Quark gluon plasma which does not behave as it should

The first interesting findings from LHC have been reported. The full article is here. In some proton-proton collisions more than hundred particles are produced suggesting a single object from which they are produced. Since the density of matter approaches to that observed in heavy ion collisions for five years ago at RHIC, a formation of quark gluon plasma and its subsequent decay is what one would expect. The observations are not however quite what QCD plasma picture would allow to expect. What is so striking is the evolution of long range correlations between particles in events containing more than 90 particles as the transverse momentum of the particles increases in the range 1-3 GeV (see the excellent description of the correlations by Lubos).

One studies correlation function for two particles as a function of two variables. The first variable is the difference Δ φ for the emission angles and second is essentially the difference for the velocities described relativistically by the difference Δ η for hyperbolic angles. As the transverse momentum p_T increases the correlation function develops structure. Around origin of Δ η axis a widening plateau develops near Δ Φ=0. Also a wide ridge with almost constant value as function of Δ η develops near Δ φ=π. What this means that particles tend to move collinearly and or in opposite directions. In the latter case their velocity differences are large since they move in opposite directions so that a long ridge develops in Δ η direction.

Ideal QCD plasma would predict no correlations between particles and therefore no structures like this. The radiation of particles would be like blackbody radiation with no correlations between photons. The description in terms of string like object proposed also by Lubos on basis of analysis of the graph showing the distributions as an explanation of correlations looks attractive. The decay of a string like structure producing particles at its both ends moving nearly parallel to the string to opposite directions could be in question.

Since the densities of particles approach those at RHIC, I would bet that the explanation (whatever it is!) of the hydrodynamical behavior observed at RHIC for some years ago should apply also now. When RHIC was in blogs, I constructed a primitive poor man's model for RHIC events and found that I had mentioned stringy structures - among many other things that I would not perhaps mention anymore;-). The introduction of string like objects was natural since in TGD framework even ordinary nuclei are string like objects with nucleons connected by color flux tubes (see this): this predicts a lot of new nuclear physics for which there is some evidence. The basic idea was that in the high density hadronic color flux tubes associated with the colliding nucleon connect to form long highly entangled hadronic strings containing quark gluon plasma. The decay of these structure would explain the strange correlations.

Note: TGD is not string theory although I talk a lot about strings like objects: these objects are three-dimensional and they are an essential element of almost all physics predicted by TGD. Even elementary particles should look string like objects in electro-weak length scales (Kähler magnetic flux tubes with magnetic charges at their ends).

Let us list the main assumptions of the model for the RHIC events and those observed now. Consider first the "macroscopic description".

1. A critical system associated with confinement-deconfinement transition of the quark-gluon plasma formed in the collision and inhibiting long range correlations would be in question.

2. The proposed hydrodynamic space-time description was in terms of a scaled variant of what I call critical cosmology defining a universal space-time correlate for criticality: the specific property of this cosmology is that the mass contained by comoving volume approaches to zero at the the initial moment so that Big Bang begins as a silent whisper and is not so scaring;-). Criticality means flat 3-space instead of Lobatchevski space and means breaking of Lorentz invariance to SO(4). Breaking of Lorentz invariance was indeed observed for particle distributions but now I am not so sure whether it has much to do with this.

The microscopic level the description would be like follows.

1. A highly entangled long hadronic string like object (color-magnetic flux tube) would be formed at high density of nucleons via the fusion of ordinary hadronic color-magnetic flux tubes to much longer one and containing quark gluon plasma. In QCD world plasma would not be at flux tube.

2. This entangled string like object would straighten and split to hadrons in the subsequent "cosmological evolution" and yield large numbers of almost collinear particles. The initial situation should be apart from scaling similar as in cosmology where a highly entangled soup of cosmic strings (magnetic flux tubes) precedes the space-time as we understand it. Maybe ordinary cosmology could provide analogy as galaxies arranged to form linear structures?

3. This structure would have also black hole like aspects but in totally different sense as the 10-D hadronic black-hole proposed by Nastase to describe the findings. Note that M-theorists identify black holes as highly entangled strings: in TGD 1-D strings are replaced by 3-D string like objects.

A comment about exact Yangian symmetry realized in terms of bound states of partons

The formulation for the earlier view about how bound states allow to realize super Yangian (and super-conformal symmetry) was not yet quite correct. The correct formulation is extremely simple.

1. The condition is that the total super momentum for the bound state must have a vanishing super part. Note that for single parton states it is satisfied only by putting super-momentum component to zero by hand, which looks somewhat strange. In TGD framework all observed particles are bound states of partons assignable to wormhole throats.
2. Since super-momenta are quadratic in super-spinors λ this gives a linear constraint on super-parameters of form X= ∑ λ_iη_i=0 , where the sum is over the partons forming the bound state. A condition of same form follows from the condition that external super-momenta sum up to zero for scattering amplitudes: in this case one could say that one has zero energy bound state with S-matrix or its generalization to M-matrix giving the entanglement coefficients.
3. The action of the super-components on the total (super-) momentum p= ∑λ_iλ*_i (here * refers to tilde) appearing as argument of the scattering amplitude gives something proportional to these quantities and therefore vanishes. Hence bound state property and non-locality is essential for having super-conformal invariance and also IR cutoff since the mass of the bound state brings in the mass/length scale.

For details see the pdf article What could be the generalization of Yangian symmetry of N=4 SYM in TGD framework? or the new chapter Yangian Symmetry, Twistors, and TGD.

M4×CP2 from pairs of twistors and twistorial formulation of TGD

In two previous postings (see this and this) I have considered the recent dramatic progress in twistor program from my humble vantage point defined by TGD. I am well aware of my technical limitations in these issuess but despite this I want to summarize some observations about the relationship between TGD and twistors which I should have discovered long time ago. There are also questions,ideas, and comments about the relationship to twistor string theory and M-theory and F-theory like structures. A more detailed discussion can be found at the end of the pdf article What could be the generalization of Yangian symmetry of N=4 SYM in TGD framework?. I apologize possible alert readers for the large amplitude critical fluctuations in the detailed formulation of the idea that the classical dynamics of TGD could allow description in terms of twistors.

The basic observations are following.

1. Causal diamonds defined as intersections of future and past light-cones correspond to the Penrose diagrams lifted to representations of conformally compactified Minkowski space obtained by replacing the points of CD with spheres. The points of CD are representable by pairs of twistors and light-like points at boundaries of CD by single twistor.

2. A pair of twistors defines a point defines a complex line of twistor space identifiable as a point of conformally compactified Minkowski space and thus of CD. Twistor itself is in turn mapped to CP₂ in the dual twistor space CP₃ by assigning to it the complex 2-plane defined by it via the linear equation Z•W=0 (in projective coordinates). Therefore the space CP₃× CP₃ is mapped naturally to M⁴×CP₂ described in terms of the dual of CP₃× CP₃. It is however enough to use single pair CP₃× CP₃ if one wants to describe space-time surfaces as holomorphic surfaces. This suggests a deep relationship between the imbedding space of TGD and twistors which I have failed to realize hitherto!

3. One can lift the partonic 2-surfaces at boundaries of CD to 4-D sphere bundles in twistor space CP₃. This suggest that the twistor strings of Witten have a generalization in TGD framework to 6-D holomorphic surfaces in the product of twistor space and its dual. One can start from 12-D CP₃ ×CP₃, where the first CP₃ represents projective twistors with projectively flat metric with signature (2,4) obtained from 8-D twistor space with signature (4,4). Second CP₃ has Euclidian metric allowing Calabi-Yau structure. Canonically imbedded CP₂ should have the standard metric with SU(3) actings as holonomies of the Calabi-Yau CP₃ acting and as isometries of CP₂.

