Congratulations to myself

While enjoying a few hours of leisure time yesterday, I realized that the further generalization of the notion of imbedding space (see for instance this) inspired by the hierarchy of Planck constants has led to a really spectacular progress in the understanding of TGD. Therefore I felt that it is good to try to declare for the History what have been communicated through Her humble correspondent to the humanity;-). Lists are the manner that She uses to make things clear to me and to bore the readers and this posting will not be an exception.

1. I have now a rather clear view about the eigenmodes of the modified Dirac operator. The generalized eigenvalue (I used plural earlier) which is a scalar field has an interpretation as Higgs vacuum expectation value. Why singular must be used should have been understood long time ago from the orthogonality requirement for various modes and lack of correlation between longitudinal (light-like) and transversal degrees of freedom occuring in the modified Dirac operator: this is what makes possible eigenvalues which are functions. The scale of the vacuum expectation is proportional to log(p) corresponding to the hierarchy of p-adic length scales.

2. The construction leads to an identification of number theoretical braids as a set of points for which Higgs potential has minimum value: the first and quite not correct identification was as quantum critical points in the intersection of all sectors of the generalized imbedding space forming a book like structure. This book is rather thick having infinite number of pages and the value of Planck constant gives a partial oage numbering: the back of the book is the 4-D quantum critical manifold M²× S²₂ of H. You have actually library of identical copies of this book obtained by applying Poincare and color isometries of H. The points in the intersection with critical sub-manifold, the back of the book, however correspond to zeros of Higgs which are unstable by their quantum criticality.

   By the way, it seems that physical thinking is the image of what happens in physics: first the simplest extremum, then the stable extremum. What is really beautiful is the realization of the breaking of the exact quantum criticality (corresponds to vacuum extremals) in terms of Higgs mechanism. Higgs as God particle: I begin to take this phrase more seriously!

3. Higgs potential can be taken to be the negative of the modulus squared of Higgs or any monotonically increasing function of it: the potential as such does not matter since it is only an auxiliary quantity and diffeo-invariance is involved so that only the extrema matter. This is new and sounds of course a little bit strange. The point is however that Higgs modulus is essentially the distance of point of the partonic 2-surface from the quantum critical 2-sphere of CP₂ which of course has minima. Partonic 2-surface induces the dynamics and in case of classical gauge fields and metric. One further example of the power of induction mechanism.

   Higgs potential has this utterly simple expression only in the local complex coordinate and Higgs actually is the universal local complex coordinate satisfying the number theoretical needs. At the critical line surrounding zero, the modulus of this coordinate begins to decrease and in order to avoid assignment of several points of partonic 2-surface to single coordinate value one must introduce coordinate patches. Earth's surface is very good metaphor for Higgs landscape with mountain tops is zeros and valley bottoms as minima defining the number theoretic braid: the modulus of Higgs corresponds to the height coordinate as a local coordinate. In global coordinates, naturally the complex coordinate of S², the minima become explicit and one can say that the minima of Higgs potential emerge through the geometry of space-time sheet.

4. Dirac determinant is identified as the product of the complex values of Higgs at the points of braids associated with the partonic 2-surface: in QFT it would be ill-defined product over all points for eigenvalues of Dirac operator. For years I have been conjecturing that Dirac determinant reduces to the product of exponents of Kähler action and Chern-Simons action. Determinant should reduce to unity for vacuum extremals of Kahler action. The pleasant news is that the construction of Higgs vacuum expectation is consistent with this condition: this is a highly, highly non-trivial result. Everything is of course finite and discrete which is rather ironic taking into account that the dynamics of arena is the infinite-dimensional world of classical worlds.

5. A concrete connection with the non-commutative geometry approach suggests itself: the non-commutative geometry of quantum critical geodesic sphere of CP₂ would be natural manner to describe the anticommutativity of induced spinor field and its conjugate at only the points of number theoretic braid rather than along line (string) as implied by the holomorphy of the spinor mode basis. Non-commutativity is equivalent with the reduction of basis to a finite sub-basis implied by the discretization of partonic 2-surface to collection of number theoretic braids defined by Higgs minima.

   The underlying philosophy is the description of the finite quantum measurement resolution in terms of Jones inclusions and leading naturally to non-commutative Hilbert space, Connes tensor product, etc... What is also new is that braiding can be seen as being induced by the motion of Higgs minima (here also the projections on the geodesic 2-sphere could define the braiding).

6. Higgs expectation has a purely geometric interpretation. The very construction of the Higgs expectation allows to assign a 4-D space-time sheet with a light-like 3-surface. The additional dimension comes via the lines connecting partonic 2-surface to the quantum critical geodesic sphere which are mathematically orbits of Kähler charged particle with a dynamically generated charge. The emergence of the additional dimension naturally is a real victory. That this space-time sheet is indeed an extremal of Kähler action remains to be demonstrated. This would not be a big surprise.

7. The zeta function defined by the values of Higgs at the points of braids associated with the partonic 2-surface codes for geometric data about it so that the earlier speculation is now become a firm fact. The zeros of zeta are excellent candidates for what I call super-canonical conformal weights and many of them should reside at the critical line of zeta since the conformal weights for the function basis depending on the radial coordinate of lightcone boundary have naturally real part equal to 1/2. A very elegant and economic reduction of most of basic physics of quantum TGD to the properties of the modified Dirac operator takes place. This means also a concrete realization of the number theoretic ideas.

8. One fascinating and emotionally stressing conjecture (I believe it just now but I cannot predict what I think after one minute) is the self referentiality of the zeta function stating that the eigenvalues λ (complex values of Higgs at minima) defining the zeta by replacing natural number n in Riemann zeta are proportional to the inverses of zeta at the points defining the number theoretical braid and continue to a map from geodesic sphere to geodesic sphere by replacing λ with general complex point of sphere. The weak form of this conjecture is satisfied by the zeros s= -2m, m=1,2,... of Riemann Zeta in the sense that zeta(s) indeed vanishes at points -2λ, &lambda= m. Now one would have stronger form holding true not only at origin but also at the points associated with the number theoretical braids and perhaps at entire sphere.

9. Also many other things remain to be done to prove the long list of purely mathematical conjectures characterizing preferred extremals of Kähler action analogous to Bohr orbits. These conjectures are obvious once TGD view about physics is expected but mathematically very non-trivial. This is one of the frustrating things in this business of re-creating universe on computer screen: you simply know that if I your philosophy is this and this then that and that which is mathematically something highly non-trivial must be true. But you have no idea about how to prove it. I wonder whether She has experienced similar frustrations and why she does not want to tell me the proof;-).

For more details see the chapter Construction of Quantum Theory: Symmetries of "Towards S-matrix".