I have proposed a realization of unitarity in twistor approach without loops and with discrete coupling constant evolution dictated by number theory (see this). The proposal relies crucially on the identification quantum numbers in M8 picture as like-light quaternionic 8-momenta and the assumption that also many-particle states are massles. The 8-momenta are also complex already at classical level with corresponding imaginary unit i commuting with octonionic imaginary units Ik of M8.
The essential assumption was that the 8-momenta of also many-particle states are light-like. It is easy to see that this cannot make sense if single particle states have light-like 8-momenta unless they are also parallel. For a moment I thought that complexification of single particle 8-momenta might help but it did not.
Next came the realization that BCFW construction actually gives analogs of zero energy states having complex light-like momenta. The single particle momenta are not however light-like anymore. In TGD these states can be assigned with the interior regions of causal diamonds and have interpretation as resonances/bound states with complex momenta. The following tries to articular this more precisely.
It is indeed easy to see that many-particle states with light-like single particle momenta cannot have light-like momenta unless the single-particle momenta are parallel so that in non-parallel case one must give up light-likeness condition also in complex sense.
- In BCFW approach the expression of residue integral as sum of poles in the variable z associated with the amplitude obtained by the deformation pi→ pi+zri of momenta (∑ ri=0, ri• rj=0) leads to a decomposition of the tree scattering amplitude to a sum of products of amplitudes in resonance channels with complex momenta at poles. The products involve 1/P2 factor giving pole and the analog of cut in unitary condition. Proof of tree level unitarity is achieved by using complexified momenta as a mere formal trick and complex momenta are an auxiliary notion. The complex massless poles are associated with groups I of particles whereas the momenta of particles inside I are complex and non-light-like.
- Could BCFW deformation give a description of massless bound states massless particles so that the complexification of the momenta would describe the effect of bound state formation on the single particle states by making them non-light-like? This makes sense if one assumes that all 8-momenta - also external - are complex. The classical charges are indeed complex already classically since Kähler coupling strength is complex (see this). A possible interpretation for the imaginary part is in terms of decay width characterizing the life-time of the particle and defining a length of four-vector.
- The basic question in the construction of scattering amplitudes is what happens inside CD for the external particles with light-like momenta. The BCFW deformation leading to factorization suggests an answer to the question. The factorized channel pair corresponds to two CDs inside which analogs of M and N-M particle bound states of external massless particles would be formed by the deformation pi→ pi+zri making particle momenta non-light-like. The allowed values of z would correspond to the physical poles. The factorization of BCFW scattering amplitude would correspond to a decomposition to products of bound state amplitudes for pairs of CDs. The analogs of bound states for zero energy states would be in question. BCFW factorization could be continued down to the lowest level below which no factorization is possible.
- One can of course worry about the non-uniqueness of the BCFW deformation. For instance, the light-like momenta ri must be parallel (ri=λi r) but the direction of r is free. Also the choice of λi is free to a high extent. BCFW expression for the amplitude as a residue integral over z is however unique. What could this non-uniqueness mean?
Suppose one accepts the number theoretic vision that scattering amplitudes are representations for sequences of algebraic manipulations. These representations are bound to be highly non-unique since very many sequences can connect the same initial and final expressions. The space-time surface associated with given representation of the scattering amplitude is not unique since each computation corresponds to different space-time surface. There however exists a representation with maximal simplicity.
Could these two kinds of non-uniqueness relate?
See the chapter The Recent View about Twistorialization in TGD Framework or the article with the same title.
- The condition of light-likeness in complex sense allows the vanishing of real and imaginary mass squared for individual particles
Im(pi)= λi Re(pi) ,
Real and imaginary parts are parallel and light-like in 8-D sense.
- The remaining two conditions come from the vanishing of the real and imaginary parts of the total mass squared:
∑i≠ j Re(pi)• Re(pj)-Im(pi)• Im(pj) =0 ,
∑i≠ j Re(pi)• Im(pj)=0 .
By using proportionality of Re(pi) and Im(pi) one can express the conditions in terms of the real momenta
∑i≠ j (1-λiλj) Re(pi)• Re(pj) =0 ,
∑i≠ j λj Re(pi)• Re(pj)=0 .
For positive/negative energy part of zero energy state the sign of time component of momentum is fixed and therefore λi have fixed sign. Suppose that λi have fixed sign. Since the inner products pi• pj of time-like vectors with fixed sign of time compomemet are all positive or negative the second term can vanish only if one has pi•pj=0. If the sign of λi can vary, one can satisfy the condition linear in λi but not the first condition as is easy to see in 2-particle case.
- States with light-like parallel 8-momenta are allowed and one can ask whether this kind of states might be realized inside magnetic flux tubes identified as carriers of dark matter in TGD sense. The parallel light-like momenta in 8-D sense would give rise to a state analogous to super-conductivity. Could this be true also for quarks inside hadrons assumed to move in parallel in QCD based model. This also brings in mind the earlier intuitive proposal that the momenta of fermions and antifermions associated with partonic 2-surfaces must be parallel so that the propagators for the states containing altogether n fermions and antifermions would behave like 1/(p2)n/2 and would not correspond to ordinary particles.