Experimental evidence for sterile neutrino?

Many physicists are somewhat disappointed to the results from LHC: the expected discovery of Higgs has been seen as the main achievement of LHC hitherto. Much more was expected. To my opinion there is no reason for disappointment. The exclusion of the standard SUSY at expected energy scale is very far reaching negative result. Also the fact that Higgs mass is too small to be stable without fine tuning is of great theoretical importance. The negative results concerning heavy dark matter candidates are precious guidelines for theoreticians. The non-QCD like behavior in heavy ion collisions and proton-ion collisions is bypassed my mentioning something about AdS/CFT correspondence and non-perturbative QCD effects. I tend to see these effects as direct evidence for M₈₉ hadron physics (see this).

In any case, something interesting has emerged quite recently. Resonaances tells that the recent analysis of X-ray spectrum of galactic clusters claims the presence of monochromatic 3.5 keV photon line. The proposed interpretation is as a decay product of sterile 7 keV neutrino transforming first to a left-handed neutrino and then decaying to photon and neutrino via a loop involving W boson and electron. This is of course only one of the many interpretations. Even the existence of line is highly questionable.

One of the poorly understood aspects of TGD is right-handed neutrino, which is obviously the TGD counterpart of the inert neutrino.

1. The old idea is that covariantly constant right handed neutrino could generate N=2 super-symmetry in TGD Universe. In fact, all modes of induced spinor field would generate super-conformal symmetries but electroweak interactions would break these symmetries for the modes carrying non-vanishing electroweak quantum numbers: they vanish for ν_R. This picture is now well-established at the level of WCW geometry (see this): super-conformal generators are labelled angular momentum and color representations plus two conformal weights: the conformal weight assignable to the light-like radial coordinate of light-cone boundary and the conformal weight assignable to string coordinate. It seems that these conformal weights are independent. The third integer labelling the states would label genuinely Yangian generators: it would tell the poly-locality of the generator with locus defined by partonic 2-surface: generators acting on single partonic 2-surface, 2 partonic 2-surfaces, ...
   
   
2. It would seem that even the SUSY generated by ν_R must be badly broken unless one is able to invent dramatically different interpretation of SUSY. The scale of SUSY breaking and thus the value of the mass of right-handed neutrino remains open also in TGD. In lack of better one could of course argue that the mass scale must be CP₂ mass scale because right-handed neutrino mixes considerably with the left-handed neutrino (and thus becomes massive) only in this scale. But why this argument does not apply also to left handed neutrino which must also mix with the right-handed one!
   
   
3. One can of course criticize the proposed notion of SUSY: wonder whether fermion + extremely weakly interacting ν_R at same wormhole throat (or interior of 3-surface) can behave as single coherent entity as far spin is considered (see this)?
   
   
4. The condition that the modes of induced spinor field have a well-defined electromagnetic charge eigenvalue (see this) requires that they are localized at 2-D string world sheets or partonic 2-surfaces: without this condition classical W boson fields would mix the em charged and neutral modes with each other. Right-handed neutrino is an exception since it has no electroweak couplings. Unless right-handed neutrino is covariantly constant, the modified gamma matrices can however mix the right-handed neutrino with the left handed one and this can induce transformation to charged mode. This does not happen if each modified gamma matrix can be written as a linear combination of either M⁴ or CP₂ gamma matrices and modified Dirac equation is satisfied separately by M⁴ and CP₂ parts of the modified Dirac equation.
   
   
5. Is the localization of the modes other than covariantly constant neutrino to string world sheets a consequence of dynamics or should one assume this as a separate condition? If one wants similar localization in space-time regions of Euclidian signature - for which CP₂ type vacuum extremal is a good representative - one must assume it as a separate condition. In number theoretic formulation string world sheets/partonic 2-surfaces would be commutative/co-commutative sub-manifolds of space-time surfaces which in turn would be associative or co-associative sub-manifolds of imbedding space possessing (hyper-)octonionic tangent space structure. For this option also
   right-handed neutrino would be localized to string world sheets. Right-handed neutrino would be covariantly constant only in 2-D sense.
   

   One can consider the possibility that ν_R is de-localized to the entire 4-D space-time sheet. This would certainly modify the interpretation of SUSY since the number of degrees of freedom would be reduced for ν_R.
   
   

6. Non-covariantly constant right-handed neutrinos could mix with left-handed neutrinos but not with charged leptons if the localization to string world sheets is assumed for modes carrying non-vanishing electroweak quantum numbers. This would make possible the decay of right-handed to neutrino plus photon, and one cannot exclude the possibility that ν_R has mass 7 keV.
   

   Could this imply that particles and their spartners differ by this mass only? Could it be possible that practically unbroken SUSY could be there and we would not have observed it? Could one imagine that sfermions have annihilated leaving only states consisting of fundamental fermions? But shouldn't the total rate for the annihilation of photons to hadrons be two times the observed one? This option does not sound plausible.
   

   What if one assumes that given sparticle is charactrized by the same p-adic prime as corresponding particle but is dark in the sense that it corresponds to non-standard value of Planck constant. In this case sfermions would not appear in the same vertex with fermions and one could escape the most obvious contradictions with experimental facts. This leads to the notion of shadron: shadrons would be (see this) obtained by replacing quarks with dark squarks with nearly identical masses. I have asked whether so called X and Y bosons having no natural place in standard model of hadron could be this kind of creatures.
   
   

The interpretation of 3.5 keV photons as decay products of right-handed neutrinos is of course totally ad hoc. Another TGD inspired interpretation would be as photons resulting from the decays of excited nuclei to their ground state.

1. Nuclear string model (see this) predicts that nuclei are string like objects formed from nucleons connected by color magnetic flux tubes having quark and antiquark at their ends. These flux tubes are long and define the "magnetic body" of nucleus. Quark and antiquark have opposite em charges for ordinary nuclei. When they have different charges one obtains exotic state: this predicts entire spectrum of exotic nuclei for which statistic is different from what proton and neutron numbers deduced from em charge and atomic weight would suggest. Exotic nuclei and large values of Planck constant could make also possible cold fusion
   (see this).
   
   
2. What the mass difference between these states is, is not of course obvious. There is however an experimental finding (see Analysis of Gamma Radiation from a Radon Source: Indications of a Solar Influence) that nuclear decay rates oscillate with a period of year and the rates correlate with the distance from Sun. A possible explanation is that the gamma rays from Sun in few keV range excite the exotic nuclear states with different decay rate so that the average decay rate oscillates. Note that nuclear excitation energies in keV range would also make possible interaction of nuclei with atoms and molecules (see this).
   
   
3. This allows to consider the possibility that the decays of exotic nuclei in galactic clusters generates 3.5 keV photons. The obvious question is why the spectrum would be concentrated at 3.5 keV in this case (second question is whether the energy is really concentrated at 3.5 keV: a lot of theory is involved with the analysis of the experiments). Do the energies of excited states depend on the color bond only so that they would be essentially same for all nuclei? Or does single excitation dominate in the spectrum? Or is this due to the fact that the thermal radiation leaking from the core of stars excites predominantly single state? Could E=3.5 keV correspond to the maximum intensity for thermal radiation in stellar core? If so, the temperature of the exciting radiation would be about T≈ E/3≈ 1.2× 10⁷ K. This in the temperature around which formation of Helium by nuclear fusion has begun: the temperature at solar core is around 1.57× 10⁷ K.
   
   

For background see the chapter SUSY in TGD Universe of "p-Adic Length Scale Hypothesis".

Class field theory and TGD: does TGD reduce to number theory?

The intriguing general result of class field theory) -something extremely abstract for physicist's brain - is that the the maximal Abelian extension for rationals is homomorphic with the multiplicative group of ideles. This correspondence plays a key role in Langlands correspondence (see this,this, this, and this).

Does this mean that it is not absolutely necessary to introduce p-adic numbers? This is actually not so. The Galois group of the maximal abelian extension is rather complex objects (absolute Galois group, AGG, defines as the Galois group of algebraic numbers is even more complex!). The ring Z of adeles defining the group of ideles as its invertible elements homeomorphic to the Galois group of maximal Abelian extension is profinite group. This means that it is totally disconnected space as also p-adic integers and numbers are. What is intriguing that p-dic integers are however a continuous structure in the sense that differential calculus is possible. A concrete example is provided by 2-adic units consisting of bit sequences which can have literally infinite non-vanishing bits. This space is formally discrete but one can construct differential calculus since the situation is not democratic. The higher the pinary digit in the expansion is, the less significant it is, and p-adic norm approaching to zero expresses the reduction of the insignificance.

1. Could TGD based physics reduce to a representation theory for the Galois groups of quaternions and octonions?

Number theoretical vision about TGD raises questions about whether adeles and ideles could be helpful in the formulation of TGD. I have already earlier considered the idea that quantum TGD could reduce to a representation theory of appropriate Galois groups. I proceed to make questions.

1. Could real physics and various p-adic physics on one hand, and number theoretic physics based on maximal Abelian extension of rational octonions and quaternions on one hand, define equivalent formulations of physics?
   
   
2. Besides various p-adic physics all classical number fields (reals, complex numbers, quaternions, and octonions) are central in the number theoretical vision about TGD. The technical problem is that p-adic quaternions and octonions exist only as a ring unless one poses some additional conditions. Is it possible to pose such conditions so that one could define what might be called quaternionic and octonionic adeles and ideles?
   

   It will be found that this is the case: p-adic quaternions/octonions would be products of rational quaternions/octonions with a p-adic unit. This definition applies also to algebraic extensions of rationals and makes it possible to define the notion of derivative for corresponding adeles. Furthermore, the rational quaternions define non-commutative automorphisms of quaternions and rational octonions at least formally define a non-associative analog of group of octonionic automorphisms (see this).
   
   

3. I have already earlier considered the idea about Galois group as the ultimate symmetry group of physics. The representations of Galois group of maximal Abelian extension (or even that for algebraic numbers) would define the quantum states. The representation space could be group algebra of the Galois group and in Abelian case equivalently the group algebra of ideles or adeles. One would have wave functions in the space of ideles.
   

   The Galois group of maximal Abelian extension would be the Cartan subgroup of the absolute Galois group of algebraic numbers associated with given extension of rationals and it would be natural to classify the quantum states by the corresponding quantum numbers (number theoretic observables).
   

   If octonionic and quaternionic (associative) adeles make sense, the associativity condition would reduce the analogs of wave functions to those at 4-dimensional associative sub-manifolds of octonionic adeles identifable as space-time surfaces so that also space-time physics in various number fields would result as representations of Galois group in the maximal Abelian Galois group of rational octonions/quaternions. TGD would reduce to classical number theory!
   
