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.

1. 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.

2. M²⊂ M⁴ appears as a central element of TGD based Feynman diagrammatics and M² 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 M² mass squared. Since the throats are massless this means that the transversal momentum squared equal to CP₂ contribution plus conformal weight contribution to mass squared.

3. 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.

4. 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.

5. 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/P² 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.

1. 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 CP₂ 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.

2. 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 M⁴× CP₂ sense, and p-adic thermodynamics indeed suggests that this is true in some sense.

1. For massless-ness in M⁴× CP₂ sense the standard twistor description should fail for massive excitations having mass scale of order 10⁴ 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.

2. Second possibility is that massless states form composites which in the case of fermions have the mass spectrum determined by CP₂ 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 CP₂ mass squared. In this case, one would have massless-ness in M⁴ sense but composite would be massless in M⁴× CP₂ sense. In this case twistorial description would work.

3. The third and the most attractive option is based on the fact that its is M² 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 M² to longitudinal parts of the parton momenta.

   One can therefore consider the possibility that M² momentum square obeys p-adic thermodynamics. M² momentum appears also in the solutions of the modified Dirac equation so that this identification looks physically very natural. M² momentum characterizes naturally also massless extremals (topological light rays) and is in this case massless. Therefore throats could be massless but M² 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 M² is performed to get the full amplitude so that there is no breaking of Poincare invariance.

There are also other complications.

1. 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.

2. 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 M² 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.

3. Each wormhole throat carries a braid and braid strands are carriers of four-momentum.
   1. The four momenta are parallel and only the M² projection of the momentum appears in the fermionic propagator. To obtain Lorentz invariance one must integrate over boosts of M² and this corresponds to integrating over the moduli space of causal diamond (CD) inside which the generalized Feynman diagrams reside.

   2. 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.

   3. 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/p²ⁿ 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.

In the following It will be assume that all braid strands appearing in the lines are massless and have parallel four-momenta and that M² momentum squared is given by p-adic thermodynamics and actually mass squared vanishes. It is also assumed that M² 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.

1. 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 M² 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.

2. 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.

3. 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.

4. 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.

1. 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.

2. 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.

3. 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 M². 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.

4. The basic conjecture is that the basic building bricks are Yangian invariants. Not only for the conformal group of M⁴ 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 M² mass square is proportional to this p-adic prime p, propagator behaves as 1/P²∝ 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.

1. At the fundamental level all elementary particles are massless and only their composites forming physical particles are massive.

2. M² 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.

3. p-Adic physics discretizes coupling constant flow. Once the p-adic length scale of the particle is fixed its M² 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.

1. 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 CP₂ length scale, the equation would become a difference equation for integer valued variable.

2. p-Adicity would suggest that the scales of CDs come as prime multiples of CP₂ scale.

3. 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 CP₂ size scale.

1. 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.

2. 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.

3. One tends to think that the diagrams are imbedded in M² 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.

4. 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.

Reader interested in background can consult to the article Algebraic braids, sub-manifold braid theory, and generalized Feynman diagrams and the new chapter Generalized Feynman Diagrams as Generalized Braids of "Towards M-Matrix".