How quaternionicity of space-time could be consistent with Hermitian/Hamilton-Jacobi structure?

The recent progress in the understanding of preferred extremals of Kähler action suggests also an interesting connection to the number theoretic vision about field equations. In particular, it might be possible to understand how one can have Hermitian/Harmilton-Jacobi structure simultaneously with quaternionic structure and how quaternionic structure is possible for the Minkowskian signature of the induced metric.

One can imagine two manners of introducing octonionic and quaternionic structures. The first one is based on the introduction of octonionic representation of gamma matrices and second on the notion of octonion real-analycity.

1. If quaternionic structure is defined in terms of the octonionic representation of the imbedding space gamma matrices, there seems to be no obvious problems since one considers automatically complexification of quaternions represented in terms of gamma matrices. For the approach based on the notion of quaternion real analyticity, one is forced to use Wick rotation to define the quaternionic structure in Minkowskian regions or to introduce what I have called hyper-quaternionic structure by imbedding the space-time surface to a sub-space $M^8$ of complexified octonions. This is admittedly artificial.
   
   
2. The octonionic representation effectively replaces SO(7,1) as tangent space group with G₂ and means selection of preferred M²⊂ M⁴ having interpretation complex plane of octonionic space. A more general condition is that the tangent space of space-time surface at each point contains preferred sub-space M²(x)⊂ M⁴ forming an integrable distribution. The same condition is involved with the definition of Hamilton-Jacobi structure. What puts bells ringing is that the modified Dirac equation for the octonionic representation of gamma matrices allows the conservation of electromagnetic charge in the proposed sense a observed for years ago. One can ask whether the conditions on the charged part of energy momentum tensor could relate to the reduction of SO(7,1) to G₂.
   
   
3. Octonionic gamma matrices appear also in the proposal stating that space-time surfaces are quaternionic in the sense that tangent space of the space-time surface is quaternionic in the sense that induced octonionic gamma matrices generate a quaternionic sub-space at a given point of space-time time. Besides this the already mentioned additional condition stating that the tangent space contains preferred sub-space M²⊂ M⁴ or integrable distribution of this kind of sub-spaces is required. It must be emphasized that induced rather than modified gamma matrices are natural in these conditions.
   

The definition of quaternionicity in terms of gamma matrices looks more promising. This however raises two questions.

1. Could the quaternionicity of the space-time surface together with a preferred distribution of tangent planes M²(x)⊂ M⁴ or E²(x)⊂ CP₂ be equivalent with the reduction of the field equations to the analogs of minimal surface equations stating that certain components of the induced metric in complex/Hamilton-Jacobi coordinates vanish in turn guaranteeing that field equations reduce to algebraic identifies following from the fact that energy momentum tensor and second fundamental form have no common components? This should be the case if one requires that the two solution ansätze are equivalent.
   
   
2. Can the conditions for the modified Dirac equation select complex of co-complex 2-sub-manifold of space-time surface identified as quaternionic or co-quaternionic 4-surface? Could the conditions stating the vanishing of charged energy momentum currents state that the spinor fields are localized to complex or co-complex (hyper-complex or co-hypercomplex) 2-surfaces?
   

One should assign to the space-time sheets both quaternionic and Hermitian or Hamilton-Jacobi structure. There are two structures involved. Euclidian metric is an essential aspect of what it is to be quaternionic or octonions. It however seems that one can assign to the induced metric only Hermitian or Hamilton-Jacobi structure. This leads to a serious of innocent questions.

1. Could these two structures be associated with energy momentum tensor and metric respectively? Or perhaps vice versa? Anti-commutators of the modified gamma matrices define an effective metric as
   

   G_αβ=T_αμT^μ_β .
   

   This effective metric should have a deep physical and mathematical meaning but this meaning has remained a mystery. Note that iT takes the role of Kähler form for Kähler metric in this expression with imaginary unit plus symmetry replacing antisymmetry. The neutral and charged parts of T would be analogous to quaternionic imaginary units and real part but one would have sum over the squares of all of them rather than only single square as for Kähler metric.
   
   

2. Could G be assigned with the quaternionic structure and induced metric to the Hermitian/Hamilton-Jacobi structures? Could the neutral and charged components of the energy momentum tensor somehow correspond to quaternionic units?
   

The basic potential problem with the assignment of quaternionic structure to the induced gamma matrices is the signature of the metric in Minkowskian regions.

1. If quaternionic structrures is defined in terms of the octonionic representation of the imbedding space gamma matrices, there seems to be no obvious problems since one considers automatically complexification of quaternions.
   
2. For the approach based on the notion of quaternion real analyticity, one is forced to use Wick rotation to define the quaternionic structure or to introduce hyper-quaternionic structure by imbedding the space-time surface to a sub-space M⁸ of complexified octonions. This is admittedly artificial.
   

Could one pose the additional requirement that the signature of the effective metric G defined by the modified gamma marices (and to be distinguished from Einstein tensor) is Euclidian in the sense that all four eigenvalues of this tensor would have same sign.

1. For the induced metric the projections of gamma matrices are given by
   

   Γ_α=Γ^ae_aα , e_aα= e_ak∂_α h^k .
   

   For the modified gamma matrices their analogs would be given by
   

   Γ_α=Γ^a E_aα , E_aα= e_aμT^μ_α .
   

   Projection would be followed by multiplication with energy momentum tensor. One cannot induce G from any metric defined in the imbedding space but the notion of tangent space quaternionicity is well-defined.
   
   

2. What quaternionic structure for G could mean? One can imagine several options.
   
   
   1. For the ordinary complex structure metric has vanishing diagonal components and the infinitesimal line element ds²=g_zz*dzdz*. Could this formula generalize to
      

      ds²=g_QQ*dQdQ*?
      

      The generalization would be a direct generalization of conformal invariance to 4-D context stating that 4-metric is quaternion-conformally equivalent to flat metric. This would give additional strong condition on energy momentum tensor:
      

      G= T_αμT^μ_β= T²δ_αβ .
      

      The proportionality to Euclidian metric means in Minkowskian realm that the G is of form G= T²(2u_αu_β -g_αβ. Here u is time-like vector field satisfying u^αu_α=1 and having interpretation as local four-velocity (in Robertson-Walker cosmology similar situation is encountered). The eigen value problem in the form G^α_βx^β= λ x^α makes sense and eigenvectors would be u with eigenvalue λ=T² and three vectors orthogonal to u with eigenvalue -T². This requires integrable flow defined by u and defining a preferred time coordinate. In number theoretic vision this kind of time coordinate is introduced and corresponds to the direction assignable to the octonionic real unit. Note that the vanishing of charged projections of the energy momentum tensor does not imply a reduction of the rank of T so that this option might work.
      
      

   2. Quaternionicity could mean also the structure of hyper-Kähler manifold. Metric and Kähler form for Kähler manifold are generalized to metric representing quaternion real unit and three covariantly constant Kähler forms I_i obeying the multiplication rules for quaternions. The necessary condition is that the holonomy group equals to SU(2) identifiable as automorphism group of quaternions. One can also define quaternionic structure: there would exist three antisymmetric tensors, whose squares give the negative of the metric. CP₂ allows quaternionic structure in this sense and only one of these forms is covariantly constant.
      

      Could space-time surface allow Hyper-Kähler or quaternionic structure somehow induced from that of CP₂? This does not work for G. G is quadratic in energy momentum tensor and therefore involves four power of J rather than being square of projection of J or two other quaternionic imaginary units of CP₂. One can of course ask whether the induced quaternionic units could obey the multiplication of quaternionic units and have same square given by the projection of CP₂ metric. In this case CP₂ metric would define the effective metric and would be indeed Euclidian. For the ansatz for preferred extremals with Minkowskian signature CP₂ projection is at most 3-dimensional but also in this case the imaginary units might allow a realization as projections.
      

For more details see the chapter Does the Modified Dirac Equation Define the Fundamental Action Principle? or the article.

Under what conditions electric charge is conserved for modified Dirac equation?

One might think that talking about the conservation of electric charge at 21st century is a waste of time. In TGD framework this is certainly not the case.

1. In quantum field theories there are two manners to define em charge: as electric flux over 2-D surface sufficiently far from the source region or in the case of spinor field quantum mechanically as combination of fermion number and vectorial isospin. The latter definition is quantum mechanically more appropriate.
   
   
2. There is however a problem. In standard approach to gauge theory Dirac equation in presence of charged classical gauge fields does not conserve electric charge: electron is transformed to neutrino and vice versa. Quantization solves the problem since the non-conservation can be interpreted in terms of emission of gauge bosons. In TGD framework this does not work since one does not have path integral quantization anymore. Preferred extremals carry classical gauge fields and the question whether em charge is conserved arises. Heuristic picture suggests that em charge must be conserved. This condition might be actually one of the conditions defining what it is to be a preferred extremal. It is not however trivial whether this kind of additional condition can be posed.
   

What does the conservation of em charge imply in the case of the modified Dirac equation? The obvious guess that the em charged part of the modified Dirac operator must annihilate the solutions, turns out to be correct as the following argument demonstrates.

1. Em charge as coupling matrix can be defined as a linear combination Q= aI+bI₃, I₃=J_klΣ^kl, where I is unit matrix and I₃ vectorial isospin matrix, J_kl is the Kähler form of CP₂, Σ^kl denotes sigma matrices, and a and b are numerical constants different for quarks and leptons. Q is covariantly constant in M⁴× CP₂ and its covariant derivatives at space-time surface are also well-defined and vanish.
   
   
2. The modes of the modified Dirac equation should be eigen modes of Q. This is the case if the modified Dirac operator D commutes with Q. The covariant constancy of Q can be used to derive the condition
   

   [D,Q] Ψ= D₁Ψ=0 , D= Γ^μD_μ ,
   

   D₁=[D,Q]=Γ^μ₁D_μ , Γ^μ₁= [ Γ^μ,Q] .
   

   
   Note that here Γ^μ denotes modified gamma matrices rather than ordinary induced gamma matrices defined by contractions of energy momentum tensor with induced gamma matrices.
   Covariant constancy of J is absolutely essential: without it the resulting conditions would not be so simple.
   