   

4. Absolute Galois group is the Galois group of the maximal algebraic extension and as such a poorly defined concept. One can however consider the hierarchy of all finite-dimensional algebraic extensions (including non-Abelian ones) and maximal Abelian extensions associated with these and obtain in this manner a hierarchy of physics defined as representations of these Galois groups homomorphic with the corresponding idele groups.
   
   
5. In this approach the symmetries of the theory would have automatically adelic representations and one might hope about connection with Langlands program.
   

2. Adelic variant of space-time dynamics and spinorial dynamics?

As an innocent novice I can continue to pose stupid questions. Now about adelic variant of the space-time dynamics based on the generalization of Kähler action discussed already earlier but without mentioning adeles (see this).

1. Could one think that adeles or ideles could extend reals in the formulation of the theory: note that reals are included as Cartesian factor to adeles. Could one speak about adelic or even idelic space-time surfaces endowed with adelic or idelic coordinates? Could one formulate variational principle in terms of adeles so that exponent of action would be product of actions exponents associated with various factors with Neper number replaced by p for Z_p. The minimal interpretation would be that in adelic picture one collects under the same umbrella real physics and various p-adic physics.
   
   
2. Number theoretic vision suggests that 4:th/8:th Cartesian powers of adeles have interpretation as adelic variants of quaternions/ octonions. If so, one can ask whether adelic quaternions and octonions could have some number theretical meaning. Note that adelic quaternions and octonions are not number fields without additional assumptions since the moduli squared for a p-adic analog of quaternion and octonion can vanish so that the inverse fails to exist.
   

   If one can pose a condition guaranteing the existence of inverse, one could define the multiplicative group of ideles for quaternions. For octonions one would obtain non-associative analog of the multiplicative group. If this kind of structures exist then four-dimensional associative/co-associative sub-manifolds in the space of non-associative ideles define associative/co-associative ideles and one would end up with ideles formed by associative and
   co-associative space-time surfaces.
   
   

3. What about equations for space-time surfaces. Do field equations reduce to separate field equations for each factor? Can one pose as an additional condition the constraint that p-adic surfaces provide in some sense cognitive representations of real space-time surfaces: this idea is formulated more precisely in terms of p-adic manifold concept (see this). Or is this correspondence an outcome of evolution?
   

   Physical intuition would suggest that in most p-adic factors space-time surface corresponds to a point, or at least to a vacuum extremal. One can consider also the possibility that same algebraic equation describes the surface in various factors of the adele. Could this hold true in the intersection of real and p-adic worlds for which rationals appear in the polynomials defining the preferred extremals.
   
   

4. To define field equations one must have the notion of derivative. Derivative is an operation involving division and can be tricky since adeles are not number field. If one can guarantee that the p-adic variants of octonions and quaternions are number fields, there are good hopes about well-defined derivative. Derivative as limiting value df/dx= lim ( f(x+dx)-f(x))/dx for a function decomposing to Cartesian product of real function f(x) and p-adic valued functions f_p(x_p) would require that f_p(x) is non-constant only for a finite number of primes: this is in accordance with the physical picture that only finite number of p-adic primes are active and define "cognitive representations" of real space-time surface. The second condition is that dx is proportional to product dx × ∏ dx_p of differentials dx and dx_p, which are rational numbers. dx goes to xero as a real number but not p-adically for any of the primes involved. dx_p in turn goes to zero p-adically only for Q_p.
   
   
5. The idea about rationals as points commont to all number fields is central in number theoretical vision. This vision is realized for adeles in the minimal sense that the action of rationals is well-defined in all Cartesian factors of the adeles. Number theoretical vision allows also to talk about common rational points of real and various p-adic space-time surfaces in preferred coordinate choices made possible by symmetries of the imbedding space, and one ends up to the vision about life as something residing in the intersection of real and p-adic number fields. It is not clear whether and how adeles could allow to formulate this idea.
   
   
6. For adelic variants of imbedding space spinors Cartesian product of real and p-adc variants of imbedding spaces is mapped to their tensor product. This gives justification for the physical vision that various p-adic physics appear as tensor factors. Does this mean that the generalized induced spinors are infinite tensor products of real and various p-adic spinors and Clifford algebra generated by induced gamma matrices is obtained by tensor product construction? Does the generalization of massless Dirac equation reduce to a sum of d'Alembertians for the factors? Does each of them annihilate the appropriate spinor? If only finite number of Cartesian factors corresponds to a space-time surface which is not vacuum extremal vanishing induced Kähler form, Kähler Dirac equation is non-trivial only in finite number of adelic factors.
   

3. Objections

The basic idea is that apporopriately defined invertible quaternionic/octonionic adeles can be regarded as elements of Galois group assignable to quaternions/octonions. The best manner to proceed is to invent objections against this idea.

1. The first objection is that p-adic quaternions and octonions do not make sense since p-adic variants of quaternions and octonions do not exist in general. The reason is that the p-adic norm squared ∑ x_i² for p-adic variant of quaternion, octonion, or even complex number can vanish so that its inverse does not exist.
   
   
2. Second objection is that automorphisms of the ring of quaternions (octonions) in the maximal Abelian extension are products of transformations of the subgroup of SO(3) (G₂) represented by matrices with elements in the extension and in the Galois group of the extension itself. Ideles separate out as 1-dimensional Cartesian factor from this group so that one does not obtain 4-field (8-fold) Cartesian power of this Galois group.
   

If the p-adic variants of quaternions/octonions are be rational quaternions/octonions multiplied by p-adic number, these objections can be circumvented.

1. This condition indeed allows to construct the inverse of p-adic quaternion/octonion as a product of inverses for rational quaternion/octonion and p-adic number! The reason is that the solutions to ∑ x_i²=0 involve always p-adic numbers with an infinite number of pinary digits - at least one and the identification excludes this possibility.
   
   
2. This restriction would give a rather precise content for the idea of rational physics since all p-adic space-time surfaces would have a rational backbone in well-defined sense.
   
   
3. One can interpret also the quaternionicity/octonionicity in terms of Galois group. The 7-dimensional non-associative counterparts for octonionic automorphisms act as transformations x→ gxg^-1. Therefore octonions represent this group like structure and the p-adic octonions would have interpretation as combination of octonionic automorphisms with those of rationals.
   

   Adelic variants of of octonions would represent a generalization of these transformations so that they would act in all number fields. Quaternionic 4-surfaces would define associative local sub-groups of this group-like structure. Thus a generalization of symmetry concept reducing for solutions of field equations to the standard one would allow to realize the vision about the reduction of physics to number theory.
   
   

For background see the chapter About Absolute Galois group of "TGD as Generalized Number Theory".

Why TGD?

Hamed kindly reminded me about article "Why TGD?" that I wrote
recently: why not mention it in blog article.

The article is as an attempt to provide a popular summary about TGD, its motivations, and basic implications. This is of course mission impossible as such since TGD is something at the top of centuries of evolution which has led from Newton to standard model. This means that there is a background of highly refined conceptual thinking about Universe so that even the best computer graphics and animations do not help much. One can still try - at least to create some inspiring impressions. The artice approaches the challenge by answering the most frequently asked questions. Why TGD? How TGD could help to solve the problems of recent day theoretical physics? What are the basic principles of TGD? What are the basic guidelines in the construction of TGD?

These are examples of this kind of questions which I try to answer in the article using the only language that I can talk. This language is a dialect used by elementary particle physicists, quantum field theorists, and other people applying modern physics. At the level of practice involves technically heavy mathematics but since it relies on very beautiful and simple basic concepts, one can do with a minimum of formulas, and reader can always to to Wikipedia if it seems that more details are needed. I hope that reader could catch the basic idea: technical details are not important, it is principles and concepts which really matter. And I almost forgot: problems! TGD itself and almost every new idea in the development of TGD has been inspired by a problem.

Why TGD?

The first question is "Why TGD?". The attempt to answer this question requires overall view about the recent state of theoretical physics.

Obviously standard physics plagued by some problems. These problems are deeply rooted in basic philosophical - one might even say ideological - assumptions which boil down to -isms like reductionism, materialism, determinism, and locality.

Thermodynamics, special relativity, and general relativity involve also postulates, which can be questioned. In thermodynamics second law in its recent form and the assumption about fixed arrow of thermodynamical time can be questions since it is hard to understand biological evolution in this framework. Clearly, the relationship between the geometric time of physics and experienced time is poorly understood. In general relativity the beautiful symmetries of special relativity are in principle lost and by Noether's theorem this means also the
loss of classical conservation laws, even the definitions of energy and momentum are in principle lost. In quantum physics the basic problem is that the non-determinism of quantum measurement theory is in conflict with the determinism of Schrödinger equation.

Standard model is believed to summarize the recent understanding of physics. The attempts to extrapolate physics beyond standard model are based on naive length scale reductionism and have products Grand Unified Theories (GUTs), supersymmetric gauge theories (SUSYs). The attempts to include gravitation under same
theoretical umbrella with electroweak and strong interactions has led to super-string models and M-theory. These programs have not been successful, and the recent dead end culminating in the landscape problem of super string theories and M-theory could have its origins in the basic ontological assumptions about the nature of space-time and quantum.

How could TGD help?

The second question is "Could TGD provide a way out of the dead alley and how?". The claim is that is the case. The new view about space-time as 4-D surface in certain fixed 8-D space-time is the starting point motivated by the energy problem of general relativity and means in certain sense fusion of the basic ideas of special and general relativities.

This basic idea has gradually led to several other ideas. Consider only the identification of dark matter as phases
of ordinary matter characterized by non-standard value of Planck constant, extension of physics by including physics in p-adic number fields and assumed to describe correlates of cognition and intentionality, and zero energy ontology (ZEO) in which quantum states are identified as counterparts of physical events. These new elements generalize considerably the view about space-time and quantum and give good hopes about possibility to understand living systems and consciousness in the framework of physics.

Two basic visions about TGD

There are two basic visions about TGD as a mathematical theory. The first vision is a generalization of Einstein's geometrization program from space-time level to the level of "world of classical worlds" identified as space of 4-surfaces. There are good reasons to expect that the mere mathematical existence of this infinite-dimensional geometry fixes it highly uniquely and therefore also physics. This hope inspired also string model enthusiasts before the landscape problem forcing to give up hopes about predictability.

Second vision corresponds to a vision about TGD as a generalized number theory having three separate threads.

1. The inspiration for the first thread came from the need to fuse various p-adic physics and real physics to single coherent whole in terms of principle that might be called number theoretical universality.
   
   
2. Second thread was based on the observation that classical number fields (reals, complex numbers, quaternions, and octonions) have dimensions which correspond to those appearing in TGD. This led to the vision that basic laws of both classical and quantum physics could reduce to the requirements of associativity and commutativity.
   
