How to define 3-D analogs of Mandelbrot fractals?

In New Scientists there was an article about 3-D counterparts of Mandelbrot fractals. It is not at all obvious how to define them. Quite impressive analogs of Mandelbrot set have been found using so called hypercomplex numbers (which can have any dimension but do not define number field but only ring) and replacing the canonical map z→ z² +c with a more general map (see this). c must be restricted to a 3-D hyperplane to obtain 3-D Mandelbrot set.

It occurred to me that there exists an amazingly simple manner to generate analogs of the Mandelbrot sets in 3 dimensions. One still considers maps of the complex plane to itself but assumes that the analytic function depends on one complex parameter c and one real parameter b so that the parameter space spanned by pairs (c,b) is 3-dimensional. Consider two examples:

1. f: z→bz³ + z²+ c, b real,

   for which b=0 cross section gives Mandelbrot fractal. The first two iteration steps are z=0→c→ bc³ + c²+ c →...

2. f: z→z³ + bz²+ c, b real, suggested by Thom's catastrophe theory. The first two iteration steps are z=0→c→ c³ +bc²+ c →...

Both options might produce something interesting. One could also construct dynamical 3-D Mandelbrots by allowing b to be complex and interpreting real or imaginary part of b of the modulus of b as time coordinate.

One can deduce some general features of these fractals.

1. These examples represent 3-D generalization of Multibrot fractals defined by z→ z^d (this video demonstrates how Multibrot fractals vary as the integer d varies). Obviously these sets look qualitatively very much like Mandelbrot fractal. Hence each b=constant cross section of these 3-D fractals is expected must have the qualitative look of Mandelbrot fractal.

2. The boundary of Mandelbrot fractal is now a 2-D surface of 3-D space spanned by points (b,c) and corresponds to the points (b,c) for which the iteration of f applied to z=0 does not lead to infinity. The line c=0 belongs in both cases to the fractal since z=0 is fixed point of iteration in this case. As the value of b grows the cross section must get increasingly thinner for both negative and positive values of b with large enough magnitude rougly as c < b^-1/3 for the first case and c<b^-1/2 by looking what happens in first iterations. The direction of b is clearly in a special position reflecting the 2-D character of the basic iteration process. This holds true for any analytic functions.

3. To guess what the boundaries of these fractals could look like, one can try to imagine what one obtains as one builds a small pile of slightly deformed 2-D Multibrot fractals above b =0 Multibrot fractal or any piece of it (see the images here). One expects complex caves inside caves structure. The caves are expected to be high. Also a fractal hierarchy of stalagmite like structures is expected. Their tips would reflect a disappearence or appearance of a new structure in 2-D Multibrot fractal as the height coordinate b varies. The objection is that the boundary curve in the case of Mandelbrot fractal- and maybe also of Multibrots- consists of single component. Stalagmites would correspond to an appearence of disjoint component to the curve would not be possible. In a given resolution however there are always invisible thin hairs connected to thicker regions so that only apparent stalagmites due to finite resolution would be possible. Being inside b> 0 part of the fractal might create same spiritual feelings that one experiences in Gothic Cathedral.

4. It is good to hear what pessistic has to say. In high enough b-resolution the variation of the Multibrot curve as a function of b could be so slow that the fractal looks like a cylinder having Multibrot as intersection and there would be no fractality in b-direction. Therefore the question is whether the fractality in c-plane implies fractality in b-direction. In other words, is the fractal curve for a fixed value of b critical against variations of b.

3-D graphics skills would be needed to disetangle the huge complexity of the situation. Probably this is far from a trivial challenge. In any case, the visualization as a pile of 2-D Multibrots could be used in the construction of these fractals and would make possible discretization in b-direction and the use of existing 2-D algoriths as such. Maybe some Mandelbrot artist might try look what these fractals look like. In New Scientists there was an article about 3-D counterparts of Mandelbrot fractals. It is not at all obvious how to define them. Quite impressive analogs of Mandelbrot set have been found using so called hypercomplex numbers (which can have any dimension but do not define number field but only ring) and replacing the canonical map z? z² +c with a more general map (see this). c must be restricted to a 3-D hyperplane to obtain 3-D Mandelbrot set.

It occurred to me that there exists an amazingly simple manner to generate analogs of the Mandelbrot sets in 3 dimensions. One still considers maps of the complex plane to itself but assumes that the analytic function depends on one complex parameter c and one real parameter b so that the parameter space spanned by pairs (c,b) is 3-dimensional. Consider two examples:

1. f: z?bz³ + z²+ c, b real,

   for which b=0 cross section gives Mandelbrot fractal. The first two iteration steps are z=0?c? bc³ + c²+ c ?...

2. f: z?z³ + bz²+ c, b real, suggested by Thom's catastrophe theory. The first two iteration steps are z=0?c? c³ +bc²+ c ?...

Both options might produce something interesting. One could also construct dynamical 3-D Mandelbrots by allowing b to be complex and interpreting real or imaginary part of b of the modulus of b as time coordinate.

One can deduce some general features of these fractals.

1. These examples represent 3-D generalization of Multibrot fractals defined by z&rarr; z^d (this video demonstrates how Multibrot fractals vary as the integer d varies). Obviously these sets look qualitatively very much like Mandelbrot fractal. Hence each b=constant cross section of these 3-D fractals is expected must have the qualitative look of Mandelbrot fractal.

2. The boundary of Mandelbrot fractal is now a 2-D surface of 3-D space spanned by points (b,c) and corresponds to the points (b,c) for which the iteration of f applied to z=0 does not lead to infinity. The line c=0 belongs in both cases to the fractal since z=0 is fixed point of iteration in this case. As the value of b grows the cross section must get increasingly thinner for both negative and positive values of b with large enough magnitude rougly as c < b^-1/3 for the first case and c<b^-1/2 by looking what happens in first iterations. The direction of b is clearly in a special position reflecting the 2-D character of the basic iteration process. This holds true for any analytic functions.

3. To guess what the boundaries of these fractals could look like, one can try to imagine what one obtains as one builds a small pile of slightly deformed 2-D Multibrot fractals above b =0 Multibrot fractal or any piece of it (see the images here). One expects complex caves inside caves structure. The caves are expected to be high. Also a fractal hierarchy of stalagmite like structures is expected. Their tips would reflect a disappearence or appearance of a new structure in 2-D Multibrot fractal as the height coordinate b varies. The objection is that the boundary curve in the case of Mandelbrot fractal- and maybe also of Multibrots- consists of single component. Stalagmites would correspond to an appearence of disjoint component to the curve would not be possible. In a given resolution however there are always invisible thin hairs connected to thicker regions so that only apparent stalagmites due to finite resolution would be possible. Being inside b> 0 part of the fractal might create same spiritual feelings that one experiences in Gothic Cathedral.

4. It is good to hear what pessistic has to say. In high enough b-resolution the variation of the Multibrot curve as a function of b could be so slow that the fractal looks like a cylinder having Multibrot as intersection and there would be no fractality in b-direction. Therefore the question is whether the fractality in c-plane implies fractality in b-direction. In other words, is the fractal curve for a fixed value of b critical against variations of b.

3-D graphics skills would be needed to disetangle the huge complexity of the situation. Probably this is far from a trivial challenge. In any case, the visualization as a pile of 2-D Multibrots could be used in the construction of these fractals and would make possible discretization in b-direction and the use of existing 2-D algoriths as such. Maybe some Mandelbrot artist might try look what these fractals look like.

Addition: Paul Nylander kindly reproduced a picture of az³+bz²+c type 3-D fractal. As you see it is from outside and the local cylinder likeness is obvious from the picture which suggests that there is no genuine fractality in b-direction. This hides the complexity of the boundary of Mandelbrot which can be seen only by going inside.

He also told that also Rudy Rucker has suggested a similar approach.

Also Janne (see the discussion) worked with 2-D sections (see the discussion section and his graph) and thinks that fractality in b-dimension is not true. He sent also a nice animation about quaternionic Julia set which also has a local cylinder like structure. This animation demonstrates the 3-D complexity below the surface hidden by the views from outside by using 2-D cross sections. I would guess that this animation catches much about the (b,c)-fractals. The graphics allowing to see the fractal from the point of view of observer at floor of infinitely high fractal Gothic Cathedral might be a fascinating challenge;-).