   It is easy to find that also [D₁,Q]Ψ=0 and its higher iterates [D_n,Q]Ψ=0, D_n= [D_n-1,Q] must be true. The solutions of the modified Dirac equation would have an additional symmetry.
   
   

3. The commutator D₁=[D,Q] reduces to a sum of terms involving the commutators of the vectorial isospin I₃=J_klΣ^kl with the CP₂ part of the gamma matrices:
   

   D₁=[Q‚ D]= [I₃,Γ_r] ∂_μs^r T^αμ .
   

   In standard complex coordinates in which U(2) acts linearly the complexified gamma matrices can be chosen to be eigenstates of vectorial isospin. Only the charged flat space complexified gamma matrices Γ^A denoted by Γ⁺ and Γ^- possessing charges +1 and -1 contribute to the right hand side. Therefore the additional Dirac equation D₁Ψ=0 states
   

   D₁Ψ=[Q‚ D]Ψ= I₃(A)e_Ar Γ^A ∂_μs^r T^αμD_αΨ
   

   = (e_+r Γ⁺-e_-rΓ^-)∂_μs^r T^αμD_αΨ =0 .
   

   The next condition is
   

   D₂Ψ=[Q‚ D₁]Ψ=(e_+r Γ⁺+e_-rΓ^-) ∂_μs^r T^αμD_αΨ =0 .
   

   The remaining conditions give nothing new.
   
   

4. These equations imply two separate equations for the two charged gamma matrices
   

   D₊Ψ = Γ⁺T₊^αD_αΨ=0
   

   D_-Ψ = Γ^-T_-^αD_αΨ=0
   

   T_+/-^α= e_+/-r ∂_μs^r T^αμ .
   

   These conditions state what one might have expected: the charged part of the modified Dirac operator annihilates separately the solutions. The reason is that the classical W fields are proportional to e_r+/-.
   

   The above equations can be generalized to define a decomposition of the energy momentum tensor to charged and neutral components in terms of vierbein projections. The equations state that the analogs of the modified Dirac equation defined by charged components of the energy momentum tensor are satisfied separately.
   
   

5. In complex coordinate one expects that the two equations are complex conjugates of each other for Euclidian signature. For the Minkowskian signature an analogous condition should hold true. The dynamics enters the game in an essential manner: whether the equations can be satisfied depends on the coefficients a and b in the expression T= aG+bg implied by Einstein's equations in turn guaranteeing that the solution ansatz generalizing minimal surface solutions holds true (see this).
   
   
6. As a result one obtains three separate Dirac equations corresponding to the the neutral part D₀Ψ=0 and charged parts D_+/-Ψ=0 of the modified Dirac equation. By acting on the equations with these Dirac operators one obtains that also the commutators [D₊,D_-], [D₀,D_+/-] and also higher commutators obtained from these annihilate the induced spinor field model. Therefore entire - possibly- infinite-dimensional - algebra would annihilate the induced spinor fields. In string model the counterpart of Dirac equation when quantized gives rise to Super-Virasoro conditions. This analogy would suggest that modified Dirac equation gives rise to the analog of Super-Virasoro conditions in 4-D case. But what the higher conditions mean? Obviously these conditions resemble Virasoro conditions L_n|phys>=0 and their supersymmetric generalizations and might indeed correspond to a generalization of these conditions just as the field equations for preferred extremals could correspond to the Virasoro conditions if one takes seriously the analogy with the quantized string.
   

What could this additional symmetry mean from the point of view of the solutions of the modified Dirac equation? The field equations for the preferred extremals of Kähler action reduce to purely algebraic conditions in the same manner as the field equations for the minimal surfaces in string model. Could this happen also for the modified Dirac equation and could the condition on charged part of the Dirac operator help to achieve this?

1. CP₂ type vacuum extremals serve as a convenient test case. In this case the modified Dirac equation reduces to the ordinary Dirac equation in CP₂. One can construct the solutions of the ordinary Dirac equation from covariantly constant right-handed neutrino spinor playing the role of fermionic vacuum annihilated by the second half of complexified gamma matrices. Dirac equation reduces to Laplace equation for a scalar function and solution can be constructed from this "vacuum" by multiplying with the spherical harmonics of CP₂ and applying Dirac operator (see this). Similar construction works quite generally thanks to the existence of covariantly constant right handed neutrino spinor. Spherical harmonics of CP₂ are only replaced with those of space-time surface possessing either hermitian structure of Hamilton-Jacobi structure (corresponding to Euclidian and Minkowskian signatures of the induced metric, see this).
   
   
2. Could the properties of the preferred extremals make it possible to satisfy the additional conditions for the modified Dirac operator? The identical vanishing of the charged components of the energy momentum tensor would be an obvious manner to reduce the conditions as algebraic identifies. Note however that it could be an un-necessarily strong condition. In any case, the vanishing of charged components of the energy momentum tensor looks like a natural weakening of the conditions stating the vanishing of charged components of the induced gauge field.
   

There is also an interesting potential connection to the number theoretic vision about field equations.

1. The octonionic representation effectively replaces SO(7,1) as tangent space group with G₂ and means selection of preferred M²⊂ M⁴ having interpretation complex plane of octonionic space (see this). A more general condition is that the tangent space of space-time surface at each point contains preferred sub-space M²(x)⊂ M⁴ forming an integrable distribution. The same condition is involved with the definition of Hamilton-Jacobi structure. What puts bells ringing is that the modified Dirac equation for the octonionic representation of gamma matrices allows the conservation of electromagnetic charge in the proposed sense a observed for years ago. One can ask whether the conditions on the charged part of energy momentum tensor could relate to the reduction of SO(7,1) to G₂.
   
   
2. Octonionic gamma matrices appear also in the proposal stating that space-time surfaces are quaternionic in the sense that tangent space of the space-time surface is quaternionic in the sense that induced octonionic gamma matrices generate a quaternionic sub-space at a given point of space-time time. Besides this the already mentioned additional condition stating that the tangent space contains preferred sub-space M²⊂ M⁴ or integrable distribution of this kind of sub-spaces is required. It must be emphasized that induced rather than modified gamma matrices are natural in these conditions.
   

This raises two questions.

1. Could the quaternionicity of the space-time surface be equivalent with the reduction of the field equations to "stringy" field equations (minimal surface equations) stating that certain components of the induced metric in complex/Hamilton-Jacobi coordinates vanish in turn guaranteeing that field equations reduce to algebraic identifies following from the fact that energy momentum tensor and second fundamental form have no common components? This should be the case if one requires that the two solution ansätze are equivalent.
   
   
2. Could the above conditions for the modified Dirac equation imply that the space-time surface is quaternionic? Or could these conditions be seen as consistency conditions making the modified Dirac equation associated with the ordinary and octonionic gamma matrices equivalent at space-time level in the sense that the induced gamma matrices or modified gamma matrices are equivalent?
   

Giant dark matter bridge between galaxy clusters discovered

I have talked a lot about magnetic flux tubes carrying dark matter identified as phases of matter with non-standard value of Planck constant. They define basic structures of TGD Universe in all scales. The formation of galaxies and even stars involves these structures in an essential manner. They play also a key role also in biology and even bio-molecular physics.

Dark magnetic energy of flux tubes can be identified dark energy and this energy replaces the energy of inflaton field inflationary scenario. Ordinary and dark matter are created in the decay of this magnetic energy to particles just as ordinary particles are created in the decay of inflaton vacuum. Magnetic tension along flux tubes generates negative "pressure" inducing accelerated cosmic expansion, the magnetic fields filling the Universe have an explanation in terms of a gradual thickening of primordial cosmic strings having 2-D string orbit as M^{4 projection, etc., etc..}

For these reasons, it is highly rewarding to learn that experimentalists are now discovering these structures. See the article
Giant Dark Matter Bridge Between Galaxy Clusters Discovered telling about dark matter "strings".

Comment about sociology of discovery

The recent situation in particle physics is fascinating sociologically. Theoreticians who have worked their professional life with Higgs paradigm, want desperately to believe that it is Higgs. Easy to understand. We are humans after all.

The twice too high rate for decays to gamma pairs reported again and again by both ATLAS and CMS is an excellent example about social forces in action. It is just reported that data are consistent with standard model Higgs. Or when the blogger sees the two-gamma discrepancy, he just says "The rate to gamma pairs is two times too high. Interesting!", and forgets that this has been the situation from the very beginning one year ago so that the explanation as a statistical fluke does not work anymore. After a while he continues without hesitation "Congratulations! Higgs has been discovered!".

This looks strange at first. I can understand this only as an example of phenomenon which we all know so well: when you go to lift in which all people standard with back towards door, you do the same within 15 seconds! As a theoretician I am a lucky one: I am working outside the academic circles and able to avoid these horrible social pressures that would force me to groupthink.

Much more than Higgs is actually in the game. Scalar vacuum expectation value defines the basic paradigm of modern theoretical physics. Extremely strong social forces are fighting to keep VEV paradigm alive since everything done during last four decades in theoretical physics and cosmology relies on it. We would lose M-theory, standard SUSY, and inflationary cosmology without this paradigm (one can of course wonder whether the loss would so fatal after all!;-)).

Thanks for God that the data processing at CERN is blind!

Addition: After the first wave of hype the attitudes towards the discovery of new particle are becoming more objective. "Higgs" is transforming to "Higgs like": two photon decays occur too fast for standard model Higgs, and we do not even know whether it is scalar or pseudo-scalar. May we will know this before the end of the year. Also Lubos wrote a nice article also instructing how to use Phil Gibbs's combo java applet to predict what ATLAS and CMS will report together next.

At the eve of discovery

The discovery of 125 GeV boson seems to be more or less established fact. Most journalists and quite (too) many bloggers have firmly decided to identify this resonance as Higgs and are irritated because experimentalists have not been willing to declare a discovery of Higgs yet. (Too) many physics bloggers are more like fans of football team rather than cool science journalists.

Lubos identifies the new state with Higgs without hesitation and has even seen the discovery of this Higgs-without-doubt as a victory of the superstring model: I could not follow the logic. But Lubos has also known for long time from super-secret sources that also standard SUSY has been more or less discovered. Peter Woit is equally uncritical. Phil Gibbs and Jester are exceptions in the choir. They have written about nice articles about the difficulties of the Higgs interpretation if one takes fully seriously the existing data. Higgs decays to two photons should take place via WW intermediate state but the rate for the direct WW production is consistent with zero.