   
3. Third thread emerged from the observation that the notion of prime (and integer, rational, and algebraic number) can be generalized so that infinite primes are possible. One ends up to a construction principle allowing to construct infinite hierarchy of infinite primes using the primes of the previous level as building bricks at new level. Rather surprisingly, this procedure is structurally identical with a repeated second quantization of supersymmetric arithmetic quantum field theory for which elementary bosons and fermions are labelled by primes. Besides free many-particle states also the analogs of bound states are obtained and this means the situation really fascinating since it raises the hope that the really hard part of quantum field theories -
   understanding of bound states - could have number theoretical solution.
   
   

It is not yet clear whether both great visions are needed or whether either of them is in principle enough. In any case their combination has provided a lot of insights about what quantum TGD could be.

Guidelines in the construction of TGD

The construction of new physical theory is slow and painful task but leads gradually to an identification of basic guiding principles helping to make quicker progress. There are many such guiding principles.

• "Physics is uniquely determined by the existence of WCW" is is a conjecture but motivates highly interesting questions. For instance: "Why M⁴× CP₂ a unique choice for the imbedding space?", "Why space-time dimension must be 4?", etc...
  
  
• Number theoretical Universality is a guiding principle in attempts to realize number theoretical vision, in particular the fusion of real physics and various p-adic physics to single structure.
  
  
• The construction of physical theories is nowadays to a high degree guesses about the symmetries of the theory and deduction of consequences. The very notion of symmetry has been generalized in this process. Super-conformal symmetries play even more powerful role in TGD than in super-string models and gigantic symmetries of WCW in fact guarantee its existence.
  
  
• Quantum classical correspondence is of special importance in TGD. The reason is that where classical theory is not anymore an approximation but in well-defined sense exact part of quantum theory.
  

There are also more technical guidelines.

• Strong form of General Coordinate invariance (GCI) is very strong assumption. Already GCI leads to the assumption that Kähler function is Kähler action for a preferred extremal defining the counterpart of Bohr orbit. Even in a form allowing the failure of strict determinism this assumption is very powerful. Strong form of general coordinate invariance requires that the light-like 3-surfaces representing partonic orbits and space-like 3-surfaces at the ends of causal diamonds are physically equivalent. This implies effective 2-dimensionality: the intersections of these two kinds of 3-surfaces and 4-D tangent space data at them should code for quantum states.
  
  
• Quantum criticality states that Universe is analogous to a critical system meaning that it has maximal structural richness. One could also say that Universe is at the boundary line between chaos and order. The original motivation was that quantum criticality fixes the basic coupling constant dictating quantum dynamics essentially uniquely.
  
  
• The notion of finite measurement resolution has also become an important guide-line. Usually this notion is regarded as ugly duckling of theoretical physics which must be tolerated but the mathematics of von Neumann algebras seems to raise its status to that of beautiful swan.
  
  
• What I have used to call weak form of electric-magnetic duality is a TGD version of electric-magnetic duality discovered by Olive and Montonen. It makes it possible to realize strong form of holography implied actually by strong for of General Coordinate Invariance. Weak form of electric magnetic duality in turn encourages the conjecture that TGD reduces to almost topological QFT. This would mean enormous mathematical simplification.
  
  
• TGD leads to a realization of counterparts of Feynman diagrams at the level of space-time geometry and topology: I talk about generalized Feynman diagrams. The highly non-trivial challenge is to give them precise mathematical content. Twistor revolution has made possible a considerable progress in this respect and led to a vision about twistor Grassmannian description of stringy variants of Feynman diagrams. In TGD context string like objects are not something emerging in Planck length scale but already in scales of elementary particle physics. The irony is that although TGD is not string theory, string like objects and genuine string world sheets emerge naturally from TGD in all length scales. Even TGD view about nuclear physics predicts string like objects.

For details see the new article Why TGD?.

Orch-Or theory of Penrose and Hameroff and new experimental findings about microtubules

The latest news in quantum biology is the claim about corroboration of the the Orch OR theory of Penrose
Hameroff (see this). To my humble opinion the news suffer from rather heavy hyping. If the observation of the group lead by Anirban Bandyopadhyay about detection of quantum vibration in microtubule scale - their lengths vary up to 50 μm - can be replicated, one can speak about breakthrough in quantum consciousness. The results do not however prove Orch OR, which involves poorly defined vision about quantum gravitational description of state function reduction so that most predictions are just order of magnitude estimates relying on Uncertainty Principle.

The biological half of the theory relies on microtubules and for this side of the theory the claimed finding would of course be a victory. Indeed, there is a meeting in Amsterdam devoted to Orch OR theory of consciousness motivated by this finding (see this) Unfortunately, I could not find any article about the findings of Bandyopadhyay in web. I managed however to find two years old Youtube talk of Bandyopahdyay summarizing earlier experimental results supporting the vision about microtubules as macroscopic quantum systems (see this) to be discussed below.

The findings reported in the talk give support for the general TGD inspired view about TQC and allow rather detailed model in the case of microtubules. The idea is that flux tubes from a 2-D coordinate grid consisting of parallel flux tubes in two different directions: say helical Fibonacci flux tubes and their mirror images. Crossing points would be associated with tubulins and the conformational state of tubulin could define a bit coding whether the braid strands defining coordinate lines are braided or not (swap or not). In this manner any bit pattern at microtubule defines a particular TQC program. If also conformations are quantum superposed one as "quantum-quantum computation". The flux tubes could form lattice of type A whereas microtubules form always a lattice of type B - a heavy objection against Penrose-Hameroff model. This picture generalizes in the fractal universe of TGD. One can form layers of 2-D coordinate grids and connect them by vertical flux tubes to obtain 3-D grid defining TQC. Brain is known to have grid like architecture and neurons could by quantum computation produce bit/qubit defining swap or not/superposition of swap and not-swap for a larger scale TQC. One would have fractal of TQCs.

One also ends up with a very crazy idea which turned out to have lifetime longer than the time it took to type it.

1. If one piles up 2-D TQC:s one obtains 3-D TQC. In crossings one must have 4 bits to specify whether to swap or not since there are three planes for TQC and 4 pairs of crossing strands (12,13,23,23).
   
   
2. If one further piles 3D TQC:s in 4-D one obtains 4-D one making sense in zero energy ontology because failure of strict non-determinism is basic element of TGD. Single crossing would in 4-D would involve crossings of four lines in orthogonal dimensions. TGD predicts also space-time regions with Euclidian signature in all scales (lines of generalised Feynman diagrams). I have proposed that any system corresponds to an Euclidian space-time sheet having its size and shape and behaving like quantum system. In these regions the fourth piling might really make sense!
   
   
3. in 4-D This would make 6 crossing pairs corresponding to 6 planes in which particular TQC takes place - for which one must tell whether to swap or not (12,13,14,23,24,34). This makes 6 bits. DNA codons correspond to 6 bits! Could codons define crossing points of magnetic flux tubes arriving from 4 coordinate directions- perhaps at Euclidian space-time sheets? Could the planes correspond to 3 components of magnetic field and 3 components electric field. Magnetic flux tubes and electric flux tubes in 3 directions? In Euclidian regions magnetic and electric do not differ intrinsically.
   

Addition: The TGD inspired interpretation of the experiments of Bandyopadhyay in terms of flux tube coordinate grids making possible TQC architectures with tubulin dimers defining bits defining in turn TQC program looks rather natural. Coordinate grids can be fixed on basis of the experimental findings and there are 8 grids. The interpretation is in terms of different resolutions. The grids for A and B type lattices are related by 2π twist for the second end of the basic 13-unit for microtubule. An attractive interpretation for the resonance frequencies is in terms of phase transitions between A and B type lattices. If A type lattices can be generated only in phase transitions induced by AC stimulus at resonance frequencies, one could understand their experimental absence, which is a strong objection against Penrose-Hameroff model.

This would fit very nicely with the general vision about frequencies as passwords inducing not only directed attention but activities in target - also TQCs! The increase of Planck constant could be associated with the phase transition to A-phase making possible high T_c dark super-conductivity for which evidence is observed! One can even deduce estimates for h_eff/h=n if one requires that AC photons have energy above thermal threshold: n= f_visible/f_AC would be the estimate. For biophoton energies one would obtain something like n≈ 10⁸-10⁹, which pops up in different contexts in TGD framework.

For details see the chapter Quantum Mind, Magnetic Body, and Biological Body or the article "Orch-Or theory of Penrose and Hameroff and new experimental findings about microtubules".

Are microtubules macroscopic quantum systems?

There has been a lot of buzz about the claimed discovery of quantum vibrations in microtubules: see this. I have been working for two days trying to understand the work of Anirban Bandyopadhyay. Certainly, the experimental finding- if true - would be a breakthrough for quantum consciousness but certainly not for Orch-OR. Therefore I was surprised for the heavy hyping of Penrose-Hameroff theory. The result only tells that microtubules are macroscopic quantums systems, it says nothing about Orch-OR.

I did not find any article about experiment. I found and listened the earlier talk (2111) of Anirban Bandyopadhyay whose group might discovered the quantum vibrations. The talk was also was about experiments done with microtubules and looked very interesting. See this .

I got the impression that he is excellent sharply thinking experimentalist and had identified the signatures for what he interpreted in terms of Froehlich B-E condensation, topological qubit, superconductivity like state of electrons inside microtubules, and so on. One can interpret the findings differently and I have been working with TGD interpretation assuming that basic findings are correct.

What was frustrating was the fuzzy terminology: for instance, he talked about conduction pathways at micro tubular surface as topological qubits and did not explain when asked about this issue. Excellent experimentalist need not be a theorist: was this the reason? Or was the unclarity purposeful?

Also the interpretation of experiments seems to be internally inconsistent and I got the impression that it reflects his own theories. It might rely on the earlier proposal of Penrose and Hameroff, which I did not find as freely available article. The only article mentioning conduction pathways that I found from web did not help: I have the feeling that they have no detailed model but want to give the impression that they have.

The wrong interpretation need not mean catastrophe: as a good experimentalist he is looking for signatures of these phenomena, and if he has found them, there are all reasons to take the findings under serious discussion.

I also looked for the Penrose Hameroff theory. As a professional I known that the Penrose's contribution relating to speculations about gravitation does not tolerate daylight and the only calculations are just dimensional analysis involving Uncertainty Principle. I had thought that Hameroff's microtubule contribution is on less shaky grounds. I learned within few hours that this is not the case. For instance, the assumption that microtubules in brain are of type A, is simply wrong: all microtubules in living matter seem to be of type B, in particular in brain this is the case. See this .

Therefore the proposal of Penrose and Hameroff for microtubules as topological quantum computers relying on Fibonacci conduction paths (with periodicity 3,5,8, or 13) possible only for A-type microtubules is simply wrong. The same is true about Hameroff's and indian theorists Gupta's later proposal suggesting that MAPs act as quantum gates. As a matter of fact, the abstract of the article contained horrible terminological errors: I tried to explain this by language but why Hameroff did not check the language? See this .

