimental finding as a support for nuclear string model.

Color flux tubes can certainly have charge 0 but also charges 1 and -1 are possible since the string has quark and antiquark at its ends giving uubar, ddbar, udbar, dubar with charges 0,0,-1,+1. Proton plus color flux tube with charge -1 would effectively behave as neuron. Could this kind of pseudo neutrons exist in nucleus? Or even more radically: could all neurons in the nucleus be this kind of pseudo neutrons?

The radical view conforms with the model of dark nuclei as dark proton sequences - formed for instance in Pollack effect (see this) - in which some color bonds can become also negatively charged to reduce Coulomb repulsion. Dark nuclei have scaled down binding energy and scaled up size. They can decay to ordinary nuclei liberating almost all ordinary nuclear binding energy: this could explaining "cold fusion" (see this).

See the chapter Nuclear string model of "Hyper-finite factors, p-adic length scale hypothesis, and dark matter hierarchy".

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

Articles and other material related to TGD.

Sunday, August 12, 2018

Could also RNA and protein methylation be involved with expression molecular emotions?

Some time ago I wrote an piece of text ) (see this) about learning of slime molds. The proposal was based on the vision inspired by the model of bio-harmony and stating that harmony of music of light (and maybe of also sound) realized as 3-chords of dark photons with frequencies of 12-note scale expresses and creates emotions and that each harmony corresponds to a particular mood. The painful conditioning of the slime mold would generate a negative mood which would infect DNA and induce epigenetic change. This picture conforms also with the finding that RNA can induce learning of conditionings in snails (see this). Slime mold does not have central nervous system but a natural guess would be that also synaptic learning involves similar mechanism.

One can ask whether also RNA and protein methylation could be involved with learning. If molecular moods correspond to bio-harmonies and if the conditioning by say painful stimulus involves a change of the emotional state of RNA inducing that of DNA, it must change some of the chords of the bio-harmony. Since bio-harmony is essential for communications by dark photons between dark proton triplets representing dark variants of the basic biomolecules and also between communications between bio-molecules and their dark variants, one expects that the change of the harmony occurs for all dark analogs of biomolecules and also for their ordinary biomolecules.

Some chords represented by DNA-, RNA-, and tRNA codons, and amino-acids - briefly basic bio-molecules - would be affected.

  1. In the case of DNA epigenetic modifications (see this) affect mRNA and thus also protein expression. There are two basic mechanisms involved. Methylation of C nucleotide of DNA and protein modification for histone.

    Methylation (addition of CH3 to N) of C nucleotide leads to a silencing of gene expression. Methylation occurs typically for CpG pairs and for both strands. Before embryogenesis demethylation occurs for the entire DNA (stem cell state) but cell differentiation means methylation of genes not expressed. In vertebrates 60-80 percent of CpG is methylated in somatic cells. CpG islands form an exception involving no methylation. Demethylation (see this) as the reversal of methylation occurs either spontaneously or actively.

    The effects on gene expression can be also inherited to next generations. The mechanism of inheritance is poorly understood. The epigenetic change should be also somehow communicated to the DNA of germ cells but this seems impossible. The mystery is deepened because before embryogenesis demethylation occurs for the entire genome. It is difficult to understand how the chemical storage of the information about methylation patterns to be transferred to the next generation is possible at all.

    The TGD view about emotional expression inducing epigenesis by communications via dark photons between basic biomolecules and their dark variants suggests an elegant mechanism. What would be inherited would be the emotional states represented by bio-harmonies assignable to the dark variants of biomolecules.

  2. In the case of pre-RNA post-transcriptional chemical modifications (see this) - in particular methylation, are known to occur, and they affect RNA splicing rates and change the distribution of mRNAs and thus of proteins. The modifications affect also un-translated RNA (UTR) but not the protein translation from mRNA.

  3. Protein modifications (see this) in turn affect the dynamics of proteins - in particular their properties as enzymes by affecting therefore the rates for various basic processes.

