Tuesday, August 10, 2021

The conjectured duality between number theory and geometry from TGD point of view

There was a Quanta Magazine article about the link between geometry and numbers (see this). In the following I consider this idea from the TGD view point.

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.

  1. 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.
  2. 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).

It would be very nice if this duality between number theory and geometry would be present at the level of mathematics itself.

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.

  1. Global Langlands correspondence (GLC) states that there is a connection between representations of continuous groups and Galois groups of extensions of rationals.
  2. Local LC states (LLC) states this in the case of p-adics.
There is a nice interpretation for the two LCs in terms of sensory experience and cognition in the TGD inspired theory of consciousness.
  1. 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.
  2. Sensory world, "real" world corresponds to representation of continuous groups/Galois groups of rationals: this would be GLC.
  3. "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.
  4. 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).

  1. 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.
  2. 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.

    Articles and other material related to TGD.

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