Now Penrose and Hameroff propose explanation for the origin of EEG based on beat phenomenon for MHz frequencies: it seems that they are trying to mimic what I have done in TGD and get the honour for discovering quantum theory of consciousness;-).

I must say that I am confused. Certainly the discovery that microtubules are macroscopic quantum systems would be wonderful if true, and the claimed earlier findings of Indian experimentalist support the TGD based picture based on the identification of braids as magnetic flux tubes. But can I trust to experimentalist without name when I cannot trust on namy theorists?

To sum up, these are my first impressions after two day's listening and reading. I might have misinterpreted and misunderstood. I am writing a summary about TGD based interpretation of the earlier experiments of the group of Bandyopadhyay -assuming that experimental findings are correct (I do not believe that the interpretation is correct) - and give a link to the article later.

Addition: I wrote a short analysis of the talk of Bandyopadhyay.

Little but important step in the understanding the arrow of time

"Fractality from your blog" posed an interesting question about possible asymmetry between boundaries of causal diamond CD. The answer to the question led to recall once again the incomplete understanding of details about how the arrow of time emerges in zero energy ontology (ZEO).

The basic vision is following.

1. CDs form a fractal scale hierarchy. Zero energy states possess a wave function in moduli degrees of freedom characterizing sizes of CDs as well telling what Lorentz boost leaving boundary invariant are allowed for them. Boosts form by number theoretic constraints a discrete subgroup of Lorentz group defining analogs of lattices generated by boosts instead of translations.
   
   
2. The arrow of subjective time maps to that of geometric time somehow. The origin of arrow comes from the fact that state function reductions can occur to either boundary of given CD and reduction creates time-asymmetric state since second boundary of CD is in a quantum superposition of different sizes and there is a superposition of many-particle states with different particles numbers and quantum number distributions.
   
   
3. Subjective existence corresponds to a sequence of moments of consciousness: state function reductions at opposite boundaries of CDs. State function reduction reduction localizes either boundary but the second boundary is in a quantum superposition of several locations and size scales for CD. This predicts that the arrow of time is not constant. In fact, there is considerable evidence for the variation of the arrow of time in living systems and Fantappie introduced long time ago the notion of syntropy to describe his view about the situation.
   
   
4. The first very naive proposal was that state function reductions occur alternately to the two boundaries of CD. This assumption would be indeed natural if one considered single fixed CD rather than superposition CDs with different size and state function reduction localizing their either boundary: restriction to single CD was what I indeed did first.
   
   
5. This assumption leads to the question about why do we do not observe this alternation of the arrow of time all the time in our personal experience. Some people actually claim to have actually experienced a temporary change of the arrow of time: I belong to them and I can tell that the experience is frightening. But why do we experience the arrow of time as stable in the standard state of consciousness?
   

One possible way to solve the problem - perhaps the simplest one - is that state function reduction to the same boundary of CD can occur many times repeatedly. This solution is so absolutely trivial that I could perhaps use this triviality to defend myself for not realizing it immediately!

I made this totally trivial observation only after I had realized that also in this process the wave function in the moduli space of CDs change in these reductions. Zeno effect in ordinary measurement theory relies on the possibility of repeated state function reductions. In the ordinary quantum measurement theory repeated state function reductions do not affect the state in this kind of sequence but in ZEO the wave function in the moduli space labelling different CDs with the same boundary could change in each quantum jump. It would be natural that this sequence of quantum jumps give rise to the experience about flow of time? This option would allow the size scale of CD associated with human consciousness be rather short, say .1 seconds. It would allow to understand why we do not observe continual change of arrow of time.

Maybe living systems are working hardly to keep the personal arrow of time changed - living creatures try to prevent kettle from boiling by staring at it intensely. Maybe it would be extremely difficult to live against the collective arrow of time.

For details and background see the chapter About nature of time of "TGD Inspired Theory of Consciousness".

Recent View about Kähler Geometry and Spin Structure of "World of Classical Worlds"

The construction of Kähler geometry of WCW ("world of classical worlds") is fundamental to TGD program. I ended up with the idea about physics as WCW geometry around 1985 and made a breakthrough around 1990, when I realized that Kähler function for WCW could correspond to Kähler action for its preferred extremals defining the analogs of Bohr orbits so that classical theory with Bohr rules would become an exact part of quantum theory and path integral would be replaced with genuine integral over WCW. The motivating construction was that for loop spaces leading to a unique Kähler geometry. The geometry for the space of 3-D objects is even more complex than that for loops and the vision still is that the geometry of WCW is unique from the mere existence of Riemann connection.

The basic idea is that WCW is union of symmetric spaces G/H labelled by zero modes which do not contribute to the WCW metric. There have been many open questions and it seems the details of the earlier approach must be modified at the level of detailed identifications and interpretations. What is satisfying that the overall coherence of the picture has increased dramatically and connections with string model and applications of TGD as WCW geometry to particle physics are now very concrete.

1. A longstanding question has been whether one could assign Equivalence Principle (EP) with the coset representation formed by the super-Virasoro representation assigned to G and H in such a manner that the four- momenta associated with the representations and identified as inertial and gravitational four-momenta would be identical. This does not seem to be the case. The recent view will be that EP reduces to the view that the classical four- momentum associated with Kähler action is equivalent with that assignable to modified Dirac action supersymmetrically related to Kähler action: quantum classical correspondence (QCC) would be in question. Also strong form of general coordinate invariance implying strong form of holography in turn implying that the super-symplectic representations assignable to space-like and light-like 3-surfaces are equivalent could imply EP with gravitational and inertial four-momenta assigned to these two representations.
   
   
2. The detailed identification of groups G and H and corresponding algebras has been a longstanding problem. Symplectic algebra associated with δM⁴_+/-× CP₂ (δM⁴_+/- is light-cone boundary - or more precisely, with the boundary of causal diamond (CD) defined as Cartesian product of CP₂ with intersection of future and past direct light cones of M⁴ has Kac-Moody type structure with light-like radial coordinate replacing complex coordinate z. Virasoro algebra would correspond to radial diffeomorphisms.
   

   I have also introduced Kac-Moody algebra assigned to the isometries and localized with respect to internal coordinates of the light-like 3-surfaces at which the signature of the induced metric changes from Minkowskian to Euclidian and which serve as natural correlates for elementary particles (in very general sense!). This kind of localization by force could be however argued to be rather ad hoc as opposed to the inherent localization of the symplectic algebra containing the symplectic algebra of isometries as sub-algebra. It turns out that one obtains direct sum of representations of symplectic algebra and Kac-Moody algebra of isometries naturally as required by the success of p-adic mass calculations.
   
   

3. The dynamics of Kähler action is not visible in the earlier construction. The construction also expressed WCW Hamiltonians as 2-D integrals over partonic 2-surfaces. Although strong form of general coordinate invariance (GCI) implies strong form of holography meaning that partonic 2-surfaces and their 4-D tangent space data should code for quantum physics, this kind of outcome seems too strong. The progress in the understanding of the solutions of modified Dirac equation led however to the conclusion that spinor modes other than right-handed neutrino are localized at string world sheets with strings connecting different partonic 2-surfaces.
   

   This leads to a modification of earlier construction in which WCW super-Hamiltonians were essentially 2-D flux integrals. Now they are 2-D flux integrals with super-Hamiltonian replaced Noether super charged for the deformations in G and obtained by integrating over string at each point of partonic 2-surface. Each spinor mode gives rise to super current and the modes of right-handed neutrino and other fermions differ in an essential manner. Right-handed neutrino would correspond to symplectic algebra and other modes to the Kac-Moody algebra and one obtains the crucial 5 tensor factors of Super Virasoro required by p-adic mass calculations.
   

   The matrix elements of WCW metric between Killing vectors are expressible as anticommutators of super-Hamiltonians identifiable as contractions of WCW gamma matrices with these vectors and give Poisson brackets of corresponding Hamiltonians. The anti-commutation relates of induced spinor fields are dictated by this condition. Everything is 3-dimensional although one expects that symplectic transformations localized within interior of X³ act as gauge symmetries so that in this sense effective 2-dimensionality is achieved. The components of WCW metric are labelled by standard model quantum numbers so that the connection with physics is extremely intimate.
   
   

4. An open question in the earlier visions was whether finite measurement resolution is realized as discretization at the level of fundamental dynamics. This would mean that only certain string world sheets from the slicing by string world sheets and partonic 2-surfaces are possible. The requirement that anti-commutations are consistent suggests that string world sheets correspond to surfaces for which Kähler magnetic field is constant along string in well defined sense (J_μνε^μνg^1/2 remains constant along string). It however turns that by a suitable choice of coordinates of 3-surface one can guarantee that this quantity is constant so that no additional constraint results.

See the new chapter Recent View about Kähler Geometry and Spin Structure of "World of Classical Worlds of "Quantum TGD as Infinite-dimensional Spinor Geometry" or the article with the same title.

Magnetic hysteresis, super string sociology, catastrophe theory, Higgs mechanism, and ferromagnets in many-sheeted space-time

Ferromagnets are simple but fascinating systems and spontaneous magnetization represents a situation analogous to Higgs mechanism. In the following I will discuss the hysteresis phenomenon for ferromagnets on basis of thermodynamical description: hysteresis model applies also to phenomenon of mass hysteria in sociology. This all is standard but the understanding of ferromagnets in many-sheeted space-time involves some new elements: in particular the possibility that the magnetic fluxes of permanent magnets consist of monopole fluxes.

Standard definitions

Let us begin with basic definitions. The magnetic field H is defined in terms of magnetization M and magnetic induction B (often called just magnetic field) via he formula H= B/μ₀-M quite generally. For historical reasons different units are used for B and H but in vacuum they are essentially one and the same thing. In the case of magnets the situation changes.

Paramagnets and diamagnets are linear: M= χ_m H in good approximation. The sign of χ_m determines whether the behavior is diamagnetic (induced magnetization reduces the external field inside magnet) or whether its paramagnetic (induce magnetization increases the external field inside magnet). For the ferromagnets the behavior
is non-linear and M is non-vanishing for vanishing H.

The behavior at the boundary between magnet and external world is important for understanding magnets. If the exterior of the magnet is vacuum, one has H=μ₀M and the fields H and B are one and the same thing. Inside magnet magnetization implies that the two fields are not same. By the absence of magnetic charges the normal component of magnetic field is continuous. Therefore the normal component satisfies B_n(in)= μ₀H_in+M= B_n(out)=μ₀H_out.