At the eve of LHC

Linear hadron collider (LHC) at CERN is expected to start soon to produce data about particle physics at energies much higher than reached hitherto. The colliding protons would have cm energy of 7 TeV per particle. Also collisions of lead nuclei with energy of 574 TeV per nucleus will be studied. The startup of LHC is a remarkable event and there has been a lot of fuss about it during last weeks in blogs. The reader can get some idea about the scale of the experimental apparatus by looking at the pictures at Tommaso's page. LHC makes me optimistic about the ultimate fate of humankind: the ability of humankind to co-operate to create something like looks like a miracle.

LHC is often seen as a kind of savior of the particle physics. As the results from LHC finally start to flow all questions will be answered, a new wave of creativity will propagate through the theoretical physics community, and the deep principles behind M-theory will be finally understood. From TGD perspective these expectations look somewhat over-optimistic, reflecting what I see as a distortion of perspective. This distortion is probably due to the basic belief that everything -including of course also consciousness- reduces to the dance of elementary particles which in turn reduces to the wiggling of the tiny Planck scale strings.

The basic distinction between TGD and more standard theories is indeed the replacement of the Planck length scale reductionism with a fractal view about Universe (many-sheeted space-time, p-adic length scale hierarchy and dark matter hierarchy corresponding to a hierarchy of Planck constants). As a consequence, TGD predicts a lot of new particle physics in all length scales instead of some exotic effects in LHC. There indeed exists a rich spectrum of anomalies giving support for this physics and the book p-Adic Length Scale Hypothesis and Dark Matter Hierarchy is about these predictions. Some of these anomalies (leptopion anomaly) date back to seventies and could have been a treasure trove of new ideas for young and imaginative theoreticians. Sadly, the colleagues who have decided that low energy physics (that is physics below string length scale) reduces to some GUT cannot but forget these findings.

The new physics derives from several sources: TGD based explanation of elementary particle quantum numbers differing in several aspects from the picture provided by standard model and its standard generalizations, the fractal view about particle massivation based on p-adic thermodynamics and p-adic length scale hypothesis, and the identification of dark matter in terms of a hierarchy of Planck constants. In the following I discuss these predictions. A warning is in order here: "predicts" is too strong expression but I will use it as a shorthand for a longer expressions involving a lot of "assuming that"'s.

The new physics related to p-adic thermodynamics and p-adic length scale hypothesis

The basic prediction of a proto type GUT is a desert beginning at intermediate boson mass scale and continuing up to the unification scale about 10^-4 Planck masses. In TGD framework this brave hypothesis looks like an over-simplification - to put it mildly;-). What interests me is not this hypothesis which is in sharp contrast with all wisdom that human kind has gained after Newton's times but the phychology behind it. This desire to have reached the final knowledge tells something very deep about conciousness itself.

1. p-adic thermodynamics combined with p-adic length scale hypothesis replaces Higgs mechanism as a mechanism of particle massivation. The geometric realization of zero energy in terms of a hierarchy of diamonds identified as intersections of future and past directed lightcones gives a good justification for the latter. This makes possible scaled up variants of particles with masses coming as half octaves of the basic mass. There are surprisingly many pieces of evidence for this prediction. These scaled up versions of quarks are essential for TGD based mass formula for the light hadrons. The evidence for two Ω_b:s with 100 MeV mass difference could be a second example about this phenomenon and TGD explains the mass difference within experimental uncertainties. No one of course takes seriously TGD based explanation and there is a hot debate going on between CDF and D0 collaborations about this (see Tommaso's posting).

2. Scaled up copies of entire hadron physics are possible (see this). This means that the space-time sheet of gluons, which corresponds to Mersenne prime M₁₀₇ for ordinary hadron physics can be replaced with that corresponding to some other Mersenne prime, for instance M₈₉. For this particular scaled up copy of hadron physics QCD Lambda is scaled up by a factor 512. p-Adically scaled up variants of quarks would topologically condense at the gluonic space-time sheet characterized by M₈₉. There are good reasons to expect that this space-time sheet is dark (large hbar would reduce the value of color coupling strength and perturbation theory would work in the resulting anyonlike phase, see this). One can imagine also hadron physics in electron length scale characterized by M₁₂₇ and the proposal is that this hadron physics is highly relevant for nuclear physics (see this).

3. Higgs like particles are possible but they can give only a small additional contribution to particle masses. The experimental determinations seem to converge to two different Higgs masses which forces to ask whether Higgs might exist in two different p-adic mass scales depending on the type of reaction in which it is detected. I have briefly discussed this possibility here and in this posting with predictions for the masses of Higgs.

TGD based explanation of standard model quantum numbers

Consider next the new physics relates to TGD based explanation of standard model quantum numbers (for a summary see this). The geometry of H=M⁴× CP₂ allows a geometrization of standard model quantum numbers and family replication phenomenon has a topological explanation.

1. The conserved quark and lepton numbers correspond to different chiralities for H-spinors so that proton is automatically stable (apart from a possible decay to lighter scaled down variants of hadrons with decay signatures totally different from those for GUT proton decays).

2. Color corresponds to CP₂ partial waves in TGD framework. Super-conformal invariance is essential for getting correct correlation between electroweak and color quantum numbers (see this). If this is the case also in reality, paper basket will become the proper place for most of the phenomenology done during the 35 years in GUTs and string models after the emergence of the first GUT around 1974. Colored excitations of leptons and quarks are possible. Lepto-hadron hypothesis relates to this and the evidence for colored excitations of electrons, muons, and tau lepton have been accumulating (see this). The first evidence dates back to seventies and last evidence I have discussed quite recently (see this).

3. The third new element is the topological explanation of family replication phenomenon in terms of the topology of wormhole throat serving as carrier of elementary particle numbers. Fermions correspond to single wormhole throat (CP₂ type vacuum extremal topologically condensed to space-time sheet with Minkowki signature of the induced metric). Gauge bosons correspond to pairs of wormhole throats assignable to wormhole contacts connecting two space-time sheets. There is good argument explaining why only the three lowest genera (sphere,torus, and sphere with two handles) correspond to light fermions (see this).

Consider now the basic predictions at LHC.

1. The topological explanation of family replication phenomenon implies dynamical SU(3) symmetry with fermionic triplets identified as three genera e,μ,τ and corresponding neutrinos and three generations of U and D type quarks. In the bosonic sector one obtains wormhole contacts in octet and singlet representations of this SU(3). The known gauge bosons correspond to singlet and octets would be waiting for their discovery. Characteristic flavor violations are predicted. There are some indications for the the existence of some members of octet counterparts of Z⁰ (see Tommaso's posting, my own posting, and this).

2. Colored excitations of ordinary quarks and are in principle possible and one cannot exclude their presence at even lower energies. One can imagine even scaled up variants of ordinary leptons and their colored excitations. It would require a collective effort to say something more concrete about the masses of these higher color excitations.

   Space-time supersymmetry in TGD framework

   The TGD based views about super-conformal symmetry and space-time supersymmetry are in many aspects different from those of string models and standard SUSYs. Majorana spinors are replaced by Weyl spinors and the oscillator operators assignable to the induced spinor fields generate a super symmetry algebra, which can be infinite-dimensional. This algebra is also associated with super-conformal algebras of quantum TGD and it remains to be seen what the implications are. The standard formalism of SUSY theories fails in TGD context but during last weeks I have been working with a formalism applying in this kind of situation and proposed also a bilocal QFT type description of gravitational interactions

   Positive energy chiral super-fields are Taylor series analytic in theta parameters. Negative energy chiral super fields are obtained by replacing thetas with derivatives with respect to thetas and by applying this operator to the product of all thetas. Vector super fields are normal ordered polynomials of derivatives of thetas and thetas operators slashed between positive and negative energy super-fields. They represent a very straighforward generalization of gauge potentials and minimal substitution fixes the interactions with chiral super fields. The corresponding kinetic term emerges via chiral loops. The model predicts that only the monomials of theta parameters with degree d=2 (spins J=0,1/2,1) behave as ordinary particles whereas higher monomials define forces which correspond to confining potentials below Compton length and contact interactions in longer length scales (see this and this).

