### Proposal for a twistorial description of generalized Feynman graphs

Listening of the lectures of Nima Arkani-Hamed is always an inspiring experience and so also at this time. The first recorded lectures was mostly about the basic "philosophical" ideas behind the approach and the second lecture continued discussion of the key points about twistor kinematics which I should already have in my backbone but do not. The lectures stimulated again the feeling that the generalized Feynman diagrammatics has all the needed elements to allow a twistorial description. It should be possible t to interpret the diagrams as the analogs of twistorial diagrams.

A couple of new ideas emerged as a result of concentrate effort to build bridge to the twistorial approach.

- Generalized Feynman diagrams involve only massless states at wormhole throats so that twistorial description makes sense for the kinematical variables. One should identify the counterparts of the lines and vertices of the twistor diagrams constructed from planar polygons and counterparts of the region momenta.
- M
^{2}⊂ M^{4}appears as a central element of TGD based Feynman diagrammatics and M^{2}projection of the four momentum appears in propagator and also in the modified Dirac equation. I realized that p-adic mass calculations must give the thermal expectation value of the M^{2}mass squared. Since the throats are massless this means that the transversal momentum squared equal to CP_{2}contribution plus conformal weight contribution to mass squared. - It is not too surprising that a very beautiful interpretation in terms of the analogs of twistorial diagrams becomes possible. The idea is to interpret wormhole contacts as pairs of lines of twistor diagrams carrying on mass shell momenta. In this manner triangles with truncated apexes with double line representing the wormhole throats become the basic objects in generalized Feynman diagrammatics. The somewhat mysterious region momenta of twistor approach correspond to momentum exchanges at the wormhole contacts defining the vertices. A reasonable expectation is that the Yangian invariants used to construct the amplitudes of
*N*=4 SUSY can be used as basic building bricks also now. - Renormalization group is not understood in the usual twistor approach and p-adic considerations and quantization of the size of causal diamond (CD) suggests that the old proposal about discretization of coupling constant evolution to p-adic length scale evolution makes sense. A very concrete realization of the evolution indeed suggest itself and would mean the replacement of each triangle with the quantum superposition of amplitudes associated with triangles with smaller size scale and contained with the original triangle characterized by the size scale of corresponding CD containing it. In fact the incoming and outgoing particles of of vertex could be located at the light-like boundaries of CD.
- The approach should be also number theoretically universal and this suggests that the amplitudes should be expressible in terms of quantum rationals and rational functions having quantum rationals as coefficients of powers of the arguments. Quantum rationals are characterized by p-adic prime p and p-adic momentum with mass squared interpreted as p-adic integer appears in the propagator. This means that the propagator proportional to 1/P
^{2}is proportional to 1/p when mass squared is divisible by p, which means that one has pole like contribution. The real counterpart of propagator in canonical identification is proportional to p. This would select the all CD characterized by n divisible by p as analogs of poles.

**What generalized Feynman diagrams could be?**

Let us first list briefly what these generalized Feynman diagrams emerge and what they should be.

- Zero energy ontology and the closely related notion of causal diamond (CD are absolutely essential for the whole approach. U-matrix between zero energy states is unitary but does not correspond to the S-matrix. Rather, U-matrix has as its orthonormal rows M-matrices which are "complex" square roots of density matrices representable as a product of a Hermitian square root of density matrix and unitary and universal S-matrix commuting with it so that the Lie algebra of these Hermitian matrices acts as symmetries of S-matrix. One can allow all M-matrices obtained by allowing integer powers of S-matrix and obtains the analog of Kac-Moody algebra. The powers of S correspond to CD with temporal distance between its tips coming as integer multiple of CP
_{2}size scale. The goal is to construct M-matrices and these could be non-unitary because of the presence of the hermitian square root of density matrix. - If is assumed that M-matrix elements can be constructed in terms of generalized Feynman diagrams. What generalized Feynman diagrams strictly speaking are is left open. The basic properties of generalized Feynman diagrams - in particular the property that only massless on mass shell states but with both signs of energy appear- however suggest strongly that they are much more like twistor diagrams and that twistorial method used to sum up Feynman diagrams apply.

**The lines of the generalized Feynman diagrams**

Generalized Feynman diagrams are constructed using solely diagrams containing on mass shell massless particles in both external and internal lines. Massless-ness could mean also massless-ness in M^{4}× CP_{2} sense, and p-adic thermodynamics indeed suggests that this is true in some sense.

