Is pair creation really understood in the twistorial picture?
Twistorialization leads to a beautiful picture about scattering amplitudes at the level of M8 (see this). In the simplest picture, scattering would be just a re-organization of Galois singlets to new Galois singlets. Fundamental fermions would move as free particles.
The components of the 4-momentum of virtual fundamental fermion with mass m would be algebraic integers and therefore complex. The real projection of 4-momentum would be mapped by M8-H duality to a geodesic of M4 starting from either vertex of the causal diamond (CD) . Uncertainty Principle at classical level requires inversion so that one has a= ℏeff/m, where ab denotes light-cone proper time assignable to either half-cone of CD and m is the mass assignable to the point of the mass shell H3⊂ M4⊂ M8.
The geodesic would intersect the a=ℏeff/m 3-surface and also other mass shells and the opposite light-cone boundaries of CDs involved. The mass shells and CDs containing them would have a common center but Uncertainty Principle at quantum level requires that for each CD and its contents there is an analog of plane wave in CD cm degrees of freedom.
One can however criticize this framework. Does it really allow us to understand pair creation at the level of the space-time surfaces X4⊂ H?
- All elementary particles consist of fundamental fermions in the proposed picture. Conservation of fermion number allows pair creation occurring for instance in the emission of a boson as fermion-antifermion pair in f→ f+b vertex.
- The problem is that if only non-space-like geodesics of H are allowed, both fermion and antifermion numbers are conserved separately so that pair creation does not look possible. Pair creation is both the central idea and source of divergence problems in QFTs.
- One can identify also a second problem: what are the anticommutation relations for the fermionic oscillator operators labelled by tachyonic and complex valued momenta? Is it possible to analytically continue the anticommutators to complexified M4⊂ H and M4⊂ M8? Only the first problem will be considered in the following.
Is it possible to understand pair creation in the proposed picture based on twistor scattering amplitudes or should one somehow bring the bff 3-vertex or actually ff fbar fbar vertex to the theory at the level of quark lines? This vertex leads to a non-renormalizable theory and is out of consideration.
One can first try to identify the key ingredients of the possible solution of the problem.
- Crossing symmetry is fundamental in QFTs and also in TGD. For non-trivial scattering amplitudes, crossing moves particles between initial and final states. How should one define the crossing at the space-time level in the TGD framework? What could the transfer of the end of a geodesic line at the boundary of CDs to the opposite boundary mean geometrically?
- At the level of H, particles have CP2 type extremals – wormhole contacts – as building bricks. They have an Euclidean signature (of the induced metric) and connect two space-time sheets with a Minkowskian signature.
The opposite throats of the wormhole contacts correspond to the boundaries between Euclidean and Minkowskian regions and their orbits are light-like. Their light-like boundaries, orbits of partonic 2-surfaces, are assumed to carry fundamental fermions. Partonic orbits allow light-like geodesics as possible representation of massless fundamental fermions.
Elementary particles consist of at least two wormhole contacts. This is necessary because the wormhole contacts behave like Kähler magnetic charges and one must have closed magnetic field lines. At both space-time sheets, the particle could look like a monopole pair.
- If fundamental fermion lines are geodesics at the light-like partonic orbits, they can be light-like but are space-like if there is motion in transversal degrees of freedom.
- Consider a geodesic carrying a fundamental fermion, starting from a partonic 2-surface at either light-like boundary of CD. As a free fermion, it would propagate to the opposite boundary of CD along the wormhole throat.
What happens if the fermion goes through a topological 3-vertex? Could it turn backwards in time at the vertex by transforming first to a space-like geodesic inside the wormhole contact leading to the opposite throat and return back to the original boundary of CD? It could return along the opposite throat or the throat of a second wormhole contact emerging from the 3-vertex. Could this kind of process be regarded as a bifurcation so that it would correspond to a classical non-determinism as a correlate of quantum non-determinism?
By Lorentz invariance, the definition of conserved mass squared as an eigenvalue of d’Alembertian could be possible. For light-like 3-surfaces the d’Alembertian reduces to the d’Alembertian for the Euclidean partonic 2-surface having only non-positive eigenvalues. If this process is possible and conserves M4 mass squared, the geodesic must be space-like and therefore tachyonic.
The non-conservation of M4 momentum at single particle level (but not classically at n-particle level) would be due to the interaction with the classical fields.
- The description of massivation in terms of p-adic thermodynamics suggests that at the fundamental level massive particles involve non-observable tachyonic contribution to the mass squared assignable to the wormhole contact, which cancels the non-tachyonic contribution.
All articles, and for the most general option all quantum states could be massless in this sense, and the massivation would be due the restriction of the consideration to the non-tachyonic part of the mass squared assignable to the Minkowskian regions of X4.
Massivation could solve the problem and p-adic thermodynamics indeed predicts that even photons have a very small thermal mass. Massless gauge bosons (and particles in general) would be possible in the sense that the positive mass squared is compensated by equally small tachyonic contribution.
For a summary of earlier postings see Latest progress in TGD.
Source: http://matpitka.blogspot.com/2022/08/is-pair-creation-really-understood-in.html
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