   The simplest super-symmetry breaking scenario assumes same mass formulas for the members of the super-multiplet. p-Adic length scales can how differ so that an elegant mechanism of super-symmetry breaking results (see this). The masses of super-partners of known particles are obtained by scaling them with some power k of 2^1/2: unfortunately the possible values of k cannot be predicted yet. This is however a strong and easily testable prediction because of the exponential sensitivity to k. The early evidence for supersymmetry about which Tommaso told in his posting allows fix the masses of sHiggs,selectron, and superpartner of Z⁰ (see this and this).

Dark matter and hierarchy of Planck constants

The ultra-reductionistic belief is that any progress in the understanding of fundamental interactions requires increasingly higher energies and that galaxy sized accelerators are needed to to reach the physics at unification energies.

The TGD inspired generalization of quantum theory by introducing an infinite hierarchy of Planck constants and the interpretation of dark matter as a hierarchy of phases with non-standard values of Planck constant changes this view dramatically. The geometrical realization is in terms of the book like structure of the generalized 8-D imbedding space with pages characterized partially by the values of Planck constant. Particles at different pages of the book are dark relative to each other in the sense of having no local interactions. Interactions via classical fields and exchange of gauge bosons leaking between different pages and thus suffering phase transition changing Planck constant are possible. Hence dark matter would not be so dark as usually thought. Zoomed up variants of ordinary particles with arbitrary long (or short) Compton lengths but same mass are possible. The phases with a large value of Planck constant correspond to long length scales and the most important applications are in biology and even in astrophysics.

1. LHC does not seem to the best place to search for dark matter and dark energy unless phases with Planck constant smaller than its standard value are also present. In TGD framework dark energy would correspond to magnetic flux tubes with gigantic values of Planck constant and Compton lengths of ordinary particles would be cosmological so that LHC would be the last place to search for dark energy.

2. It might be however possible to study the phase transition from dark to ordinary matter at LHC if confined valence quarks in ordinary hadrons are in a phase with large Planck constant. This would reduce the value of color coupling strength proportional 1/hbar and guarantee that perturbation theory works for the resulting anyonlike phases (see this). The phase transition to a non-confining phase might explain the findings of RHIC summarized using the notion of color glass (see this). This liquid like phase would be quantum critical phase intermediate between confined phase with large have and quark gluon plasma with standard value of hbar. The collisions of lead nuclei at LHC are expected to provide further information in this respect.

3. There is also the hope that LHC might help to understand the weakness of gravity. Some variants of string model predict large extra dimensions and even production of mini black holes at LHC. TGD does not support these hopes. The new physics implied by the gigantic values of gravitational Planck constant predict becomes manifest in astrophysical length scales. Allais effect is one possible manifestation of this new gravitational physics (see this). Quite recent measurements from LIGO have reached the resolution which allows to detect the gravitational flux from some objects and the findings suggests that the detected graviton flux is below the predicted value. The possible failure to detect gravitons might be due to the fact that gravitons arrive as large hbar gravitons having very larges energies and decay to bunches of ordinary gravitons interpreted as external perturbations.

What after LHC?

TGD replaces reductionism with fractality. This challenges the usual belief that the progress in particle physics requires higher collision energies and larger accelerators. There are two kind of fractalities: p-adic fractality meaning a hierarchy of mass and corresponding length scales and fractality associated with hierarchy of Planck constants meaning a hierarchy of length scales with a fixed mass scale.

1. The p-adically scaled up versions of hadron physics and lepto-hadron physics with both large and low mass scales are possible in TGD Universe. This might make it possible to gain information about QCD type physics at ultrahigh energies by studying scaled down counterparts of the ordinary hadron physics at low energies. Gigantic accelerators would not be needed anymore.

2. Large hbar means small dissipation and large hbar table top particle accelerators utilizing strong electric fields might be possible some day. The recent discovery of Fermi telescope about gamma rays of .511 MeV emerging from lightnings gives support for both electro-pion production and dissipationless acceleration mechanism in large hbar phase (see this). The strong electric fields created by the positively charged thunder cloud would induce an acceleration of dark electrons to relativistic energies and their collisions would take place at the cloud producing electro-pions decaying to pairs of gamma rays. If thunder cloud can act as large hbar particle accelerator, it should be possible for us to build it. If colleagues were some day psychologically mature to to take TGD seriously, LHC could become one of the last dinosaurs and experimental particle physics might transform from an activity of gigantic Pentagon like organizations to something possible for gifted and devoted individuals in home laboratory.

Is the QFT type description of gravitational interactions possible in TGD framework?

During the last month I have developed a formulation for the super-symmetric QFT limit of quantum TGD based on the generalization of chiral and vector super-fields appropriate for N=∞ supersymmetry. The next question concerns the possibility to describe gravitational interactions using QFT like formalism. The physical picture is following.

1. In TGD Universe graviton is necessarily a bi-local object and the emission and absorption of graviton are bi-local processess involving two wormhole contacts: a pair of particles rather than single particle emits or absorbs graviton. This is definitely something new and defies a description in terms of QFT limit using point like particles. Graviton like states would be entangled states of vector bosons at both ends of string and propagating collinearly so that gravitation could be regarded as a square of YM interactions in a rather concrete sense.

2. The connection with strings is via the assignment of wormhole contacts at the ends of a stringy curve. Stringy diagrams would not however describe graviton emission. Rather, a generalization of the vertex of Feynman diagram would be in question in the sense that three string world sheets would be glued together along their 1-dimensional ends in the vertex. This generalizes similar description for gauge interactions using Feynman diagrams. In the microscopic description point like particles are replaced with 2-D partonic surfaces so that in gravitational case one has stringy 3-surfaces at vertices.

This picture provides strong constraints if one wants to describe gravitation using a generalization of QFT type theory.

1. At QFT limit one can hope a description as a bi-local process using a bi-local generalization of the QFT limit so that stringy degrees of freedom need not be described explicitly. There are hopes about success, since these degrees of freedom have been taken into account in the spectrum of modes of the induced spinor field and reflect themselves as quantum numbers labeling fermionic oscillator operators. Also modified gamma matrices feed information about space-time surface to the theory.

2. The hypothesis is that a generalization of the super-symmetric QFT limit to its bi-local variant allows to describe the emission and absorption of gravitons. Also now the kinetic part for gravitational action emerges so that only the counterpart of T^αβδ g_αβ interaction term appears in the fundamental action.

3. The idea about gravitation as a square of YM theory generalizes in the sense that graviton is a pair of gauge bosons at the ends of string d and bosonic propagators determine graviton propagator. The fact that bosonic loop is dimensionless implies that IR cutoff defined by the size of largest CD must be actively involved. The requirement of gauge invariance fixes uniquely the form of the bi-local gravitational action. The form is remarkably simple.

4. A huge number of graviton like states together with their super-partners are predicted (super-symmetry is N=∞ SUSY in the general case!). Most of them are massive. The ordinary graviton must correspond to electroweak gauge group U(1) for which charge is essentially fermion number (quantum counterpart of Kähler gauge potential of CP₂). This means that gravitational Weinberg angle vanishes.

I feel it fair to say that the addition of the measurement interaction term to the modified Dirac action has led to a profound understanding of Quantum TGD at mathematical level - consider only the generalization of the super-fields as an example. It remains to be seen how far-reaching the implications are for the more technical understanding of quantum TGD proper.

For more details the interested reader can consult the five-page excerpt Is the QFT type description of gravitational interactions possible? from the new chapter Does the QFT Limit of TGD Have Space-time Super-Symmetry? of the book "Towards M-Matrix".