- For massless-ness in M
^{4}× CP_{2}sense the standard twistor description should fail for massive excitations having mass scale of order 10^{4}Planck masses. At external lines massless states form massive on mass shell particles. In the following this possible difficulty will be neglected. Stringy picture suggests that this problem cannot be fatal. - Second possibility is that massless states form composites which in the case of fermions have the mass spectrum determined by CP
_{2}Dirac operator and and that that physical states correspond to states of super-conformal representations with ground states weight determined by the sum of vacuum conformal weight and the contribution of CP_{2}mass squared. In this case, one would have massless-ness in M^{4}sense but composite would be massless in M^{4}× CP_{2}sense. In this case twistorial description would work. - The third and the most attractive option is based on the fact that its is M
^{2}momentum that appears in the propagators. The picture behind p-adic mass calculations is string picture inspired by hadronic string model and in hadron physics one can assign M^{2}to longitudinal parts of the parton momenta.One can therefore consider the possibility that M

^{2}momentum square obeys p-adic thermodynamics. M^{2}momentum appears also in the solutions of the modified Dirac equation so that this identification looks physically very natural. M^{2}momentum characterizes naturally also massless extremals (topological light rays) and is in this case massless. Therefore throats could be massless but M^{2}momentum identifiable as the physical momentum would be predicted by p-adic thermodynamics and its p-adic norm could correspond to the scale of CD.Mathematically this option is certainly the most attractive one and it might be also physically acceptable since integration over moduli characterizing M

^{2}is performed to get the full amplitude so that there is no breaking of Poincare invariance.

There are also other complications.

- Massless wormhole throats carry magnetic charges bind to form magnetically neutral composite particles consisting of wormholes connected by magnetic flux tubes. The wormhole throat at the other end of the wormhole carries opposite magnetic charge and neutrino pair canceling the electro-weak isospin of the physical particle. This complication is completely analogous to the appearance of the color magnetic flux tubes in TGD description of hadrons and will be neglected for a moment.
- Free fermions correspond to single wormhole throats and the ground state is massless for them. Topologically condensed fermions carry mass and the ground states has developed mass by p-adic thermodynamics. Above considerations suggests that the correct interpretation of p-adic thermal mass squared is as M
^{2}mass squared and that the free fermions are still massless! Bosons are always pairs of wormhole throats. It is convenient to denote bosons and topologically condensed fermions by a pair of parallel lines very close to each other and free fermion by single line. - Each wormhole throat carries a braid and braid strands are carriers of four-momentum.
- The four momenta are parallel and only the M
^{2}projection of the momentum appears in the fermionic propagator. To obtain Lorentz invariance one must integrate over boosts of M^{2}and this corresponds to integrating over the moduli space of causal diamond (CD) inside which the generalized Feynman diagrams reside. - Each line gives rise to a propagator. The sign of the energy for the wormhole throat can be negative so that one obtains also space-like momentum exchanges.
- It is not quite clear whether one can allow also purely bosonic braid strands. The dependence of the over all propagator factor on longitudinal momentum is 1/p
^{2n}so that throats carrying 1 or 2 fermionic strands (or single purely bosonic strand) are in preferred position and braid strand numbers larger than 2 give rise to something different than ordinary elementary particle. It is probably not an accident that quantum phases q=exp(i2π/n) give rise to bosonic and fermionic statistics for n=1,2 and to braid statistics for n>2. States with n≥ 3 are expected to be anyonic. This also reduces the large super symmetry generated by fermionic oscillator operators at the partonic 2-surfaces effectively to*N*=1 SUSY.

- The four momenta are parallel and only the M

In the following It will be assume that all braid strands appearing in the lines are massless and have parallel four-momenta and that M^{2} momentum squared is given by p-adic thermodynamics and actually mass squared vanishes. It is also assumed that M^{2} momenta of the throats of the wormhole throats are paralleI in accordance with the classical idea that wormhole throats move in parallel. It is convenient to denote graphically the wormhole throat by a pair of parallel lines very close to each other.

**Vertices**

The following proposal for vertices neglects the fact that physical elementary particles are constructed from wormhole throat pairs connected by magnetic flux tubes. It is however easy to generalizes the proposal to that case.

