Friday, December 31, 2021

About TGD counterparts of twistor amplitudes

The twistor lift of TGD, in which H=M4 × CP2 is replaced with the product of twistor spaces T(M4) and T(CP2), and space-time surface X4⊂ H with its 6-D twistor space as 6-surface X6 ⊂ T(M4)× T(CP2), is now a rather well-established notion and M8-H duality predicts it at the level of M8.

Number theoretical vision involves M8-H duality. At the level of ⊂H⊂, the quark mass spectrum is determined by the Dirac equation in ⊂H⊂. In M8 mass squared spectrum is determined by the roots of the polynomial P determining space-time surface and are in general complex. By Galois confinement the momenta are integer valued when p-adic mass is used as a unit and mass squared spectrum is also integer valued. This raises hope about a generalization of the twistorial construction of scattering amplitudes to TGD context.

It is always best to start from a problem and the basic problem of the twistor approach is that physical particles are not massless.  

  1. The intuitive TGD based proposal has been that since quark spinors are massless in H, the masslessness in the 8-D sense could somehow solve the problems caused by the massivation in  the construction of twistor scattering amplitudes. However, no obvious mechanism has been identified. One step in this direction was the realization that in H quarks propagate with well-defined chiralities and only the square of Dirac equation is satisfied. For a quark of given helicity the  spinor can be identified as helicity spinor.
  2.  M8 quark momenta are in general complex as algebraic integers. They are the counterparts of the area momenta xi of momentum twistor space whereas H momenta are identified as ordinary momenta. Total momenta  of Galois confined states  have as components ordinary integers.
  3.   The  M8 counterpart of  the 8-D massless condition in H is the restriction of momenta to mass shells m2= rn determined as roots  of P. The M8 counterpart of Dirac equation in H is octonionic Dirac equation, which is algebraic as everything in M8 and analogous to massless Dirac equation. The solution is a helicity spinor λ associated with the massive momentum. 
The outcome is an extremely simple proposal for the scattering amplitudes.
  1. Vertices correspond to trilinears of Galois confined many-quark states as states  of super symplectic algebra acting as isometries of the "world of classical worlds" (WCW). Quarks are on-shell with H  momentum p and M8 momenta xi,xi+1, pi=xi+1-xi. Dirac operator xkiγk restricted to fixed helicity L,R appears as a vertex factor and has an  interpretation as a residue of a pole from an on-mass-shell propagator D so that a correspondence with twistorial construction becomes obvious. D  is expressible in terms of the helicity spinors of given chirality and gives two independent holomorphic factors as in case of massless theories.
  2. MHV construction utilizing k=2 MHV amplitudes as building bricks does not seem to be needed at the level of a single space-time surface. One can of course ask, whether the M8 quark lines could be regarded as analogs of lines connecting different MHV diagrams replaced with Galois singlets. The scattering amplitudes would be rational functions in accordance with the number theoretic vision. The absence of logarithmic radiative corrections is not a problem: the coupling constant evolution would be discrete and defined by the hierarchy of extensions of rationals.
  3. The scattering amplitudes for a single 4-surface X4 are determined by a polynomial. The integration over WCW is replaced with a summation of polynomials characterized by rational coefficients. Monic polynomials are highly suggestive. A connection with p-adicization emerges via the identification of the p-adic prime as one of the ramified primes of P. Only (monic) polynomials having a common p-adic prime are allowed in the sum. The counterpart of the vacuum functional exp(-K) is naturally identified as the discriminant D of the extension associated with P and p-adic coupling constant evolution emerges from the identification of exp(-K) with D.
See the article About TGD counterparts of twistor amplitudes or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD. 

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