QFT limit of TGD and space-time supersymmetry

The understanding of the QFT limit of TGD is now a twenty year old challenge. How to feed information about classical physics characterized by Kähler action has been the basic question. The conflict with Poincare invariance destroying all hopes about practical calculations looks unavoidable. Zero energy ontology and the addition of measurement interaction depending on momenta and color charges to modified Dirac action led to a resolution of this dilemma. The point is that the momenta act on the tip of causal diamond rather than space-time coordinates, which therefore appear as external parameters like the couplings in Hamiltonian. QFT in infinitely slowly varying background fields is the counterpart in ordinary QFT but in TGD there is no need to pose this restriction. One obtains for each space-time point its own QFT limit. A weighted integral over amplitudes corresponding to these limits is performed in analogy with what is done in the theory of spin glasses at the level of statistical physics. As a matter fact, TGD Universe is 4-D quantum spin glass.

This led also to the realization that space-time supersymmetry can be realized at the fundamental level as anticommutation relations of the fermionic oscillator operators associated with the modes of the induced spinor field. The next task was to construct the counterpart of SUSY QFT limit for TGD. Here the problem was that the value of N for the super-symmetry in question is large or even infinite so that the standard notions of chiral and vector superfields fail. N=∞ limit how forced to find the correct formalism. It is considerably simpler than the standard one since chiral condition is replaced with Grassman analyticity. Bosonic emergence is unavoidable in this framework relying strongly on zero energy ontology and on the identification of fermions (bosons) and their super-partners in terms of wormhole throats (contacts). The finiteness of the theory follows by extending the standard argument stating that fermion and sfermion loops cancel each other in SUSY. One prediction is a hierarchy of exotic particles with propagators behaving like 1/pⁿ. For boson exchanges with n=2m the corresponding interaction potentials behave like exp(-mr)r^2m-3. For massless case n=4 gives linear confining force possibly highly relevant for QCD. Also the information about space-time surface (corresponds to maximum of Kähler function) can be feeded to the theory by using modified gamma matrices defined by Kähler action without losing Poincare invariance.

I dare to regard the resulting formalism combining the ideas about the generalization of twistors and about bosonic emergence with other basic ideas of TGD as the final breakthrough. The resulting formalism makes possible concrete calculational recipes. During last months I have often experienced a deep and to me strange feeling of relief. After 32 years of work my great mission has been realized to a high extent! The practical part of me of course starts to worry whether this 59 old me can survive this kind of strange feelings of relief without total collapse;-)? Probably so! There is so much to do and one thing to do is to look how much of this formalism generalizes to TGD proper.

I attach below also the abstract of the new chapter Does the QFT Limit of TGD Have Space-time Super-Symmetry?, which can be found in the book "Towards M-Matrix".

Contrary to the original expectations, TGD seems to allow a generalization of the space-time super-symmetry. This became clear with the increased understanding of the modified Dirac action. The introduction of a measurement interaction term to the action allows to understand how stringy propagator results and provides profound insights about physics predicted by TGD.

The appearance of the momentum and color quantum numbers in the measurement interaction couples space-time degrees of freedom to quantum numbers and allows also to define SUSY algebra at fundamental level as anti-commutation relations of fermionic oscillator operators. Depending on the situation a finite-dimensional SUSY algebra or the fermionic part of super-conformal algebra with an infinite number of oscillator operators results. The addition of a fermion in particular mode would define particular super-symmetry. Zero energy ontology implies that fermions as wormhole throats correspond to chiral super-fields assignable to positive or negative energy SUSY algebra whereas bosons as wormhole contacts with two throats correspond to the direct sum of positive and negative energy algebra and fields which are chiral or antichiral with respect to both positive and negative energy theta parameters. This super-symmetry is badly broken due to the dynamics of the modified Dirac operator which also mixes M⁴ chiralities inducing massivation. Since righthanded neutrino has no electro-weak couplings the breaking of the corresponding super-symmetry should be weakest.

The question is whether this SUSY has a realization as a SUSY algebra at space-time level and whether the QFT limit of TGD could be formulated as a generalization of SUSY QFT. There are several problems involved.

1. In TGD framework super-symmetry means addition of fermion to the state and since the number of spinor modes is larger states with large spin and fermion numbers are obtained. This picture does not fit to the standard view about super-symmetry. In particular, the identification of theta parameters as Majorana spinors and super-charges as Hermitian operators is not possible.

2. The belief that Majorana spinors are somehow an intrinsic aspect of super-symmetry is however only a belief. Weyl spinors meaning complex theta parameters are also possible. Theta parameters can also carry fermion number meaning only the supercharges carry fermion number and are non-hermitian. The the general classification of super-symmetric theories indeed demonstrates that for D=8 Weyl spinors and complex and non-hermitian super-charges are possible. The original motivation for Majorana spinors might come from MSSM assuming that right handed neutrino does not exist. This belief might have also led to string theories in D=10 and D=11 as the only possible candidates for TOE after it turned out that chiral anomalies cancel.

3. The massivation of particles is basic problem of both SUSYs and twistor approach. The fact that particles which are massive in M⁴ sense can be interpreted as massless particles in M⁴×CP₂ suggests a manner to understand super-symmetry breaking and massivation in TGD framework. The octonionic realization of twistors is a very attractive possibility in this framework and quaternionicity condition guaranteing associativity leads to twistors which are almost equivalent with ordinary 4-D twistors.

4. The first approach is based on an approximation assuming only the super-multiplets generated by right-handed neutrino or both right-handed neutrino and its antineutrino. The assumption that right-handed neutrino has fermion number opposite to that of the fermion associated with the wormhole throat implies that bosons correspond to N=(1,1) SUSY and fermions to N=1 SUSY identifiable also as a short representation of N=(1,1) SUSY algebra trivial with respect to positive or negative energy algebra. This means a deviation from the standard view but the standard SUSY gauge theory formalism seems to apply in this case.

5. A more ambitious approach would put the modes of induced spinor fields up to some cutoff into super-multiplets. At the level next to the one described above the lowest modes of the induced spinor fields would be included. The very large value of N means that N > 32 SUSY cannot define the QFT limit of TGD for higher cutoffs. One must generalize SUSYs gauge theories to arbitrary value of N but there are reasons to expect that the formalism becomes rather complex. More ambitious approach working at TGD however suggest a more general manner to avoid this problem.
   1. One of the key predictions of TGD is that gauge bosons and Higgs can be regarded as bound states of fermion and antifermion located at opposite throats of a wormhole contact. This implies bosonic emergence meaning that it QFT limit can be defined in terms of Dirac action. The resulting theory was discussed in detail in and it was shown that bosonic propagators and vertices can be constructed as fermionic loops so that all coupling constant follow as predictions. One must however pose cutoffs in mass squared and hyperbolic angle assignable to the momenta of fermions appearing in the loops in order to obtain finite theory and to avoid massivation of bosons. The resulting coupling constant evolution is consistent with low energy phenomenology if the cutoffs in hyperbolic angle as a function of p-adic length scale is chosen suitably.

   2. The generalization of bosonic emergence that the TGD counterpart of SUSY is obtained by the replacement of Dirac action with action for chiral super-field coupled to vector field as the action defining the theory so that the propagators of bosons and all their super-counterparts would emerge as fermionic loops.

   3. The huge super-symmetries give excellent hopes about the cancelation of infinities so that this approach would work even without the cutoffs in mass squared and hyperbolic angle assignable to the momenta of fermions appearing in the loops. Cutoffs have a physical motivation in zero energy ontology but it could be an excellent approximation to take them to infinity. Alternatively, super-symmetric dynamics provides cutoffs dynamically.

6. The condition that N=∞ variants for chiral and vector superfields exist fixes completely the identification of these fields in zero energy ontology.
   1. In this framework chiral fields are generalizations of induced spinor fields and vector fields those of gauge potentials obtained by replacing them with their super-space counterparts. Chiral condition reduces to analyticity in theta parameters thanks to the different definition of hermitian conjugation in zero energy ontology (q is mapped to a derivative with respect to theta rather than to [`(q)]) and conjugated super-field acts on the product of all theta parameters.

   2. Chiral action is a straightforward generalization of the Dirac action coupled to gauge potentials. The counterpart of YM action can emerge only radiatively as an effective action so that the notion emergence is now unavoidable and indeed basic prediction of TGD.