- Conservation of momentum holds in each vertex but only for the total momentum assignable to the wormhole contact rather than for each throat. The latter condition would force all partons to have parallel massless four-momenta and the S-matrix would be more or less trivial. Conservation of four-momentum, the massless on mass shell conditions for 4-momenta of wormhole throatas and on mass shell conditions M
^{2}momentum squared given by stringy mass squared spectrum are extremely powerful and it is quite possible that one obtains in a given resolution defined by the largest and smallest causal diamonds finite number of diagrams. - I have already earlier developed argments strongly suggesting that that only three-vertices are fundamental kenociteallb/elvafu. The three vertex at the level of wormhole throats means gluing of the ends of the generalized line along 2-D partonic two surface defining their ends so that diagrams are generalization of Feynman diagrams rather than 4-D generalizations of string diagrams so that a generalization of a a trouser diagram does not describe particle decay). The vertex can be BFF or BBB vertex or a variant of this kind of vertex obtained by replacing some B:s and F:s with their super-partners obtained by adding right handed neutrino or antineutrino on the wormhole throat carrying fermion number. Massless on mass shell conditions hold true for wormhole throats in internal lines but they are not on mass shell as a massive particles like external lines.
- What happens in the vertex is momentum exchange between different wormhole throats regarded as braids with strands carrying parallel momenta. This momentum exchange in general corresponds to a non-vanishing mass squared and can be graphically described as a line connecting two vertices of a triangle defined by the particles emerging into the vertex. To each vertex of the triangle either massless fermion line or pair of lines describing topologically condensed fermion or boson enters. The lines connecting the vertices of the triangle carry the analogs of region momenta kenociteallb/Yangian, which are in general massive but the differences of two adjacent region momenta are massless. The outcome is nothing but the analog of the twistor diagram. 3- vertices are fundamental and one would obtain only 3-gons and the Feynman graph would be a collection of 3-gons such that from each line emerges an internal or external line.
- A more detailed graphical description utilizes double lines. For FFB vertices with free fermions one would have 4-gon containing a pair of vertices very near to each other corresponding to the outgoing boson wormhole decribed by double line. This is obtained by truncating the bosonic vertex of 3-gon and attaching bosonic double line to it. For topologically condensed fermions and BBB vertex one would have 6-gon obtained by truncating all apices of a 3-gon.

Some comments about the diagrammatics is in order.

- On mass shell conditions and momentum conservation conditions are extremely powerful so that one has excellent reasons to expect that in a given resolution defined by the largest and smallest CD involves the number of contributing diagrams is finite.
- The resulting diagrams are very much like twistor diagrams in
*N*=4 D=4 SYM for which also three-vertex and its conjugate are the fundamental building bricks from which tree amplitudes are constructed: from tree amplitudes one in turn obtains loop amplitudes by using the recursion formulas. Since all momenta are massless, one can indeed use twistor formalism. For topologically condensed fermions one just forms all possible diagrams consisting of 6-gons for which the truncated apices are connected by double lines and takes care that n lines are taken to be incoming lines. - The lines can cross, and this corresponds to the analog of non-planar diagram. I have proposed a knot-theoretic description of this situation based on the generalized braiding matrix appearing in integrable QFTs defined in M
^{2}. By using a representation for the braiding operation which can be used to eliminate the crossings of the lines one could transform all diagrams to planar diagrams for which one could apply existing construction recipe. - The basic conjecture is that the basic building bricks are Yangian invariants. Not only for the conformal group of M
^{4}but also for the super-conformal algebra should have an extension to Yangian. This Yangian should be related to the symmetry algebra generated by the M-matrices and analogous to Kac-Moody algebra. For this Yangian points as vertices of the momentum polygon are replaced with partonic 2-surfaces.

**Generalization of the diagrammatics to apply to the physical particles**

The previous discussion has neglected the fact that the physical particles are not wormhole contacts. Topologically condensed elementary fermions and bosons indeed correspond to magnetic flux pairs at different space-time sheets with wormhole contacts at the ends. How could one describe this situation in terms of the generalization Feynman diagrams?

