Number theoretical universality and quantum arithmetics, renormalization, and relation of TGD to N=4 SYM

In the previous posting I already discussed the proposal for how twistorial construction could generalize to apply to generalized Feynman diagrams in TGD framework. During last days I have made a further progress in understanding of the number theoretical aspects of the proposed construction.

In particular, I finally have simple and general justification for the hypothesis that length scales coming as powers of two and p-adic length scales associated p-adic primes near powers of 2 are very special. The explanation is extremely simple: quantum arithmetics is characterized by prime p and for p=2 all odd quantum integers are identical with ordinary integers so that only powers of two mapped to their inverses distinguish 2-adic quantum arithematics from the ordinary one.

I have also corrected some erratic statements in the view about coupling constant evolutiont and compared the approach to that of N=4 SYM developed by Nima Arkani Hamed and others. Therefore this posting contains some material overlapping with the previous posting.

Number theoretical universality

The construction of the amplitudes should be number theoretically universal meaning that amplitudes should make sense also in p-adic number fields or perhaps in adelic sense in the tensor product of p-adic numbers fields. Quantum arithmetics is characterized by p-adic prime and canonical identification mapping p-adic amplitudes to real amplitudes is expected to make number theoretical 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 achieved 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 squared 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 propagator poles. Note that the infrared singularity is moved and the largest p-adic prime appearing as divisor of integer characterizing the largest CD indeed serves as a physical IR cufoff.

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 get only finite number of diagrams in a given IR and UV resolution?

In gauge theory one obtains infinite number of diagrams. In zero energy ontology the overall important additional constraint comes from on mass shell conditions at internal lines and external lines and from the requirement that the M² momentum squared is quantized for super-conformal representation in terms of stringy mass squared spectrum.

This condition alone does not however imply that the number of diagrams is finite. If forward scattering diagram is non-vanishing also scattering without on mass shell massive conditions on final state lines is possible. One can construct diagrams representing a repeated n→ n scattering and combining these amplitudes with non-forward scattering amplitude one obtains infinite number of scattering diagrams with fixed initial and final states. Number theoretic universality however requires that the number of the contributing diagrams must be finite unless some analytic miracles happens.

The finite number of diagrams could be achieved if one gives for the vision about CDs within CDs a more concrete metric meaning. In spirit of Uncertainty Principle, the size scale of the CD defined by the temporal distance between its tips could correspond to the inverse of the momentum scale defined as its inverse. A further condition would be that the sub-CDs and their Lorentz boosts are indeed within the CD and do not overlap. Obviously the number of diagrams representing repeated n-n scattering forward scattering is finite if these assumptions are made. This would also suggest a scale hierarchy in powers of 2 for CDs: the reason is that given CD with scale T=nT(CP₂) can contain two non-overlapping sub-CDs with the same rest frame only if sub-CD has size scale smaller than nT_CP2/2. This applies also to the Lorentz boosts of the sub-CDs.

Amplitudes would be constructed by labeling the CDs by integer n defining its size scale. p-Adicity suggests that the factorization of n to primes must be important and if n=p condition holds true, a new resonant like contribution appears corresponding to p-adic diagrams involving propagator.

Should one allow all M² momenta in the loops in all scales or should one restrict the M² momenta to have a particular mass squared scale determined somehow by the size of CD involved? If this kind of constraint is posed it must be posed in mathematically elegant manner and it is not clear how to to this.

Is this kind of constraint really necessary? Quantum arithmetics for the length scale characterized by p-adic prime p would make M² mass squared values divisible by p to almost poles of the propagators, and this might be enough to effectively select the particular p and corresponding momentum scale and CD scale. Consider only the Mersenne prime M₁₂₇=2¹²⁷-1 as a concrete example.

How to realize the number theoretic universality?

One should be able to realized the p-adicity in some elegant manner. One must certainly allow different p-adic primes in the same diagram and here adelic structure seems unavoidable as tensor product of amplitudes in different p-adic number fields or rather - their quantum arithmetic counterparts characterized by a preferred prime p and mapped to reals by the substitution p→ 1/p. What does this demand?

1. One must be able to glue amplitudes in different p-adic number fields together so that the lines in some case must have dual interpretation as lines of two p-adic number fields. It also seems that one must be able to assign p-adic prime and quantum arithmetics characterized by a given prime p to to a given propagator line. This prime is probably not arbitrarily and it will be found that it should not be larger than the largest prime dividing n characterizing the CD considered.