Hysteresis for ferromagnets

For ferromagnets the relationship between H M is non-linear and many-valued, and one has hysteris so that the effects of increasing and decreasing the magnetic field are different in saturation. The hysteresis cycle characterizes this behavior quite nicely.

1. The plot can be understood easily when one realizes that magnetization M corresponds microscopically to the density of magnetic dipoles typically realized as electrons spins. The field H corresponds essentially to the difference between external magnetic field and that caused by magnetization. As one starts from H=0 to increase the magnitude of H, M gradually increases as for paramagnets but above some critical value M starts to increase and reaches the saturation value. The interpretation is that electron spins change they direction in phase transitions producing magnetized regions. This behavior can be understood only quantum mechanically and can be seen as an outcome of exchange interaction for spins relating to fermion statistics. Essentially collective behavior is in question: when the number of spins in direction of external field is large enough all other spins "follow the fashion".
   
   
2. One can also gradually decrease the value of H in saturation. Now M decreases smoothly as some spins here and there change their direction but no phase transition like reduction of M occurs since "follow the fashion" behavior is possible only when the number of reversed spins is high enough. Therefore one proceeds along the lower part of the hysteresis cycle: inside the cycle the behavior is reversible for small enough changes of H and irreversibily occurs only near the ends.
   
   
3. At some point one has a situation in which the added external field vanishines: H=0. There is a net magnetization and the normal component of external field is essentially this magnetization at the surface of the magnet. This field is known as remanescence. This corresponds to a magnetization without any added external field H.
   
   
4. As H is reduced further so that its sign changes, eventually "follow the fashion" behavior sets in and most spins change their direction to opposite and the direction of magnetization changes. The magnitude of the magnetic field at which this occurs is known as coercivity. Coercivity and remanescence characterize the magnetic in the first approximation.
   
   
5. The mathematical description of Higgs mechanism is very similar to the idealized description of spontaneous magnetization. In spontaneous magnetization one has however large number of magnetized regions which means that the direction of magnetization does not change instantaneously since the analog of overheating and overcooling are possible for all these regions.
   

Hysteresis (and hysteria) in sociology

The hysteresis model applies also in sociology. Superstring fashion represents an excellent example about hysteresis phenomenon in the sociology of science. There are two kinds of people involved: those who really understand (H) and those who just follow the fashion (M). Some fraction p of those who understand, believes in superstrings. The remaining fraction 1-p does not. H dould correspond mathematically to the difference (p-p₀)/p₀, where p₀ is a critical propability analogous to remanesence in ferromagnetism. As p increases a sudden phase transition occurs at critical fraction satisfying p-p₀=Δ p_cr, and everyone suddenly believes that super strings are not only a promising theory but even the only possible one. Δ p_cr is analogous to coercivity.

The return along upper part of hysteris curve means that the per cent of the believers who really understand starts to decrease as people who have not suffered the fate of becoming a string guru are learning about the deep problems of the theory and start to worry about them publicly. Eventually the number of the non-gurus who understand and do not believe becomes so high that the followers of fashion make the decision to not believe anymore on superstrings, and everyone suddenly "knows" that super strings are pseudoscience. Except certain billionaire who decides to make super string theorists millionaires;-) . This billionaire does not follow the fashion, which is nice. He does not even follow those who understand, which is not so nice. To my humble opinion, it would be more rational to build an institute supporting the study of alternatives for super string models than drowning theoreticians to money which is not even their primary interest.

Hysteresis in magnets cannot be understood microscopically without introducing quantum effects. One can of course ask, whether also hysteresis involves quantum mechanics but now in macroscopic length and time scales applying in sociology.

Ferromagnetism, catastrophe theory, and Higgs mechanism

The model for magnetism can be also understood in terms of catastrophe theory using cusp catastrophe. This applies to Higgs mechanism (not the manner in which particle massivation is described in TGD) too. Catastrophe theory brings in also the temperature as external parameter besides H. This is the simplest interesting catastrophe and involves two parameters/control variables: (temperature T and added external field H) plus one behavior variable (magnetization M). Above certain critical temperature (Curie temperature) there is no magnetization. Below Curie temperature there is a magnetization in certain range for field H. As the magnitude of H is increased, one reaches at critical magnitude of H (coercivity) a situation when the only possible manner to continue is to jump from the boundary of the cusp to the sheet below or above it. A sudden phase transition like change changing the sign of magnetization takes place.

Above we have assumed that the external contribution H to the field is in the same direction as magnetization. What happens if this is not the case? And what happens when the external field H changes the direction of magnetization: is it enough apply the basic criterion to the vector sum H= B/μ₀-M. Presumably this is the case. If it were not, it would be mentioned in Wikipedia!

Catastrophe theory provides a rough description and does not say anything about the microscopic mechanism of hysteresis. In the case of Higgs mechanism one can expect the same to be true: there should exist a microscopic thermodynamical description replacing Higgs mechanism, which is only capable of reproducing the particle masses but not predicting them and suffers from un-naturality. In TGD framework p-adic thermodynamics provides this microscopic thermodynamical description.

Since QFT is in rough sense square root of thermodynamics, one can argue that also p-adic thermodynamics must have also square root and characterizes single particle states rather than ensembles. This is the case. In zero energy one can say that quantum theory is square root of thermodynamics. U-matrix characterizes the physics. I call the rows M_i of unitary U-matrix M-matrices, which are products of mutually orthogonal hermitian square roots P_i^1/2 of density matrices P_i with a unitary S-matrix S. One has M_iM_j^†= P_iδ_ij and Tr(M_iM_j^†)=δ_ij. U defines negentropic entanglement in time direction. In p-adic thermodynamics the Hermitian matrix P_i^1/2 corresponds to a diagonal matrix defined by the square roots of eigenvalues of mass squared operator expressible as vibrational part of scaling generator L₀ of conformal algebra. The unitary matrix S is the counterpart of the ordinary S-matrix.

Permanent magnet in many-sheeted space-time

How could one understand permanent magnet in the framework of many-sheeted space-time?

1. It seems that one must consider two space-time sheets. That of magnet at which magnetization M alone defines the magnetic field B₁=μ₀M and that of external world at which external added field defines the magnetic field B₂=μ₀ H_in. Test particle has topological sum contacts on both space-time sheets and effectively experiences the sum B₁+B₂: the fields do not however sum at same space-time sheet - only their effects on spins are additive so that '+' is replaced with a set theoretic union inTGD. This applies quite generally and saves TGD from the catastrophe caused otherwise by the fact that linear superposition of fields is not possible at given space-time sheet even approximately.
   
   
2. The spins inside the space-time sheet of magnet also experience B₂ and when B₂ is above critical value (coercivity), a phase transition changing the direction of spins occurs and magnetization is generated. It seems that TGD does not add much to the model of how spontaneous magnetization takes place and what is behind the hysteresis.
   
   
3. The notion of magnetic field in TGD differs from that in Maxwell's theory. In TGD Universe it is quite possible that the flux tubes have closed cross section (sphere) instead of disk and that they carry a monopole flux. One would have magnetic flux tube without a circulating current at its boundary generating the magnetic field inside it as in the case of electromagnets, which are typically flux tubes with current carrying helical wire. Could the static magnetic field of permanent magnet consist of monopole flux tubes with a closed rather than disk-like cross section requiring no circular currents at their boundaries?
   

   If this kind of flux tubes are possible, one could understand why cosmos is populated by magnetic fields in all scales. In Maxwell's theory this is not possible since coherent currents defining flux tubes are not possible in primordial cosmology. In superconductors the flux quanta associated with this monopole like fields would become visible as they penerated the super-conductor.
   

Negentropic entanglement, NMP, braiding and TQC

Negentropic entanglement for which number theoretic entropy characterized by p-adic prime is negative so that entanglement carries information, is in key role in TGD inspired theory of consciousness and quantum biology.

1. The key feature of negentropic entanglement is that density matrix is proportional to unit matrix so that the assumption that state function reduction corresponds to the measurement of density matrix does not imply state function reduction to one-dimensional sub-space. This special kind of degenerate density matrix emerges naturally for the hierarchy h_eff=nh interpreted in terms of a hierarchy of dark matter phases. I have already earlier considered explicit realizations of negentropic entanglement assuming that the entanglement matrix is invariant under the group of unitary or orthogonal transformations (also subgroups of unitary group can be considered - say symplectic group). One can however consider much more general options and this leads to a connection with topological quantum computation (TQC).
   

2. Entanglement matrices E equal to n^-1/2 factor times unitary matrix U (as a special case to orthogonal matrix O) defines a density matrix given by ρ=UU^†/n= Id_n/n, which is group invariant. One has negentropic entanglement (NE) respected by state function reduction if Negentropy Maximization Principle (NMP) is assumed. This would give huge number of negentropically entangled states providing a representation for some unitary group or its subgroup (such as symplectic group). In principle any unitary representation of any Lie group would allow representation in terms of NE.
   

3. In physics as generalized number theory vision, a natural condition is that the matrix elements of E belong to the algebraic extension of p-adic numbers used so that discreted algebraic subgroups of unitary or orthogonal group are selected. This realizes evolutionary hierarchy as a hierarchy of p-adic number fields and their algebraic extensions, and one can imagine that evolution of cognition proceeds by the generation of negentropically entangled systems with increasing algebraic dimensions and increasing dimension reflecting itself as an increase of the largest prime power dividing n and defining the p-adic prime in question.
   

4. One fascinating implication is the ability of TGD Universe to emulate itself like Turing machine: unitary S-matrix codes for scattering amplitudes and therefore for physics and negentropically entangled subsystem could represent sub-matrix for S-matrix as rules representing "the laws of physics" in the approximation that the world corresponds to n-dimension Hilbert space. Also the limit n → ∞ makes sense, especially so in the p-adic context were real infinity can correspond to finite number in the sense of p-adic norm. Here also dimensions n given as products of powers of infinite primes can be formally considered.
   

One can consider various restrictions on E.

1. In 2-particle case the stronger condition that E is group invariant implies that unitary matrix is identity matrix apart from an overall phase factor: U= exp(iφ)Id. In orthogonal case the phase factor is +/- 1. For n-particle NE one can consider group invariant states by using n-dimensional permutation tensor ε_i1,...in.
   
   
2. One can give up the group invariance of E and consider only the weaker condition that permutation is represented as transposition of entanglement matrix: C_ij→ C_ij. Symmetry/antisymmetry under particle exchange would correspond to C_ji=ε C_ij, ε=+/- 1. This would give in orthogonal case OO^T= O²=Id and UU^*= Id in the unitary case.
   