Tomorrow we will know much more. My expectation is that experimentalists will not talk about discovery of Higgs. The announcement about a discovery of new particle looks however quite plausible to me.

In this kind of situation theoretician with visions deviating from mainstream thinking of course feels excitement and stress. I am not an exception to this rule. What if the production rate and branching ratios are those predicted by standard model? Is my vision wrong in this case? How it could be wrong? Can I modify it without losing something essential?

To begin with, to me a rather plausible interpretation for 125 GeV state would be as pion-like state of scaled up copy of hadron physics. Two-photon decay would be due to axial anomaly and involve only photon loop instead of W loop as in the case of Higgs. I could be wrong. Fermi has produced 5 sigma evidence for the existence of 145 GeV state and also indications for a lighter state around 125 GeV (see the previous posting). The heavier state is also a good candidate for pion-like state. Therefore I must be ready to consider also alternative interpretations for 125 GeV state. Could one imagine that it has
some Higgs like characteristics?

Recall that standard model Higgs has two functions. Higgs vacuum expectation gives masses for fermions and weak gauge bosons and Higgs gives longitudinal components for massive gauge bosons. Could one have Higgs like states performing only one or none of these functions?

1. In TGD framework fermion massivation by Higgs vacuum expectation is replaced by p-adic thermodynamics giving the dominant contribution to the masses. One cannot however exclude scalar vacuum expectations giving small corrections to fermion masses. p-Adic thermodynamics as a microscopic mechanism of fermion massivation is so beautiful and predictive that it beats massivation based on Higgs expectation, which in TGD framework can be seen as a phenomenological parametrization at best.
   
   
2. In the case of weak gauge bosons p-adic temperature would be probably smaller (T=1/2 instead of T=1 for fermions) and the analog of Higgs expectation could give a significant or even dominating contribution to weak gauge boson masses. There are however conceptual problems. What Higgs expectation means? Does it characterize coherent state? Does this expectation have classical space-time correlate as gauge bosons have?
   

What about the second function of Higgs as provider of longitudinal polarizations for massive gauge bosons?

1. TGD allows to imagine the existence of Higgs like states (see the previous posting). They generalize the notions of scalar and pseudo-scalar in Minkowski space to vector and pseudo-vector in 8-D imbedding space with components only in CP₂ directions defining the analogs of polarizations. These states appear always as singlet and charged triplet and are very much analogous to 1+3 formed by electroweak gauge bosons.
   
   
2. In standard model the three components of standard model Higgs also provide the longitudinal components of weak bosons W and Z. Zero energy ontology allows to understand the massivation of spin 1 bosons as something unavoidable without Higgs like particle. And, I do not have any elegant proposal how the possible scalar 1+3 could transform to longitudinal components of weak bosons and single neutral Higgs. Thus I have tendency to conclude that if Higgs like states exist in TGD Universe they appear as full multiplets 1+3.
   

I could of course be wrong! Maybe Higgs could after all serve as provider of longitudinal polarizations. Could one imagine the classical counterparts of gauge bosons eating Higgs components in classical TGD? To get some perspective, consider modified Dirac equation for induced spinors at preferred extremals of Kähler action (see the previous posting for a considerable progress in this respect).

1. For the TGD counterparts of induced Dirac equation both gamma matrices and gauge potentials appearing in the modified Dirac equation are induced from those of imbedding space by simply projecting them to the space-time surface. This implies that induced gamma matrices contain also CP₂ part. This gives rise to new kind of couplings proportional to the contraction of gauge potential with CP₂ part of induced gamma matrices.
   

   Induced gamma matrices are actually replaced by modified gamma matrices defined by Kähler action to obtain supersymmetry and internal consistency of the theory but the conclusion remains the same. Modified gamma matrices are proportional to Maxwell energy momentum tensor expressible in terms of Einstein equations using Einstein tensor and metric for the proposed ansatz for preferred extremals. Could these couplings involving energy momentum tensor and thus mass mimic Higgs couplings?
   
   
   
   

2. Quantum classical correspondence requires the existence of classical counterparts of quanta, also Higgs. My inability to imagine any convincing candidate has been one reason for my skepticism concerning Higgs like states. While writing this I however decided to try once again. I failed but learned that em charge as isospin like quantum number is conserved in TGD classically - something very non-trivial that I have taken as granted and shown to be true only for the octonionic representation of imnedding space gamma matrices.
   
   
   1. An important aspect of the standard model Higgs mechanism is that it respects em charge leaving photons massless. In standard model the conservation of em charge defined as isospin like quantum number is non-trivial since the presence of classical gauge fields induces transitions between different charge states of fermions. In second quantization this problem is circumvented by replacing classical gauge fields with quantized ones. The so called unitary gauge defined by a gauge transformation depending on Higgs fields allows to express the action in terms of physical (in general massive) fields and makes charge conservation explicit.
      
      
   2. How the conservation of em charge is obtained in TGD? Doesn't one have the same problem but as much worse variant since classical long range electro-weak gauge fields are unavoidable in TGD? Can one get rid of the problem? Could it make sense to speak about unitary gauge also in TGD framework? Could one turn around this idea to derive classical Higgs from the possibly existing gauge transformation to unitary gauge? The answer is negative. There is no need for the unitary gauge. Em charge is conserved automatically! This sounds miraculous but is one of the wonderful aspects of CP₂ geometry.
      
      
   3. The point is that the Kähler form of CP₂ is covariantly constant and one can identify covariantly constant em charge as a matrix of form Q=aI+bJ_klΣ^kl: the coefficients a and b are different for quarks and leptons (different chiralities of H-spinors). This matrix is covariantly constant also with respect to the induced spinor structure and commutes with Dirac operator (be it the TGD counterpart of the ordinary massless Dirac operator or modified Dirac operator). Therefore one can choose the modes of induced spinor field to have a well-defined em charge at each point of space-time surface. That em charge in this spinorial sense is conserved in the classical gauge fields defined by the induced spinor connection is highly non-trivial and not true in standard model - a strong argument in favor of TGD. The covariantly constant Kähler form of CP₂ is responsible for the conservation of em charge and derives from the supersymmetry generated by covariantly constant right-handed neutrino.
      

To sum up: situation could not be more exciting for a theoretician whose views about particle physics are more than just the usual text book wisdom.

Addition: According to leaked video a new particle is found. The particle has a clear sharp peak in the diphoton channel, clear signal in the ZZ channel, inconclusive behavior in other channels, and extra tests are needed to find out whether it deviated from the Standard Model.

Therefore it seems that the interpretation as pionlike state of scaled up hadron physics predicted by TGD is not excluded. By listening the talk of Joe Candela one learns that situation concerning interpretation is still open. Note that chairman Candela emphasizes the possiblity of entirely new physics, of new space-time dimensions, and of new particles. One could take this as a signal that something else than Higgs is in question and there already might exist unpublished data about new particle candidates. New hadron physics predicts just this! One of key questions concerns the rate for the decays to gamma pair: is it still too high by factor of order two for standard model Higgs interpretation to make sense? We will learn about this today!

For the video and discussion about it see the blog of Phil Gibbs.

Addition:Phil Gibbs gave the first data about CMS reprsentations. The two-gamma signal is more than two times too high still. To me this suggests that something else than standard model Higgs is in question. In the image text Phil says "Discovery" but does not specify what has been discovered. Maybe this is a wise choice;-).

Addition: Conclusion: the discovery is seen by most commentators as a discovery of standard model Higgs although the fact is that the production rate for two-gamma states is still two times too high. Interesting food for thoughts of a social psychologist;-). Sometimes thinkers feel themselves very lonely in the middle of crowd;-).

Addition: The state above about conservation of electromagnetic charge in spinorial sense was not quite correct. The constancy of right handed neutrino is needed but is not enough.

1. I did later a more precise analysis and the conclusion was that the solutions of the modified Dirac equations for which em charge is conserved in classical induced gauge fields, are restricted to 2-dimensional surfaces of space-time surface. String model in 4-D space-time becomes part of TGD: this has been of course conjectured for a long time ago.
   
   
2. This also means that their ends at light-like 3-surfaces and at 3-D space-like 3-surfaces at ends of space-time surface define braid strands. Just this picture was derived earlier from finite measurement resolution. Also 3-D modified Dirac at light-like 3-surfaces was shown to lead to the same conclusion (localization of solutions to braid strands). Number theoretical vision suggests the same picture. Imbedding space would be octonionic, space-time surfaces associative or co-associative (quaternionic of co-quaternionic) and partonic 2-surfaces and string world sheets commugative or co-commutative (complex or hyper-complex).
   
   
3. A further outcome is a rather detailed understanding of super-conformal symmetries in TGD framework having implications also for the space-time SUSY in TGD sense. In particular, the role of right-handed neutrino in the extension of super-conformal symmetries to 4-D context is understood.
   
   
4. Also Yangian symmetry emerges naturally as well as understanding how Super-Kac-Moody symmetry for string world sheets is dual for super gauge symmetry.
   

For details see this.

Metabolic energy, ATP, and negentropic entanglement

The ideas about the detailed relationship between metabolism and negentropic entanglement are still in a state of turmoil. Let us sum up those concepts and ideas which look reasonably reliable.

1. Negentropic entanglement is the first basic notion. There is a strong tendency to consider the presence of a magnetic flux tube connecting two objects and carrying negentropically entangled quantum state as a fundamental structure giving rise to a directed attention. Negentropic entanglement would be basic element of conscious cognition, and one can assign to it various attributes like experience of understanding. The mildest assumption is that negentropic entanglement is associated with the flux tube. A stronger assumption is that it is between states assignable to the ends of the flux tube identifiable as observer and target of attention.
   

   An analogy with Orch OR is suggestive. The period of negentropic entanglement - period of directed attention - would correspond to Orch Or and its end to state function reduction.
   