4. Also the first CP₃ with (2,4) signature of conformal metric could allow a generalization of Calabi-Yau structure to Minkowskian signature (the Ricci tensor vanishes by conformal flatness) so that one would have something resembling F-theory with 2 time dimensions in the Cartesian product CP₃ ×CP₃ of twistors and dual twistors. The lifts of space-time surfaces in this space would be holomorphic 6-surfaces of this 12-D space. The consistency with M⁴ ×CP₂ picture requires that these surfaces are sphere bundles with spheres projected to points of CD. Also the projections to the second CP₃ must be consistent with the CP₂ picture so that the holomorphic equations for complex plane are satisfied. The remaining two holomorphic equations would determine the dynamics of the space-time surface.

   The sphere bundle postulate reduces CP₃×CP₃ to CP₃×CP₂ and therefore leads to an analog of M-theory with two time directions. The further conditions imply that there is only one physical time-like direction. It must be emphasized that the super-symmetry in TGD framework is not same as in M-theory. For instance, the separate conservation of baryon and lepton numbers is a crucial distinction.

5. Grassmannians have been suggested to have an identification in terms of the moduli spaces of twistor strings having representation as holomorphic surfaces. This conjecture should generalize appropriately so that partonic 2-surfaces with 4-D tangent space data or equivalently space-time surfaces (with certain restrictions) would bring in Grassmannians as part of the moduli spaces of their lifts to 6-D holomorphic surfaces in CP₃×CP₃.

Addition: Witten related the degree d of the algebraic curve describing twistor string, its genus g, the number k of negative helicity gluons, and the number l of loops by the formula

d=k-1+l g≤ l.

One should generalize the definition of the genus so that it applies to 6-D surfaces. For projective complex varieties of complex dimension n this definition indeed makes sense. Algebraic genus is expressible in terms of the dimensions of the spaces of closed holomorphic forms known as Hodge numbers h^p,q as

g= ∑ (-1)^n-kh^k,0 .

The first guess is that the formula of Witten generalizes by replacing genus with its algebraic counterpart. This requires that the allowed holomorphic surfaces are projective curves of twistor space, that is described in terms of homogenous polynomials of the 4+4 projective coordinates of CP₃ ×CP₃.

I do not bother to type further but give link to short file in which these observations are described in more detail. For an overall view about the proposed generalization of Yangian symmetry see the pdf article What could be the generalization of Yangian symmetry of N=4 SYM in TGD framework? or the new chapter Yangian Symmetry, Twistors, and TGD.

Exact Yangian symmetry, non-trivial scattering amplitudes, no IR singularities: only a dream?

I have work hardly to understand in more detail the formulation of the scattering amplidudes in terms of Yangian invariants defined by Grasmannian integrals from the recent article of Nima Arkani-Hamed and collaborators and its predecessor The S-matrix in Twistor Space.

The exact super Yangian invariance would be extremely attractive constraint on the theory and I proposed in the earlier posting a generalization of this symmetry to TGD framework obtained by replacing finite-dimensional conformal algebra with various infinite-dimensional super-conformal appearing in TGD. It seems that in momentum degrees of freedom this symmetry gives the super-conformal Yangian symmetry of N=4 SYM so that an appropriate generalization of the Grassmannian approach should work also in TGD context.

The problem is that this symmetry applied to scattering amplitudes allows only the trivial Yangian invariant which is constant (this does not however mean physical triviality, only the triviality of loop contributions). The argument demonstrating this is very simple and can be found at the end of the recent article by Nima and others. Non-triviality of the scattering amplitudes is due to the infrared singularities spoiling the Yangian invariance. The basic idea of the approach of Nima and others was indeed the idea that the leading IR singularities code for the scattering amplitudes. As a romantic soul I cannot avoid the feeling that something is wrong in QFT approach. Note that the IR singularities of the scattering amplitudes are also physically problematic and one must develop a procedure for eliminating them. This however leads difficulties with Yangian aesthetics.

The question is whether one can circumvent this objection in TGD framework, where the physical particles appearing as incoming and outgoing states of particle reactions are bound states of fundamental massless fermions assigned with ligh-like wormhole throats whereas virtual particles are massive on mass shell particles with both positive and negative energies so that loop integrals reduce to sums subject to very powerful constraints from on mass shell property (zero energy ontology). The first good news is that the study of the simple special case shows that at least in this case the algebraic discretization of virtual masses and even of virtual four-momenta suggested by the p-adicization (number theoretical universality) and modified Dirac equation does not spoil the applicability of the Grassmannian approach involving residue integrals along complex contours: a discrete version of loop integral in momentum space is obtained. The number theoretical beauty of the residue integrals is that they make sense also p-adically unlike Riemannian integral.

It might be possible to achieve Yangian invariance and non-trivial scattering amplitudes in TGD framework and the bound state masses bring in also the natural infared cutoff as a simple modification of the killer argument of Nima and others to a less lethal form implies.

The basic point is that the action of super-generators on bound states with constraints between momentum components and thus between components of components of super-twistors (in particular their super parts) associated with different wormhole throats carrying fermion quantum numbers can annihilate the amplitude without it being constant. The non-locality of the bound states is in accordance with the non-locality of Yangian symmetry and bound state mass scale brings in naturally a physical IR cutoff.

This argument and a vision about how the Grassmannian integral approach might generalize to TGD framework can be found in the article What could be the generalization of Yangian symmetry of N=4 SUSY in TGD framework? or the new chapter Yangian Symmetry, Twistors, and TGD.

The reader interested in other articles written during this year about quantum TGD can find them here.

Hawking and God

There has been a lot of discussion about Hawking's new book The Grand Design. Lubos applaudes Hawking for believing in M-theory but not so much for deducing the non-existence of God from this belief.

Not Even Wrong in turn strongly criticizes Hawking for his belief on M-theory. I cannot but agree with his criticism. The fact is that M-theory has gained no experimental support hitherto and the standard media hype nowadays is that after these forty years superstring theory has finally been able to make a prediction. M-theory of course contains many mathematical ingredients of the next theory but involves spontaneous compactification as ad hoc element responsible for the landscape problem. The need for spontaneous compactification is in turn due to the wrong identification of fundamental objects as strings. The dead end is admitted also by many of its main proponents.The quirk of psychology of vanity is that in many brilliant minds the catastrophic weakness of M-theory of not being able to predict has gradually transformed to its greatest virtue. Sad that Hawking wants to advocate this kind of give-up-the-attempts-to-predict-anything philosophy after the absolutely fantastic successes of theoretical physics during the last century.

In viXra the comment of cosmologist Lawrence Krauss about Hawking's book related to the notion of energy in General Relativity is discussed but Hawking's basic claim is not discussed. I glue below the main part of my comment in this blog relating to the notion of God against which Hawking is fighting against.

Before doing it I have however a request to make. "Do not classify me!". Neither as an atheist nor as a proponent of some religion. With all respect to the proponents of these views, I regard these views as inconsistent with what we already known from fundamental physics and its deepest problems. Indeed, my own view point has developed from an atttempt to resolve one of the most pressing questions of recent day quantum theory: what state function means physically and for world view and how it should be described mathematically.

From what I have understood from a discussion in Lubos Motl's blog I understand that Hawking's view about God is badly in need of updating. It is essentially the God allowed by classical deterministic physics. God dictated the initial conditions of Big Bang and lost interest on the Universe after that. This because Godly intervention would break the laws of classical physics. In quantum measurement theory we encounter the same problem: quantum measurement apparently breaks the determinism of Schroedinger equation. Now we cannot however claim that state function collapse or something equivalent with it does not occur. The irrational manner to get rid of the problem is to say that there is no objective reality at all.

In TGD inspired theory of consciousness can be seen as a generalization of quantum measurement theory in order to overcome this difficulty. It leads to a quantal view about divine as ability to recreate the whole 4-D Universe (or more precisely, their quantum superposition) again and again. This allows to understand biological evolution as something genuine and generalize the concept of evolution. Zero energy ontology means that physical states correspond to pairs of positive and negative energy states so that symmetries and conservation laws do not restrict the free will of quantum jump. Every physical state is in principle reachable from a given physical state by quantum jumps. Free will is completely consistent with the determinism of the laws of classical physics since the free will of quantum jump is outside the space-time and Hilbert space: entire time evolution of Schroedinger equation is replaced with a new one. Consistency with physics does not anymore exclude divine.