   In the unitary case particle exchange could be also identified as hermitian conjugation C_ij→ C_ji^* and one would have also now U²=Id. Euclidian gamma matrices γ_i define unitary and hermitian generators of Clifford algebra having dimension 2^2m for n=2m and n=2m+1. It is relatively easy to verify that the squares of completely anti-symmetrized products of k gamma matrices representing exterior algebra normalized by factor 1/k!^1/2 are equal to unit matrix. For k=n the antisymmetrized product gives essentially permutation symbol times the product ∏_k γ_k. In this manner one can construct entanglement matrices representing negentropic bi-partite entanglement.
   
   

3. The possibility of taking tensor products ε_ij..k...nγ_i⊗ γ_j..⊗ γ_k of k gamma matrices means that one can has also co-product of gamma matrices. What is interesting is that quantum groups important in topological quantum computation as well as the Yangian algebra associated with twistor Grassmann approach to scattering amplitudes possess co-algebra structure. TGD leads also to the proposal that this structure plays a central role in the construction of scattering amplitudes. Physically the co-product is time reversal of product representing fusion of particles.
   
   
4. One can go even further. In 2-dimensional QFTs braid statistics replaces ordinary statistics. The natural question is what braid statistics could correspond to at the level of NE. Braiding matrix is unitary so that it defines NE. Braiding as a flow replaces the particle exchange and lifts permutation group to braid group serving as its infinite covering. The allowed unitary matrices representing braiding in tensor product are constructed using braiding matrix R representing the exchange for two braid strands? The well-known Yang-Baxter equation for R defined in tensor product as an invertible element (see this) expresses the associativity of braiding operation. Concretely it states that the two braidings leading from 123 to 321 produce the same result. Entanglement matrices constructed R as basic operation would correspond to unitary matrices providing a representation for braids and each braid would give rise to one particular NE.
   

   This would give a direct connection with TQC for which the entanglement matrix defines a density matrix proportional to n× n unit matrix: R defines the basic gate (see this). Braids would provide a concrete representation for NE giving rise to "Akashic records". I have indeed proposed the interpretation of braidings as fundamental memory representations much before the vision about Akashic records. This kind of entanglement matrix need not represent only time-like entanglement but can be also associated also with space-like entanglement. The connection with braiding matrices supports the view that magnetic flux tubes are carriers of negentropically entangled matter and also suggests that this kind of entanglement between - say - DNA and nuclear or cell membrane gives rise to TQC.
   

Some comments concerning the covering space degrees of freedom associated with h_eff=nh viz. ordinary degrees of freedom are in order.

1. Negentropic entanglement with n entangled states would correspond naturally to h_eff=nh and is assigned with "many-particle" states, which can be localized to the sheets of covering but one cannot exclude similar entanglement in other degrees of freedom. Group invariance leaves only group singlets and states which are not singlets are allowed only in special cases. For instance for SU(2) the state kenovert |j,m >kenorangle= | 1,0 > represented as 2-particle state of 2 spin 1/2 particles is negentropically entangled whereas the states | j,m >= |1,+/- 1 > are pure.
   
   
2. Negentropic entanglement associated with h_eff=nh could factorize as tensor product from other degrees of freedom. Negentropic entanglement would be localised to the covering space degrees of freedom but there would be entropic entanglement in the ordinary degrees of freedom - say spin. The large value of h_eff would however scale up the quantum coherence time and length also in the ordinary degrees of freedom. For entanglement matrix this would correspond to a direct sum proportional to unitary matrices so that also density matrix would be a direct sum of matrices p_n E_n= p_n Id_n/n , ∑ p_n=1 correspond ing to various values of "other quantum numbers", and state function reduction could take place to any subspace in the decomposition. Also more general entanglement matrices for which the dimensions of direct summands vary, are possible.
   
   
   
3. One can argue that NMP does not allow halting of quantum computation. The counter argument would be that the halting is not needed if it is indeed possible to deduce the structure of negentropically entangled state by an interaction free quantum measurement replacing the state function reduction with "externalised" state function reduction. One could speak of interaction free TQC. This TQC would be reading of "Akashic records". NE should be able to induce a conscious experience about the outcome of TQC which in the ordinary framework is represented by the reduction probabilities for various possible outcomes.
   

   One could also counter argue that NMP allows the transfer of NE from the system so that TQC halts. NMP allows this if some another system receives at least the negentropy contained by NE. The interpretation would be as the increase of information obtained by a conscious observer about the outcome of halted quantum computation. It am not able to imagine how this could happen at the level of details.
   
   

For details and background see the section "Updates since 2012" of chapter "Negentropy Maximization Principle" and and the article ""Negentropic entanglement, NMP, braiding and topological quantum computation"

Naturalness, fine-tuning, second law, and love

In previous posting we have had a very nice discussion with Hamed about various aspects of "world of classical worlds" realized as a space of space-time surfaces (very rough statement). I thought that the following comment - intended to be a further response relating to the uniqueness of "world of classical worlds" (WCW) - deserves the status of a separate blog posting.

So called naturalness is the physicist's manner to say "of course" as Bee very neatly expresses the gist of this notion. After the results from LHC we know that standard model is not natural and SUSY in standard form cannot help. What one should think about theories whose predictions change dramatically when some parameter is varied only slightly? Certainly these kind of theories are not very useful. Many people are however ready to accept that theory could be of this kind. I see this as giving up. If theory in unstable under small allowed variations of its parameters it is definitely wrong and this is extremely valuable guideline.

The essential attribute appearing in this claim is " allowed". In some situations one cannot allow any variations. Indeed, classical number fields, groups, etc. are extremely rigid mathematical structures, which do not allow any variations of their structure. In the same manner, in TGD WCW is really something God given (or mathematics given). There are no parameters to be varied and fine-tune.

This is of course the great idea of TGD: physics follows from the mere existence of an infinite-dimensional geometry of WCW - more precisely from the existence of Riemann connection, which forces WCW to be a union of infinite-D symmetric spaces with maximal isometries having interpretation as conformal symmetries or analogs of them (symplectic symmetries of boundary of causal diamond). Among other things, this condition fixes imbedding space uniquely to M⁴ × CP₂ (also the condition that the twistor spaces associated with the Cartesian factors exist and are Kähler manifolds forces the same conclusion) allows to avoid the landscape difficulty of super string models, which is actually much more general problem.

Anthropic principle is second aspect related to fine-tuning. Fine-tuning of certain parameters essential for life seems to be present in physics. The idea behind the anthropic principle is that our own existence allows to deduce the values of fine-tuned parameters. This hypothesis involves however many implicit assumptions. For instance, one assumes that life as we know it recently is the only possible form of life and that there cannot be any other kinds of life forms realized for the other values of the key parameters.

The TDG based explanation for the fine-tuning relies on Negentropy Maximization Principle. NMP implies evolution accompanied by increasing negentropy resources realized in terms of negentropic entanglement. It also means finite-tuning: slowly varying parameters approach quantum jump by quantum jump values, which make possible maximal negentropy resources. The evolutionary self-organization process implied by NMP would lead to highly unique final states. Since NMP is more or less the mirror image of second law, the self-organization process could be mathematically analogous to the approach to thermodynamical equilibrium or thermodynamical non-equilibrium state (in presence of energy feed) polishing out all unessential details and leaving only the gem.

It is often said that life generates huge amounts of entropy. Looks at first just the opposite for what NMP predicts! Of course, the entropy in question is ensemble entropy and has as such nothing do with number theoretic entropy characterizing negentropic entanglement having interpretation as a quantum correlate for rule realized as superposition of its instances. NMP implies second law for ordinary entanglement and this might be enough (NMP however also predicts varying arrow of thermodynamical time, something highly non-trivial although considerable support for this exists in living matter!).

I have also considered the pessimistic conjecture that second law holds true in the following sense. If negentropy associated with negentropic entanglement is generated, it is somehow accompanied by a generation of entropy at least compensating for it. I do not believe this conjecture: the two notions do not simply apply in the same context and I do not have any mechanism for how the entropy would be generated. One can however consider the following natural correlation. Systems with a large number of degenerate states (same energy) in thermodynamical equilibrium have large entropy. The same highly degenerate systems can however entangle negentropically and this gives rise to a large entanglement negentropy. Entropy can transforms to negentropy! Love - for which I have suggested negentropic entanglement to serve as a quantum correlate - makes jewels from dirt!

About the notion of four-momentum in TGD framework

The starting point of TGD was the energy problem of General Relativity (see this). The solution of the problem was proposed in terms of sub-manifold gravity and based on the lifting of the isometries of space-time surface to those of M⁴× CP₂ in which space-times are realized as 4-surfaces so that Poincare transformations act on space-time surface as an 4-D analog of rigid body rather than moving points at space-time surface. It however turned out that the situation is not at all so simple.

There are several conceptual hurdles and I have considered several solutions for them. The basic source of problems has been Equivalence Principle (EP): what does EP mean in TGD framework (see this and this)? A related problem has been the interpretation of gravitational and inertial masses, or more generally the corresponding 4-momenta. In General Relativity based cosmology gravitational mass is not conserved and this seems to be in conflict with the conservation of Noether charges. The resolution is in terms of zero energy ontology (ZEO), which however forces to modify slightly the original view about the action of Poincare transformations.

A further problem has been quantum classical correspondence (QCC): are quantal four-momenta associated with super conformal representations and classical four-momenta associated as Noether charges with Kähler action for preferred extremals identical? Could inertial-gravitational duality - that is EP - be actually equivalent with QCC? Or are EP and QCC independent dualities. A powerful experimental input comes p-adic mass calculations (see this) giving excellent predictions provided the number of tensor factors of super-Virasoro representations is five, and this input together with Occam's razor strongly favors QCC=EP identification.

Twistor Grassmannian approach has meant a technical revolution in quantum field theory (for attempts to understand and generalize the approach in TGD framework (see this and this). This approach seems to be extremely well suited to TGD and I have considered a generalization of this approach from N=4 SUSY to TGD framework by replacing point like particles with string world sheets in TGD sense and super-conformal algebra with its TGD version: the fundamental objects are now massless fermions which can be regarded as on mass shell particles also in internal lines (but with unphysical helicity). The approach solves old problems related to the realization of stringy amplitudes in TGD framework, and avoids some problems of twistorial QFT (IR divergences and the problems due to non-planar diagrams). The Yangian variant of 4-D conformal symmetry is crucial for the approach in N=4 SUSY, and implies the recently introduced notion of amplituhedon (see this). A Yangian generalization of various super-conformal algebras seems more or less a "must" in TGD framework. As a consequence, four-momentum is expected to have characteristic multilocal contributions identifiable as multipart on contributions now and possibly relevant for the understanding of bound states such as hadrons.