2. Should one assign p-adic prime to a given vertex?
   1. Suppose first that bare 3-vertices reduce to algebraic numbers containing no rational factors. This would guarantee that they are same in both real and p-adic sense. Propagators would be however quantum rationals and depend on p and have almost pole when the integer valued mass squared is proportional to p.

   2. The radiative corrections to the vertex would involve propagators and this suggests that they bring in the dependence on p giving rise to p-adic coupling constant evolution for the real counterparts of the amplitudes obtained by canonical identification.
      1. Should also vertices obey p-adic quantum arithmetics for some p? What about a vertex in which particles characterized by different p-adic primes enter? Which prime defines the vertex or should the vertex somehow be multi-p p-adic? It seems that vertex cannot contain any prime as such although it could depend on incoming p-adic primes in algebraic or transcendental manner.

      2. Could the radiative corrections sum up to algebraic number depending on the incoming p-adic primes? Or are the corrections transcendental as ordinary perturbation theory suggests and involve powers of π and logarithm of mass squared and basically logarithms of some primes requiring infinite-dimensional transcendental extension of p-adic numbers? If radiative corrections depend only on the logarithms of these primes p-adic coupling constant evolution would be obtained. The requirement that radiative vertex corrections vanish does not look physically plausible.

   3. Only CDs corresponding to integers m< n would be possible as sub-CDs. A geometrically attractive possibility is that CD characterized by integer n allows only propagator lines which correspond to prime factors of integers not larger than the largest prime dividing n in their quantum arithmetics. Bare vertices in turn could contain only primes larger than the maximal prime dividing n. This would simplify the situation considerably- This could give rise to coupling constant evolution even in the case that the radiative corrections are vanishing since the rational factors possibly present in vertices would drop away as n would increase.

   4. Integers n=2^k give rise to an objection. They would allow only 2-adic propagators and vertices containing no powers of 2. For p=2 the quantum arithmetics reduces to ordinary arithmetics and ordinary rationals correspond to p=2 apart from the fact that powers of 2 mapped to their inverses in the canonical identification. This is not a problem and might relate to the fact that primes near powers of 2 are physically preferred. Indeed, the CDs with n=2^k would be in a unique position number theoretically. This would conform with the original - and as such wrong - hypothesis that only these time scales are possible for CDs. The preferred role of powers of two supports also p-adic length scale hypothesis.

These observations give rather strong clues concerning the construction of the amplitudes. Consider a CD with time scale characterized by integer n.

1. For given CD all sub-CDs with m<n are allowed and all p-adicities corresponding to the primes appearing as prime factors of given m are possible. m=2^k are in a preferred position since p=2 quantum rationals not containing 2 reduce to ordinary rationals.

2. The geometric condition that sub-CDs and their boosts remain inside CD and do not overlap together with momentum conservation and on-mass-shell conditions on internal lines implies that only a finite number of generalized Feynman diagrams are possible for given CD. This is essential for number theoretical universality. To each sub-CD one must assign its moduli spaces including its not-too-large boosts. Also the planes M² associated with sub-CDs should be regarded as independent and one should integrate over their moduli.

3. The construction of amplitudes with a given resolution would be a process involving a finite number of steps. The notion of renormalization group evolution suggests a generalization as a change of the amplitude induced by adding CDs with size smaller than smallest CDs and their boosts in a given resolution.

4. It is not clear whether increase of the upper length scale interpreted as IR cutoff makes sense in the similar manner although physical intuition would encourage this expectation.

How to understand renormalization flow in twistor context?

In twistor context 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 of CP₂ scale 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. The proposed realization of p-adicity indeed puts CDs characterized by p-adic primes p in a special position since they correspond to the emergence of a vertex corresponding to p-adic prime p which depends on p in the sense that the radiative corrections to 3-vertex can give it a dependence on log(p). This requires infinite-D transcendental extension of p-adic numbers.

   As far as coupling constant evolution in strict sense is considered, a natural looking choice is evolution of vertices as a function of p-adic primes of the particles arriving to the vertex since radiative corresponds are expected to depend on their logarithms.

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. The fact that quantum arithmetics for p=2 reduces to ordinary arithmetics when quantum integers do not contain 2 raises with size scales coming as powers of 2 in a special position and also supports p-adic length scale hypothesis.