    As already noticed, protein modifications are important in epigenesis by histone modification. Wikipedia article mentions lys acetylization by adding CH3=O group (see this), lys and arg methylation (see this), ser and thr phosphorylation, lys ubiquintination and sumoylation. For N-terminus (H2 group in the start of protein) the process is irreversible and new amino acid residues emerge. Methylation in C terminus (O=C-OH end of protein) can increase chemical repertoire. Note that the methylation occurs at the ends of the protein just like it tends to occur in the case of RNA as will be found.

RNA modifications deserve to be discussed in more detail. This field of study is known as epitranscriptomics (see this). These chemical modifications does not affect protein expression except in the case that they affect the rates of various alternative pre-RNA splicing so that the distribution of alternative protein outcomes changes. Clearly, the effect is somewhat like the effect of mood on overall activity. There are also many other modifications of RNA. One of the is A-I de-amination which changes in RNA but does not affect protein expression.

The methylation of RNA is the most common and best understood modification of RNA.

  1. The modelling of the methylation of both DNA and RNA is based on writer-reader-eraser model. Writing corresponds to methylation. Reading corresponds to attachment of enzymes involved in the splicing or protein synthesis with higher rate to methylated sites. Demethylation is example of erasing.

  2. Methylation is known to occur for various variants of RNA (ribosomal rRNA, tRNA, mRNA, and small nuclear RNA snRNA related to metabolic machinery) after transcription. The biochemical modifications of RNA are called epitranscriptomes (see this). N6-Methyladenosine (m6A) is the most common and best understood modification of RNA. m6A tells that nitrogen in position 6 of adenosine (A) is methylated by adding group CH3. m6A sites are often located in the last exon near the end of mRNA, in untranslated RNA (UTR) at 3' end, and inside long exons.

    It has been found that 3 members of so called YTH domain protein family acting as readers have larger affinity to bind to methylated sites. One of them shortens the lifetime of mRNA after translation.

  3. Methylation in general shortens the UTR (un-translated regions) of mRNA in its 5' and 3' ends (head and tail of mRNA) ). One speaks of alternative poly-adenylation (APA, see this) of the tail of the mRNA: poly-adenylation (PA) adds A-sequences to the end of mRNA affecting its dynamics: shortening of UTRs means shortening of PAs.

  4. Methylation affects the rates in the dynamics of translation but does not affect the product of translation itself. A-sequences shields mRNA and during its life cycle its length is reduced somewhat like telomere (see this) consisting of a repeated sequence TTAGGG and also shortening during the life cycle of DNA. APA affects rates for the dynamics of translation. Also stem loops of pre-RNA can be methylated and this can increase the rate of an alternative splicing and thus change relative rates of alternative gene expressions.

The basis question is which of the following options is correct.
  1. The chemical modification of the basic biomolecules required by the preservation of resonance condition. In this case the modification would be associated with all codons and mean a drastic change of both DNA and RNA and also amino-acids. The modifications, in particular methylation, are however associated with with highly restricted portions of DNA and RNA. On particular, only A nucleotide of RNA is methylated. Hence this option is definitely excluded.

  2. The basic bio-molecules have several resonance frequencies corresponding to various moods so that chemical modifications are not needed for preserving the resonance conditions. This was assumed about the emotional effect of RNA to DNA in the earlier considerations. Chemical modifications could be seen as emotional expression of dark variants of bio-molecules.

    This option conforms with the above facts about RNA methylation. Only UTRs at the ends of RNA and associated with the stem loops are sensitive to modifications and the interpretation is that these allow the emotional expression of RNA. Note that somewhat similar situation is encountered in the case of microtubules for which the other end is highly dynamical. One can ask whether the shortening of the A-sequences and telomeres could be seen as outcome of expression of negative emotions.

What inspired this piece of text was a highly interesting popular article "Methyl marks on RNA discovered to be key to brain cell connections" about methylation of RNA in brain (see this). The research article (see this) by Daria Merkuvjev et al has title "Synaptic N6-methyladenosine (m6A) epitranscriptome reveals functional partitioning of localized transcripts".