The natural guess is that one just puts two copies of diagrams above each other so that the triangles are replaced with small cylinders with cross section given by the triangle and the edges of this triangular cylinder representing magnetic flux tubes. It is natural to allow momentum exchanges also at the other end of the cylinder: for ordinary elementary particle these ends carry only neutrino pairs so that the contribution to interactions is screening at small momenta. Also momentum exchanges long the direction of the cylinder should be allowed and would correspond to the non-perturbative low energy degrees of freedom in the case of hadrons. This momentum exchange assignable to flux tube would be between the truncated triangle rather than separately along the three vertical edges of the triangular cylinder.

**Number theoretical universality and quantum arithmetics**

The approach should be also number theoretically universal meaning that amplitudes should make sense also in p-adic number fields. Quantum arithmetics is characterized by p-adic prime and canonical identification mapping p-adic amplitudes to real amplitudes is expected to make the universality possible.

This is achieved if the amplitudes should be expressible in terms of quantum rationals and rational functions having quantum rationals as coefficients of powers of the arguments. This would be achieve by simply mapping ordinary rationals to quantum rationals if they appear as coefficients of polynomials appearing in rational functions.

Quantum rationals are characterized by p-adic prime p and p-adic momentum with mass squared interpreted as p-adic integer appears in the propagator. If M^{2} mass square is proportional to this p-adic prime p, propagator behaves as 1/P^{2}∝ 1/p, which means that one has pole like contribution for these on mass shell longitudinal masses. p-Adic mass calculations indeed give mass squared proportional to p. The real counterpart of propagator in canonical identification is proportional to p. This would select the all CD characterized by n divisible by p as analogs of poles.

It would seem that one must allow different p-adic primes in the generalized Feynman diagram since physical particles are in general characterized by different p-adic primes. This would require the analog of tensor product for different quantum rationals analogous to adeles. These numbers would be mapped to real (or complex) numbers by canonical identification.

**How to understand renormalization flow in twistor context?**

In twistor contex the notion of mass renormalization is not straightforward since everything is massless. In TGD framework p-adic mass scale hypothesis suggests a solution to the problem.

- At the fundamental level all elementary particles are massless and only their composites forming physical particles are massive.
- M
^{2}mass squared is given by p-adic mass calculations and should correspond to the mass squared of the physical particle. There are contributions from magnetic flux tubes and in the case of baryons this contribution dominates. - p-Adic physics discretizes coupling constant flow. Once the p-adic length scale of the particle is fixed its M
^{2}momentum squared is fixed and massless takes care of the rest.

Consider now how renormalization flow would emerge in this picture. At the level of generalized Feynman diagrams the change of the IR (UV) resolution scale means that the maximal size of the CDs involve increases (the minimal size of the sides decreases).

Concerning the question what CD scales should be allowed, the situation is not completely clear.

- The most general assumption allows integer multiples and would guarantee that the products of hermitian matrices and powers of S-matrix commuting with them define Kac-Moody type algebra assignable to M-matrices. If one uses in renormalization group evolution equation CDs corresponding to integer multiples of CP
_{2}length scale, the equation would become a difference equation for integer valued variable. - p-Adicity would suggest that the scales of CDs come as prime multiples of CP
_{2}scale. - p-Adic length scale hypothesis would allow only p-adic length scales near powers of two. There are excellent reasons to expect that these scales are selected by a kind of evolutionary process favoring those scales for CDs for which particles are maximally stable.

Renormalization group equations are based on studying what happens in an infinitesimal reduction of UV resolution scale would mean. Now the change cannot be infinitesimal but must correspond to a change in the scale of CD by one unit defined by CP_{2} size scale.

- The decrease of UV cutoff means that the vertex amplitudes associated with smallest truncated 3-polygons in the diagram are replaced with the sum of all amplitudes in which smaller polygons down to the cutoff size and having 3-external legs appear. The change of the total amplitude in this replacements define renormalization group equation. Conservation of four-momentum and on mass shell conditions suggest that only finite number of terms are allowed.
- The increase of UV cutoff means that the size of the largest CD increases. The physical interpretation would be in terms of the time scale in which one observes the process. If this time scale is too long, the process is not visible. For instances, the study of strong interactions between quarks requires short enough scale for CD. At long scales one only observes hadrons and in even longer scales atomic nuclei and atoms.
- One tends to think that the diagrams are imbedded in M
^{2}allowing identification as 2-plane in Minkowski space-time. This in turn would suggest that the step increasing UV resolution corresponds of replacement of triangles with graphs consisting of smaller triangles contained by them and having no intersections. This interpretation is attractive but might not be needed. Essential conditions are momentum conservation and on mass shell conditions. - One could also allow the UV scale to depend on the particle. This scale should correspond to the p-adic mass scales assignable to the stable particle. In hadron physics this kind of renormalization is standard operation.