   

2. The identification of the increments zero point kinetic energies as universal metabolic energy quanta is one of the oldest hypothesis of TGD inspired theory of consciousness. Zero point kinetic energy is associated with the zero point motion of particle at space-time sheet. The finite size of the space-time sheet gives rise to this energy for which non-relativistic parametrization is E₀= k× 3hbar²π²/2mL²(k). L(k)=2^(-k+151)/2 L(151), L(151) ≈ 10 nm is the p-adic length scale of the space-time sheet, and k is a numerical factor not far from unity. Particle in 3-D box gives k=1.
   

   As particle is transferred to a larger space-time sheet the zero point kinetic energy is reduced, and the difference is liberated as usable metabolic energy. For proton the size scale of this space-time sheet could be atomic size scale k=137. For electron it could electron Cooper pair k=149 (prime) corresponding to a lipid layer of cell membrane could be in question. Entire hierarchy of metabolic energy quanta is predicted and the energy scale depends on the particle mass and p-adic length scale and geometric factors characterizing the shape of space-time sheet only.
   

   One can ask whether the high energy phosphate bond in the phosphate of ATP molecule contains this kind of smaller space-time sheet and in the transition ATP → ADP, electron or proton drops from this kind of space-time sheet. The following considerations show that this hypothesis is not necessary, and that one can also modify the identification of the fundamental metabolic energy quantum as zero point kinetic energy without losing anything. Therefore the details of the scenario are far from being fully nailed down.
   
   

3. Magnetic flux tubes are carriers of charged particle and the hypothesis is that cyclotron Bose-Einstein condensates for fermionic Cooper pairs and bosonic ions are relevant for consciousness. In particular, cyclotron transitions in which bosons in these condensates are excited would be important for the generation of conscious experiences. The hierarchy of Planck constants and the fact that cyclotron energy is proportional to hbar allows to have arbitrarily high cyclotron energies in given magnetic field. This is essential in the model for the effects of ELF em fields on living matter (see this).
   
   
4. Becker's findings about the relevance of DC currents for healing of wounds lead to an idea about how electromagnetic radiation interacts with the charged particles at magnetic flux tubes (see this). What would happen is that charged particles experience the electric and magnetic fields of the radiation field described in terms of massless extremals. Electric field would generate acceleration in the direct of the flux tube and could excite Becker currents which would give rise to biological effects - healing of wound in the simplest case. The proposal has been that this process gives rise to what could be seen as a loading of metabolic batteries.
   

   The combination of this view with the notion of cyclotron BE condensate leads to a slightly more complex picture. Radiation field can excite single boson states both in transversal and longitudinal degrees of freedom. Transversal ones correspond to cyclotron states with energies E_c,n= (n+1/2)E_c, E_c= hbar qB/m and the energies of excitations are of form nE_c. Longitudinal degrees of freedom correspond to a particle in 1-D box -possibly in presence of longitudinal electric field: a simple model for the states was derived in the model for Becker's DC currents.
   

   In the absence of longitudinal electric field the energy spectrum is E_n=n²E₀, E₀=hbar²π²/2mL², L the length of the flux tube. Longitudinal excitations correspond to energies (n_f-n_i)²E₀ and would classically correspond to the acceleration in the electric field component parallel to the flux tube giving rise to Becker currents. For both excitations negentropically entangled states result very naturally as superpositions of single particle excitations and possibly also multi-particle excitations. Both incoming photons and liberation of metabolic energy quantum as photon can induce the excitation.
   

   One could reinterpret the idea about universal metabolic energy quanta by interpreting it as increment of longitudinal energies at flux tube. For the excitation n=1→ 2 the energy would be 3hbar²π²/2mL² which is same as zero point kinetic energy for a particle in 3-D box of side L. Quantitative prediction is therefore same as that of the original model. One can of course consider also the original option that the transfer of particles from the flux tube to a larger space-time sheet indeed liberates metabolic energy.
   

Let us now try to weave these ideas to an internally consistent picture. It is perhaps best to proceed by making questions.

1. Could one assign negentropic entanglement with high energy phosphate bond?If so, the period of negentropic entanglement (having Orch OR as a counterpart) would correspond to the presence of ATP and the end of this period to ATP→ ADP. I have considered this possibility earlier. The problem is that it is difficult to understand how negentropic entanglement could be assigned simultaneously both to ATP and to the magnetic flux tube whose length and thickness are proportional to hbar and therefore varies. One should treat ATP and flux tube as single basic structure and this does not sound convincing since the scales of flux tubes are expected to be much longer than the size scale of ATP molecule. Therefore the safest assumption would be that ATP is just what it is believed to be: provider of metabolic energy only. One can leave also open the question whether high energy phosphate bond can be interpreted in terms of zero point kinetic energy or not.
   
   
2. Could the non-local excitations of cyclotron Bose-Einstein condensates by large hbar photon give rise to the negentropically entangled states? Excitation of cyclotron BE condensate requires energy so that metabolic energy is required. ATP could provide this energy. Cyclotron energy quantum is given by E_c= hbar qB/m , q and m are charged and mass of the boson. As already found, the energy of boson is sum of two contributions: energy E_n∝ n² associated with free longitudinal motion and magnetic energy E_c,n∝ n+1/2. Longitudinal excitations could be assigned to the generation of Becker currents. This proposal would integrate metabolism, negentropy generation, and quantum like behavior of ELF em fields in living matter to single picture.
   
   
3. Could it be that ATP - instead of being a carrier of negentropic entanglement as suggested earlier - only provides the metabolic energy quantum transformed to cyclotron energy quantum or longitudinal energy quantum when negentropic entanglement is generated by exciting the cyclotron BE condensate? Or could ATP carry both metabolic energy and negentropic entanglement and both of them are transferred to the magnetic flux tube
   in ATP → ADP process?
   
   
   1. Cyclotron energies are quite too small for this to make sense for the ordinary value of Planck constant. The nominal value of the metabolic energy quantum is E₀=0.5 eV which by E₀= h₀f₀ corresponds to frequency f₀= 5 × 10¹³ Hz in near infrared. The value of electron's cyclotron frequency in the endogenous magnetic field B_end=.2× 10^-4 Tesla postulated to explain the effects of ELF em fields on vertebrate brain is f_c,e≈ 6× 10⁵ Hz. If metabolic energy quantum is to excite cyclotron state (n→ n+1), one must have E_c=E₀.
      

      Even for electron E_c is much below E₀ small for B=B_end and hbar=hbar₀. One can however scale both B from B_end and hbar from hbar₀. Requiring E_c(hbar,B)= E₀ and using E_c= hf gives f_c,e/f₀=r₁r, r₁= B/B_end and r=hbar/hbar₀, where hbar₀ denotes the standard value of Planck constant. This gives r₁r≈ (5/6)× 10⁸.
      
      

   2. There are many manners to achieve the desired upwards scaling of cyclotron energies. Magnetic flux quantization gives further constraints. One could require that magnetic flux is quantized, and that for hbar=hbar₀ the flux quantum has radius of order L(151) (1 nm, cell membrane thickness) corresponding to the thickness of a flux tube assignable to single DNA nucleotide.
      

      The radius of flux quantum corresponds to the magnetic length r_B= (hbar/qB)^1/2. In the scaling B_end→ 1 Tesla (r₁= 2.5× 10⁴), magnetic length scales as r_B≈ 2.5 μm → 11 nm. From the condition r₁r= (5/6)× 10⁸ one has for the scaling of Planck constant r ∼ 3.3× 10³. The scaling of the flux tubes length of L(151) would give flux tube length of order 3× 10 μ m, which corresponds to cell size so that a flux tube connecting DNA and cell membrane could be in question. Note that the scaling of hbar does not affect zero point kinetic energy in the longitudinal direction since L scales as hbar.
      
      

   3. For flux tube length L(151) and for hbar=hbar₀ the energy of the lowest longitudinal excitation is same order of magnitude as metabolic energy quantum so that the excitation of longitudinal states could be in key role in the generation of Becker's currents. There is evidence about non-local excitations of electrons in photosynthesis (see also the blog posting), which suggests that the longitudinal energy excitations for Cooper pairs could indeed play the role of fundamental metabolic energy quantum transferred to the the energy of high energy phosphate bond of ATP. This interpretation leaves open the structure of high energy phosphate bond and there is no absolute need to assign zero point kinetic energy with it.
      

      Longitudinal energies are negligible, one must require flux tube length to be considerably longer than L(151) for the ordinary value of hbar. Longitudinal energies are significant only for electron for given flux tube length. Indeed, Becker currents are known to be carried by electrons.
      
      

   4. If one allows ionic Bose Einstein condensates the value of Planck constant must be scaled up by the mass ratio m_I/m_e, where m_I and m_e are the masses of ion and electron. For proton this would give scaling ratio r=2¹¹ and one would end up with the hierarchy of Planck constants coming as powers of 2¹¹ suggests years ago. What is remarkable that in cyclotron degrees of freedom also protons and ions can play a signification role: the quantal effects of ELF em fields on vertebrate brain suggest that this is the case.
      

4. What happens if one has just electrons rather than Cooper pairs? In both transversal and longitudinal degrees of freedom one would have the analog of Fermi sphere with electron states filled up to some maximum values integers characterizing cyclotron energy and longitudinal momentum. Transitions induce also now negentropic entanglement. For cyclotron states the energy increment would be E_c so that basic metabolic energy quantum can induce the transitions. In longitudinal degrees of freedom the minimal energy increment would be (2N+1)E₀, where N characterizes the populated state with maximal longitudinal momentum. This energy should be equal to the metabolic energy quantum. This can be arranged but is not so natural. Experimental work is sooner or later bound to reveal whether electrons or their Cooper pairs are in question.
   

The option developed above is perhaps the most elegant found hitherto: it would raise the BE condensates of electronic and ionic Cooper pairs in a special position, it would lead to explicit proposal for what negentropic entanglement is, and it would requires no modification of the ideas related to ATP, even the standard view about ATP can be kept. It suggests that electronic cyclotron BE condensates are essential also for the understanding of photosynthesis. The absorption of dark photon would generate a non-local excitation of BE condensate of electron Cooper pairs- also a negentropically entangled state. The energy gain in this process could be interpreted as a fundamental metabolic energy quantum, and the subsequent steps in photosynthesis would only take care of the storage of the energy to ATP. Also the metabolic energy liberated in ATP→ ADP could be realized universally as IR dark photon absorbed by cyclotron BE condensate at magnetic flux tube so that dark photon beams would become the key actors of metabolism and negentropy generation. Note that a maximal negentropy gain is obtained if the number of Cooper pairs in the condensate is power of prime. Relatively small primes in the scale defined by the p-adic length scales assignable to elementary particles would be in question.