The researchers isolated brain cells from adult mice and compared epitranscriptomes found at synapses to those elsewhere in the cells. At more than 4,000 spots on the genome, the mRNA at the synapse was methylated more often. In more than half of genes the epitranscriptomes were found in genes coding for proteins found mostly in synapses. If the methylation was disrupted, the brain cells did not function normally. It was concluded that the methylation probably makes signalling faster.

These findings conform with the idea about representation of molecular emotions as bio-harmony. Synaptic contacts are the places where emotions should be expressed to give rise to learning by conditioning realized in terms of changed synaptic strengths. Methylation would induced as emotional expression due to the changing of the 3-chords of the harmony.

See the article Emotions as sensory percepts about the state of magnetic body?, a shorter article Could also RNA and protein methylation of RNA be involved with the expression of molecular emotions? or the chapter of "TGD based view about consciousness, living matter, and remote mental interactions" with the same title.

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

Articles and other material related to TGD.

Thursday, August 09, 2018

Are space-time surfaces minimal surfaces everywhere except at 2-D interaction vertices?

The action S determining space-time surfaces as preferred extremals follows from twistor lift and equals to the sum of volume term Vol and Kähler action SK. The field equation is a geometric generalization of d'Alembert (Laplace) equation in Minkowskian (Eucidian) regions of space-time surface coupled with induced Kähler form analogous to Maxwell field. Generalization of equations of motion for particle by replacing it with 3-D surface is in question and the orbit of particle defines a region of space-time surface.


  1. Zero energy ontology (ZEO) suggests that the external particles arriving to the boundaries of given causal diamond (CD) are like free massless particles and correspond to minimal surfaces as a generalization of light-like geodesic. This dynamic reduces to mere algebraic conditions and there is no dependence on the coupling parameters of S. In contrast to this, in the interaction regions inside CDs there could be a coupling between Vol and SK due to the non-vanishing divergences of energy momentum currents associated with the two terms in action cancelling each other.

  2. Similar algebraic picture emerges from M8-H duality at the level of M8 and from what is known about preferred extremals of S assumed to satisfy infinite number of super-symplectic gauge conditions at the 3-surfaces defining the ends of space-time surface at the opposite boundaries of CD.

  3. At M8 side of M8-H duality associativity is realized as quaternionicity of either tangent or normal space of the space-time surface. The condition that there is 2-D integral distribution of sub-spaces of tangent spaces defining a distribution of complex planes as subspaces of octonionic tangent space implies the map of the space-time surface in M8 to that of H. Given point m8 of M8 is mapped to a point of M4× CP2 as a pair of points (m4,s) formed by M4 ⊂ M8 projection m4 of m8 point and by CP2 point s parameterizing the tangent space or the normal space of X4⊂ M8.

  4. If associativity or even the condition about the existence of the integrable distribution of 2-planes fails, the map to M4× CP2 is lost. One could cope with the situation since the gauge conditions at the boundaries of CD would allow to construct preferred extremal connecting the 3-surfaces at the boundaries of CD if this kind of surface exists at all. One can however wonder whether giving up the map M8→ H is necessary.

  5. Number theoretic dynamics in M8 involves no action principle and no coupling constants, just the associativity and the integrable distribution of complex planes M2(x) of complexified octonions. This suggests that also the dynamics at the level of H involves coupling constants only via boundary conditions. This is the case for the minimal surface solutions suggesting that M8-H duality maps the surfaces satisfying the above mentioned conditions to minimal surfaces. The universal dynamics conforms also with quantum criticality.

  6. One can argue that the dependence of field equations on coupling parameters in interactions leading to a perturbative series in coupling parameters in the interior of the space-time surface spoils the extremely beautiful purely algebraic picture about the construction of solutions of field equations using conformal invariance assignable to quantum criticality. Classical perturbation series is also in conflict with the vision that the TGD counterparts twistorial Grassmannian amplitudes do not involve any loop contributions coming as powers of coupling constant parameters.