## 12 Comments:

So, Matti,

You insist on quantum descriptions- but what then is original beyond Dirac's four spinors- I mean something is generalized is these work (but only so well for these decades since him) as your last post that wants to make sense out of four waves... this four or five fold pattern in nature (recall Einstein tried 5 dimensions for one rather cylindrical unified field.)

And these can be reduced to 2+1 formulism from the 3+1, much like Feymann diagram so reduced and see as a general matrix to be expanded again into your 2x2 view (a debate of exclusion.) Now, in the quantum formulation we have different ways to view things that amount to almost the same thing- in its way it explains why we wind up with three dimensional space and two types of particles.

Mass in not in the equations- that is what some are looking for as well the nature of gravity. To base things on mass-less is merely to interpret Dirac's "nilpotent" algebra rather than "indempotent" forms of models.

If you cut a knot, as if a string, does it have more than 2 end points- if twistors are only complex duplications and numbers so to justify the 2 or 4 formulisms (of which in one form Dirac uses five...) it is not enough, qm mechanics is not enough.

Now, the Mersenne's as you use them may be enough but that is a wide field to explore- but it is to me an original approach.

I understand someone independent of the academia being creative and free to read and speculate- but I see no reason to make a great deal over idols of the day like Arkani-Hamed which all the bloggers seems to have done even when in disagreement. His is another near idea along the way like some of Hawking's.

There are other ways to explain quantization than that from Dirac in that differences in space and time and matter and charge are those of that great foundational difference between the continuous and the discontinuous in the search for some measure.

Maybe the old Egyptians had it right- we should not always rationalize our fractions- we simplify but lose information- and the lost information is not clearly lost- nor does it prove anything.

The PeSla

Dear Pesla,

some comments. First about twistors and related things.

a) Why I appreciate Nima is that he is theoretician with a tight contact with reality. Also clarity of thinking, enthusiasm, and courage and ability to imagine belong to his virtues. Before knowing one must imagine.

b) Second point is that the work with twistors lead to the realization of Yangian invariance and discovery of an infinite hierarchy of them. Their generalization emerges naturally in TGD framework. N=4 SYM is the most useful toy model ever discovered. If I want to formulate TGD as concrete rules some day, I must use all the wisdom that already exists.

c) The idea that all massive states have massless building bricks is extremely powerful. Much much more than "nilpotent Dirac algebra". Generalization of Yangian invariance, twistors, new view about Feynman diagrams as twistor diagrams implied by zero energy ontology,... Together with zero energy ontology this identification might resolve also the basic problems of twistor approach: how to get rid of infrared divergences, how massive states emerge and can be described, how to describe non-planar diagrams, how to understand renormalization group and coupling constant evolution...

To be continued...

Some comments about Dirac spinors.

a) I - and I think all theoreticians nowadays- use the term "Dirac spinor" in more general sense than Dirac. This may or not be regarded "original": physics means to me much more than "originality". Some things have been understood and the mathematical description of spin is one of them: it is waste of time to try to invent ad hoc descriptions of spin.

Same applies to electroweak quantum numbers. The attempts to reduce them to - say - knot or braid topology is simply waste of time. There is huge data basis of empirical facts demonstrating that group theory is behind electroweak symmetries. Braids are extremely interesting part of also TGD but in the case of elementary particles (1 or 2 braids only) they do not bring in anything interesting.

The interesting new things emerge at the level anyonic physics involving braids with more than two strands. They define new kind of entities not identifiable as elementary particles since the propagators from these states do not behave like they should behave for elementary particles.

b) Dirac spinors in TGD framework are 8+8-component spinors of M^4xCP_2 and describe electroweak isospin besides spin and give automatically rise to quarks and leptons with correct electroweak quantum numbers. At the level of "world of classical worlds" - WCW- the counterparts of Dirac spinors are fermionic Fock states and spinor fields in WCW describe all quantum states. Fermi statistics finds a geometrization in terms of WCW gamma matrices expressible as combinations of fermionic oscillator operators. Quite a leap conceptually but mathematically very natural.

c) Mersenne primes are only specific p-adic primes which are of special importance in p-adic physics. What is important are the p-adic topologies as the natural topologies for the correlates of cognition. They have also deep connection with particle physics and number theoretical universality realized in terms of quantum arithmetics becomes the deep principle posing constraints on quantum physics. These specific primes give only grasp to the reality via applications. Twistorial approach allows imagine to see how the number theoretically universal amplitudes should be constructed using quantum arithmetics .