   3. The propagators associated with the monomials of n theta parameters behave as 1/pⁿ so that only J=0,1/2,1 states propagate in normal manner and correspond to normal particles. The presence of monomials with number of thetas higher than 2 is necessary for the propagation of bosons since by the standard argument fermion and scalar loops cancel each other by super-symmetry. This picture conforms with the identification of graviton as a bound state of wormhole throats at opposite ends of string like object.

   4. This formulation allows also to use modified gamma matrices in the measurement interaction defining the counterpart of super variant of Dirac operator. Poincare invariance is not lost since momenta and color charges act on the tip of CD rather than the coordinates of the space-time sheet. Hence what is usually regarded as a quantum theory in the background defined by classical fields follows as exact theory. This feeds all data about space-time sheet associated with the maximum of Kähler function. In this approach WCW as a Kähler manifold is replaced by a cartesian power of CP₂, which is indeed quaternionic Kähler manifold. The replacement of light-like 3-surfaces with number theoretic braids when finite measurement resolution is introduced, leads to a similar replacement.

   5. Quantum TGD as a "complex square root" of thermodynamics approach suggests that one should take a superposition of the amplitudes defined by the points of a coherence region (identified in terms of the slicing associated with a given wormhole throat) by weighting the points with the Kähler action density. The situation would be highly analogous to a spin glass system since the modified gamma matrices defining the propagators would be analogous to the parameters of spin glass Hamiltonian allowed to have a spatial dependence. This would predict the proportionality of the coupling strengths to Kähler coupling strength and bring in the dependence on the size of CD coming as a power of 2 and give rise to p-adic coupling constant evolution. Since TGD Universe is analogous to 4-D spin glass, also a sum over different preferred extremals assignable to a given coherence regions and weighted by exp(K) is probably needed.

7. In TGD Universe graviton is necessarily a bi-local object and the emission and absorption of graviton are bi-local processes involving two wormhole contacts: a pair of particles rather than single particle emits graviton. This is definitely something new and defies a description in terms of QFT limit using point like particles. Graviton like states would be entangled states of vector bosons at both ends of stringy curve so that gravitation could be regarded as a square of YM interactions in rather concrete sense. The notion of emergence would suggest that graviton propagator is defined by a bosonic loop. Since bosonic loop is dimensionless, IR cutoff defined by the largest CD present must be actively involved. At QFT limit one can hope a description as a bi-local process using a bi-local generalization of the QFT limit. It turns out that surprisingly simple candidate for the bi-local action exists.

The latest discovery of Fermi telescope: electro-pions from lightnings?

Lubos Motl wrote a posting about the most recent discovery of Fermi space telescope.

It was discovered already years ago that lightnings are accompanied by gamma rays. For instance, the strong electric fields created by a positively charged region of cloud could accelerate electron from both downwards and upwards to this region. The problem is that atmosphere is not empty and dissipation would restrict the energies to be much lower than gamma ray energies which are in MeV range. Note that the temperatures in lightning are about 3× 10⁴ K and correspond to electron energy of 2.6 eV which is by a factor 10⁵ smaller than electron mass and gamma ray energy scale!

Situation changes if dissipation is absent so that electrons are accelerated without any energy losses. The alert reader of earlier postings can guess what I am going to say next;-)! Electrons reside in large hbar quantum phase at magnetic flux tubes so that dissipative losses are small and electrons can reach relativistic energies. This is the explanation that I provided years ago for the gamma rays associated with lightnings.

Fermi however observed also something completely new. There is also a peaking of gamma rays around energy .511 MeV. This requires a different mechanism. One such mechanism is a decay of some exotic particle to two gamma rays produced in a collision of electrons. This brings in exotic particles that I call lepto-hadrons. They represent one of the basic predictions of quantum TGD distinguishing it from standard model and its standard extensions (also string models). Basically color excited states of leptons are in question forming color bound states about which simplest examples are leptopions, in particular electro-pion whose mass is just twice the electron mass so that its decays wold produce gamma rays with energy .511 MeV. Leptohadron hypothesis is discussed extensively here, and the article predicting the particles was published already in 1990 (after this publishing became in practice impossible due to the censorship by string hegemony and blackmailing activities of finnish colleagues).

Amusingly, just year ago there was an intense debate going on about the evidence discovered by CDF for a new particle (see this and the subsequent posts). This particle could be identified as one of the exotic particles predicted by leptohadron hypothess. The interpretation was that CDF had found evidence for colored excitation of τ lepton and associated leptopion like particles. There was an intense debate and - quite predictably - the anomaly was forgotten after the explanation based on Nima Arkani Hamed's theory failed (Lubos already predicted Nobel prize for Nima!) and the only quantitative and working explanation had turned out to be the one based on TGD. This also led to oppressive actions in Finland: Helsinki University did not allow anymore to keep my homepage in University computer anymore and refused also to redirect visitors to the new address. Situation had got really dangerous and local powerholders had simply no other choice than the tactic of burned bridges applied to web links.

After this short sidetrack to the sociology of science (charming-isn't it?!) let us return to the leptopion associated with electron - electropion. It has mass slightly above 2m_e and decays to a pair of gamma rays with energy .511 MeV. The first evidence for leptopions was found surprisingly early- already in seventies in heavy ion collisions- just above the Coulomb wall. I constructed a model for these events around 1990. By general anomaly considerations it became clear that electropions are created when heavy nuclei collide near Coulomb wall. What is essential is the presence of mutually non-orthogonal electric and magnetic fields during the collision. The production amplitude is essentially the Fourier transform of the "instanton density" E·B. There are many other anomalies supporting this model- in particular, orthopositronium decay anomaly. There is also evidence for muo-pions and CDF provided it for tau-pions. All these anomalies have been forgotten- presumably for the simple reason that they do not fit to standard model and its standard extensions, which have become the prevailing ideology.

But experiment strikes back mercilessly! Now it seems that Fermi finds leptopions in lightning strikes! This must be a horrible nightmare for a theoretician firmly decided what can exist and what not! If these disgusting electro-pions are there, collisions of highly energetic particles lasting for time of about hbar/MeV are expected. The natural candidates for the colliding charged particles are electrons. The center of mass system -the system in which total momentum of colliding electron pair vanishes- should be in good approximation at rest with respect to Fermi space telescope. Otherwise the energy of gamma rays would be higher or lower than .511 MeV. The only possibility that I can imagine is that the second electron comes from below and second from above the positively charged region of the thunder cloud. Both arrive as dark electrons with a large value of hbar and are accelerated to relativistic energies since dissipation is very small. They could collide as dark electrons (the more probable option as will be found below) or suffer a phase transition transforming them to ordinary electrons before the collision. Electropion coherent state is created in the strong E·B created for a a period of time of order hbar₀/MeV. This state annihilates rapidly to pairs of gamma rays which are ordinary or transform to ordinary ones depending on whether electrons where dark or not.

What the phase transition of dark electrons to ordinary electrons means, needs some explaining. The generalized imbedding space is obtained by glueing almost copies of 8-D imbedding space M⁴×CP₂ along their common back to get a book like structure. Particles at different pages of the book are dark with respect to each other in the sense that they have no local interactions. This is enough to explain what is actually known about dark matter. Particles at different pages can however interact via classical fields and photon exchange (for instance). The phase transition of electron from dark to visible form preceding the collision of dark electrons would simply mean the leakage from large hbar page to the "visible" page with ordinary value of Planck constant.

Alert reader might be ready to ask the obvious question. Why not to test the hypothesis in laboratory? It should not be too difficult to allow two electrons to collide with a relativistic energy and find whether gamma pairs with energy .511 MeV are produced in rest system. Maybe gamma ray pairs have been missed for some reason? If not (the probable option), then colored electrons and leptopions are always dark. This would explain why the colored leptons do not contribute to the decay widths of weak gauge bosons which pose very strong constraints for the existence of light exotic particles.

For more details about leptohadron hypothesis see the chapter Recent Status of Leptohadron Hypothesis of "p-Adic Length Scale Hypothesis And Dark Matter Hierarchy".

An experimental breakthough in quantum understanding of telepathy?

Telepathy by quantum entanglement is one of the basic ideas of TGD inspired consciousness. This requires some new physics.

