Tuesday, June 20, 2023

Master formula for the scattering amplitudes: finally?

Most pieces that have been identified over the years in order to develop a master formula for the scattering amplitudes are as such more or less correct but always partially misunderstood. Maybe the time is finally ripe for the fusion of these pieces to a single coherent whole. I will try to list the pieces into a story in the following.
  1. The vacuum functional, which is the exponential Kähler function defined by the classical bosonic action defining the preferred extremal a an analog of Bohr orbit, is the starting point. Physically, the Kähler function corresponds to the bosonic action (e.g. EYM) in field theories.

    Because holography is almost unique, it replaces the path integral by a sum over 4-D Bohr trajectories as functional integral over 3-surfaces plus discrete sum.

  2. However, the fermionic part of the action is missing. I have proposed a long time ago a super symmetrization of WCW K hler function by adding to it what I call modified Dirac action. It relies on modified gamma matrices modified gamma matrices Γα, which are contractions ΓkTα k of H gamma matrices Γk with the canonical momentum currents T= ∂ L/∂αhk defined by the Lagrangian L. Modified Dirac action is therefore determined by the bosonic action from the requirement of supersymmetry. This supersymmetry is however quite different from the SUSY associated with the standard model and it assigns to fermonic Noether currents their super counterparts.

    Bosonic field equations for the space-time surface actually follow as hermiticity conditions for the modified Dirac equation. These equations also guarantee the conservation of fermion number(s). The overall super symmetrized action that defines super symmetrized Kähler function in WCW would be unambiguous. One would get exactly the same master formula as in quantum field theories, but without the path integral.

  3. The overall super symmetrized action is sum of contributions assignable to the space-time surface itself, its 3-D light-like parton orbits as boundaries between Minkowskian regions and Euclidian wormhole contact, 2-D string world sheets and their 1-D boundaries as orbits of point-like fermions. These 1-D boundaries are the most important and analogous to the lines of ordinary Feynman diagrams. One obtains a dimensional hierarchy.
  4. One can assign to these objects of varying dimension actions defined in terms of the induced geometry and spinor structure. The supersymmetric actions for the preferred extremals analogous to Bohr orbit in turn give contributions to the super symmetrized Kähler function as an analogue of the YM action so that, apart from the reduction of path integral to a sum over 4-D Bohr orbits, there is a very close analogy with the standard quantum field theory.
However, some problems are encountered.
  1. It seems natural to assume that a modified Dirac equation holds true. I have presented an argument for how it indeed emerges from the induction for the second quantized spinor field in H restricted to the space-time surface assuming modified Dirac action.

    The problem is, however, that the fermionic action, which should define vertex for fermion pair creation, disappears completely if Dirac's equation holds everywhere! One would not obtain interaction vertices in which pairs of fermions arise from classical induced fields. Something goes wrong.

  2. If one gives up the modified Dirac equation, the fermionic action does not disappear? In this case, one should construct a Dirac propagator for the modified Dirac operator. This is an impossible task in practice.

    Moreover, the construction of the propagator is not even necessary and in conflict with the fact that the induced spinor fields are second quantized spinors of H restricted to the space-time surface and the propagators are therefore well-defined and calculable and define the propagation at the space-time surface.

    Should we conclude that the modified Dirac equation cannot hold everywhere? What these, presumably lower-dimensional regions of space-time surface, are and could they give the interaction vertices as topological vertices?

The key question is how to obtain emission of fermion pairs and bosons as their bound states?
  1. I have previously derived a topological description for reaction vertices. The fundamental 1 → 2 vertex (for example e → e+ gamma) generalizes the basic vertex of Feynman diagrams, where a fermion emits a boson or a boson decays into a pair of fermions. Three lines meet at the ends.

    In TGD, this vertex can topologically correspond to the decomposition of a 3-surface into two 3-surfaces, the decomposition of a partonic 2-surface into two, the decomposition of a string into two, and finally, the turning of the fermion line backwards from time. One can say that the n-surfaces are glued together along their n-1-dimensional ends, just like the 1-surfaces are glued at the vertex in the Feynman diagram.

  2. In the earlier posting, I already discussed how to identify vertex for fermion-antifermion pair creation as a V-shaped turning point of a 1-D fermion line. The fermion line turns back in time and fermion becomes an antifermion. Now, however, the quantized boson field at the vertex is replaced by a classical boson field. This description is basically the same as in the ordinary path integral where the gauge potentials are classical.

    The problem was that if the modified Dirac equation holds everywhere, there are no pair creation vertices. The solution of the problem is that the modified Dirac equation at the V-shaped vertex cannot hold true.

    What this means physically is that fermion and antifermion numbers are not separately conserved in the vertex. The modified Dirac action for the fermion line can be transformed to the change of antifermion number as operator (or fermion number at the vertex) expressible as the change of the antifermion part of the fermion number. This is expressible as the discontinuity of a corresponding part of the conserved current at the vertex. This picture conforms with the appearance of gauge currents in gauge theory vertices. Notice that modified gamma matrices determined by the bosonic action appear in the current.

  3. This argument was limited to 1-D objects but can be generalized to higher-dimensional defects by assuming that the modified Dirac equation holds true everywhere except at defects represented as vertices, which become surfaces. The modified Dirac action reduces to an integral of the discontinuity of say antifermion current at the vertex, i.e. the change of the antifermion charge as an operator.
What remains to be understood more precisely is the connection with the exotic smooth structures possible only in 4-D space-time.
  1. As already explained, towards the end of last year I realized that this V-shaped defect could correspond to a point-like defect of an exotic 4-D smooth structure. In general relativity as also in TGD, causal loops are associated with these defects. In TGD, the causal anomaly would mean that the direction of time is reversed in the vertex since antifermions and fermions can be thought of as moving in opposite directions of time. What is so remarkable is that this interpretation is possible only in 4-D space-time; in higher dimensions irreducible exotic smooth structures are impossible.
  2. The next step is to ask whether a generalization is possible. Exotic smooth structures reduce to standard ones except in a set of defects having measure zero. The interpretation is that the dimension of defects, in the case that they are surfaces, is less than 4. Also non-point-like defects might be possible in contrast to what I assumed at first. If not, then only the direction of fermion lines could change.

    If the generalization is possible, also 1-D, 2-D, and 3-D defects are possible. In the 1→ 2 vertex the orbit of an n<4- dimensional surface would turn back in the direction of time. These are exactly the various topological vertices that I have previously arrived at, but guided by a physical intuition.

    An entire hierarchy of particles of different dimensions is possible. As a matter of fact, in topological condensed matter physics, they are commonplace. One talks about bulk states, boundary states, edge states and point-like singularities.

    All in all, exotic smooth structures would give vertices without vertices assuming only free fermions fields and no primary boson fields! And this is possible only in space-time dimension 4!

See the article Exotic smooth structures at space-time surfaces and master formula for scattering amplitudes in TGD} , the earlier article Intersection form for 4-manifolds, knots and 2-knots, smooth exotics, and TGD or the chapter Does M8 H duality reduce classical TGD to octonionic algebraic geometry?: Part II .

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

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