To be continued...

.... and few words about four wave interaction.

Four-wave interaction is nonlinear interaction of laser waves - as such it is not something that unified theorists is usually interested in. Modulation is second non-linear interaction involving two waves familiar from first year courses in physics.

Why I am talking about this kind of basic things is that the linear superposition of fields corresponds in TGD only to the superposition of their effects: particle has topological sum contacts to the two or more space-time sheets and experiences the sum of the fields carried by them.

This is a profound difference at the basic ontological level, and it is more than interesting to see whether it really works: can one describe effects like amplitude modulation and four-wave interaction in this framework? One can!

Cheers,

Matti

Dear Matti,

I don’t understand difference between 3-surface and topological field quantum in a well form. They both are many sheeted. Are they same?

Does Earth have two space-time sheets, one for outer surface and one for inner surface? And we glue to outer surface.

“Elementary particles interpreted as CP2 type extremals topologically condensed simultaneously to the two space-time sheets involved.” Then electrons in atom topologically condensed between two sheets, one for atom and other?

8+8 component spinors left me with ?

Dirac spinors acting on a field?

Is this a good text? http://www.math.sunysb.edu/rtg/Images/07.04.30.14.30.RTGSpin.pdf

Note the Massive and massless representations in the middle of text.

In the massless case we cannot go to the rest frame as this would require boosting up to warp 1. We can however always rotate to to the frame in which the massless state of energy E is traveling in the positive z-direction.

As the representation is supposed to be irreducible, it must be one-dimensional. In this case, the eigenvalue of Sz (again half-integer) called the helicity and there are only two of them: The massless state is making either a left-handed corkscrew around its axis of propagation or a right-handed one.

Note that helicity is a good quantum number in the massless case because we cannot change the handedness of the screw unless we boost past the state.

...the 2-component spinors are the left- and right-handed parts

of the original Dirac spinor.

To Hamed:

I mean with space-time surface any 4-surface.

*It could be "massless extremal" having 4-D M^4 projection and would correspond to our ideas about physics QFT in M^4. This topological light ray would be example of topological field quantum and would provide space-time correlate for say laser beam. It could carry topologically condensed particles. Massless extremal would be more like a macroscopic correlate for em and other gauge fields and also for gravitational radiation.

Also magnetic and electric flux quanta or space-time sheets carrying magnetic and electric fields simultaneously would be examples of topological field quanta. They have 4-D M^4 projection so that they also correspond to physics QFT in M^4 intuition. Why I call them quanta is that they typically have finite size. For instance, an attempt to imbed constant magnetic field as induced gauge fields give rise to a flux tube (say) with finite radius. Space-time ends at certain radius for these imbeddings. The induced metric becomes singular and a good guess is that the boundary is light-like and four metric is degenerate (effectively 3-D) at it. This kind of surface can also act as wormhole throat at which region of space-time with Euclidian signature of induced metric begins.

This is quite a general phenomenon since 4-D CP_2 is compact whereas the space of the values of em gauge potentials at given point is non-compact: essentially M^4. This implies that the representation of arbitrary gauge potential of M^4 as a space-time sheet (map from M^4 to CP_2) cannot be global. Boundaries are generated. This leads to the quantization of classical fields iin the sense of the splitting of the space-time surface to pieces.

Preferred extremal property also corresponds to Bohr quantization for fields. In fact, it seems that space-time sheet very generally decomposes to regions having interpretation as massless quanta. This would not allow superposition and here the idea that parallel space-time sheets to which charged particles have simultaneous topological sum contacts, gives rise to a linear superposition of the effects of classical fields although classical fields not superpose as in ordinary gauge theories.

Space-time surface could also be CP_2 type vacuum extremal with 1-D light-like random curve as M^4 projection: free elementary particles would be like this and the randomness of the light-like curve would basically relate to massivation since motion with finite length scale resolution would be sub-luminal. The roles of M^4 and CP_2 have changed and one has more like QFT with M^4 valued field in CP_2.