Accepting this view means also a new view about relationship between experienced time and geometric time. They are not one and same thing as should be clear already from the fact that subjective time is irreversible and geometric time reversible. Their identification can however make sense approximately and locally applying to one particular system from which the contents of consciousness of one particular conscious entity is about. Everywhere in 8-D Universe there are space-time sheets about which a contents of sensory consciousness of a particular conscious entity comes from.

In this framework there is no sense in asserting that consciousness is a kind of 3-D time=constant slice moving towards geometric future. The time slice idea is also in conflict with General Coordinate Invariance since a special time coordinate would be relevant for consciousness. And our conscious experience is not about time=constant snapshot. We have memories- even sensory ones- and the experiments of Libet demonstrated that our volitional act induces neural activity in the geometric past. The contents of our conscious experience is about 4-D space-time region, and the challenge is to understand why our sensory experience is localized to about .1 second wide interval of geometric time in the usual wake up state of consciousness.

For these reasons I do not find the classical physics view about God selecting initial conditions very interesting. Hawking should find himself more demanding challenges than killing for all practical purposes already dead God of classical mechanics;-)!

Could the notion of hyper-determinant be useful in TGD framework?

Hyperdeterminants have stimulated interesting discussions in viXra blog and also Kea has talked about them. The notion is new to me but so interesting from TGD point of view that I cannot resist the temptation of making fool of myself by declaring why it looks so interesting. This gives also an excellent opportunity to demonstrate my profound ignorance about the notion;-). Instead of typing all my ignorance in html, I give a link to pdf article Could the notion of hyper-determinant be useful in TGD framework?. Addition: I decided to glue the response to a comment by Phil Gibbs summarizing my motivations for getting interested in hyper-determinants.

1. Why the equations stating the vanishing of n:th variation of Kähler action are interesting in TGD framework is due to the infinite vacuum degeneracy of Kähler action making possible an infinite hierarchy of criticalities: one can say that TGD Universe is quantum critical. Criticality means a hierarchy of vanishing n:th variations. Phase transitions inside phase transitions inside.... This property is responsible for a lot of new physics and mathematics involved with TGD.
2. The equations for n:th variation of Kähler action formulated in terms of functional derivatives are formally of this form and the existence of solution means vanishing of a generalized hyper-determinant. In standard QFT vanishing n≥3:th variations are not terribly interesting and even their existence is questionable. Vanishing second variations correspond to zero modes and vanishing of Gaussian determinant.
3. n:th variations correspond formally to infinite tensor product with same dimension for all tensor factors and in this case there should be no restrictions on the number of tensor factors. The definition of hyper-determinant in this case is of course highly non-trivial. Already functional (Gaussian) determinants are tricky objects. What makes hyper-determinant so interesting from TGD view point is that it applies to multilinear equations involving homogeneous polynomials. Something between linear and genuinely non-linear and solvable.

What hopes one has for genuine multilinearity, which seems to be almost synonymous to non-locality?

1. In the general case multilinearity requires non-locality and in purely local non-linear field theories there are not must hopes about multilinearity. The field equations for n:th variation should not contain powers of the same imbedding space coordinate or same derivative of it at same point. This is certainly not the case for a typical action principle. If the equations are genuinely multilinear in some basis for the deformations of space-time surface they are solvable and generalized hyper-determinant should tell whether this is the case. Its vanishing would also code for criticality for a higher order phase transition.
2. When one constructs perturbation theory for a functional integral using exponent of Kähler function, one considers Kähler function identified as Kähler action for a preferred extremal. Formally this is a non-local functional of the data about 3-surface but actually reduces to 3-D Chern-Simons Kähler action with constraints characterizing weak form of electric magnetic duality. By effective 2-dimensionality Chern-Simons action is however a non-local functional of data about partonic 2-surface and its tangent space. n:th variation for 3-surface and 4-surface reduce to a non-local function of n:th variation of partonic 2-surface and its tangent space data. This is just what genuine multilinearity means so that multilinearity seems to hold true!
3. This also relates to the local divergences of quantum field theories. They are present just because of higher order purely local couplings. Now they are absent if non-locality implying multilinearity holds true so that the functional integral over partonic 2-surfaces plus tangent space data should be free of infinities. Hence multilinearity might be behind integrability and absence of divergences. Maybe this relates also to the Yangian algebras which are non-local.

This is how it looks like at this moment.

What could be the generalization of Yangian symmetry of N=4 SUSY in TGD framework?

Lubos told for some time ago about last impressive steps in the understanding of N=4 maximally sypersymmetric YM theory possessing 4-D super-conformal symmetry. This theory is related by AdS/CFT duality to certain string theory in AdS₅× S⁵ background. Second stringy representation was discovered by Witten and based on Calabi-Yau manifold defined by twistors.

Note: I have added to the original posting few sections about a concrete Grassmannian realization of twistorial approach in TGD framework, and also a proposal for the physical interpretation of the Cartan algebra of Yangian algebra allowing to understand at the fundamental level how the mass spectrum of n-particle bound states could be understood in terms of the n-local charges of the Yangian algebra. I have not included all the material to this posting, and for the reader interested about what M⁸-H duality is and how it relates to the proposed generalization of Yangian symmetry, I recommend the pdf article What could be the generalization of Yangian symmetry of N=4 SYM in TGD framework?.

Background

I am outsider as far as concrete calculations in N=4 SUSY are considered and the following discussion of the background probably makes this obvious. I am ashamed;-).

The developments initiated by Witten with his Perturbative Gauge Theory As a String Theory In Twistor Space and led to Britto-Cachazo-Feng-Witten (BCFW) recursion relations for tree level amplitudes. The progress inspired the idea that the theory might be completely integrable meaning the existence of infinite-dimensional un-usual symmetry. This symmetry would be so called Yangian symmetry assigned to the super counterpart of the conformal group of 4-D Minkowski space.

Drumond, Henn, and Plefka represent in the article Yangian symmetry of scattering amplitudes in N = 4 super Yang-Mills theory an argument suggesting that the Yangian invariance of the scattering amplitudes ins an intrinsic property of planar N=4 super Yang Mills at least at tree level.

The last step in the progress was taken by Arkani-Hamed, Bourjaily, Cachazo, Carot-Huot, and Trnka and represented in the article The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM. At same day there was also the article of Rutger Boels entitled On BCFW shifts of integrands and integrals in the archive. Arkani-Hamed and others argue that a full Yangian symmetry of the theory allows to generalize the BCFW recursion relation for tree amplitudes to all loop orders at planar limit (planar means that Feynman diagram allows imbedding to plane without intersecting lines). On mass shell scattering amplitudes are in question.

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 Yangian Symmetry in D=4 superconformal Yang-Mills theory. 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 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.

How to generalize Yangian symmetry in TGD framework?

As far as concrete calculations are considered, I have nothing to say. I am just perplexed. 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 and Virasoro algebras 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 ontains 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 partonic 2-surface. A coset construction is applied to these two Virasoro algebras so that the differences of the corresponding Super-Virasoro generators and Kac-Moody generators annihilate physical states. This implies that the corresponding four-momenta are same: this expresses the equivalence of gravitational and inertial masses. 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.

Is there any hope about description in terms of Grassmannians?

At technical level the successes of the twistor approach rely on the observation that the amplitudes can be expressed in terms of very simple integrals over sub-manifolds of the space consisting k-dimensional planes of n-dimensional space defined by delta function appearing in the integrand. These integrals define super-conformal Yangian invariants appearing in twistorial amplitudes and the belief is that by a proper choice of the surfaces of the twistor space one can construct all invariants. One can construct also the counterparts of loop corrections by starting from tree diagrams and annihilating pair of particles by connecting the lines and quantum entangling the states at the ends in the manner dictated by the integration over loop momentum. These operations can be defined as operations for Grassmann integrals in general changing the values of n and k. This description looks extremely powerful and elegant and nosta importantly involves only the external momenta.

