Reduction of standard model structure to CP_2 geometry and other key ideas of TGD

Originally this article was an appendix meant to be a purely technical summary of basic facts about CP₂ but in its recent form it tries to briefly summarize those basic visions about TGD which I dare to regarded stabilized. I have added links to illustrations making it easier to build mental images about what is involved and represented briefly the key arguments. This chapter is hoped to help the reader to get fast grasp about the concepts of TGD.

The basic properties of embedding space and related spaces are discussed and the relationship of CP₂ to the standard model is summarized. The basic vision is simple: the geometry of the embedding space H=M⁴× CP₂ geometrizes standard model symmetries and quantum numbers. The assumption that space-time surfaces are basic objects, brings in dynamics as dynamics of 3-D surfaces based on the induced geometry. Second quantization of free spinor fields of H induces quantization at the level of H, which means a dramatic simplification.

The notions of induction of metric and spinor connection, and of spinor structure are discussed. Many-sheeted space-time and related notions such as topological field quantization and the relationship many-sheeted space-time to that of GRT space-time are discussed as well as the recent view about induced spinor fields and the emergence of fermionic strings. Also the relationship to string models is discussed briefly.

Various topics related to p-adic numbers are summarized with a brief definition of p-adic manifold and the idea about generalization of the number concept by gluing real and p-adic number fields to a larger book like structure analogous to adele (see this). In the recent view of quantum TGD (see this), both notions reduce to physics as number theory vision, which relies on M⁸-H duality (see this and this ) and is complementary to the physics as geometry vision.

Zero energy ontology (ZEO) (see this) has become a central part of quantum TGD and leads to a TGD inspired theory of consciousness as a generalization of quantum measurement theory having quantum biology as an application. Also these aspects of TGD are briefly discussed.

The preparation of this article led to one very interesting question. The twistor lift of TGD (see this and this) leads to the proposal that the preferred extremals of the 4-D dimensionally reduced 6-D Kähler action reduces to the sum of 4-D Kähler action and volume term. These extremals are analogues of soap films spanned by frames: minimal surfaces with singularities. Outside the frames, these minimal 4-surfaces are simultaneous extremals of Kähler action. This is guaranteed if the holomorphicity of string world sheets generalizes to the 4-D case. The interpretation is in terms of quantum criticality.

These surfaces are actually extremals for a very large class of actions. Does it make sense to ask which 4-D action is the correct one? The 4-D action defines a Kähler function of a Kähler metric of "world of classical worlds". Do different actions define different Kähler metrics or are the metrics actually identical when some constraints on the couplings are posed. If the WCW metrics defined by different actions are equivalent, the Kähler functions differ by an addition of a gradient of a complex function to the Kähler gauge potential defined by Kähler function.

The number theoretic vision of TGD based on M⁸ predicts a discrete coupling constant evolution with levels labelled by degrees of rational polynomials and corresponding extensions of rationals characterized by Galois groups and by ramified primes defining p-adic length scales (see this). Hierarchies of composite polynomials define inclusion sequences of extensions. These sequences would correspond to discrete coupling constant evolutions.

What could be the counterparts of these evolutions at the level of H=M⁴× CP₂ and WCW? Could they be characterized by the values of coupling parameters defining the action defining the Kähler function of WCW but giving rise to the same Kähler metric of WCW? Could the coupling constant evolution correspond to a sequence of U(1) gauge transformations of WCW and identifiable as symplectic transformations of WCW?

WCW could decompose to sub-WCWs corresponding to different actions and coupling constant evolutions, a kind of theory space. These theories would not be equivalent. Rather, a theory corresponding to a given composite would contain as subtheories the theories corresponding to lower polynomial composites. A possible interpretation would be as hierarchies of effective field theories. The choice of the action in an optimal manner in a given scale could be seen as a choice of the most appropriate effective field theory.

See the article Reduction of standard model structure to CP_2 geometry and other key ideas of TGD.

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