1. Scale dependent notion of four-momentum in zero energy ontology

Quite generally, General Relativity does not allow to identify four-momentum as Noether charges but in GRT based cosmology one can speak of non-conserved mass cosmo, which seems to be in conflict with the conservation of four-momentum in TGD framework. The solution of the problem comes in terms of zero energy ontology (ZEO) (see this and this), which transforms four-momentum to a scale dependent notion: to each causal diamond (CD) one can assign four-momentum assigned with say positive energy part of the quantum state defined as a quantum superposition of 4-surfaces inside CD.

ZEO is necessary also for the fusion of real and various p-adic physics to single coherent whole. ZEO also allows maximal "free will" in quantum jump since every zero energy state can be created from vacuum and at the same time allows consistency with the conservation laws. ZEO has rather dramatic implications: in particular the arrow of thermodynamical time is predicted to vary so that second law must be generalized. This has especially important implications in living matter, where this kind of variation is observed.

More precisely, this superposition corresponds to a spinor field in the "world of classical worlds" (WCW) (see this): its components - WCW spinors - correspond to elements of fermionic Fock basis for a given 4-surface - or by holography implied by general coordinate invariance (GCI) - for 3-surface having components at both ends of CD. Strong form of GGI implies strong form of holography (SH) so that partonic 2-surfaces at the ends of space-time surface plus their 4-D tangent space data are enough to fix the quantum state. The classical dynamics in the interior is necessary for the translation of the outcomes of quantum measurements to the language of physics based on classical fields, which in turn is reduced to sub-manifold geometry in the extension of the geometrization program of physics provided by TGD.

Holography is very much reminiscent of QCC suggesting trinity: GCI-holography-QCC. Strong form of holography has strongly stringy flavor: string world sheets connecting the wormhole throats appearing as basic building bricks of particles emerge from the dynamics of induced spinor fields if one requires that the fermionic mode carries well-defined electromagnetic charge (see this).

2. Are the classical and quantal four-momenta identical?

One key question concerns the classical and quantum counterparts of four-momentum. In TGD framework classical theory is an exact part of quantum theory. Classical four-momentum corresponds to Noether charge for preferred extremals of Kähler action. Quantal four-momentum in turn is assigned with the quantum superposition of space-time sheets assigned with CD - actually WCW spinor field analogous to ordinary spinor field carrying fermionic degrees of freedom as analogs of spin. Quantal four-momentum emerges just as it does in super string models - that is as a parameter associated with the representations of super-conformal algebras. The precise action of translations in the representation remains poorly specified. Note that quantal four-momentum does not emerge as Noether charge: at at least it is not at all obvious that this could be the case.

Are these classical and quantal four-momenta identical as QCC would suggest? If so, the Noether four-momentum should be same for all space-time surfaces in the superposition. QCC suggests that also the classical correlation functions for various general coordinate invariant local quantities are same as corresponding quantal correlation functions and thus same for all 4-surfaces in quantum superposition - this at least in the measurement resolution used. This would be an extremely powerful constraint on the quantum states and to a high extend could determined the U-, M-, and S-matrices.

QCC seems to be more or less equivalent with SH stating that in some respects the descriptions based on classical physics defined by Kähler action in the interior of space-time surface and the quantal description in terms of quantum states assignable to the intersections of space-like 3-surfaces at the boundaries of CD and light-like 3-surfaces at which the signature of induced metric changes. SH means effective 2-dimensionality since the four-dimensional tangent space data at partonic 2-surfaces matters. SH could be interpreted as Kac-Mody and symplectic symmetries meaning that apart from central extension they act almost like gauge symmetries in the interiors of space-like 3-surfaces at the ends of CD and in the interiors of light-like 3-surfaces representing orbits of partonic 2-surfaces. Gauge conditions are replaced with Super Virasoro conditions. The word "almost" is of course extremely important.

3. What does Equivalence Principle (EP) mean in TGD?

EP states the equivalence of gravitational and inertial masses in Newtonian theory. A possible generalization would be equivalence of gravitational and inertial four-momenta. In GRT this correspondence cannot be realized in mathematically rigorous manner since these notions are poorly defined and EP reduces to a purely local statement in terms of Einstein's equations. What about TGD? What could EP mean in TGD framework?

1. Is EP realized at both quantum and space-time level? This option requires the identification of inertial and gravitational four-momenta at both quantum and classical level. QCC would require the identification of quantal and classical counterparts of both gravitational and inertial four-momenta. This would give three independent equivalences, say P_I,class=P_I,quant, P_gr,class=P_gr,quant, P_gr,class=P_I,quant, which imply the remaining ones.
   

   Consider the condition P_gr,class=P_I,class. At classical level the condition that the standard energy momentum tensor associated with Kähler action has a vanishing divergence is guaranteed if Einstein's equations with cosmological term are satisfied. If preferred extremals satisfy this condition they are constant curvature spaces for non-vanishing cosmological constant. A more general solution ansatz involves several functions analogous to cosmological constant corresponding to the decomposition of energy momentum tensor to terms proportional to Einstein tensor and several lower-dimensional projection operators (see this). It must be emphasized that field equations are extremely non-linear and one must also consider preferred extremals (which could be identified in terms of space-time regions having so called Hamilton-Jacobi structure): hence these proposals are guesses motivated by what is known about exact solutions of field equations.
   

   Consider next P_gr,class=P_I,class. At quantum level I have proposed coset representations for the pair of super-symplectic algebras assignable to the light-like boundaries of CD and the Super Kac-Moody algebra assignable to the light-like 3-surfaces defining the orbits of partonic 2-surfaces as realization of Equivalence Principle. For coset representation the differences of super-conformal generators would annihilate the physical states so that one can argue that the corresponding four-momenta are identical. One could even say that one obtains coset representation for the "vibrational" parts of the super-conformal algebras in question. I must admit that the notion of coset representation creates uneasy feeling in my stomach. Note however that coset representations occur naturally for the subalgebras of symplectic algebra and Super Kac-Moody algebra and are naturally induced by finite measurement resolution.
   
   

2. Does EP reduce to one aspect of QCC? This would require that classical Noether four-momentum identified as inertial momentum equals to the quantal four-momentum assignable to the states of super-conformal representations and identifiable as gravitational four-momentum. There would be only one independent condition: P_class== P_I,class=P_gr,quant== P_quant.
   

   Holography realized as AdS/CFT correspondence states the equivalence of descriptions in terms of gravitation realized in terms of strings in 10-D spacetime and gauge fields at the boundary of AdS. What is disturbing is that this picture is not completely equivalent with the proposed one. In this case the super-conformal algebra would be direct sum of super-symplectic and super Kac-Moody parts.
   

Which of the options looks more plausible? The success of p-adic mass calculations (see this) have motivated theuse of them as a guideline in attempts to understand TGD. The basic outcome was that elementary particle spectrum can be understood if Super Virasoro algebra has five tensor factors. Can one decide the fate of the two approaches to EP using this number as an input?

1. For the coset option the situation is unclear. Even the definition of coset representation is problematic. If Super Kac-Moody generators vanish at partonic 2-surfaces one would have just direct sum Super-Virasoro algebras and coset representations would reduce to that for symplectic group containing only single tensor factor. If Super-Kac Moody generators do not vanish at partonic 2-surfaces. one must extend the symplectic generators by making them local with respect to partonic 2-surface in order to get a closed algebra. The imbedding of Kac-Moody to Sympl might be well-defined since isometries form subgroup of symplectic transformations. But is it possible to speak about a direct sum of Super Virasoro algebras in this case? It seems that only the inclusion electroweak part to symplectic part represented in terms of fermionic currents allows this and would bring in two tensor factors so that one would have 3 tensor factors.
   If one counts fermionic tensor factors assignable to 2-D transversal part of Kac-Moody algebra one would have 5 tensor factors. This seems however tricky.
   
   
2. For the P_class,I= P_quant,gr option the number of tensor factors is naturally five. Four tensor factors come from Super Kac-Moody and correspond to translational Kac-Moody type degrees of freedom in M⁴, to color degrees of freedom and to electroweak degrees of freedom (SU(2)× U(1)). One tensor factor comes from the symplectic degrees of freedom in Δ CD× CP₂ (note that Hamiltonians include also products of δ CD and CP₂ Hamiltonians so that one does not have direct sum!). Therefore this option seems to be slightly favored. Note that Kac-Moody generators are local isometries localized with respect to the coordinates of light-like 3-surface and vanish at partonic 2-surface so that one can speak about direct sum of algebras.
   
   

Clearly, both experimental input and Occam's razor seem to favor the option reducing Equivalence Principle to Quantum Classical Correspondence.

For this option however the GRT inspired interpretation of Equivalence Principle at space-time level remains to be understood. Is it needed at all? The condition that the energy momentum tensor of Kähler action has a vanishing divergence leads in General Relativity to Einstein equations with cosmological term. In TGD framework preferred extremals satisfying the analogs of Einstein's equations with several cosmological constant like parameters can be considered.

Should one give up this idea, which indeed might be wrong? Could the divergence of of energy momentum tensor vanish only asymptotically as was the original proposal? Or should one try to generalize the interpretation? QCC states that quantum physics has classical correlate at space-time level and implies EP. Could also quantum classical correspondence itself have a correlate at space-time level. If so, space-time surface would able to represent abstractions as statements about statements about.... as the many-sheeted structure and the vision about TGD physics as analog of Turing machine able to mimic any other Turing machine suggest.g machine suggests.

4. How translations are represented at the level of WCW?

The four-momentum components appearing in the formulas of super conformal generators correspond to infinitesimal translations. In TGD framework one must be able to identify these infinitesimal translations precisely. As a matter of fact, finite measurement resolution implies that it is probably too much to assume infinitesimal translations. Rather, finite exponentials of translation generators are involved and translations are discretized. This does not have practical signficance since for optimal resolution the discretization step is about CP₂ length scale.

Where and how do these translations act at the level of WCW? ZEO provides a possible answer to this question.

1. Discrete Lorentz transformations and time translations act in the space of CDs: inertial four-momentum

Quantum state corresponds also to wave function in moduli space of CDs. The moduli space is obtained from given CD by making all boosts for its non-fixed boundary: boosts correspond to a discrete subgroup of Lorentz group and define a lattice-like structure at the hyperboloid for which proper time distance from the second tip of CD is fixed to T_n=n× T(CP₂). The quantization of cosmic redshift for which there is evidence, could relate to this lattice generalizing ordinary 3-D lattices from Euclidian to hyperbolic space by replacing translations with boosts (velocities).

The additional degree of freedom comes from the fact that the integer n>0 obtains all positive values. One has wave functions in the moduli space defined as a pile of these lattices defined at the hyperboloid with constant value of T(CP₂): one can say that the points of this pile of lattices correspond to Lorentz boosts and scalings of CDs defining sub-WCW:s.