Thus both M8-H duality, number theoretic vision, quantum criticality, twistor lift of TGD reducing dynamics to the condition about the existence of induced twistor structure, and the proposal for the construction of twistor scattering amplitudes suggest an extremely simple picture about the situation. The divergences of the energy momentum currents of Vol and SK would be non-vanishing only at discrete points at partonic 2-surfaces defining generalized vertices so that minimal surface equations would hold almost everywhere as the original proposal indeed stated.
  1. The fact that all the known extremals of field equations for S are minimal surfaces conforms with the idea. This might be due to the fact that these extremals are especially easy to construct but could be also true quite generally apart from singular points. The divergences of the energy momentum currents associated with SK and Vol vanish separately: this follows from the analog of holomorphy reducing the field equations to purely algebraic conditions.

    It is essential that Kähler current jK vanishes or is light-like so that its contraction with the gradients of the imbedding space coordinates vanishes. Second condition is that in transversal degrees of freedom energy momentum tensor is tensor of form (1,1) in the complex sense and second fundamental form consists of parts of type (1,1) and (-1-1). In longitudinal degrees of freedom the trace Hk of the second fundamental form Hkαβ= Dβαhk vanishes.

  2. Minimal surface equations are an analog of massless field equation but one would like to have also the analog of massless particle. The 3-D light-like boundaries between Minkowskian and Euclidian space-time regions are indeed analogs of massless particles as are also the string like word sheets, whose exact identification is not yet fully settled. In any case, they are crucial for the construction of scattering amplitudes in TGD based generalization of twistor Grassmannian approach. At M8 side these points could correspond to singularities at which Galois group of the extension of rationals has a subgroup leaving the point invariant. The points at which roots of polynomial as function of parameters co-incide would serve as an analog.

    The intersections of string world sheets with the orbits of partonic 2-surface are 1-D light-like curves X1L defining fermion lines. The twistor Grassmannian proposal is that the ends of the fermion lines at partonic 2-surfaces defining vertices provide the information needed to construct scattering amplitudes so that information theoretically the construction of scattering amplitudes would reduce to an analog of quantum field theory for point-like particles.

  3. Number theoretic vision reduces coupling constant evolution to a discrete evolution. This implies that twistor scattering amplitudes for given values of discretized coupling constants involve no radiative corrections. The cuts for the scattering amplitudes would be replaced by sequences of poles. This is unavoidable also because there is number theoretical discretization of momenta from the condition that their components belong to an extension of rationals defining the adele.

What could the reduction of cuts to poles for twistorial scattering amplitudes at the level of momentum space mean at space-time level?
  1. Poles of an analytic function are co-dimension 2 objects. d'Alembert/Laplace equations holding true in Minkowskian/Euclidian signatures express the analogs of analyticity in 4-D case. Co-dimension 2 rule forces to ask whether partonic 2-surfaces defining the vertices and string world sheets could serve analogs of poles at space-time level? In fact, the light-like orbits X3L of partonic 2-surfaces allow a generalization of 2-D conformal invariance since they are metrically 2-D so that X3L and string world sheets could serve in the role of poles.

    X3L could be seen as analogs of orbits of bubbles in hydrodynamical flow in accordance with the hydrodynamical interpretations. Particle reactions would correspond to fusions and decays of these bubbles. Strings would connect these bubbles and give rise to tensor networks and serve as space-time correlates for entanglement. Reaction vertices would correspond to common ends for the incoming and outgoing bubbles. They would be analogous to the lines of Feynman diagram meeting at vertex: now vertex would be however 2-D partonic 2-surface.

  2. What can one say about the singularities associated with the light-like orbits of partonic 2-surfaces? The divergence of the Kähler part TK of energy momentum current T is proportional to a sum of contractions of Kähler current jK with gradients ∇ hk of H coordinates. jK need not be vanishing: it is enough that its contraction with ∇ hk vanishes and this is true if jK is light-like. This is the case for so called massless extremals (MEs). For the other known extremals jK vanishes.

    Could the Kähler current jK be light-like and non-vanishing and singular at X3L and at string world sheets? This condition would provide the long sought-for precise physical identification of string world sheets. Minimal surface equations would hold true also at these surface. Even more: jK could be non-vanishing and thus also singular only at the 1-D intersections X1L of string world sheets with X3L - I have called these curves fermionic lines?

