The problem is that at the level of M8 analogous to momentum space polynomials define the space-time surfaces and number theoretically periodic functions are not a natural or even possible basis. At the level of H the space-time surfaces can be regarded orbits of 3-surfaces representing particles and are dynamical so that Fourier analysis is natural.
That this is the case, is suggested by the fact that M8-H duality maps normal spaces of space-time surface to points of CP2. Normal space is essentially the velocity space for surface deformations normal to the surface, which define the dynamical degrees of freedom by general coordinate invariance. Therefore dynamics enters to the picture. It turns out that the conjecture finds support.
Consider now the topic of this posting. Number theoretic vision about TGD is forced by the need to to describe the correlates of cognition. It is not totally surprising that these considerations lead to new insights related to the notion of cognitive measurement involving a cascade of measurements in the group algebra of Galois group as a possible model for analysis as a cognitive process (see this, this and this).
- The dimension n of the extension of rationals as the degree of the polynomial P=Pn1∘ Pn2∘ ... is the product of degrees of degrees ni: n=∏ini and one has a hierarchy of Galois groups Gi associated with Pni∘.... Gi+1 is a normal subgroup of Gi so that the coset space Hi=Gi/Gi+1 is a group of order ni. The groups Hi are simple and do not have this kind of decomposition: simple finite groups appearing as building bricks of finite groups are classified. Simple groups are primes for finite groups.
- The wave function in group algebra L(G) of Galois group G of P has a representation as an entangled state in the product of simple group algebras L(Hi). Since the Galois groups act on the space-time surfaces in M8 they do so also in H. One obtains wave functions in the space of space-time surfaces. G has decomposition to a product (not Cartesian in general) of simple groups. In the same manner, L(G) has a representation of entangled states assignable to L(Hi) (see this and this).
- Cognitive measurement would reduce the entanglement between L(H1) and L(H2), the between L(H2) and L(H3) and so on. The outcome would be an unentangled product of wave functions in L(Hi) in the product L(H1)× L(H2)× .... This cascade of cognitive measurements has an interpretation as a quantum correlate for analysis as factorization of a Galois group to its prime factors defined by simple Galois groups. Similar interpretation applies in M4 degrees of freedom.
- This decomposition could correspond to a replacement of P with a product ∏i Pi of polynomials with degrees n= n1n2..., which is irreducible and defines a union of separate surfaces without any correlations. This process is very much analogous to analysis.
- The analysis cannot occur for simple Galois groups associated with extensions having no decomposition to simpler extensions. They could be regarded as correlates for irreducible primal ideas. In Eastern philosophies the notion of state empty of thoughts could correspond to these cognitive states in which SSFRs cannot occur.
- An analogous process should make sense also in the gravitational sector and would mean the splitting of K=nA appearing as a factor ngr=Kp to prime factors so that the sizes of CDs involved with the resulting structure would be reduced. Note that ep(1/K) is the root of e defining the trascdental infinite-D extension rationals which has finite dimension Kp for p-adic number field Qp. This process would reduce to a simultaneous measurement cascade in hyperbolic and trigonometric Abelian extensions. The IR cutoffs having interpretation as coherence lengths would decrease in the process as expected. Nature would be performing ordinary prime factorization in the gravitational degrees of freedom.
- For the algebraic EQs, the geometric description would be as a decay of n-sheeted 4-surface with respect to M4 to a union of ni-sheeted 4-surfaces by SSFRs. This would take place for flux tubes mediating all kinds of interactions.
In gravitational degrees of freedom, that is for trascendental EQs, the states with ngr=Kp having bundles of Kp flux tubes would deca to flux tubes bundles of ngr,i=Kip, where Ki is a prime dividing K. The quantity log(K) would be conserved in the process and is analogous to the corresponding conserved quantity in arithmetic quantum field theories and relates to the notion of infinite prime inspired by TGD \citeallbvisionc.
- This picture leads to ask whether one could speak of cognitive analogs of particle reactions representing interactions of "thought bubbles" i.e. space-time surfaces as correlates of cognition. The incoming and outgoing states would correspond to a Cartesian product of simple subgroups: G=∏×i Hi. In this composition the order of factors does not matter and the situation is analogous to a many particle system without interactions. The non-commutativity in general case leads to ask whether quantum groups might provide a natural description of the situation.
See the article Is M8-H duality consistent with Fourier analysis at the level of M4× CP2? or the chapter Breakthrough in understanding of M8-H duality.
For a summary of earlier postings see Latest progress in TGD.