The interpretation in terms of group which is product of the group of shifts T_n(CP₂)→ T_n+m(CP₂) and discrete Lorentz boosts is natural. This group has same Cartesian product structure as Galilean group of Newtonian mechanics. This would give a discrete rest energy and by Lorentz boosts discrete set of four-momenta giving a contribution to the four-momentum appearing in the super-conformal representation.

What is important that each state function reduction would mean localisation of either boundary of CD (that is its tip). This localization is analogous to the localization of particle in position measurement in E³ but now discrete Lorentz boosts and discrete translations T_n-->T_n+m replace translations. Since the second end of CD is necessary del-ocalized in moduli space, one has kind of flip-flop: localization at second end implies de-localization at the second end. Could the localization of the second end (tip) of CD in moduli space correspond to our experience that momentum and position can be measured simultaneously? This apparent classicality would be an illusion made possible by ZEO.

The flip-flop character of state function reduction process implies also the alternation of the direction of the thermodynamical time: the asymmetry between the two ends of CDs would induce the quantum arrow of time. This picture also allows to understand what the experience growth of geometric time means in terms of CDs.

2. The action of translations at space-time sheets

The action of imbedding space translations on space-time surfaces possibly becoming trivial at partonic 2-surfaces or reducing to action at δ CD induces action on space-time sheet which becomes ordinary translation far enough from end end of space-time surface. The four-momentum in question is very naturally that associated with Kähler action and would therefore correspond to inertial momentum for P_I,class=P_quant,gr option. Indeed, one cannot assign quantal four-momentum to Kähler action as an operator since canonical quantization badly fails. In finite measurement infinitesimal translations are replaced with their exponentials for P_I,class=P_quant,gr option.

What looks like a problem is that ordinary translations in the general case lead out from given CD near its boundaries. In the interior one expects that the translation acts like ordinary translation. The Lie-algebra structure of Poincare algebra including sums of translation generators with positive coefficient for time translation is preserved if only timelike superpositions if generators are allowed also the commutators of time-like translation generators with boost generators give time like translations. This defines a Lie-algebraic formulation for the arrow of geometric time. The action of time translation on preferred etxremal would be ordinary translation plus continuation of the translated preferred extremal backwards in time to the boundary of CD. The transversal space-like translations could be made Kac-Moody algebra by multiplying them with functions which vanish at δ CD.

A possible interpretation would be that P_quant,gr corresponds to the momentum assignable to the moduli degrees of freedom and P_cl,I to that assignable to the time like translations. P_quant,gr=P_cl,I would code for QCC. Geometrically quantum classical correspondence would state that timelike translation shift both the interior of space-time surface and second boundary of CD to the geometric future/past while keeping the second boundary of space-time surface and CD fixed.

5. Yangian and four-momentum

Yangian symmetry implies the marvellous results of twistor Grassmannian approach to N=4 SUSY culminating in the notion of amplituhedron which promises to give a nice projective geometry interpretation for the scattering amplitudes (see this). Yangian symmetry is a multilocal generalization of ordinary symmetry based on the notion of co-product and implies that Lie algebra generates receive also multilocal contributions. I have discussed these topics from slightly different point of view (see this).

1. Yangian symmetry

The notion equivalent to that of Yangian was originally introduced by Faddeev and his group in the study of integrable systems. Yangians are Hopf algebras which can be assigned with Lie algebras as the deformations of their universal enveloping algebras. The elegant but rather cryptic looking definition is in terms of the modification of the relations for generating elements (see this). Besides ordinary product in the enveloping algebra there is co-product Δ which maps the elements of the enveloping algebra to its tensor product with itself. One can visualize product and co-product is in terms of particle reactions. Particle annihilation is analogous to annihilation of two particle so single one and co-product is analogous to the decay of particle to two. Δ allows to construct higher generators of the algebra.

Lie-algebra can mean here ordinary finite-dimensional simple Lie algebra, Kac-Moody algebra or Virasoro algebra. In the case of SUSY it means conformal algebra of M⁴- or rather its super counterpart. Witten, Nappi and Dolan have described the notion of Yangian for super-conformal algebra in very elegant and and concrete manner in the article kenoem Yangian Symmetry in D=4 superconformal Yang-Mills theory (see this). Also Yangians for gauge groups are discussed.

In the general case Yangian resembles Kac-Moody algebra with discrete index n replaced with a continuous one. Discrete index poses conditions on the Lie group and its representation (adjoint representation in the case of N=4 SUSY). One of the conditions conditions is that the tensor product R⊗ R^* for representations involved contains adjoint representation only once. This condition is non-trivial. For SU(n) these conditions are satisfied for any representation. In the case of SU(2) the basic branching rule for the tensor product of representations implies that the condition is satisfied for the product of any representations.

Yangian algebra with a discrete basis is in many respects analogous to Kac-Moody algebra. Now however the generators are labelled by non-negative integers labeling the light-like incoming and outgoing momenta of scattering amplitude whereas in in the case of Kac-Moody algebra also negative values are allowed. Note that only the generators with non-negative conformal weight appear in the construction of states of Kac-Moody and Virasoro representations so that the extension to Yangian makes sense.

The generating elements are labelled by the generators of ordinary conformal transformations acting in M⁴ and their duals acting in momentum space. These two sets of elements can be labelled by conformal weights n=0 and n=1 and and their mutual commutation relations are same as for Kac-Moody algebra. The commutators of n=1 generators with themselves are however something different for a non-vanishing deformation parameter h. Serre's relations characterize the difference and involve the deformation parameter h. Under repeated commutations the generating elements generate infinite-dimensional symmetric algebra, the Yangian. For h=0 one obtains just one half of the Virasoro algebra or Kac-Moody algebra. The generators with n>0 are n+1-local in the sense that they involve n+1-forms of local generators assignable to the ordered set of incoming particles of the scattering amplitude. This non-locality generalizes the notion of local symmetry and is claimed to be powerful enough to fix the scattering amplitudes completely.

2. How to generalize Yangian symmetry in TGD framework?

As far as concrete calculations are considered, it is not much to say. It is however possible to keep discussion at general level and still say something interesting (as I hope!). The key question is whether it could be possible to generalize the proposed Yangian symmetry and geometric picture behind it to TGD framework.

1. The first thing to notice is that the Yangian symmetry of N=4 SUSY in question is quite too limited since it allows only single representation of the gauge group and requires massless particles. One must allow all representations and massive particles so that the representation of symmetry algebra must involve states with different
   masses, in principle arbitrary spin and arbitrary internal quantum numbers. The candidates are obvious: Kac-Moody algebras (see this) and Virasoro algebras (see this) and their super counterparts. Yangians indeed exist for arbitrary super Lie algebras. In TGD framework conformal algebra of Minkowski space reduces to Poincare algebra and its extension to Kac-Moody allows to have also massive states.
   

2. The formal generalization looks surprisingly straightforward at the formal level. In zero energy ontology one replaces point like particles with partonic two-surfaces appearing at the ends of light-like orbits of wormhole throats located to the future and past light-like boundaries of causal diamond (CD× CP₂ or briefly CD). Here CD is defined as the intersection of future and past directed light-cones. The polygon with light-like momenta is naturally replaced with a polygon with more general momenta in zero energy ontology and having partonic surfaces as its vertices. Non-point-likeness forces to replace the finite-dimensional super Lie-algebra with infinite-dimensional Kac-Moody algebras and corresponding super-Virasoro algebras assignable to partonic 2-surfaces.
   

3. This description replaces disjoint holomorphic surfaces in twistor space with partonic 2-surfaces at the boundaries of CD×CP₂ so that there seems to be a close analogy with Cachazo-Svrcek-Witten picture. These surfaces are connected by either light-like orbits of partonic 2-surface or space-like 3-surfaces at the ends of CD so that one indeed obtains the analog of polygon.
   

What does this then mean concretely (if this word can be used in this kind of context)?

1. At least it means that ordinary Super Kac-Moody and Super Virasoro algebras associated with isometries of M⁴ ×CP₂ annihilating the scattering amplitudes must be extended to a co-algebras with a non-trivial deformation parameter. Kac-Moody group is thus the product of Poincare and color groups. This algebra acts as deformations of the light-like 3-surfaces representing the light-like orbits of particles which are extremals of Chern-Simon action with the constraint that weak form of electric-magnetic duality holds true. I know so little about the mathematical side that I cannot tell whether the condition that the product of the representations of Super-Kac-Moody and Super-Virasoro algebras contains adjoint representation only once, holds true in this case. In any case, it would allow all representations of finite-dimensional Lie group in vertices whereas N=4 SUSY would allow only the adjoint.
   
   
2. Besides this ordinary kind of Kac-Moody algebra there is the analog of Super-Kac-Moody algebra associated with the light-cone boundary which is metrically 3-dimensional. The finite-dimensional Lie group is in this case replaced with infinite-dimensional group of symplectomorphisms of δ M⁴_+/- made local with respect to the internal coordinates of the partonic 2-surface. A generalization of the Equivalence Principle is in
   question. This picture also justifies p-adic thermodynamics applied to either symplectic or isometry Super-Virasoro and giving thermal contribution to the vacuum conformal and thus to mass squared.
   
   
3. The construction of TGD leads also to other super-conformal algebras and the natural guess is that the Yangians of all these algebras annihilate the scattering amplitudes.
   
   
4. Obviously, already the starting point symmetries look formidable but they still act on single partonic surface only. The discrete Yangian associated with this algebra associated with the closed polygon defined by the incoming momenta and the negatives of the outgoing momenta acts in multi-local manner on scattering amplitudes. It might make sense to speak about polygons defined also by other conserved quantum numbers so that one would have generalized light-like curves in the sense that state are massless in 8-D sense.

3. Could Yangian symmetry provide a new view about conserved quantum numbers?

The Yangian algebra has some properties which suggest a new kind of description for bound states. The Cartan algebra generators of n=0 and n=1 levels of Yangian algebra commute. Since the co-product Δ maps n=0 generators to n=1 generators and these in turn to generators with high value of n, it seems that they commute also with n≥ 1 generators. This applies to four-momentum, color isospin and color hyper charge, and also to the Virasoro generator L₀ acting on Kac-Moody algebra of isometries and defining mass squared operator.

Could one identify total four momentum and Cartan algebra quantum numbers as sum of contributions from various levels? If so, the four momentum and mass squared would involve besides the local term assignable to wormhole throats also n-local contributions. The interpretation in terms of n-parton bound states would be extremely attractive. n-local contribution would involve interaction energy. For instance, string like object would correspond to n=1 level and give n=2-local contribution to the momentum. For baryonic valence quarks one would have 3-local contribution corresponding to n=2 level. The Yangian view about quantum numbers could give a rigorous formulation for the idea that massive particles are bound states of massless particles.

See the article About the Notion of Four-momentum in TGD Framework. For background see the chapter TGD and GRT of "Physics in Many-Sheeted Space-time".