    What it means that jK is singular - that is has 2-D delta function singularity at string world sheets? jK is defined as divergence of the induced Kähler form J so that one can use the standard definition of derivative to define jK at string world sheet as the limiting value jKα= (Div+- J)α = limΔ xn→ 0 (J+α n- J-α n)/Δ xn, where xn is a coordinate normal to the string world sheet. If J is not light-like, it gives rise to isometry currents with non-vanishing divergence at string world sheet. This current should be light like to guarantee that energy momentum currents are divergenceless. This is guaranteed if the isometry currents T&n; A are continuous through the string world sheet.

  3. If the light-like jK at partonic orbits is localized at fermionic lines X1L, the divergences of energy momentum currents could be non-vanishing and singular only at the vertices defined at partonic 2-surfaces at which fermionic lines X1L meet. The divergences of energy momentum tensors TK of SK and TVol of Vol would be non-vanishing only at these vertices. They should of course cancel each other: Div TK=-Div TVol.

  4. Div TK should be non-vanishing and singular only at the intersections of string world sheets and partonic 2-surfaces defining the vertices as the ends of fermion lines. How to translate this statement to a more precise mathematical form? How to precisely define the notions of divergence at the singularity?

    The physical picture is that there is a sharing of conserved isometry charges of the incoming partonic orbit i=1 determined TK between 2 outgoing partonic orbits labelled by j=2,3 . This implies charge transfer from i=1 to the partonic orbits j=2,3 such that the sum of transfers sum up to to the total incoming charge. This must correspond to a non-vanishing divergence proportional to delta function. The transfer of the isometry charge for given pair i,j of partonic orbits that is Divi→ j TK must be determined as the limiting value of the quantity Δi→ j TKα,A/Δ xα as Δ xα approaches zero. Here Δi→ j TKα,A is the difference of the components of the isometry currents between partonic orbits i and j at the vertex. The outcome is proportional delta function.

  5. Similar description applies also to the volume term. Now the trace of the second fundamental form would have delta function singularity coming from Div TK. The condition Div TK= -Div TVol would bring in the dependence of the boundary conditions on coupling parameters so that space-time surface would depend on the coupling constants in accordance with quantum-classical correspondence. The manner how the coupling constants make themselves visible in the properties of space-time surface would be extremely delicate.

This picture conforms with the vision about scattering amplitudes at both M8 and H sides of M8-H duality.
  1. M8 dynamics based on algebraic equations for space-time surfaces leads to the proposal that scattering amplitudes can be constructed using the data only at the points of space-time surface with M8 coordinates in the extension of the rationals defining the adele. I call this discrete set of points cognitive representation.

  2. At H side the information theoretic interpretation would be that all information needed to construct scattering amplitudes would come from points at which the divergences of the energy momentum tensors of SK and Vol are non-vanishing and singular.

Both pictures would realize extremely strong form of holography, much stronger than the strong form of holography that stated that only partonic 2-surfaces and string world sheets are needed.

See the article The Recent View about Twistorialization in TGD Framework or the shorter article Further comments about classical field equations in TGD framework, or the chapter chapterwith the same title.

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

Articles and other material related to TGD.

Wednesday, August 08, 2018

Three dualities of the field equations of TGD

The basic field equations of TGD allow several dualities. There are 3 of them at the level of basic field equations (and several other dualities such as M8-M4× CP2 duality).

  1. The first duality is the analog of particle-field duality. The spacetime surface describing the particle (3-surface of M4× CP2 instead of point-like particle) corresponds to the particle aspect and the fields inside it geometrized in terms of sub-manifold geometry in terms of quantities characterizing geometry of M4× CP2 to the field aspect. Particle orbit serves as wave guide for field, one might say.

  2. Second duality is particle-spacetime duality. Particle identified as 3-D surface means that particle orbit is space-time surface glued to a larger space-time surface by topological sum contacts. It depends on the scale used, whether it is more appropriate to talk about particle or of space-time.