Magnetic body, DNA replication, mitosis, meiosis, and fertilization

If magnetic body uses biological body as a motor instrument and sensory receptor, the natural question is whether basic process such as mitosis, meiosis. could be induced by more fundamental processes for the magnetic body. One can argue that if magnetic flux tubes are responsible for making living organism and even population a kind of Indra's net, cell division should be induced by magnetic body and should produce automatically this Indra's net.

As a matter of fact, cell division brings strongly in mind division of magnetic dipole but also the reconnection of magnetic flux tubes can be considered as a basic mechanism. At least the following basic mechanisms can be considered.

1. Consider a pair of magnetic flux tubes with opposite fluxes connecting objects A and B. The division of A+B to A and B would be induced by a reconnection process for the members of the pair producing two loops associated with A and B but no connection between A and B anymore. The problem of this option is that the flux tube connection defined in this manner might not be stable enough.
   
   
2. Magnetic dipole would correspond to a flux tube at the core of the dipole field itself decomposing to flux tubes with weaker magnetic flux at its ends. The division to two dipoles would correspond to a formation of segment in which flux tube decomposes into several flux tubes, which need not be parallel anymore. Two new dipole ends are formed and the old dipole ends remain connected so that the repetition of this process would yield a kind of Indra's net predicting that all cells of living organism are connected by the flux tubes to single coherent whole.
   

   The division of flux tube to several flux tubes could also correspond to the increase of Planck constant by integer factor n along a segment of flux tube. The resulting n flux sheets would corresponds to the sheets of the covering. The length of the segment would be scaled up by n.
   
   

3. If one has pair of dipoles A-B and C-D with same total flux, a reconnection leading to A-D and C-B is possible.
   

Could biochemical processes associated with cell division be induced by some of the listed processes? The two latter options would predict that the cells produced in cell division remain connected by magnetic flux tubes. The division of dipole creates two new dipole ends connected by short flux tube. The already existing ends remain connected by "long" flux tubes carrying weak magnetic fields as compared to that carried by the dipole itself. Also the processes of meiosis and fertilization could respect the presence of long flux tubes connecting the cells participating in the process so that flux tube connections could also exist between parents and offspring. The members of population could form a kind of super-organism. Remote interactions between DNA and other biomolecules of closely related members of species and even shared use of DNA (and its TGD variant "dark DNA") can be imagined.

1. Consider first DNA replication and reshuffling taking place in meiosis essential for the sexual reproduction in eukariotes. The dividing nucleus (of form MMFF) is ordinary nucleus and contains two pairs of chromosomes coming both mother (MM) and father (FF). Division produces four haploid cells containing only two chromosomes (AB) with A and B obtained by reshuffling the DNAs of mother and father to obtained 4 unique chromosome pairs. In sexual reproduction these cells fuse to form diploid cells (MMFF).
   
   
   1. The reshuffling of a pair MF of DNA strands from father and mother could be induced by a repeated reconnection process for flux tubes parallel to DNA strands. The simplest reconnection for strands A-B and C-D produces strands A-X-D and C-Y-B where A-Z and C-Y are pieces of A-B and C-D with same number of codons.
      
      
   2. The replication of DNA takes place for all four chromosomes before reshuffling. One obtains a nucleus containing 4 pairs of doubled chromosomes. This double nucleus divides to two daugher nuclei containing 2 doubled chromosomes each.These divide further to two nuclei each containing only two chromosomes each (AB).
      

      The DNA reshuffling could correspond to a multiple reconnection process if the two DNA strands are accompanied by by long magnetic dipoles (flux tubes). Note that in absence of additional restrictions many combinations (28) are possible.
      
      

   3. After replication and reshuffling the division of the nucleus two two intermedaires could be induced either by splitting of a flux tube connecting pairs of doubled chromosomes to flux tubes not anymore parallel to each other. The flux could diverge to a larger volume in this segment. Second possibility is that the increase of Planck constant increases the length of segment and at the same time divides the flux into sub-fluxes. Dipole field flux tube would give long flux tubes and split dipole shorter flux tubes connecting the resulting cells together.
      
      
   4. Also the chromosome pairs of the resulting intermediate nuclei could be connected to each other by flux tubes to form a connected structure A-B-C-D and reconnection process could divide it to A-B plus C-D (say) and lead to a division of the nucleus producing 4 ordinary daughter nuclei.
      
      
2. In mitosis the initial nucleus corresponds to MMFF and DNA replication leads to pairs of doubled chromosomes but without re-shuffling. One doubled pair from mother and one pair from father the members of doubled chromosomes are connected by a kind of bridge. In the mitosis proper the doubled chromosome pairs are split and two chromosome pairs containing one chromosome from father and mother are formed. After this division leads to two diploid cells similar to the dividing cell.
   
   
3. In fertilization gametes from father and mother fuse together to form a single cell with two pairs of chromosomes from both father and mother. The question is how the two gametes are able to find each other. The reconnection of closed magnetic flux tubes associated with the gametes could lead to a formation of bridges connection the two gametes and a phase transition reducing the value of Planck constant could lead the two gametes near each other and make possible the fusion.
   
   

DNA replication is clearly the fundamental process, and the question is whether also this step could be reduced to a reconnection for a pair flux tubes: first would connect the separated DNA strands and second one free nucleotide and its conjugate.

1. Suppose that there are flux tubes connecting nucleotides of DNA and corresponding nucleotides of the conjugate strand: they could be rather short flux tubes of length shorter than 1 nm in the normal situation but could grow longer when DNA strands separate. This might involve a phase transition increasing temporarily the value of Planck constant assignable to these flux tubes and increasing the length of the segment and of connecting flux tube and therefore the distance of DNA strands.
   
   
2. There are also free DNA nucleotides and their conjugates in the environment which can be used in the replication process as building bricks. If also free nucleotides and their conjugates are connected in a pairwise manner by similar flux tubes and if the value of magnetic flux characterizes a given pair then reconnection could take place for these two kinds of flux tubes and lead to a correct pairing of DNA strand with conjugate nucleotides. Same would happen for the conjugate strand. The reduction of Planck constant would lead to a pair of ordinary DNA double strands.
   
   
3. The details of the dynamics would be determined by other factors but the outcome would be fixed by the nucleotide-conjugate pairing and dependence of the flux on the nucleotide pair. In particular, conservation of magnetic flux would guarantee that the nucleotides can be assigned only with their conjugates.
   

These arguments suggest that reconnection of magnetic flux tubes, temporary change of the Planck constant, and coding of nucleotide-conjugate pairs by magnetic flux could be key element of meiosis, mitosis, and reshuffling of chromosomes in meiosis. Also higher level processes - such as cell division and fertilization - could involve reconnection process as a fundamental step. These mechanisms would appear in several length scales corresponding to DNA, nucleus, and cell length scale. In an approach based on mere chemistry, this must be assumed as a result of reaction kinetics.

DNA, speech, music, and ordinary sound

Peter Gariaev's group has made rather dramatic claims about DNA during years. The reported findings have served as inspiration in the development of TGD based view about living matter (see this, this, this, this and this).

1. The group has proposed that the statistical distributions of nucleotides and codons in the intronic portion of DNA resemble the distribution of letters and words in the natural languages. For instance, it is proposed that Zipf law applying to natural languages applies to the distributions of codons in the intronic portion of DNA. One can study the popularity of the words in natural languages and order them against their popularity. Zipf law states that the integer characterizing popularity is in constant proportion to the number of times it appears in given long enough text.
   
   
2. It has been also claimed that DNA can be reprogrammed using modulated laser light or even radio waves. I understand that reprogramming means a modified gene expression. Gariaev's group indeed proposes that the meaning of the third nucleotide (having a rather low significance in the DNA-aminoacid correspondence) in the genetic codon depends on the context giving rise to a context dependent translation to amino-acids. This is certainly a well-known fact for certain variants of the genetic code. This context dependence might make possible the re-programming. The notion of dark DNA allows to consider much more radical possibility based on the transcription of dark DNA to mRNA followed by translation to aminoacds. This could effectively replaced genes with new ones.
   
   
3. Also the modulation of the laser light by speech is claimed to have the re-programming effect. The broad band em wave spectrum resulting in the scattering of red laser light on DNA is reported to have rather dramatic biological effects. The long wave length part of this spectrum can be recorded and transformed to sound waves and these sound waves are claimed to have the same biological effects as the light. The proposal is that acoustic solitons propagating along DNA represent this effect on DNA.
   

I do not have the competence to make statements about the plausibility of these claims. TGD view about quantum biology makes also rather strong claims. The natural question is however whether a justification for the claims of Gariaev and collaborators could be found in TGD framework? In particular, can one say about possible effects of sound on DNA. One intriguing fact about sound perception is that music and speech have meaning whereas generic sounds to not. Could one say something interesting about how this meaning is generated at the level of DNA?

Basic picture

Before continuing it is good to restate the basic TGD inspired ideas about the generation of meaning.

1. The generation of the negentropic entanglement is the correlate for the experience of the meaning. In the model inspired by Becker's findings discussed in the earlier posting, the generation of negentropic entanglement involves a generation of supra currents along flux tubes moving in the electric field parallel to them. This is a critical phenomenon taking place when the voltage along the flux tube is near critical value. The generation of nerve pulse near critical value of the resting potential is one example of this criticality. Becker's direct currents involved with the healing of wounds is another example.
   
   

   
   The flow of the supra current gives rise to the acceleration of charges along the flux tubes and generation of Cooper pairs or even many-electrons systems at smaller space-time sheets in negentropically entangled state and carrying metabolic energy quantum as zero point kinetic energy. The period of negentropic entanglement gives rise to a conscious experience to which one can assign various attributes such as understanding, attention, and so on. Negentropic entanglement would measure the information contained by a rule having as instances the state pairs in the quantum superposition defining the entangled state. When the period of negentropic entanglement ceases, the metabolic energy is liberated.
   