The obvious question is whether one could use similar invariants in TGD framework to construct the momentum dependence of amplitudes.

1. The first thing to notice is that the super algebras in question act on infinite-dimensional representations and basically in the world of classical worlds assigned to the partonic 2-surfaces correlated by the fact that they are associated with the same space-time surface. This does not promise anything very practical. On the other hand, one can hope that everything related to other than M⁴ degrees of freedom could be treated like color degrees of freedom in N=4 SYM and would boil down to indices labeling the quantum states. The Yangian conditions coming from isometry quantum numbers, color quantum numbers, and electroweak quantum numbers are of course expected to be highly non-trivial and could fix the coefficients of various singlets resulting in the tensor product of incoming and outgoing states.

2. The fact that incoming particles can be also massive seems to exclude the use of the twistor space. The following observation however raises hopes. The Dirac propagator for wormhole throat is massless propagator but for what I call pseudo momentum. It is still unclear how this momentum relates to the actual four-momentum. Could it be actually equal to it? The recent view about pseudo-momentum does not support this view but it is better to keep mind open. In any case this finding suggests that twistorial approach could work in in more or less standard form. What would be needed is a representation for massive incoming particles as bound states of massless partons. In particular, the massive states of super-conformal representations should allow this kind of description.

Could zero energy ontology allow to achieve this dream?

1. As far as divergence cancellation is considered, zero energy ontology suggests a totally new approach producing the basic nice aspects of QFT approach, in particular unitarity and coupling constant evolution. The big idea related to zero energy ontology is that all virtual particle particles correspond to wormhole throats, which are pairs of on mass shell particles. If their momentum directions are different, one obtains time-like continuum of virtual momenta and if the signs of energy are opposite one obtains also space-like virtual momenta. The on mass shell property for virtual partons (massive in general) implies extremely strong constraints on loops and one expect that only very few loops remain and that they are finite since loop integration reduces to integration over much lower-dimensional space than in the QFT approach. There are also excellent hopes about Cutkoski rules.

2. Could zero energy ontology make also possible to construct massive incoming particles from massless ones? Could one construct the representations of the super conformal algebras using only massless states so that at the fundamental level incoming particles would be massless and one could apply twistor formalism and build the momentum dependence of amplitudes using Grassmannian integrals.

   One could indeed construct on mass shell massive states from massless states with momenta along the same line but with three-momenta at opposite directions. Mass squared is given by M²= 4E² in the coordinate frame, where the momenta are opposite and of same magnitude. One could also argue that partonic 2-surfaces carrying quantum numbers of fermions and their superpartners serve as the analogs of point like massless particles and that topologically condensed fermions and gauge bosons plus their superpartners correspond to pairs of wormhole throats. Stringy objects would correspond to pairs of wormhole throats at the same space-time sheet in accordance with the fact that space-time sheet allows a slicing by string worlds sheets with ends at different wormhole throats and definining time like braiding.

The weak form of electric magnetic duality indeed supports this picture. To understand how, one must explain a little bit what the weak form of electric magnetic duality means.

1. Elementary particles correspond to light-like orbits of partonic 2-surfaces identified as 3-D surfaces at which the signature of the induced metric of space-time surface changes from Euclidian to Minkowskian and 4-D metric is therefore degenerate. The analogy with black hole horizon is obvious but only partial. Weak form of electric-magnetic duality states that the Kähler electric field at the wormhole throat and also at space-like 3-surfaces defining the ends of the space-time surface at the upper and lower light-like boundaries of the causal diamond is proportonial to Kähler magnetic field so that Kähler electric flux is proportional Kähler magnetic flux. This implies classical quantization of Kähler electric charge and fixes the value of the proportionality constant.

2. There are also much more profound implications. The vision about TGD as almost topological QFT suggests that Kähler function defining the Kähler geometry of the "world of classical worlds" (WCW) and identified as Kähler action for its preferred extremal reduces to the 3-D Chern-Simons action evaluted at wormhole throats and possible boundary components. Chern-Simons action would be subject to constraints. Wormhole throats and space-like 3-surfaces would represent extremals of Chern-Simons action restricted by the constraint force stating electric-magnetic duality (and realized in terms of Lagrange multipliers as usual).

   If one assumes that Kähler current and other conserved currents are proportional to current defining Beltrami flow whose flow lines by definition define coordinate curves of a globally defined coordinate, the Coulombic term of Kähler action vanishes and it reduces to Chern-Simons action if the weak form of electric-magnetic duality holds true. One obtains almost topological QFT. The absolutely essential attribute "almost" comes from the fact that Chern-Simons action is subject to constraints. As a consequence, one obtains non-vanishing four-momenta and WCW geometry is non-trivial in M⁴ degrees of freedom. Otherwise one would have only topological QFT not terribly interesting physically.

Consider now the question how one could understand stringy objects as bound states of massless particles.

1. The observed elementary particles are not Kähler monopoles and there much exist a mechanism neutralizing the monopole charge. The only possibility seems to be that there is opposite Kähler magnetic charge at second wormhole throat. The assumption is that in the case of color neutral particles this throat is at a distance of order intermediate gauge boson Compton length. This throat would carry weak isospin neutralizing that of the fermion and only electromagnetic charge would be visible at longer length scales. One could speak of electro-weak confinement. Also color confinement could be realized in analogous manner by requiring the cancellation of monopole charge for many-parton states only. What comes out are string like objects defined by Kähler magnetic fluxes and having magnetic monopoles at ends. Also more general objects with three strings branching from the vertex appear in the case of baryons. The natural guess is that the partons at the ends of strings and more general objects are massless for incoming particles but that the 3-momenta are in opposite directions so that stringy mass spectrum and representations of relevant super-conformal algebras are obtained. This description brings in mind the description of hadrons in terms of partons moving in parallel apart from transversal momentum about which only momentum squared is taken as observable.

2. Quite generally, one expects for the preferred extremals of Kähler action the slicing of space-time surface with string world sheets with stringy curves connecting wormhole throats. The ends of the stringy curves can be identified as light-like braid strands. Note that the strings themselves define a space-like braiding and the two braidings are in some sense dual. This has a concrete application in TGD inspired quantum biology, where time-like braiding defines topological quantum computer programs and the space-like braidings induced by it its storage into memory. Stringlike objects defining representations of super-conformal algebras must correspond to states involving at least two wormhole throats. Magnetic flux tubes connecting the ends of magnetically charged throats provide a particular realization of stringy on mass shell states. This would give rise to massless propagation at the parton level. The stringy quantization condition for mass squared would read as 4E²= n in suitable units for the representations of super-conformal algebra associated with the isometries. For pairs of throats of the same wormhole contact stringy spectrum does not seem plausible since the wormhole contact is in the direction of CP₂. One can however expect generation of small mass as deviation of vacuum conformal weight from half integer in the case of gauge bosons.

If this picture is correct, one might be able to determine the momentum dependence of the scattering amplitudes by replacing free fermions with pairs of monopoles at the ends of string and topologically condensed fermions gauge bosons with pairs of this kind of objects with wormhole throat replaced by a pair of wormhole throats. This would mean suitable number of doublings of the Grassmannian integrations with additional constraints on the incoming momenta posed by the mass shell conditions for massive states.

Could zero energy ontology make possible full Yangian symmetry?

The partons in the loops are on mass shell particles have a discrete mass spectrum but both signs of energy are possible for opposite wormhole throats. This implies that in the rules for constructing loop amplitudes from tree amplitudes, propagator entanglement is restricted to that corresponding to pairs of partonic on mass shell states with both signs of energy. As emphasized by Arkani Hamed and collaborators, it is the Grassmannian integrands and leading order singularities of N=4 SYM, which possess the full Yangian symmetry. The full integral over the loop momenta breaks the Yangian symmetry and brings in IR singularities. Zero energy ontologist finds it natural to ask whether QFT approach shows its inadequacy both via the UV divergences and via the loss of full Yangian symmetry. The restriction of virtual partons to discrete mass shells with positive or negative sign of energy imposes extremely powerful restrictions on loop integrals and resembles the restriction to leading order singularities. Could this restriction guarantee full Yangian symmetry and remove also IR singularities?

