The preparation of an article about number theoretic aspects of TGD forced to go through various related ideas and led to a considerable integration of the ideas. In this note idea about the symplectic structure of M4 is discussed although it is not directly related to number theoretic aspects of TGD.
- Twistor lift of TGD suggests strongly a symmetry between M4 and CP2. In particular, M4 should have the analog of symplectic structure.
- It has been already noticed that this structure could allow to understand both CP breaking and matter-antimatter asymmetry from first principles. A further study showed that it can also allow to understand electroweak symmetry breaking.
- Should assign also to M4 the analog of symplectic structure giving an additional contribution to the induced Kähler form? The symmetry between M4 and CP2 suggests this, and this term could be highly relevant for the understanding of the observed CP breaking and matter antimatter asymmetry. Poincare invariance is not lost since the needed moduli space for M4 Kähler forms would be the moduli space of CDs forced by ZEO in any case, and M4 Kähler form would serve as the correlate for fixing rest system and spin quantization axis in quantum measurement.
- Also induced spinor fields are present. The well-definedness of electro-magnetic charge for the spinor modes forces in the generic case the localization of the modes of induced spinor fields at string world sheets (and possibly to partonic 2-surfaces) at which the induced charged weak gauge fields and possibly also neutral Z0 gauge field vanish. The analogy with branes and super-symmetry force to consider two options.
Option I: The fundamental action principle for space-time surfaces contains besides 4-D action also 2-D action assignable to string world sheets, whose topological part (magnetic flux) gives rise to a coupling term to Kähler gauge potentials assignable to the 1-D boundaries of string world sheets containing also geodesic length part. Super-symplectic symmetry demands that modified Dirac action has 1-, 2-, and 4-D parts: spinor modes would exist at both string boundaries, string world sheets, and space-time interior. A possible interpretation for the interior modes would be as generators of space-time super-symmetries.
This option is not quite in the spirit of SH and string tension appears as an additional parameter. Also the conservation of em charge forces 2-D string world sheets carrying vanishing induced W fields and this is in conflict with the existence of 4-D spinor modes unless they satisfy the same condition. This looks strange.
Option II: Stringy action and its fermionic counterpart are effective actions only and justified by SH. In this case there are no problems of interpretation. SH requires only that the induced spinor fields at string world sheets determine them in the interior much like the values of analytic function at curve determine it in an open set of complex plane. At the level of quantum theory the scattering amplitudes should be determined by the data at string world sheets. If induced W fields at string world sheets are vanishing, the mixing of different charge states in the interior of X4 would not make itself visible at the level of scattering amplitudes! In this case 4-D spinor modes do not define space-time super-symmetries.
This option seems to be the only logical one. It is also simplest and means that quantum TGD would reduce to string model apart from number theoretical discretization of space-time surface bringing in dark matter as heff/h=n phases with n identifiable as factor of the order of the Galois group of extension of rationals. This would also lead to adelic physics, predict preferred extensions and identify corresponding ramified primes as preferred p-adic primes.
- Why the string world sheets coding for effective action should carry vanishing weak gauge fields? If M4 has the analog of Kähler structure, one can speak about Lagrangian sub-manifolds in the sense that the sum of the symplectic forms of M4 and CP2 projected to Lagrangian sub-manifold vanishes. Could the induced spinor fields for effective action be localized to generalized Lagrangian sub-manifolds? This would allow both string world sheets and 4-D space-time surfaces but SH would select 2-D Lagrangian manifolds. At the level of effective action the theory would be incredibly simple.
Induced spinor fields at string world sheets could obey the "dynamics of avoidance" in the sense that both the induced weak gauge fields W,Z0 and induced Kähler form (to achieve this U(1) gauge potential must be sum of M4 and CP2 parts) would vanish for the regions carrying induced spinor fields. They would coupleonly to the induced em field (!) given by the vectorial R12 part of CP2 spinor curvature for D=2,4. For D=1 at boundaries of string world sheets the coupling to gauge potentials would be non-trivial since gauge potentials need not vanish there. Spinorial dynamics would be extremely simple and would conform with the vision about symmetry breaking of weak group to electromagnetic gauge group.
The projections of canonical currents of Kähler action to string world sheets would vanish, and the projections of the 4-D modified gamma matrices would define just the induced 2-D metric. If the induced metric of space-time surface reduces to an orthogonal direct sum of string world sheet metric and metric acting in normal space, the flow defined by 4-D canonical momentum currents is parallel to string world sheet. These conditions could define the "boundary" conditions at string world sheets for SH.
For a summary of earlier postings see Latest progress in TGD.