1. Macroscopic quantum coherence is needed in scales much longer than standard quantum mechanics allows. Hierarchy of Planck constants makes this certainly possible but one cannot exclude the possibility that mere magnetic flux tube like structures is enough: I do not believe in this option.

2. The idea that entangelment of selves gives rise to telepathy is plagued by a problem: in TGD framework entanglement means a loss of consciousness at the level of both selves since it is the fusion of selves which becomes the conscious entity! The subselves of self can however entangle if this entangelement is below the measurement resolution of selves and therefore does not lead to a loss of consciousness at the level of selves. Selves experience the fusion of subselves as a fusion of corresponding mental images to single shared mental image. Stereo consciousness is essentially in question. Stereo vision represents only one particular example of this. During sleep we ourselves could be the mental images fusing to a single gigantic stereo mental image of a higher level conscious entity. This mental image would represent "human condition".

3. Any quantal idea of TGD must have also geometrical space-time counterpart by quantum classical correspondence. The geometrical correlates for selves are space-time sheets. Subselves correspond to smaller space-time sheets topologically condensed at those of selves. The geometrical correlate for the entanglement are flux tubes connecting the space-time sheets associated with the selves. In the case of subselves these subselves are too "thin" to be visible in the resolution of selves.

Only fifteen years after the explicit formulation of the idea situation seems to be mature for the experimental verification as one learns from Hammock Physicist, a blog belonging to Scientific Bloggin. James Randi Educational Foundation might be forced to pay the one Megadollar prize that it has promised to anyone who can experimentally demonstrate some paranormal power.

The experimental arragement is simple. The two subject persons - not Alice and Bob at this time;-) but Cora and Reid - two postdocs in theoretical physics - color the 3×3 squares of a larger square. Reid follows the rule that each column contains an odd number of red squares (that is 1 or 3). Cora obeys the rule that each row contains an even number of red squares. Since the total number of red squares is odd in the first case and even in the second case, colorings satisfying both rules are not possible. It this were the case Cora and Reid could make an agreement about making only these optimal colorings in a fixed order for instance.

Each coloring process equals three steps and for a given step the row (column) to be colored is selected in random manner. Alice and Bob try to perform the coloring in such a manner that the intersection of the row of Cora and column of Reid has same color. The optimal coloring strategy would yield a success rate of 89 per cent on the average. In the preliminary test Alice and Bob involving 40 turns were however able to reach 100 per cent success rate! The probability of this using optimal strategy is 1 percent.

I must say that I am shocked. Only 15 years after I began to work systematically with TGD inspired theory of consciousness, one of its most spectacular predictions might be demonstratable by an extremely simple experiment having fantastic implications -not only for our views about consciousness and biology - but also for quantum theory itself.

The eight books about TGD inspired quantum theory of consciousness and biology can be found at my homepage.

Is the perturbation theory based on TGD inspired definitions of super fields UV finite?

In the case of infinite-dimensional super-space the definition of the super-fields is not quite straightforward since the super-space integrals of finite polynomials of theta parameters always vanish so that the construction of super-symmetric action as an integral over super-space would give a trivial result. For chiral fields the integrals are formally non-vanishing but in the case that the super-field reduces to a finite polynomial of theta at y^μ=0 the non-vanishing terms in real Lagrangian involve the action of an infinite number of operators D^cα_c (^c denotes overline for D and _c dot for Weyl spinor index) implying the proportionality to an infinite power of momentum which vanishes for massless states. It seems that one should be able to add in a natural manner terms which are obtained as theta derivatives of the product of all theta parameters and that the action should consist of the products of the terms associated with mononomials of theta and monomials of derivatives with respect to theta parameters acting on the infinite product of theta parameters, call it X.

The fact that positive resp. negative energy vacuum is analogous to Dirac sea with negative resp. positive energy states filled suggests a remedy to the situation. This would mean that positive energy chiral field is just like its ordinary counterpart whereas negative energy chiral fields would be obtained by applying a polynomial of derivatives of theta to the product X=∏θ of all theta parameters. The theta integral of X is by definition equal to 1. In integral over theta parameters the monomials of theta associated with positive energy chiral field and negative energy chiral field would combine together and one would obtain desired action. In the following this approach is sketched. Devil lies in the details and detailed checks that everything works are not yet done.

This was what I wrote in the first version of this posting and I was right;-)! Devil indeed lies in the details! The calculations turned out to contain blunder (should I blame flu or market economy for the error or just admit that I have miserable calculational skills?;-)). It became clear that in TGD context the definition of super-covariant derivative reducing to ordinary partial derivative leads to much more elegant theory. In zero energy ontology super-symmetry reduces to analyticity with respect to theta parameters. In standard framework analyticity would not give kinetic terms to the chiral action but now the situation is different.

1. TGD variants of chiral super fields

Consider first the construction of chiral super-fields and of the super-counterpart of Dirac action.

1. Wormhole throats carry a collection of collinearly moving fermions with momentum appearing in the measurement interaction term identified as the total momentum. This suggests that kinetic terms behave positive powers of Dirac operator with one power for each theta parameter.

2. One must be careful with dimensions. The counterpart of Dirac operator is D = σ^k(p_k+Q_k)/M. The mass parameter M must be included for dimensional reasons and changes only the normalization of the theta parameters from that used earlier and changes the anti-commutation relations of the super-algebra in an obvious manner. The value of M is most naturally CP₂ mass defined as m(CP₂) = n× hbar₀/R, where R is the length of CP₂ geodesic and n is a numerical constant.

3. In the case of single wormhole throat one can speak about positive and negative energy chiral fields. Positive energy chiral fields are constructed as polynomials, and more generally, as Taylor series whereas negative energy chiral fields are obtained by mapping positive energy chiral fields to an operator in which each theta parameter θ is replaced with

   ∂_θD=∂_θσ^k(p_k+Q_k)/M .

   This operator acts in the product X of all theta parameters to give the negative energy counterpart of chiral field. The inclusion of sigma-matrices is necessary in order to obtain chiral symmetry at the level of H, in particular the counterpart of Dirac action. In the integral over all theta parameters defining the Lagrangian density the terms corresponding to mononomials M(θ,x) and their conjugates M(∂θ _cD^→,x) are paired and theta integrals can be carried out easily. Here → tells that the spatial derivatives appearing in D are applied to M.

4. There is an asymmetry between positive and negative energy states and the experience with the ordinary Dirac action Ψ^cD^→Ψ-Ψ^cD^←Ψ (^c denotes overline) suggests that one should add a term in which θ parameters are replaced with -Dθ so that space-time derivatives act on the positive energy chiral field and partial derivative ∂_θc appear as such. The most plausible interpretation is that the negative energy chiral field is obtained by replacing θs in the positive energy chiral field with ∂_θs and allowing to act on X. The addition of D would thus give rise to the generalization of the kinetic term.

5. Chiral condition can be posed and one can express positive energy chiral field in as an infinite powers series containing all finite powers of theta parameters whereas negative energy chiral field contains only infinite powers of θ. The interpretation is in terms of different Dirac vacuum. What one means which super-covariant derivatives is not quite clear.
   1. The usual definition of super covariant derivatives would be as

      D_iα=∂_iα+ i(θ_cD)_iα ,

      D^c_iαc=∂_iαc +i(Dθ)_iαc .

   2. A definition giving rise to the same anti-commutators would be as

   D_iα=∂_iα ,

   D^c_iαc=∂_iαc +2i(Dθ)_iαc

   In the recent case D^c does not appear at all in the chiral action since for negative energy chiral field conjugation does not correspond to θ→θ_c but to θ→ ∂_θ and 1→ X. Hence the simplest theory would result using D_α=∂_α.

   3. If one includes into the product of X of theta parameters only θs but not their conjugates, the two definitions are equivalent since the powers of θ_cDθ give nothing in theta integration. This definition of X is be possible using the definition of hermitian conjugation appropriate also for N=∞. This formalism of course works also for a finite value of N.

Consider now the resulting action obtained by performing the theta integrations. The interesting question is what form of the super-covariant derivatives one should use. The following considerations suggests that the two alternatives give almost identical -if not identical- results but that the simpler definitionD_α=∂_α is much more elegant.