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.

For more details about the proposed generalization of Yangian symmetry, see the pdf article What could be the generalization of Yangian symmetry of N=4 SYM in TGD framework? or the new chapter Yangian Symmetry, Twistors, and TGD..

Recreate life to understand how life began

Mark Williams sent a link to a very interesting article in New Scientists. It tells about success in attempts to recreate life in laboratory. Not from existing building blocks but at much more fundamental level. RNA is believed to play a fundamental role in the prebiotic evolution of RNA (so called RNA world scenario) and much of the effort has gone on attempts to induce the generation of RNA molecules in laboratory. The problem is that it is difficult to get long enough RNA molecules which would replicate. The newest experimental work however supports the idea that life emerges as a co-evolution of both RNA and of protocell membranes formed by phospholipids (fatty acids). Soap films represent a familiar example of this kind membrane.

The challenge is to get cell membranes to grow and replicate in laboratory.

1. Phospholipid layers are generated via self-organization and their development does not require any genetic apparatus. Therefore it is not too difficult to make protocell membranes to grow.

2. The first discovery was that RNA within cell membranes drives the vesicle growth. The presence of RNA makes the membrane able to steal lipids from neighbors: market economy is after all not so new discovery and means actually return to the protocell level in evolution! Congratulations, Dear Human Kind!

3. Division was the tough problem since it is not energetically favored. The solution to the problem came as an accidental discovery. It was found that the cells grew large enough they tend to become elongated and tubules began to grow from the surface of cell. Eventually the membrane becomes so elongated and filamentous that it becomes unstable against division. Clearly a critical state in which it does not cost much to build new cell membrane is in question. Maybe the generation of axons and microtubules is analogous process and also the formation of cilia at the surfaces of real cells.

These findings are very interesting from TGD point of view. DNA as topological quantum computer model (see this and this and also other chapters of the book Genes and Memes) assumes that the magnetic flux tubes connecting DNA and lipids of lipid layer and their braiding make possible a realization of topological quantum computer programs and memory based on braiding. A good metaphor is dancers in dancing hall with threads connecting their feet to the wall. The dancing pattern defines a braiding in time direction coding for topological quantum computation and the entanglement of threads codes this program to memory. The braiding can be also interpreted as a representation for the fluid flow of the lipid molecules which form a liquid crystal and induced by the flow of surrounding cellular water. In case of axons this could give rise to a memory about nerve pulse patterns involving directly the relationship between DNA and cell membrane defined by the braiding.

The flux tubes containing large hbar matter would be actually everywhere in living matter. The phase transitions reducing or increasing Planck constant would induce phase transitions taking molecules near to each other and vice versa and would explain the magic ability of bio-molecules to find each other. Bio-catalysis and DNA replication and also the phase transitions involving typically large changes in the density of cytoplasm would involve change of hbar. Same applies to sol-gel transitions and the transformation of biomatter between resting state to active state involving protein folding and un-folding. The reconnection of the flux tubes of the Indra's net formed by the magnetic flux tubes would be second non-chemical key process of biology.

In this framework it would not be too surprising that the evolution of RNA and cell membranes would take place in co-operation. Formation of membranes would make possible topological quantum computation and memory and intelligence at cell level, and this in turn would make for protocell to evolve further. Also the growth of tubules could be assigned with the generation of magnetic flux tubes above critical size. DNA is stable inside cell since cellular water can be in ordered phase in which the decay of DNA polymer by hydration does not take place. Therefore this evolution would precede the emergence of DNA in TGD Universe.

Can one define conserved Poincare charges in General Relativity?

There has been an interesting discussion in viXra blog about whether it is possible to define the notion of conserved energy, and more generally the notion of conserved Poincare charges in General Relativity. Also Lubos has participated. My conviction is that this is not possible without additional conditions on the metric (asymptotic Minkowski space property) and one must certainly give up the hopes of obtaining the conserved Poincare charges as Noether charges from standard action describing matter coupled to gravitation.

The following argument suggests that there are some hopes of getting non-conserved but well-defined Poincare charges in asymptotically Minkowskian space-time.

1. Entire Poincare algebra is needed in quantum theory and the Lie-algebraic realization in terms of space-time vector fields gives the only hope of achieving the goal. One could consider also the extension of Poincare algebra to an infinite-dimensional Lie algebra with generators approaching Poincare algebra generators asymptotically.

2. You would start with the identification of vector fields j^Ia defining infinitesimal translations, rotations, and boosts in asymptotic regions. In this region they define asymptotic Killing vector fields satisfying

   D_aj^I_b+D_bj^I_a=0

   and the currents

   (G-λ g)^abj_b

   are asymptotically divergenceless because Killing vector field property is true and G and g are divergenceless in covariant sense. If you can continue j^I to entire space-time uniquely ,you get well-defined Poincare charges, which are however not conserved.

3. You must replace Killing vector field property with something weaker and the condition that j^I define flows conserving only four-volume instead of distances is a natural generalization. This implies the condition

   ∇⋅ j^I=0

   and the infinite-dimensional Lie-algebra of volume preserving vector fields is obtained.

4. A further condition is needed and this is very natural. You must be able to define global coordinates along the flow lines of the vector fields in questions. This requires

   j^I = Ψ ∇ Φ.

   Φ defines the coordinate. This kind of vector fields are known as Beltrami fields.

5. In asymptotic region Φ would represent either a counterpart of linear M⁴ coordinate, rotation angle around some space-like axis , or hyperbolic angle around time-like axis. In the asymptotic region Ψ would be constant for translations in the asymptotic region. For the rotations around a given axis the orthogonal it would reduce to the orthogonal distance ρ from that axis. For the Lorentz boosts around given time-like axis to the orthogonal radial distance r from origin in the rest frame defined by that axis.

Let us look what volume preservation and Beltrami property give.

1. By simple calculation you obtain

   ∇² Φ +2 ∇ (log(&Psi);⋅ ∇ Φ=0.

   This is massless field equation with additional term which might relate to massivation. If one has two solutions with same Φ, one obtains the condition

   (∇ Ψ₁-∇ Ψ₂)⋅ ∇Φ=0 ,

   which suggests that you must have

   ∇ Ψ⋅ ∇ Φ=0

   quite generally.

2. The physical interpretation would be obvious. The solutions describe as special case the modes of massless gauge field. Φ defines the counterpart of a pulse propagating to local light-like direction and Ψ defines a local polarization vector orthogonal to it. There are also solutions which do not allow this interpretation and corresponds to the functions Φ and Ψ, which are relevant in the recent case.

3. The solution set is quite large for a given Φ. You can replace Φ with an arbitrary function of Φ if the additional condition

   ∇Φ⋅∇Φ=0

   having obvious interpretation holds true. Same applies to Ψ. Linear superposition holds true. You can also form the Lie-brackets for given Φ and one finds that they vanish. Therefore you have infinite-dimensional Abelian algebra. The natural interpretation is as commuting observables corresponding to polarization direction and propagation direction.

4. Can one obtain unique continuation of j^I from the asymptotic region to the interior so that unique conserved Poincare charges would exists for asymptotically Minkowskian space-time? The radiative solutions are the problem. If the condition that the radiative part vanishes in the asymptotic region implies that it vanishes everywhere, there are no problems.

   Minkowski space serves as a good test bench. In this case functions Φ(p⋅ m) are simplest propagating pulses: here p is light-like momentum. The condition that they vanish in all directions including the propagation direction in which p⋅ m is constant indeed implies that Φ vanishes. By choosing Ψ so that it vanishes far away does not allow to achieve the condition. Hence there are hopes that one can define non-conserved Poincare charges in asymptotically flat space-times. One can however imagine the presence of light-pulses which are emitted and absorbed and thus exists in a finite volume of space-time. These might course problems.

5. In the case of non-vanishing cosmological constant one would obtain infinite energy and the contribution to the charge would be the charge assignable to the vector field defining time translation. This does not favor cosmological constant.