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 addition of new details represented as bare 3-vertices represented by truncated triangle having size below the earlier length scale resolution. The addition can be done inside the original CD and inside any sub-CD would be in question taking care that the details remain inside CD. The hope is that this addition of details allows a recursive definition. Typically addition would involve attaching two sub-CDs to propagator line or two propagator lines and connecting them with propagator. The vertex in question would correspond to a p-adic prime dividing the integer characterizing the sub-CDs. Also the increase of the shortest length scale makes sense and means just the deletion of the corresponding sub-CDs. Note that also the positions of sub-CDs inside CD manner since the number of allowed boosts depends on the position. This would mean an additional complication.

2. The increase of IR cutoff length 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 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.

Comparison with N=4 SYM

The ultimate hope is to formulate all these ideas using precise formulas. This goal is still far away but one can make trials. Let us first compare the above proposal to the formalism in N=4 SYM.

1. In the construction of twistorial amplitudes the 4-D loop integrals are interpreted as residue integrals in complexified momentum space and reduces to residues around the poles. This is analogous to using "on mass shell states" defined by this poles. In TGD framework the situation is different since one explicitly assigns massless on-mass-shell fermions to braid strands and allows the sign of the energy to be both positive and negative.

2. Twistor formalism and description of momentum and helicity in terms of the twistor (λ,μ) certainly makes sense for any spin. The well-known complications relate to the necessity to use complex twistors for M⁴ signature: this would correspond to complexified space-time or momentum space. Also region momenta and associated momentum twistors are the TGD counterparts so that the basic building bricks for defining the analogs of twistorial amplitudes exist.

An important special feature is that the gauge potential is replaced with its N=4 super version.

1. This has some non-generic implications. In particular gluon helicity -1 is obtained from 1 ground state by "adding" four spartners with helicity +1/2 each. This interpretation of the two helicities of a massless particle is not possible in N<4 theories nor in TGD and the question is whether this is something deep or not remains open.

2. In TGD framework it is natural to interpret all fermion modes associated with partonic 2-surface (and corresponding light-like 3-surfaces) as generators of super-symmetry and fermions are fundamental objects instead of helicity +1 gauge bosons. Right-handed neutrino has special role since it has no electroweak or color interactions and generates SUSY for which breaking is smallest.

3. The N=2 SUSY generated by right-handed neutrino and antineutrino is broken since the propagator for states containing three fermion braid strands at the same wormhole throat behaves like 1/p³: this is already an anyon-like state. The least broken SUSY is N=1 SUSY with spartners of fermions being spin zero states. The proposal is that one could construct scattering amplitudes by using a generalize chiral super-field associated with N equal to the number of spinor modes acting on ground state that has vanishing helicity. For N=4 it has helicity +1. This would suggest that the analogs of twistorial amplitudes exist and could even have very similar formulas in terms of twistor variables.

4. The all-loop integrand for scattering amplitudes in planar N=4 SYM relies of BCFW formula allowing to sew two n-particle three amplitudes together using single analog of propagator line christened as BCFW bridge. Denote by Yn,k,l n-particle amplitudes with k positive helicity gluons and l loops. One can glue Y_nL,kL,lL and Y_nR,kR,lR by using BCFW bridge and add "entangled " removal of two external lines of Y(n+2,k+1,l-1) amplitude with n= n_L+n_R-2,k=k_L+k_R,l=l_L+l_R to get Y_n,k,l amplitude recursively by starting from just two amplitudes defining the 3-vertices. The procedure involves only residue integral over the Gl(k,n) for a quantity which is Yangian invariant. The question is whether one could apply this procedure by replacing N=4 SUSY with SUSY in TGD sense and generalizing the fundamental three particle vertices appropriately by requiring that they are Yangian invariants?

5. One can also make good guesses for the BCFW bridge and entangled removal. By looking the structure of the amplitudes obtained by the procedure from 3-amplitudes, one learns that one obtains tree diagrams for which some external lines are connected to give loop. The simplest situation would be that BCFW bridge corresponds to M² fermion propagator for a given braid strand and entangled removal corresponds to a short cut of two external lines to internal loop line. One would have just ordinary Feynman graphs but vertices connected with Yangian invariants (not that there is sum over loop corrections). It should be easy to kill this conjecture.

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".