   

2. Remote activation of DNA by analogs of laser beams is another essential piece of TGD inspired quantum biology (see this). In the proposed addressing mechanism a collection of frequencies serves as a password activating intronic portions of DNA. This would take place via a resonance for the proposed interaction between photons and dark supra currents flowing along magnetic flux tubes and perhaps also along DNA strands or flux tubes parallel to them. The superposition of electric fields of photons (massless extremals) with the electric fields parallel to flux tubes (so that massless extremals serving as correlates for laser beams would traverse the flux tube in orthogonal direction).
   
   
3. The flux tubes, and more generally flux sheets labelled by the value of Planck constant, and along which the radiation arrives would be transversal to DNA and contain DNA strands. This kind of flux tubes and sheets also define the connections to the magnetic body, and form parts of it. A given flux sheet would naturally select the portion of DNA, which is activated by the radiation: it could be a portion of intronic part of DNA activating in turn a gene. These flux tubes and sheets could be connected to the lipids of nuclear and cell membranes - also cell membranes of other cells - as assumed in the model of DNA as topological quantum computer. The sheets could also give rise to a hierarchy of genomes - besides genome one would have super-genome in which genomes are organelles are integrated by flux sheets to a large coherently expressed structure containing individual genomes like page of a book contains lines of text. These pages would be in turn organized to a book - hyper-genome as I called it. One could have also libraries, etc... There would fractal flux quanta inside flux quanta structure.
   
   

Phonons and photons In TGD Universe

Consider next phonons and their coupling to photons in TGD Universe.

1. Sound waves could quite well transform to electromagnetic radiation since living matter is piezo-crystal transforming sound to radiation and vice versa. Microwave hearing represents an example of this kind transformation. This would require that photons of given energy and varying value of Planck constant couple to phonons with the same energy, Planck constant, and frequency.
   
   
2. Whether one can assign to phonons a non-standard value of Planck constant is not quite clear, but there seems to be no reason preventing this. If so, even photons of audible sounds would have energies above thermal threshold and have direct quantal effects on living matter if they have same Planck constant as the photons with same frequency.
   
   
3. Acoustic phonons represent longitudinal waves and this would require longitudinal photons. In Maxwell's electrodynamics they are not possible but in TGD framework photon is predicted to have a small mass and also longitudinal photons are possible.
   
   
4. For general condensed matter systems one can have also optical phonons for which the polarization is orthogonal to the wave vector and these could couple to ordinary photons. The motion of the charged particles in the electromagnetic field of massless extremal (topological light ray) would be a situation in which phonons and photons accompany each other. This would make possible the piezo-electric mechanism.
   

Under these assumptions the collections of audible frequencies could also represent passwords activating the intronic portion of the genome and lead to gene expression or some other activities. If one believes on the hypothesis that DNA acts like topological quantum computer based on the braid strand connections between nucleotides in the intronic portion of DNA with the lipids of the nuclear and/or cell membranes, also topological quantum computation type processes could be activated by the collections of sound frequencies (see this).

What distinguishes speech and music from sounds without meaning?

Speech and music ares very special form of sound in that they have direct meaning. The more one thinks about these facts, the more non-trivial they look. For music - say singing - the frequency of the carrier wave is piecewise constant whereas for speech it remains constant and the amplitude modulation is important. In fact, by slowing down the recorded speech, one gets the impression that carrier frequency is actually modulated like in chirp (frequency goes down and covers a range of frequencies). What is the mechanism giving to speech and music its meaning and in this manner distinguishes them from other sounds?

Besides the frequency also phase is important for both speech and music experience. Speech and reverse speech sound quite different the intensity in frequency space is same. Therefore the relative phases associated with the Fourier coefficients of various frequencies must be important. For music simple rational multiples of the fundamental define the scale. Could it be that also the frequencies relevant to the comprehension of speech correspond to these rational multiples?

Suppose that one indeed believes on the proposed vision based on the fundamental role of negentropic entanglement in generation of meaning and takes seriously the proposed mechanisms for generating it. Can one understand why music and speech differ from general sounds and what distinguishes between them?

1. With these assumptions suitable collections of frequencies sound wave would indee activates the intronic portion of DNA by generating negentropic entanglement. Also other dark flux tubes than those assignable to DNA are involved. For instance, hair cells responsible for hearing of sounds around particular frequencies could involved flux tubes and utilize similar mechanism. Allowing only hair cells would define the conservative option. On the other hand, one could well claim that what happens in ear has nothing to do with the understanding of the speech and music, it could take place only at the level of neuronal nuclei.
   
   
2. Could the direct interaction of sound waves with magnetic flux tubes generate the experiences of speech and music? In other words, assign meaning to sounds? The criterion for sound to have an interpretation as speech or music would be that it contains the resonance frequencies needed to activate the DNA, or more generally generate dark super currents generating Cooper pairs in this manner loading metabolic energy storages. This would apply to both speech and musical sounds.
   
   
3. The pitch of the speech and musical sound can vary. We are aware of the key of the music piece and of modulations of the key and remember the starting key, and it is highly satisfactory to make a return to "home" defined by the original key. This would imply that the overall scale of the collection of frequencies can be varied and that the pitch of the speech defines a natural expectation value of this scale. For persons possessing so called absolute ear this scaling symmetry would be broken in a well-defined sense.
   
   
4. Musical scales involve frequencies coming as rational multiples of the basic frequency. Octaves - power of two multiples- of the frequency can be said to be equivalent as far musical experience is considered. One might understand the special role of rational multiples of the basic frequency if the Fourier components have same phase periodically so that the experience is invariant under discrete time translations. This requires commensurable frequencies expressible as rational multiples of the same fundamental frequency. The preferred role of p-adic primes comings as powers of two could relate to the octave phenomenon.
   
   
5. Are the relative phases of different Fourier components important for music experience? If one requires a periodical occurrence of maximal possible intensity (maximal constructive interference) then the relative phases must vanish at the values of time for maximal possible intensity. What seems essential that the presence of commensurate frequencies gives rise to time translation invariant sensation whereas speech consists of pulses.
   

Are speech and music quantum duals like position and momentum?

Frequencies are crucial for music experience. In the case of of speech the relative phases are very important as the example of reverse speech demonstrates. How a given phoneme is heard is determined to high degree by the frequency spectrum in the beginning of the phoneme (this distinguishes between consonants). Vowels are nearer to notes in vocalization. Speech consists of pulses and destructive interference between different frequencies is required to generate pulses and different pulse shapes so that phase information is important. At least the harmonics of the basic rational multiples of the fundamental are necessary for speech.

One can criticize the previous discussion in that it has been completely classical. Phase and frequency are in wave mechanics canonically conjugate variables analogous to position and momentum. Is it really possible to understand the difference between music and speech purely classically by assuming that one can assign to sound waves both frequencies and phases simultaneously - just like one assigns to a particle sharp values of both momentum and position? Or should one use either representation either in terms numbers of phonons in different modes labelled by frequencies or as coherent states of phonons with ill defined phonon numbers but well defined amplitudes? Could the coherent states serve as the analogs of classical sound waves. Speech would be as near as possible to classical sound and music would be quantal. Of course, there is a large variety of alternative choices of basis states between these two extremes as a specialist in quantum optics could tell.

Suppose that this picture is more or less correct. What could be the minimal scenario allowing to understand the differences between speech and music?

1. Only a subset of frequencies could activate DNA (or if one wants to be conservative, the hair cells) also in the case of speech. One could still pick up important frequencies for which the ratios are simples rational numbers as in the case of musical scale plus their harmonics If this assumption is correct, then speech from which all frequencies except for the harmonics of the simple rational multiples of the fundamental are removed, should be still be comprehensible as speech. The pitch of the speech would determine a good candidate for the fundamental frequency.
   
   
2. The harmonics of frequencies activating DNA would be crucial for speech. Harmonics are present also in music and their distribution allows to distinguish between different instruments and persons. The deviation of musical notes from ideal Fock states would correspond to this.
   
   
3. The naive guess is that the simple rational multiples of fundamental and the possibility of having their harmonics could be reflected in the structure of intronic portions of DNA as repetitive structures of various sizes. This cannot be the case since the wavelengths of ordinary photons would be so small that the energies would be in keV range. Neither is this expected to be the case. It is magnetic flux tubes and sheets traversing the DNA which carry the radiation and the natural lengths assignable to these flux quanta should correspond to the wave lengths. The larger, the flux quantum, the lower the frequency and the larger the value of Planck constant. Harmonics of the fundamental would appear for given flux tube length naturally.
   

   The DNA strands and flux tubes and sheets form a kind of electromagnetic music instrument with flux quanta taking the role of guitar strings and DNA strands and other structures such as lipids and possible other molecules to which flux tubes get attached taking the role of frets in guitar. This analogy suggests that for wave lengths measured in micrometers the basic frequencies correspond to the distances between "frets" defined by cell and nuclear membranes in the tissue in the scale of organism. This would relate the spectrum of resonance frequencies to the spectrum of distances between DNAs in the tissue.
   

   For wavelengths corresponding to very large values of Planck constant giving rise to frequencies in VLF and ELF range and corresponding also to audible frequencies, the preferred wave lengths would correspond to lengths of flux quanta in Earth size scale. One should understand whether the quantization of these lengths in simple rational ratios could take place for the preferred extremals.
   
   

4. Could the pulse shape associated with massless extremals (MEs, topological light rays) allow to distinguish classically between speech and music at the level of space-time correlates? Linear superposition of Fourier components in the direction of ME is possible and this allows to speak about pulse shape. It allows also the notions of coherent state and Fock state for given direction of wave vector. Essential would be the restriction of the superposition of fields in single direction of propagation to be distinguished from the superposition of the effects of fields associated with different space-time sheets on multiply topologically condensed particle. Maybe this would allow to make testable predictions.
   

This text can be found at my homepage from an article with title Quantum Model for the Direct Currents of Becker. See also the chapter Quantum Mind, Magnetic Body, and Biological Body of "TGD based view about living matter and remote mental interactions".