As a matter fact, one ends up with the Beltrami fields from a general solution ansatz for a solution of field equations in TGD. The interpretation is that one has the analog of Bohr quantization for solutions of the extremely nonlinear counterpart of Maxwell's equations coupled to classical gravitation via induced metric. Only the superposition of solutions corresponding to same function Φ is allowed. They represent pulses of various shapes and different polarizations propagating in a particular local light-like direction.This conforms with what one knows about outcomes of state function reduction. These solutions have 3- or 4-D CP₂ projection. So called massless extremals with 2-D CP₂ projection have same physical characteristics. Cosmic strings and CP₂ vacuum extremals with Euclidian signature of metric describing massless particles are also basic solutions and the topological condensation of CP₂ type vacuum extremals to a space-time sheet with Minkowskian signature of the induced metric creates around itself a solution described by Ψ and Φ meaning that particle picture implies field picture. Note that the proposed identification of gravitational charges could make sense also in TGD framework.

These Abelian algebras and perhaps large algebras generated by them via commutators might be relevant also for the construction of the solutions of field equations in General Relativity. The construction of deformations of an existing metric by adding gravitons is what comes in mind first. The scalars Ψ would define polarizations in a given background metric used to build polarization tensor and the functions Φ could be used to build the analogs of plane waves. One would obtain gravitons and also gauge bosons localized in transversal directions. The algebra formed by the Beltrami flows could thus play a role analogous to Kac-Moody algebras. What is interesting that one could always interpret a many-graviton state as a background to which one can add new kind of gravitons! This all is of course speculation but because these algebras allow a concrete interpretation as classical representations of elementary bosons, I would not find it completely surprising if an algebra related directly to the metric would play a fundamental role in quantization of General Relativity.

viXra blog

Phil Gibbs founded for a new ago a new archive christened as viXra org, where people finding it impossible to publish their works in so called respected journals and archives can upload their works. As one of these unlucky ones I am happy for this kind of opportunity. Now Phil started also a blog- Vixra blog - and there have been very stimulating discussions in a friendly atmosphere. Warmly recommended. By the way, Phil has also some articles about topics discussed in the blog in the recent issue of Prespacetime journal.

Comparison of TGD Higgs and with MSSM Higgs

There has been a lot of blog activity around Higgs lately (see this and this). There have been also rumors about indications for supersymmetry in the sense that it mitght be able to see indications for two neutral Higgses prediced by the minimal supersymmetric extension of standard model (MSSM) (see this and this).

Two out-of-topic comments before going to the main topic.

1. There is still no evidence for GUT type decays of proton have appeared from Super-Kamiokande (see this): as noticed by Phil Gibbs this negative result could be much more far reaching that finding of Higgs bosons since GUT type low energy phenomelogy is starting point of all theory building during last years. Maybe some young brains are sooner or later ready to question the existing belief system. Separate conservation of quark and lepton numbers and therefore stability of proton against GUT decays is what TGD predicts.

2. There are also some empirical motivations for speculations about the existence of fourth generation quark and Tommas dorigo is even ready to make a bet for it (see this). TGD predicts an infinite number of fermion families (they correspond to the topologies of partonic 2-surfaces) but there is a good argument that there are only three light generations. Unfortunately, the argument leaves open what "light" precisely means.

3. For some reason the Lamb shift anomaly of muonic hydrogen that I discussed in previous posting from TGD view point has stimulated very little blog activity. This is strange since the discovery challenges the basic foundations of quantum field theory and thus also of superstring models.

The notion of Higgs in TGD framework differs from that of standard model and super-symmetric extension in several respects.

1. Higgs does not give the dominating contribution to the masses of fermions (p-adic thermodynamics does it) . It might give the dominating contribution in the case of gauge bosons. Even this is not absolutely clear. A mechanism modifying the ground state conformal weight from half-integer value could also give a small contribution to the mass of the particle. Higgs is needed since the longitudinal degrees of massive gauge bosons must come somewhere and scalar particle is the only natural candidate here. The transition to unitary gauge leaving for Higgs only its magnitude as a dynamical degree of freedom is an elegant manner to describe how this happens.

2. There is no good argument excluding the existence of scalar and pseudoscalar bosons deserving the attribute "elementary" in the same sense as gauge bosons. Just the opposite. Bosonic emergence means that "elementary" Higgs particles are constructed by a recipe similar to that applying in the case of gauge bosons: that is by putting fermion and antifermion a the opposite light-like throats of a wormhole contact. This makes it also natural for Higgs particles to transform to longitudinal degrees of freedom of gauge bosons. Of course, the description of these states in terms of quantum fields is only an approximation.

3. The two complex SU(2)_V doublets are replaced with real scalar and pseudoscalar triplet and singlet (2 +2 → 2× (3+1)) so that the number of field components is same as in standard model. The Higgs possibly developing vacuum expectation is now uniquely the scalar singlet unless one allows parity breaking. The basic reason to group theoretical differences is that in TGD 4-D spinors are replaced with 8-D spinors.

4. TGD predicts super-conformal symmetry and the recent view about it predicts the analog of broken space-time supersymmetry. The modes of induced spinor field on light-like wormhole throat define the generators of super-symmetries. This supersymmetry has as the least broken sub-symmetry N=1 SUSY generated by covariantly constant right-handed neutrino. Therefore also sparticles- in particular Higgsinos- should exist.

5. The number of dynamical Higgs field components is 5 as in the minimal supersymmetric extension of the standard model. The basic difference between MSSM and TGD is that the second neutral Higgs is pseudoscalar in TGD.

Since it is boring to transform tex to html, I give a link to a short pdf file Comparison of TGD Higgs and with MSSM Higgs. For details you can see also the chapter p-Adic Mass calculations: Elementary Particle Masses of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy".

The incredibly shrinking proton

Ulla sent to the previous posting an interesting link about the discovery that the charge radius of proton deduced from the muonic version of hydrogen atom is about 4 per cent smaller than from the radius deduced from hydrogen atom and specialists tell that this cannot be true. The New Scientist article is here. The Nature article Quantum electrodynamics-a chink in the armour? is here. Nature does not have a habit of publishing rumours so that the finding must be taken very seriously.

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

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

Addition: This posting has been subject to continual modifications as I have been fighting with the model armed with my miserable calculational skills. Therefore I made a bigger updating which hopefully provides a clearer representation. The calculations are represented in a little article at my homepage.

1. Basic facts and notions

Can one say anything interesting about the possible mechanism behind the anomaly if one accepts TGD framework? How the presence of muon could reduce the charge radius of proton? Let us first list the basic facts.

1. One can say that the size of muonic hydrogen characterized by Bohr radius is by factor m_e/m_μ=211.4 smaller than for hydrogen atom and equals to 250 fm. Hydrogen atom Bohr radius is .53 Angstroms.

2. Proton contains 2 quarks with charge 2e/3 and one d quark which charge -e/. These quarks are light. The last determination of quark masses gives masses, which are m_u=2 MeV and m_d=5 MeV (I leave out the error bars). The standard view is that the contribution of quarks to proton mass is of same order of magnitude. This would mean that quarks are not too relativistic meaning that one can assign to them a size of order Compton wave length of order 4×r_e≈600 fm in the case of u quark (roughly twice the Bohr radius of muonic hydrogen) and 10×r_e≈24 fm in the case of d quark. These wavelengths are much longer than the proton charge radius and for u quark more than twice longer than the Bohr radius of the muonic hydrogen. That parts of proton would be hundreds of times larger than proton itself sounds a rather weird idea. One could of course argue that the scales in question do not correspond to anything geometric. In TGD framework this is not the way out since quantum classical correspondence requires this geometric correlate.

