1. M8-M4× CP2 duality
What makes the proposed connection so interesting from the TGD point of view is that in TGD M8-M4× CP2 duality (see for instance this, this and this) states number theoretic and geometric descriptions of physics are dual and this duality is the generalization of wave-mechanical momentum-position duality having no generalization in quantum field theory since position is not an observable in quantum field theory but mere coordinate of space-time.
- M8 picture about space-time surface provides a number theoretic description of physics based on the identification of space-time surfaces as algebraic surfaces. Dynamics is coded by the condition that the normal space of the space-time surface is associative.
- H= M4× CP2 provides a geometric description of space-time surfaces based on differential geometry, partial differential equations, and action principle. The existence of twistor lift of TGD fixes the choice of H uniquely (this).
The solutions of field equations reduce to minimal surfaces as counterparts for solutions of massless field equations and the simultaneous extremal of Kähler action implies a close connection with Maxwell's theory. Space-time surfaces would be analogous to soap films spanned by dynamically generated frames (this).
Beltrami field property implies that dissipation is absent at the space-time level and gives support for the conjecture that the QFT limit gives Einstein-YM field equations in good approximation. The absence of dissipation is also a correlate for quantum coherence implying absence of dissipation (this).
2. Adelic physics as unified description of sensory experience and cognition
Adelic physics involves both real and p-adic number fields (see for instance this). p-Adic variants of the space-time surface are an essential piece and give rise to mathematical correlates of cognition. Cognitive representations are discretizations, which consist of points of space-time surface a, whose imbedding space coordinates are in an extension of rationals characterizing a given adele are common to real and various p-adic variants of space-time, define the intersection of cognitive and sensory realities.
What is so nice from the physics point of view, is that these discretizations are unique for a given adele and adeles form an infinite evolutionary cognitive hierarchy . The p-adic geometries proposed by Scholze would be very interesting from this point of view and I wonder whether there might be something common between TGD and the work done by Scholze. Unfortunately, I do not have the needed knowledge about technicalities.
3. Langlands corresponds and TGD
Also Langlands correspondence, which I have tried to understand several times with my tiny physicist's brain, is involved.
- Global Langlands correspondence (GLC) states that there is a connection between representations of continuous groups and Galois groups of extensions of rationals.
- Local LC states (LLC) states this in the case of p-adics.
- In adelic physics real numbers and p-adic number fields define the adele. Sensory experience corresponds to reals and cognition to p-adics. Cognitive representations are in their discrete intersection and for extensions of rationals belonging to the intersection.
- Sensory world, "real" world corresponds to representation of continuous groups/Galois groups of rationals: this would be GLC.
- "p-Adic" worlds correspond to cognition and representations of p-adic variants of continuous groups and Galois groups over p-adics: this would be LLC.
- One could perhaps talk also about Adelic LC (ALC) in the TGD framework. Adelic representations would combine real and p-adic representations for all primes and give as complete a view about reality as possible.
4. Galois groups, physics and cognition
TGD provides a geometrization for the action of Galois groups (see this and this).
- Galois groups are symmetry groups of TGD since space-time surfaces are determined by polynomials with rational (possibly also algebraic) coefficients continued to octonionic polynomials Galois groups relate to each other sheets of space-time time and a very nice physical picture emerges. Physical states correspond to the representations of Galois groups and are crucial in the dark matter sector, especially important in quantum biology. Space-time surface provides them and also the fermionic Fock states realize them.
- The order n of the Galois group over rationals corresponds to an effective Planck constant heff= nh0 so that there is a direct connection to a generalization of quantum physics (see for instance this). The phases of ordinary matter with heff=nh0 behave like dark matter. n measures the algebraic complexity of space-time surfaces and serves also as a kind of IQ. Evolution means an increase of n and therefore increase of IQ.
The representation of real continuous groups assignable to the real numbers as a piece of adele would be related to the representations of Galois groups in GLC.
Also p-adic representations of groups are needed to describe cognition and these p-adic group representations and representations of p-adic Galois groups would be related by LLC.
For a summary of earlier postings see Latest progress in TGD.
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