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".