3. There is also the notion of classical radius of electron and quark. It is given by r= α hbar/m and is in the case of electron this radius is 2.8 fm whereas proton charge radius is .877 fm and smaller. The dependence on Planck constant is only apparent as it should be since classical radius is in question. For u quark the classical radius is .52 fm and smaller than proton charge radius. The constraint that the classical radii of quarks are smaller than proton charge radius gives a lower bound of quark masses: p-adic scaling of u quark mass by 2^-1/2 would give classical radius .73 fm which still satisfies the bound. TGD framework the proper generalization would be r= α_Khbar/m, where α_K is Kähler coupling strength defining the fundamental coupling constant of the theory and quantized from quantum criticality. Its value is very near or equal to fine structure constant in electron length scale.

4. The intuitive picture is that light-like 3-surfaces assignable to quarks describe random motion of partonic 2-surfaces with light-velocity. This is analogous to zitterbewegung assigned classically to the ordinary Dirac equation. The interpretation of zitterbewegung radius as classical radius looks rather natural. The notion of braid emerging from Chern-Simons Dirac equation via periodic boundary conditions means that the orbits of partonic 2-surface effectively reduces to braids carrying fermionic quantum numbers. These braids in turn define higher level braids which would move inside a structure characterizing the particle geometrically. Internal consistency suggests that the classical radius r=&alpha_Khbar/m characterizes the size scale of the zitterbewegung orbits of quarks.

   I cannot resist the temptation to emphasize the fact that Bohr orbitology is now reasonably well understood. The solutions of field equations with higher than 3-D CP₂ projection describing radiation fields allow only generalizations of plane waves but not their superpositions in accordance with the fact it is these modes that are observed. For massless extremals with 2-D CP₂ projection superposition is possible only for parallel light-like wave vectors. Furthermore, the restriction of the solutions of the Chern-Simons Dirac equation at light-like 3-surfaces to braid strands gives the analogs of Bohr orbits. Wave functions of -say electron in atom- are wave functions for the position of wormhole throat and thus for braid strands so that Bohr's theory becomes part of quantum theory.

5. In TGD framework quantum classical correspondence requires -or at least strongly suggests- that also the p-adic length scales assignable to u and d quarks have geometrical correlates. That quarks would have sizes much larger than proton itself how sounds rather paradoxical and could be used as an objection against p-adic length scale hypothesis. Topological field quantization however leads to the notion of field body as a structure consisting of flux tubes and and the identification of this geometric correlate would be in terms of Kähler (or color-, or electro-) magnetic body of proton consisting of color flux tubes beginning from space-time sheets of valence quarks and having length scale of order Compton wavelength much longer than the size of proton itself. Magnetic loops and electric flux tubes would be in question. Also secondary p-adic length cale characterizes field body. For instance, in the case of electron the causal diamond assigned to electron would correspond to the time scale of .1 seconds defining an important bio-rhythm.

2. A general formula for Lamb shift in terms of proton charge radius The charge radius of proton is determined from the Lamb shift between 2S- and 2P states of muonic hydrogen. Without this effect resulting from vacuum polarization of photon Dirac equation for hydrogoen would predict identical energies for these states. The calculation reduces to the calculation of vacuum polarization of photon inducing to the Coulomb potential and an additional vacuum polarization term. Besides this effect one must also take into account the finite size of the proton which can be coded in terms of the form factor deducible from scattering data. It is just this correction which makes it possible to determine the charge radius of proton from the Lamb shift.

1. In the article The Lamb shift Experiment in Muonic Hydrogen the basic theoretical results related to the Lamb shift in terms of the vacuum polarization of photon are discussed. Proton's charge density is in this representation is expressed in terms of proton form factor in principle deducible from the scattering data. Two special cases can be distinguished corresponding to the point like proton for which Lamb shift is non-vanishing only for S wave states and non-point like proton for which energy shift is present also for other states. The theoretical expression for the Lamb shift involves very refined calculations. Between 2P and 2S states the expression for the Lamb shift is of form

   Δ E(2P_3/2^F=2 − 2S_1/2^F=1)=a-br_p² +cr_p³= 209.968(5) − 5.2248 × r²_p + 0.0347 × r³_p meV .

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

2. The general expression of Lamb shift is given in terms of the form factor by

   E(2P-2S)=∫ (d³q/(2π)³)× (-4π α ) (F(q²)/q²) × (Π(q²)/q²)× ∫ (| Ψ_2P(r)|²-|Ψ_2S(r)|²)exp(−iq• r) dV .

   Here Π is is a scalar representing vacuum polarization due to decay of photon to virtual pairs.

The TGD inspired model predicts that the effect is due to a leakage from "standard" state to what I call flux tube state. This means a multiplication of |Ψ_2P|²_3/2 with the normalization factor 1/N of the standard state orthogonalized with respect to flux tube state. It is essential that 1/N is larger than unity so that the effect is a genuine quantum effect not understandable in terms of classical probability.

The modification of the formula is due to the normalization of the 2P and 2S states. These are in general different. The normalization factor 1/N is same for all terms in the expression of Lamb shift for a given state but in general different for 2S and 2P states. Since the lowest order term dominates by a factor of ≈ 40 over the second one, one one can conclude that the modification should affect the lowest order term by about 4 per cent. Since the second term is negative and the modification of the first term is interpreted as a modification of the second term when r_p is estimated from the standard formula, the first term must increase by about 4 per cent. This is achieved if this state is orthogonalized with respect to the flux tube state. For states Ψ₀ and Ψ_tube with unit norm this means the modification

Ψ₀→ (1/(1-| C|²)× (Ψ_i -CΨ_tube) ,

C=⟨ Ψ_tube| Ψ₀⟩ .

In the lowest order approximation one obtains

a-br²_p+cr_p³→ (1+|C|²)a-br_p²+cr_p³ .

Using instead of this expression the standard formula gives a wrong estimate r_p from the condition

a-b r²_p,1+cr_p,1³→ (1+| C|²)a-br_p²+cr_p³ .

This gives the equvalent conditions

r_p,1²= r_p²- | C|²a/b ,

P_tube≡| C|²≈ (2b/a)× r_p² × (r_p-r_p,1)/r_p) .

The resulting estimate for the leakage probability is P_tube≈ .0015. The model should be able to reproduce this probability.

3. Could the notion of field body explain the anomaly?

The large Compton radii of quarks and the notion of field body encourage the attempt to imagine a mechanism affecting the charge radius of proton as determined from electron's or muon's wave function.

1. Muon's wave function is compressed to a volume which is about 8 million times smaller than the corresponding volume in the case of electron. The Compton radius of u quark more that twice larger than the Bohr radius of muonic hydrogen so that muon should interact directly with the field bodies of u quarks. The field body of d quark would have size 24 fm which is about ten times smaller than the Bohr radius so that one can say that the volume in which muons sees the field body of d quark is only one thousandth of the total volume. The main effect would be therefore due to the two u quarks having total charge of 4e/3.

   One can say that muon begins to "see" the field bodies of u quarks and interacts directly with u quarks rather than with proton via its elecromagnetic field body. With d quarks it would still interact via protons field body to which d quark should feed its electromagnetic flux. This could be quite enough to explain why the charge radius of proton determined from the expectation value defined by its wave function wave function is smaller than for electron. One must of course notice that this brings in also direct magnetic interactions with u quarks.

2. What could be the basic mechanism for the reduction of charge radius? Could it be that the electron is caught with some probability into the flux tubes of u quarks and that Schrödinger amplitude for this kind state vanishes near the origin? The original idea was based on classical probability: the flux tube portion of state would not contribute to the charge radius and since the portion of the ordinary state would bbe smaller, and effective reduction of the charge radius would be implied. Unfortunately just the opposite occurs and this is due to the fact that the expression for the Lamb shift involves also constant term besides powers of charge radius. What happens is that the normalization factor of the standard contribution increases by the orthogonalization with the flux tube state. The effect is therefore genuinely quantum mechanical having no classical counterpart.

3. I have blundering with this model for a week and it seems that no bad misunderstandings are present anymore. Precise numerical factors are of course dangerous at this age. By the earlier general argument one should have P_tube= .0015. This value of leakage probability is obtained for z=1 and N=2 corresponding to single flux tube per u quark. If the flux tubes are in opposite directions, the leakage into 2P state vanishes by parity. Note that the leakage does not affect the value of the coefficient a in the general formula for the Lamb shift.