Could magnetic flux tubes make possible effective holograms?

The notions of massless extremals (topological light rays) and magnetic flux tubes carrying dark matter identified as phases have large value of effective Planck constant have become central to TGD inspired quantum biology and new applications emerge continually. Just some time ago I told about addressing mechanism based on collection of frequencies acting as a password. The latest application is an interpretation for what holograms - or rather effective holograms- could be in TGD framework.

The idea about living matter as hologram of some kind is not new. Peter Gariaev's approach to DNA uses the notion of hologram. In neutroscience Karl Pribram is one of the advocators of hologram concept. There is a lot of empirical support for the notion. The notion of hologram - or rather, conscious hologram - is key concept also in TGD inspired view about quantum biology.

But what (conscious) holograms really are? Are they genuine holograms or are they holograms only in the sense that the scattering of light beams from them is very much like scattering on ordinary holograms - that is like scattering from the original object. Could one imagine mechanism making possible scattering from the original object effectively represented by the hologram like structure?

Holograms as relay stations?

To proceed notice that there is rather general belief that just some objects possessed by the patient is enough for healer- in some sense this object are holograms of the patient. Usually this belief is of course regarded as primitive pars pro toto magic. This belief might however have some justification in terms of negentropic entanglement expected to be fundamental aspect of remote mental interactions. In principle negentropic quantum entanglement can take place via arbitrary number of relay stations and magnetic flux tubes connecting the entangled objects would be the quantum correlate for it. Negentropic entanglement would serve as a correlate for attention, experience of understanding, etc., and it would correlate closely with metabolism: generation of ATP and associated high energy phosphate bond would generate negentropically entangled electron Cooper pair or add electron to negentropically entangled existing many-electron system and its decay to ADP would liberate metabolic energy quantum and destroy the negentropic entanglement.

Negentropic entanglement could actually mean that objects of the external world - say living beings - can act like parts of our biological body. There is a wide variety of psychological experiments which show how illusory is our view about what our body is. Quantum entanglement of object with its target having magnetic flux tubes as geometric correlates making object a relay station. The object - call it O - would only serve as a relay station connected to say person, call it P, possesses the object. The light scattering from the O could actually transform to dark photons and travel along flux tubes to P, where it is scattered back- say from DNA- and returns back along flux tubes and leaves O. Effectively this is like scattering from a hologram of P represented by object O. The flux tube connection would make various objects in our vicinity effective holograms. This is something that one actually expects since attention- both visual and auditory - has flux tubes connecting perceiver to the target of attention as correlates.

One can consider two options since the radiation to object could transform to positive or negative energy photons. In the first case scattering could be seen as ordinary scattering from P. Negative energy photons would however represent signals traveling to the geometric past (analogs of phase conjugate laser beams) and scatter back from P as positive energy photons traveling to O. TGD based models of memory as communications with the geometric past and intentional action as a process in which negative energy signal to geometric past initiates neural activities (Libet's findings about active aspects of consciousness) involve similar mechanism. Also the remote metabolism based on sending of negative energy signals to a energy storage (analogous to population inverted laser) relies on the same mechanism.

Peter Gariaev's experiments irradiating DNA with red laser beam generate broad of radio waves, which in TGD Universe could correspond to photons with same energy but with large Planck constant. These photons have biological effects on organisms of the same species and even on closely related species. TGD based proposal is that the scattered laser beam defines a collection of frequencies serving as addresses for parts of DNA activating gene expression.

If this represents a basic mechanism of genetic expression, one can quite well imagine that an organism - call it A - whose DNA is somehow damaged, could utilize the healthy DNA of another organism B by sending to it the counterpart of laser beam which scatters and generates the superposition of dark photon beams serving as an address activating the DNA of A. A would effectively use the DNA of B and B would effectively become part of A:s biological body. Even more, B could apply remote versions of various functions of its DNA, say remote replication and (in particular) remote transcription. One can say that A and B combine their genomes and collaborate. This idea is not too surprising if one acccepts the notion of super-genome suggested strongly by the hypothesis that DNA strands can organize to magnetic flux sheets traversing them to form larger structures allowing coherent gene expression even at the level of species.

This mechanism could explain why the mere presence of healthy organisms of the species can induce the healing of organism which is not healthy. It could be the basic mechanism of healing: patient could remotely use the healthy DNA of the healer to generate signals activating her own genes.

Further comments and questions

Some further comments and questions are in order.

1. The relay station mechanism could universal in biology. The transformation of
   ordinary photons to dark photons at flux tubes defining the magnetic body of DNA is assumed in the model explaining the photos taken by Peter Gariaev and his group about DNA sample showing the presence of what looks like macroscopic flux tube structures kenociteallb/dnahologram.
   
   
2. The mechanism could also explain phantom DNA as real DNA connected by flux tubes to the chamber that contained the original DNA. The laser beam arriving to the empty chamber would travel along flux tubs to the place, where the removed DNA is, scatter and return back. This would create the scattering pattern assigned with the phantom DNA.
   
   
3. One can even ask whether the basic mechanism of homeopathy relies on relay station mechanism. Homeopathically treated water would be a collection of flux tube connections to the molecules, which were present in the first stage of the preparation process of the homeopathic remedy. Since the dark photons travel with light velocity, the times for travel of photons would be so small that the scattering of incoming light via the relay station mechanism would almost instantaneous so that the original molecules would be effectively present.
   
   
4. For instance, the de-differentiation of cells which looks to my rather mysterious phenomenon, means rejuvenation. Could one imagine that the genetic programs are replaced with those in geometric past and similar mechanism is at work. Could the rejuvenation mechanism involve scattering of the counterpart of phase conjugate laser light from non-differentiated healthy cells of the geometric past? If so, one should try to achieve the same effect directly at the level of cells. One could try to induce de-differentiation of the cells of the owner of the object serving as a relay station in the same manner. Healing of say cancer cells by de-differentiating them to omnipotent state. In the experiments involving Becker's DC current just this happened. In this microscopic situation might be can demonstrate the effect really convincingly.
   

Neutron anomaly as evidence for many-sheeted space-time

There was a very interesting article about magnetic anomaly UCN trapping. UCN is a shorthand for ultra-cold neutrons. The article had a somewhat hypish title Magnetic anomaly in UCN trapping: signal for neutrons oscillations to parallel world?. Perhaps this was explains why I did not bother to look at it at the first time I saw it.

As I saw again the popular story hyping the article, I realized that the anomaly - if real - could provide a direct evidence for the transitions of neutrons between parallel space-time sheets of many-sheeted space-time. TGD of course predicts that this phenomenon is completely general applying to all kinds of particles.

The interpretation of authors is that ultra-cold neutrons oscillate between parallel worlds- albeit in different sense as in TGD. The authors describe this oscillation using same mathematical model as describing neutrino oscillations. What would be observed would be that in statistical sense neutrons in the beam disappear and reappear periodically. The model predicts that the frequency for this is just the Larmor frequency ω = μ • B/2 for the precession of spin of neutron in magnetic field. The authors claim that just this is observed and the interpretation is somewhat outlandish looking. Standard model gauge group is doubled: all particles have exact mirror copies with same quantum numbers. This of course is extremely inelegant interpretation. Something much more elegant is needed.

TGD based description of the situation

TGD allows to understand the finding in terms of many-sheeted space-time and one ends up with a phenomenological model similar to that of authors. Now however the phenomenon is predicted to be completely general applying to all kinds of particles and does not require the weird doubling of standard model symmetries.

Imagine the presence of two space-time sheets (or even more of them) carrying magnetic fields which decompose to flux tubes.

1. Suppose that neutron is topologically condensed in one of these flux tubes. What happens when the flux tubes are "above each other" in the sense that that their Minkowski space projections intersect and the flux tubes are extremely near to each other: the distance is of order CP₂ size of order 10⁴ Planck lengths. It took long time to take seriously the obvious: neutrons topologically condense on both space-time sheets and experience the sum of the magnetic fields in these regions. This actually allows to overcome the basic objection against TGD due to the fact that all classical gauge fields are expressible in terms of CP₂ coordinates and their gradients so that enormously powerful constraints between classical gauge fields are satisfied and linear superposition of fields is lost. In many-sheeted space-time this superposition is replaced with the superposition of their effects in multiple topological condensation,
   

2. In the regions where the intersection of M⁴ projections of flux tubes is empty, topological condensation takes place on either space-time sheet.
   

   
   

3. What happens when one has neutrons propagating along flux tube 1 characterized by magnetic field B₁ arrive to a region where flux tube 2 of magnetic field B₂ resides? In the intersection region the neutrons experience the field B₁+B₂ in good approximation. The interaction energy E=μ B• σ, where B is the magnetic field and σ is the spin of neutron. In flux tube 1 has B=B₁, in flux tube 2 one has B=B₂ and in the intersection region B=B₁+B₂. It can happen that neutron arriving along flux tube 1 continues its travel along flux tube 2.
   

4. Magnetic fields in question actually consists of large number of nearly parallel flux tubes and the travel of neutron is a series of segments: B_i1→ B₁+B₂ → B_i2→ ..... As if neutron would make jumps between parallel worlds. Now these worlds are geometrically parallel rather than identifiable as copies in tensor product of standard model gauge groups.
   

A phenological description predicting the probabilities for the transitions between the parallel worlds assignable to the two magnetic fields could be based on simple Hamiltonian used to describe also neutrino mixing. Hamiltonian is sum of spin Hamiltonians H_i =μ B_i• σ and of non-diagonal mixing term ε. H= H₁⊕ H₂ + ε The diagonal term H_i are non-vanishing in the nonintersecting region i and non-diagonal describing what happens in the intersecting regions. Just this description was used by the authors of the article to parametrize the observed anomaly.

One can test this interpretation by introducing a third magnetic field. The interpretation of authors might force to introduce even third copy of standard model gauge group;-).