There are also string like objects X^2xY^2 subset M^4xCP_2. X^2 is string orbit (minimal surface) and Y^2 is complex surface of CP_2. These would correspond to string model type description. QFT picture would not make sense for their deformations and one would have something analogous to field theory in X^2xY^2.

To be continued....

..continuation of previous message to Hamed.

Earth has magnetic field and it is topologically quantized. The magnetic field decomposes to topological flux quanta: they can be flux tubes or flux sheets, say spherical shells with finite and varying thickness. One can clearly say that Earth possesses magnetic body (and field body). In Maxwellian theory this kind of "personal field body" does not make sense. Dipole field splices to flux tubes and flux walls is a good visual image.

Second question about electrons as deformations of CP_2 type vacuum extremals.

Free electron when seen at CP_2 length scale corresponds to space-time surface of this kind can indeed touch several space-time sheets simultaneously because the distance between them is of order CP_2 length scale. The analogy is two or more thin membranes which are parallel and extremely near to each other.

You could assign these membranes to Earth, Sun, galaxy, etc... You could visualize electron as an extremely tiny object between them having size of the order of their distance between them- CP_2 scale. Electron cannot avoid touching them.

When one enters to elementary particle mass scales which are much longer one has some delicacies. The wormhole throat associated with electrons own space-time sheet carries Kahler magnetic charge which must be neutralized. Also the weak isospin must be neutralized above weak scale about 10^-17 meters. Second wormhole throat carrying opposite Kahler magnetic charge and pair of neutrino and antineutrino could achieve this. Magnetic flux tube connects the two throats that electron in the scales looks like string like object and second end makes itself visible via the screening of weak interaction. Should the length of the flux tube be Compton length of electron or weak length scale? I am not sure. Weak gauge bosons themselves are similar objects and might be enough for weak screening. Could it be enough that the length of string is Compton length?

To Ulla:

To Ulla:

Yes, the text looks ok although I still wonder what "warp" has to do with representations of Poincare group.

The notion of left and right handedness mentioned in the text appears in 4-D case. In massless theories the numbers of left and right handed fermions are separately conserved.

Handedness generalizes in 8-D case to 8-D chirality which is conserved in TGD and gives rise to separate conservation of baryon and lepton numbers. M^4 handedness is not conserved anymore and this is direct indication for unavoidable massivation of the observed particles - as opposed to the fundamental building bricks of them.

Thanks, I have much to learn.

Here is a little delicate question. What is behind c as a 'force' or constant/iniertia, thinking of the very broad result it is giving in the em-radiation? And gauge forces? Holography? Mach?

Sarfatti says it may not be real numbers, and a square. Maybe many roots? http://stardrive.org/index.php?option=com_myblog&show=debate-with-jim-woodward-on-his-mach-starship-engine-theory-1-15-12-v2.html&Itemid=56

" People like us...know that the distinction between past, present, and future is only a stubbornly persistent illusion."

- Albert Einstein

Matti,

How much of other physics values can be like this concept? Chirality? Charge? Space and Mass? Gravity?

What is time that we can be said to waste it with knots and braiding theory or waste it doing endless diagonalization of matrices? (or we open the oyster to find the pearl through the p-adic, adelic ,cracks?

If we go deep enough into the depths will we at last find a stable world encompassing everything explaining all variations and anomalies? Do we want this for our grounding that our consciousness is a screen of bare charges made of shifting sand?

hammed, quantum theory cannot yet explain why separate sheets take charges or how tape pulled apart emits x-rays... or how bits of ice build up lightening. How might such things fit in the TGD framework?

The PeSla who had nothing to post today but wondering about someone a few years ago in the science chat rooms who once looked at quasics and said "I hope your are not just wasting your time diagonalizing matrices."

To my opinion Einstein is imprecise in this statement. E speaks about geometric time and what he says applies to it but not to experienced time. E however identifies subjective, experience time and geometric time. This was is mistake also in the debate with Bohr. One must distinguish between past, present, and future in the case of experienced, subjective time.

Knots and braids are certainly *not* waste of time to my opinion. I have myself used and will use a lot of time to them and I see them as basic objects. The attempt to artificially reduce elementary particle quantum numbers to knot invariants is however waste of time. Little amount of knowledge about group theory would prevent this kind of waste of time.

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