  3. The third duality is hydrodynamics-massless field theory duality Hydrodynamical equations state local conservation of Noether currents. Field equations indeed reduce to local conservation conditions of Noether currents associated with isometries of M4× CP2. One the other hand, these equations have interpretation as non-linear geometrization of massless wave equation with coupling to Maxwell fields. This realizes the ultimate dream of theoretician: symmetries dictate the dynamics completely. This is expected to be realized also at the level of scattering amplitudes and the generalization of twistor Grassmannian amplitudes could realize this in terms of Yangian symmetry.

    Hydrodynamics-wave equations duality generalizes to the fermionic sector and involves superconformal symmetry.

  4. What I call modified gamma matrices are obtained as contractions of the partial derivatives of the action defining space-time surface with respect to the gradients of imbedding space coordinate with imbedding space gamma matrices. Their divergences vanish by field equations for the space-time surface and this is necessary for the internal consistency the Dirac equation. The modified gamma matrices reduces to ordinary ones if space-time surface is M4 and one obtains ordinary massless Dirac equation.

  5. Modified Dirac equation expresses conservation of super current and actually infinite number of super currents obtained by contracting second quantized induced spinor field with the solutions of modified Dirac. This corresponds to the super-hydrodynamic aspect. On the other hand, modified Dirac equation corresponds to fermionic analog of massless wave equation as super-counterpart of the non-linear massless field equation determining space-time surface.

See the article The Recent View about Twistorialization in TGD Framework or the shorter article Further comments about classical field equations in TGD framework, or the chapter chapterwith the same title.

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

Articles and other material related to TGD.

Tuesday, August 07, 2018

About the physical interpretation of the velocity parameter in the formula for the gravitational Planck constant

Nottale's formula for the gravitational Planck constant hbargr= GMm/v0 involves parameter v0 with dimensions of velocity. I have worked with the quantum interpretation of the formula but the physical origin of v0 - or equivalently the dimensionless parameter β0=v0/c (to be used in the sequel) appearing in the formula has remained open hitherto. In the following a possible interpretation based on many-sheeted space-time concept, many-sheeted cosmology, and zero energy ontology (ZEO) is discussed.

A generalization of the Hubble formula β=L/LH for the cosmic recession velocity, where LH= c/H is Hubble length and L is radial distance to the object, is suggestive. This interpretation would suggest that some kind of expansion is present. The fact however is that stars, planetary systems, and planets do not seem to participate cosmic expansion. In TGD framework this is interpreted in terms of quantal jerk-wise expansion taking place as relative rapid expansions analogous to atomic transitions or quantum phase transitions. The TGD based variant of Expanding Earth model assumes that during Cambrian explosion the radius of Earth expanded by factor 2.

There are two measures for the size of the system. The M4 size LM4 is identifiable as the maximum of the radial M4 distance from the tip of CD associated with the center of mass of the system along the light-like geodesic at the boundary of CD. System has also size Lind defined defined in terms of the induced metric of the space-time surface, which is space-like at the boundary of CD. One has Lind<LM4. The identification β0= LM4/LH<1 does not allow the identification LH=LM4. LH would however naturally corresponds to the size of the magnetic body of the system in turn identifiable as the size of CD.

One can deduce an estimate for β0 by approximating the space-time surface near the light-cone boundary as Robertson-Walker cosmology, and expressing the mass density ρ defined as ρ=M/VM4, where VM4=(4π/3) LM43 is the M4 volume of the system. ρ can be expressed as a fraction ε2 of the critical mass density ρcr= 3H2/8π G. This leads to the formula β0= [rS/LM4]1/2 × (1/ε), where rS is Schwartschild radius.

This formula is tested for planetary system and Earth. The dark matter assignable to Earth can be identified as the innermost part of inner core with volume, which is .01 per cent of the volume of Earth. Also the consistency of the Bohr quantization for dark and ordinary matter is discussed and leads to a number theoretical condition on the ratio of the ordinary and dark masses.

See the article About the physical interpretation of the velocity parameter in the formula for the gravitational Planck constant.

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

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