1. For D_iα=∂_iα the propagators are just inverses of D^d where d is the number of theta parameters in the monomial defining the super-field component in question so that the Feynman rules for calculating bosonic propagators and vertices are very simple. Only the spinor and vector terms corresponding to degree d=1 and d=2 in theta parameters behave in the expected manner. This conforms with the collinearity. In particular, for spin 2 states the propagator would behave like p^-4 for large momenta. This conforms with the prediction that graviton cannot correspond to singlet wormhole throat but to a string like object consisting of a superposition of pairs of wormhole contacts and of wormhole throats. If this expansion makes sense, higher spin propagators would behave as increasingly higher inverse powers of momentum and would not contribute much to the high energy physics. At energies much smaller than mass scale they would give rise to contact terms proportional to a negative power of mass dictated by the number of thetas.

2. For D_iα=∂_iα+i(θ_cD) _iα the formulas become considerably more complex due to the infinite exponentials exp(i&theta_cDθ), and for N= ∞ one obtains infinite factors given essentially by N multiplying the propagators and vertices. These factors however cancel in the chiral loops defining bosonic vertices and propagators. Also a factor depending on momentum appears but cancel in these loops. The deviations from the first option are small but it seems that this option is so ugly that it can be safely forgotten.

2. TGD variant of vector super field

Chiral super-fields are certainly not all that is needed. Also interactions must be included, and this raises the question about the TGD counterpart of the vector super-field.

1. The counterpart of the chiral action would be a generalization of the Dirac action coupled to a gauge potential obtained by adding the super counterpart of the vector potential to the proposed super counterpart of Dirac action. The generalization of the vector potential would be the TGD counterpart of the vector super field. Vector particle include M⁴ scalars since Higgs behaves as CP₂ vector and H-scalars are excluded by chiral invariance.

2. Since bosons are bound states of positive and negative energy fermions at opposite wormhole throats it seems that vector super field must correspond to an operator slashed between positive and negative energy super-fields rather than ordinary vector super-field. The first guess is that vector super-field is an operator expressible as a Taylor series in which positive energy fermions correspond to the powers of θ and negative energy fermions correspond to the powers of derivatives ∂_θ. Naively, D in ∂_θD is replaced by D+V. Vector super-field must be hermitian (V=V⁺) with hermitian conjugation defined so that it maps theta parameters to the partial derivatives ∂_θ and performs complex conjugation. A better guess is that D appearing in the definition of the kinetic term is replaced with D+V where V is a hermitian super-field. This definition would be direct generalization of the minimal substitution rule.

3. It is difficult to imagine how a kinetic term for the vector super-field could be defined. This supports the idea that bosonic propagators and vertices emerge as one performs functional integral over components of the chiral fields.

4. There is also the question about gauge invariance. The super-field generalization of the non-Abelian gauge transformation formula looks more like the generalization of Dirac action to its super-counterpart: D→ D+V everywhere. Positive energy chiral field would transform as Φ₊→ exp(Λ)&Phi₊;, where Λ is a chiral field. The negative energy chiral field would transform as Φ_-→ exp(Λ⁺)&Phi_-; with hermitian conjugation (denotes by +) involving also the map of thetas to their derivatives. Each theta parameter would represent a fermion transforming under gauge symmetries in a manner dictated by its electro-weak quantum numbers (the inclusion of color quantum numbers is not quite trivial: probably they must be included as a label for quark modes). As in the case of Dirac action, the transformation formula for vector super-field would be dictated by the requirement that the derivatives of Λ coming from exp(Λ) are canceled by the derivative terms in the transformation formula for the vector super-field.

3. Is the perturbation theory UV finite?

Also for the proposed TGD inspired identifications of chiral super-fields and vector super-fields, the cancelation of UV divergences should be essentially algebraic and due to the cancelation of chiral contributions from the loops contributing to the vector super-field propagators and vertices. Also for the emerging bosonic effective action same mechanism should be at work.

The renormalization theorems state that the only renormalizations in SUSYs are wave function renormalizations. In the case of bosonic propagators loops therefore mean only the renormalization of the propagator. In the recent case only the chiral loops are included so that the situation is analogous to Abelian YM theory or N=4 super YM theory, where the beta functions for gauge couplings vanish. Hence one might hope that also now wave function renormalization is the only effect so that the radiatively generated contribution should be proportional to the standard form of the vector propagator. The worst that can occur is logarithmically diverging renormalization of the propagator which occur in many SUSYs. The challenge is to show that logarithmic divergences possibly coming from the θ^d, d=1,2, parts of the chiral super-field cancel. The condition for this cancelation is purely algebraic since the coupling to k=2 part is gradient coupling so that the leading divergences have same form. It could happen that the lowest contributions cancel but the contributions from field components with d>2 give a non-vanishing and certainly finite contribution.

It could happen that the d<1 contributions cancel exactly as they do in SUSYs but the contributions from field components with d> 2, give a non-vanishing and certainly finite contribution. If this were the case then the exotic chiral field components with propagators behaving like 1/p^d, d> 2, would make possible the propagation for the components of the vector super-fíeld.

For the proposed SUSY limit of TGD see the new chapter Does the QFT Limit of TGD Have Space-time Super-Symmetry? of the book "Towards M-Matrix".

Why SUSY would not allow fields with spin higher than two?

The recent progress in understanding QFT limit of TGD led to a question, which I would be happy to find an answer. The standard wisdom says that N=8 is absolute upper bound for the super-symmetry (spins larger than 2 are not regarded as physical). In TGD N=8 emerges naturally for space-time surfaces due to the dimension D=8 of imbedding space and the fact that imbedding space spinors with given H-chirality (quarks and leptons which color appearing as partial waves in CP₂ have 8 complex components. One obtains N=8 if one considers only the super-algebra defined by the oscillator operators associated with the lowest modes of these spinor fields at light-like 3-surfaces obtained as a solutions of the modified Dirac equation with measurement interaction term.

It is also possible to consider the supersymmetry generated by all modes of the induced spinor fields and thus with a quite large (even infinite for string like objects) number N of super generators. This supersymmetry is broken as all supersymmetries in TGD framework. This means that rather high spins are present in the analogs of scalar and vector multiplets and the Kähler potential (expected to be closely related to the Kähler function of the world of the classical worlds (WCW)) describing interaction of chiral multiplet with a vector multiplet can be constructed also for any value of N - at least formally. If one believes on the generalization of the bosonic emergence, one expects that bosonic part of the action is generated radiatively as one functionally integrates over the fields appearing in the chiral multiplet.

I tried to find material from web about possibly existing proposals for N>8 SUSY theories or D>12 SUSY theories containg higher spin fields. I found proposals for higher spin theories with N=1 for instance, but nothing else. Superstring thinking has really made its way through: D=12 (F-theory) and N=8 are the absolute upper bounds! It seems that my colleagues enjoying a monthly salary are maximally rational career builders.

The standard wisdom says that is is not possible to construct interactions for higher spin fields. Is this really true? Why wouldn't the analogs of scalar (chiral/hyper) and vector multiplets make sense for higher values of N? Why would it be impossible to define an spin 1/2 chiral super-field associated with the vector-multiplet and therefore the supersymmetric analog of YM action using standard formulas? Why the standard coupling to chiral multiplet would not make sense? Could some-one better-informed tell me the answer?

One objection against higher spins is of course the lack of the geometric interpretation. Spin 1 and Spin 2 fields allow it. Can one then imagine any geometric interpretation for higher spin components of super-fields? John Baez and others are busily developing non-Abelian generalizations of group theory, categories and geometry and speak about things that they call n-groups, n-categories, and n-geometries. Could the generalization of ordinary geometry to n-geometry in which parallel translations are performed for higher dimensional objects rather than points provide a natural interpretation for gauge fields assigned to higher spins? One would have natural hierarchy. Parallel translations of points would give rise curves, parallel translations of curves would give rise to surfaces, and so on. As as a special case the entire hierarchy of these parallel translations would be induced by ordinary parallel translation as I suggested in this blog for years ago.

Addition: At this moment one can make only guesses concerning the super-fields describing wormhole throats and contacts as particles.