Amusing co-incidence

What is so amusing that the magnetic field used in the experiments was .2 Gauss. It is exactly the nominal value of the endogenous magnetic field needed to explain the strange quantal effects of radiation at cyclotron frequencies of biologically important ions on vertebrate brain. The frequencies are extremely low - in EEG range - and corresponding thermal energies are 10 orders below thermal energy so that standard quantum mechanics predicts no effects. The explanation assumes B_end=.2 GeV containing dark variants of these ions with so large Planck constants that the cyclotron energies are above thermal energy at physiological temperatures. Why experimentalists happened to use just this .2 Gauss magnetic field which is 2/5 of the the nominal value of the Earth's magnetic field B_E=.5 Gauss? If I were a paranoid, I would swear that the experimentalists were well aware of TGD;-). Of course they were not! One cannot be aware of TGD in a company of respectable scientists and even less in respectable science journals;-)!

About deformations of known extremals of Kähler action

I have done a considerable amount of speculative guesswork to identify what I have used to call preferred extremals of Kähler action. The problem is that the mathematical problem at hand is extremely non-linear and that there is no existing mathematical literature. One must proceed by trying to guess the general constraints on the preferred extremals which look physically and mathematically plausible. The hope is that this net of constraints could eventually chrystallize to Eureka! Certainly the recent speculative picture involves also wrong guesses. The need to find explicit ansatz for the deformations of known extremals based on some common principles has become pressing. The following considerations represent an attempt to combine the existing information to achieve this.

The dream is to discover the deformations of all known extremals by guessing what is common to all of them. One might hope that the following list summarizes at least some common features.

Effective three-dimensionality at the level of action

1. Holography realized as effective 3-dimensionality also at the level of action requires that it reduces to 3-dimensional effective boundary terms. This is achieved if the contraction j^αA_α vanishes. This is true if j^α vanishes or is light-like, or if it is proportional to instanton current in which case current conservation requires that CP₂ projection of the space-time surface is 3-dimensional. The first two options for j have a realization for known extremals. The status of the third option - proportionality to instanton current - has remained unclear.
   
   
2. As I started to work again with the problem, I realized that instanton current could be replaced with a more general current j=*B∧J or concretely: j^α= ε^αβγδB_βJ_γδ, where B is vector field and CP₂ projection is 3-dimensional, which it must be in any case. The contractions of j appearing in field equations vanish automatically with this ansatz.
   
   
3. Almost topological QFT property in turn requires the reduction of effective boundary terms to Chern-Simons terms: this is achieved by boundary conditions expressing weak form of electric magnetic duality. If one generalizes the weak form of electric magnetic duality to J=Φ *J one has B=dΦ and j has a vanishing divergence for 3-D CP₂ projection. This is clearly a more general solution ansatz than the one based on proportionality of j with instanton current and would reduce the field equations in concise notation to Tr(TH^k)=0.
   
   
4. Any of the alternative properties of the Kähler current implies that the field equations reduce to Tr(TH^k)=0, where T and H^k are shorthands for Maxwellian energy momentum tensor and second fundamental form and the product of tensors is obvious generalization of matrix product involving index contraction.
   

Could Einstein's equations emerge dynamically?

For j^α satisfying one of the three conditions, the field equations have the same form as the equations for minimal surfaces except that the metric g is replaced with Maxwell energy momentum tensor T.

1. This raises the question about dynamical generation of small cosmological constant Λ: T= Λ g would reduce equations to those for minimal surfaces. For T=Λ g modified gamma matrices would reduce to induced gamma matrices and the modified Dirac operator would be proportional to ordinary Dirac operator defined by the induced gamma matrices. One can also consider weak form for T=Λ g obtained by restricting the consideration to sub-space of tangent space so that space-time surface is only "partially" minimal surface but this option is not so elegant although necessary for other than CP₂ type vacuum extremals.
   
   
2. What is remarkable is that T= Λ g implies that the divergence of T which in the general case equals to j^βJ_β^α vanishes. This is guaranteed by one of the conditions for the Kähler current. Since also Einstein tensor has a vanishing divergence, one can ask whether the condition to T= κ G+Λ g could the general condition. This would give Einstein's equations with cosmological term besides the generalization of the minimal surface equations. GRT would emerge dynamically from the non-linear Maxwell's theory although in slightly different sense as conjectured (see this)! Note that the expression for G involves also second derivatives of the imbedding space coordinates so that actually a partial differential equation is in question. If field equations reduce to purely algebraic ones, as the basic conjecture states, it is possible to have Tr(GH^k)=0 and Tr(gH^k)=0 separately so that also minimal surface equations would hold true.
   

   What is amusing that the first guess for the action of TGD was curvature scalar. It gave analogs of Einstein's equations as a definition of conserved four-momentum currents. The recent proposal would give the analog of ordinary Einstein equations as a dynamical constraint relating Maxwellian energy momentum tensor to Einstein tensor and metric.
   
   

3. Minimal surface property is physically extremely nice since field equations can be interpreted as a non-linear generalization of massless wave equation: something very natural for non-linear variant of Maxwell action. The theory would be also very "stringy" although the fundamental action would not be space-time volume. This can however hold true only for Euclidian signature. Note that for CP₂ type vacuum extremals Einstein tensor is proportional to metric so that for them the two options are equivalent. For their small deformations situation changes and it might happen that the presence of G is necessary. The GRT limit of TGD discussed in kenociteallb/tgdgrt kenocitebtart/egtgd indeed suggests that CP₂ type solutions satisfy Einstein's equations with large cosmological constant and that the small observed value of the cosmological constant is due to averaging and small volume fraction of regions of Euclidian signature (lines of generalized Feynman diagrams).
   
   
4. For massless extremals and their deformations T= Λ g cannot hold true. The reason is that for massless extremals energy momentum tensor has component T^vv which actually quite essential for field equations since one has H^k_vv=0. Hence for massless extremals and their deformations T=Λ g cannot hold true if the induced metric has Hamilton-Jacobi structure meaning that g^uu and g^vv vanish. A more general relationship of form T=κ G+Λ G can however be consistent with non-vanishing T^vv but require that deformation has at most 3-D CP₂ projection (CP₂ coordinates do not depend on v).
   
   
5. The non-determinism of vacuum extremals suggest for their non-vacuum deformations a conflict with the conservation laws. In, also massless extremals are characterized by a non-determinism with respect to the light-like coordinate but like-likeness saves the situation. This suggests that the transformation of a properly chosen time coordinate of vacuum extremal to a light-like coordinate in the induced metric combined with Einstein's equations in the induced metric of the deformation could allow to handle the non-determinism.
   

Are complex structure of CP₂ and Hamilton-Jacobi structure of M⁴ respected by the deformations?

The complex structure of CP₂ and Hamilton-Jacobi structure of M⁴ could be central for the understanding of the preferred extremal property algebraically.

1. There are reasons to believe that the Hermitian structure of the induced metric ((1,1) structure in complex coordinates) for the deformations of CP₂ type vacuum extremals could be crucial property of the preferred extremals. Also the presence of light-like direction is also an essential elements and 3-dimensionality of M⁴ projection could be essential. Hence a good guess is that allowed deformations of CP₂ type vacuum extremals are such that (2,0) and (0,2) components the induced metric and/or of the energy momentum tensor vanish. This gives rise to the conditions implying Virasoro conditions in string models in quantization:
   

   g_ξⁱξ^j=0 , g_ξ*ⁱξ*^j=0 , i,j=1,2 .
   

   
   Holomorphisms of CP₂ preserve the complex structure and Virasoro conditions are expected to generalize to 4-dimensional conditions involving two complex coordinates. This means that the generators have two integer valued indices but otherwise obey an algebra very similar to the Virasoro algebra. Also the super-conformal variant of this algebra is expected to make sense.
   
   

   These Virasoro conditions apply in the coordinate space for CP₂ type vacuum extremals. One expects similar conditions hold true also in field space, that is for M⁴ coordinates.
   
   

2. The integrable decomposition M⁴(m)=M²(m)+E²(m) of M⁴ tangent space to longitudinal and transversal parts (non-physical and physical polarizations) - Hamilton-Jacobi structure- could be a very general property of preferred extremals and very natural since non-linear Maxwellian electrodynamics is in question. This decomposition led rather early to the introduction of the analog of complex structure in terms of what I called Hamilton-Jacobi coordinates (u,v,w,w*) for M⁴. (u,v) defines a pair of light-like coordinates for the local longitudinal space M²(m) and (w,w*) complex coordinates for E²(m). The metric would not contain any cross terms between M²(m) and E²(m): g_uw=g_vw= g_uw* =g_vw*=0.
   
   

   A good guess is that the deformations of massless extremals respect this structure. This condition gives rise to the analog of the constraints leading to Virasoro conditions stating the vanishing of the non-allowed components of the induced metric. g_uu= g_vv= g_ww=g_w*w* =g_uw=g_vw= g_uw* =g_vw*=0. Again the generators of the algebra would involve two integers and the structure is that of Virasoro algebra and also generalization to super algebra is expected to make sense. The moduli space of Hamilton-Jacobi structures would be part of the moduli space of the preferred extremals and analogous to the space of all possible choices of complex coordinates. The analogs of infinitesimal holomorphic transformations would preserve the modular parameters and give rise to a 4-dimensional Minkowskian analog of Virasoro algebra. The conformal algebra acting on CP₂ coordinates acts in field degrees of freedom for Minkowskian signature.
   

Field equations as purely algebraic conditions

If the proposed picture is correct, field equations would reduce basically to purely algebraically conditions stating that the Maxwellian energy momentum tensor has no common index pairs with the second fundamental form. For the deformations of CP₂ type vacuum extremals T is a complex tensor of type (1,1) and second fundamental form H^k a tensor of type (2,0) and (0,2) so that Tr(TH^k)= is true. This requires that second light-like coordinate of M⁴ is constant so that the M⁴ projection is 3-dimensional. For Minkowskian signature of the induced metric Hamilton-Jacobi structure replaces conformal structure. Here the dependence of CP₂ coordinates on second light-like coordinate of M²(m) only plays a fundamental role. Note that now T^vv is non-vanishing (and light-like). This picture generalizes to the deformations of cosmic strings and even to the case of vacuum extremals.

For background see the chapter Basic Extremals of Kähler action of "Physics in Many-Sheeted Space-time". For details see the article About deformations of known extremals of Kähler action.