   The radius of the flux tube is by a factor 1/4 smaller than the classical radius of electron and one could argue that this makes it impossible for electron to topologically condense at the flux tube. For z=4 one would have P_tube= .015, which is 10 times too large a value. Numerical errors are possible. Note that the nucleus possess a wave function for the orientation of the flux tube. If this corresponds to S-wave state then only the leakage beween S-wave states and standard states is possible.

   Since the formulas are too painful to type here, reader can consult the calculation here or in the chapter p-Adic Mass calculations: New Physics.

4. This effect would be of course present also in the case of electron but in this case the u quarks correspond to a volume which million times smaller than the volume defined by Bohr radius so that electron does not in practice "see" the quark sub-structure of proton. The probability P for getting caught would be in a good approximation proportional to the value of |Ψ(r_u)|² and in the first approximation one would have

   P_e/P_μ ≈ (a_μ/a_e)³ =(m_e/m_μ)³≈ 10^-7

   from the proportionality &Psi_i propto 1/a_i^3/2, i=e,μ.

Article series about TGD in Prespacetime journal

I have worked out during last two months an article series to Prespacetime Journal. It covers the two basic mathematical approaches to quantum TGD and their interconnections. Physics as infinite-dimensional geometry of "world of classical worlds" (just WCW among friends) and physics as a generalized number theory. These are the two great visions.

One could of course raise also some other principle at a special status. The notion of finite measurement resolution or the reduction to almost topological quantum field theory could be taken as a basic principle. Or one could talk about hyper-finite factors as really fundamental structures. Somehow however these two approaches seem to be the most natural ones and the basic vision is that they are more or less equivalent.

The writing process was very fruitful for the simple reason that it forced to print out most of the material from the two books devoted to these approaches and to perform painstaking comparisons to identify all inconsistencies. The visual feedback led to the realization of a large number of new interconnections between various approaches. Perhaps the most important outcome was the re-incarnation of the old idea of electric-magnetic duality as something which I re-christened as weak form of electric-magnetic duality. This turned out to be a Golden Road to an integration of the existing mathematical understanding of the theory.

The basic outcome was the reduction of Kahler function to Chern-Simons term subject to the constraint given by electric-magnetic duality. This meant that the reduction to almost topological QFT- to which I had already lost my belief- had been implicitly present in quantum TGD for more than five years. The condition guaranteing the reduction was the propoportionality of the conserved Kähler current to instanton current that I had proposed more than five years ago to characterize general solution ansatz to the classical field equations: at that time the motivation was the vanishing of Lorentz force guaranteed by this condition. At that time I did not realize that the Coulomb interaction term in Kähler action also vanishes so that it reduces to a boundary term and by the weak form of electric magnetic duality to Chern-Simons term.

A much more detailed form of the solution ansatz emerged: all conserved Noether currents are proportional to the same current which by integrability conditions must be what is known as Beltrami field. The hydrodynamical interpretation is that the flow parameters associated with flow lines integrate to a global coordinate. For this kind of hydrodynamic flows one can assign to each flow line conserved quantities and the flow apparently decomposes from separate independent flow lines: the interpretation is in terms of separation of degrees of freedom. Quantum field theorists speak about integrable quantum field theories in this kind of situation.

The solutions to the Beltrami conditions have beautiful dual interpretations in terms of hydrodynamics and as modes massless radiation fields. A beautiful realization of quantum classical correspondence realizing the idea about space-time surface as a generalized Bohr orbit emerges: one obtains only the generalization of plane waves but not their superpositions. This is what QFT predicts and experiment verifies but linear Maxwell's electrodynamics fails to give!

The ansatz generalized also to the Kähler Dirac equation in the interior of space-time sheet and generalized eigenvalue equation for Chern-Simons Dirac action. Even braids and number theoretic braids pop up automatically from the modified Dirac equation: periodic boundary conditions force the basic solutions of Chern-Simons Dirac equation to concentrate at discrete sets of flow lines of the Beltrami flow defining the braidingof topological QFT. Also connections between p-adic length scale hypothesis and infinite primes and corresponding arithemetic quantum field theory emerged but this is more speculative "must-be-true" stuff. An important outcome was an explicit expression for Dirac determinant in terms of geometric data characterizing the orbits of partonic 2-surfaces giving an alternative formula for the Kähler function of WCW. The second expression is as an exponent of Chern-Simons term. The basic problem of calculating the WCW Kähler function is therefore solved at certain level of details.

This kind of process is of course dangerous since it generates also not too long-lived ideas and my proof readers must have had moments of horror (apologies and thanks to them). One such one-night adventure was indeed generated by a rather delicate error in certain mathematical statement. I spoke about Kähler function as "Chern-Simons term" instead of "Chern-Simons term subject to constraint given by the weak form of electric-magnetic duality". The shocking conclusion was that the metric of WCW seems to be trivial in Minkowski degrees of freedom in contrast to all physical and mathematical intuitions. This mistake forced to consider a modification of Kähler action by an additional term in the induced Kähler form which in principle is possible but which I had for a long time ago tested and given up for reasons forgotten. It however looked mathematically extremely beautiful since the vacuum degeneracy of Kähler action was extended dramatically. It however turned out to lead to a non-acceptable long length scale limit for gravitation. I managed to kill the idea two days before deadline and also to notice my mistake!

The birth of TGD is a holistic process taking place in the time span of 32 years involving all physics related things from Planck length scale to quantum biology to cosmology and therefore a diametric opposite of the usual scientific work in which one has a precisely defined problem and method and one can forget everything else in the Universe and be fully left-brained and extremely reductive, precise, and analytic. Therefore it is obvious that I will probably remain the only one who understands what TGD is in this left-brained scientific community. Despite this I am happy and proud albeit somewhat frustrated. Who wants to be a musician playing music which sounds heavenly in his own ears but whom no one bothers to listen? A lonely millionaire is not a happy millionaire. But maybe- someday when I have gone TGD will be standard stuff of theoretical physics;-)!

Anyone interested about TGD even when I am still here, can find the articles here.

Could neutrinos appear in several p-adic mass scales?

There are some indications that neutrinos can appear in several mass scales coming from neutrino oscillation data. These oscillations can be classified to vacuum oscillations and to solar neutrino oscillations believed to be due to the so called MSW effect in the dense matter of Sun. There are also indications that the mixing is different for neutrinos and antineutrinos.

In TGD framework padic length scale hypothesis might explain these findings. I wrote already earlier hasty impressions about the situation but found that there were too many misunderstanding involved so that I decided to remove the blog page in a desperate attempt to keep some of my respectability (of course I know that I have lost it long time ago so that the manouvre came quite too late);-).

The basic vision is that the p-adic length scale of neutrino can vary so that the mass squared scale comes as octaves. Mixing matrices would be universal. The large discrepancy between LSND and MiniBoone results contra solar neutrino results could be understood if electron and muon neutrinos have same p-adic mass scale for solar neutrinos but for LSND and MiniBoone the mass scale of either neutrino type is scaled up. The sterile neutrino suggested as an explanation of the findings would be p-adically scaled up variant of ordinary neutrino having standard weak interactions. This scaling up can be different for neutrinos and antineutrinos as suggested by the fact that the anomaly is present only for antineutrinos.

The different values of Δ m² for neutrinos and antineutrinos in MINOS experiment can be understood if the p-adic mass scale for neutrinos increases by one unit. The breaking of CP and CPT would be spontaneous and realized as a choice of different p-adic mass scales and could be understood in zero energy ontology. Similar mechanism would break supersymmetry and explain large differences between the mass scales of elementary fermions, which for same p-adic prime would have mass scales differing not too much.

I do not bother to type the formulas in html format and give a link to a short pdf file Could neutrinos appear in several p-adic mass scales?, where a serious analysis of the findings is discussed. See also the chapter p-Adic Mass calculations: New Physics of "p-Adic Length scale Hypothesis and Dark Matter Hierarchy".