1. The physical picture suggested by the notion of emergence is that kinetic terms behave negative powers of Dirac operator since wormhole throats carry a collection of collinearly moving fermions with momentum appearing in the measurement interaction term identified as the total momentum.

2. This suggests a construction of a super-field from any finite polynomial P(θ,x) of theta parameters by assigning to each monomial appearing in it the monomial P(∂_θc σ_c ^k ∂_k →,x) and applying it to the conjugate of X. Here → tells that the derivative is applied to P itself. All theta parameters can be included and _c denotes conjugation denoted by overline usually. By restricting the degree of the monomial to one half of the maximal the construction works also for a finite value of N.

3. In the analog of the chiral action monomials and their conjugates would combine to form a term involving a power of Dirac operator equal to the degree of the monomial of thetas so that kinetic terms would come as powers of σ^k∂_k.

4. Only the spinor and vector terms would behave in the expected manner and scalar term would vanish. In particular, for spin 2 - the propagator would behave like p^-4 for large momenta. This conforms with the view that graviton must correspond to a string like object consisting of a superposition of pairs of wormhole contacts and of wormhole throats rather than single wormhole throat. If this expansion makes sense, higher spin propagators would behave as increasingly higher inverse powers of momentum and would not contribute much to the high energy physics. At energies much smaller than mass scale they would give rise to contact terms proportional to a negative power of mass dictated by the number of thetas.

5. This is certainly not all that is needed since interactions must be included too. Here one might consider a generalization of Dirac action as a trilinear interaction term formed from similar "chiral field" assignable to bosons described as wormhole contacts with negative and positive energy thetas and from positive and negative energy fermionic super fields. Generalization of bosonic emergence would give purely bosonic part of action as radiative corrections. More conventional approach would add the bosonic kinetic term also the action.

For the proposed SUSY limit of TGD see the new chapter Does the QFT Limit of TGD Have Space-time Super-Symmetry? of the book "Towards M-Matrix".

Space-Time Super-Symmetry and TGD

Contrary to the original expectations, TGD seems to allow a generalization of the space-time super-symmetry. This became clear with the increased understanding of the modified Dirac action. The introduction of a measurement interaction term to the action allows to understand how stringy propagator results and provides profound insights about physics predicted by TGD (see the new chapter Does the Modified Dirac Equation Define the Fundamental Action Principle of "TGD: Physics as Infinite-Dimensional Geometry").

The appearance of the momentum and color quantum numbers in the measurement interaction couples space-time degrees of freedom to quantum numbers and allows also to define SUSY algebra at fundamental level as anti-commutation relations of fermionic oscillator operators. Depending on the situation a finite-dimensional SUSY algebra or the fermionic part of super-conformal algebra with an infinite number of oscillator operators results. The addition of a fermion in particular mode would define particular super-symmetry. Zero energy ontology implies that fermions as wormhole throats correspond to chiral super-fields assignable to positive or negative energy SUSY algebra whereas bosons as wormhole contacts with two throats correspond to the direct sum of positive and negative energy algebra and fields which are chiral or antichiral with respect to both positive and negative energy theta parameters. This super-symmetry is badly broken due to the dynamics of the modified Dirac operator which also mixes M⁴ chiralities inducing massivation. Since righthanded neutrino has no electro-weak couplings the breaking of the corresponding super-symmetry should be weakest.

The question is whether this SUSY has a realization as a SUSY algebra at space-time level and whether the QFT limit of TGD could be formulated as a generalization of SUSY QFT. There are several problems involved.

1. In TGD framework super-symmetry means addition of fermion to the state and since the number of spinor modes is larger states with large spin and fermion numbers are obtained. This picture does not fit to the standard view about super-symmetry. In particular, the identification of theta parameters as Majorana spinors and super-charges as Hermitian operators is not possible.

2. The belief that Majorana spinors are somehow an intrinsic aspect of super-symmetry is however only a belief. Weyl spinors meaning complex theta parameters are also possible. Theta parameters can also carry fermion number meaning only the supercharges carry fermion number and are non-hermitian. The the general classification of super-symmetric theories indeed demonstrates that for D=8 Weyl spinors and complex and non-hermitian super-charges are possible. The original motivation for Majorana spinors might come from MSSM assuming that right handed neutrino does not exist. This belief might have also led to string theories in D=10 and D=11 as the only possible candidates for TOE after it turned out that chiral anomalies cancel.

3. The massivation of particles is basic problem of both SUSYs and twistor approach. The fact that particles which are massive in M⁴ sense can be interpreted as massless particles in M⁴×CP₂ suggests a manner to understand super-symmetry breaking and massivation in TGD framework. The octonionic realization of twistors is a very attractive possibility in this framework and quaternionicity condition guaranteing associativity leads to twistors which are almost equivalent with ordinary 4-D twistors.

4. The first approach is based on an approximation assuming only the super-multiplets generated by right-handed neutrino or both right-handed neutrino and its antineutrino. The assumption that right-handed neutrino has fermion number opposite to that of the fermion associated with the wormhole throat implies that bosons correspond to N=(1,1) SUSY and fermions to N=1 SUSY identifiable also as a short representation of N=(1,1) SUSY algebra trivial with respect to positive or negative energy algebra. This means a deviation from the standard view but the standard SUSY gauge theory formalism seems to apply in this case.

5. A more ambitious approach would put the modes of induced spinor fields up to some cutoff into super-multiplets. At the level next to the one described above the lowest modes of the induced spinor fields would be included. The very large value of N means that N > 32 SUSY cannot define the QFT limit of TGD for higher cutoffs. One must generalize SUSYs gauge theories to arbitrary value of N but there are reasons to expect that the formalism becomes rather complex. More ambitious approach working at TGD however suggest a more general manner to avoid this problem.
   1. One of the key predictions of TGD is that gauge bosons and Higgs can be regarded as bound states of fermion and antifermion located at opposite throats of a wormhole contact. This implies bosonic emergence meaning that it QFT limit can be defined in terms of Dirac action. The resulting theory was discussed in detail in and it was shown that bosonic propagators and vertices can be constructed as fermionic loops so that all coupling constant follow as predictions. One must however pose cutoffs in mass squared and hyperbolic angle assignable to the momenta of fermions appearing in the loops in order to obtain finite theory and to avoid massivation of bosons. The resulting coupling constant evolution is consistent with low energy phenomenology if the cutoffs in hyperbolic angle as a function of p-adic length scale is chosen suitably.

   2. The generalization of bosonic emergence is natural in the sense that the TGD counterpart of SUSY is obtained by the replacement of Dirac action with action for chiral super-field coupled to vector field as the action defining the theory so that the propagators of bosons and all their super-counterparts would emerge as fermionic loops.

   3. The huge super-symmetries give excellent hopes about the cancelation of infinities so that this approach would work even without the cutoffs in mass squared and hyperbolic angle assignable to the momenta of fermions appearing in the loops. Cutoffs have a physical motivation in zero energy ontology but it could be an excellent approximation to take them to infinity. Alternatively, super-symmetric dynamics provides cutoffs dynamically.

6. The intriguing formal analogy of the Kähler potential and super-potential with the Kähler function defining the Kähler metric of WCW and determined up to a real part of analytic function of the complex coordinates of WCW. This analogy suggests that the action defining the SUSY-Kähler potential- is identifiable as the Kähler function defining WCW Kähler metric at its maximum. Super-potential in turn would correspond to a holomorphic function defining the modification of Kähler function due and the space-time sheet due to measurement interaction. This beautiful correspondence would make WCW geometry directly visible in the properties of QFT limit of TGD.

To sum up, the new chapter fuses three ideas developed during this year. The generalization of the twistor formalism via the induced octonionic twistor structure and masslessness in 8-D sense as a prerequisite for twistorialization and higher N super-symmetry, bosonic emergence, and the possibility to realize space-time super-symmetry algebra via the introduction of the measurement interaction term in the modified Dirac action. It seems that all basic prerequisite for developing quantum TGD to a calculable theory exist but a collective effort is of course needed to achieve this.

For the details see the new chapter Does the QFT Limit of TGD Have Space-time Super-Symmetry? of the book "Towards M-Matrix".