Number theoretician encounters the problem of finding rational points of an algebraic curve defined as real or complex variant in which case the curve is 2-D surface and 1-D in complex sense. The curve is defined as root of polynomials polynomials or several of them. The polynomial have typically rational coefficients but also coefficients in extension of rationals are possible.
For instance, Fermat's theorem is about whether xn+yn=1, n=1,2,3,... has rational solutions for n≥1. For n=1, and n= 2 it has and these solutions can be found. It is now known that for n>2 no solutions do exist. Quite generally, it is known that the number is finite rather than infinite in the generic case.
A more general problem is that of finding points in some algebraic extension of rationals. Also the coefficients of polynomials can be numbers in the extension of rationals.
Connection with TGD and physics of cognition
The problem is extremely difficult even for mathematicians - to say nothing about humble physicist like me with hopelessly limited mathematical skills. It is however just this problem which I encounter in TGD inspired vision about adelic physics. Recall that in TGD space-times are 4-surfaces in H=M4×CP2, preferred extremals of the variational principle defining the theory.
- In this approach p-adic physics for various primes p provide the correlates for cognition: there are several motivations for this vision. Ordinary physics describing sensory experience and the new p-adic physics describing cognition for various primes p are fused to what I called adelic physics. The adelic physics is characterized by extension of rationals inducing extensions of various p-adic number fields. The dimension n of extension characterizes kind of intelligence quotient and evolutionary level since algebraic complexity is the larger, the larger the value of n is. The connection with quantum physics comes from the conjecture that n is essentially effective Planck constant h_eff/h_0=n characterizing a hierarchy of dark matters. The larger the value of n the longer the scale of quantum coherence and the higher the evolutionary level, the more refined the cognition.
- An essential notion is that of cognitive representation. It has several realizations. One of them is the representation as a set of points common to reals and extensions of various p-adic number fields induced by the extension of rationals. These space-time points have points in the extension of rationals considered defining the adele. The coordinates are the imbedding space coordinates of a point of the space-time surface.
- The gigantic challenge is to find these points common to real number field and extensions of various p-adic number fields appearing in the adele.
- If this were not enough, one must solve an even tougher problem. In TGD the notion of "world of classical worlds" (WCW) is also a central notion. It consists of space-time surfaces in imbedding space H =M4× CP2, which are so called preferred extremals of the action principle of theory. Quantum physics would reduce to geometrization of WCW and construction of classical spinor fields in WCW and representing basically many-fermion states: only the quantum jump would be genuinely quantal in quantum theory.
There are good reasons to expect that space-time surfaces are minimal surfaces with 2-D singularities, which are string world sheets - also minimal surfaces. This gives nice geometrization of gauge theories since minimal surfaces equations are counterparts for massless field equations.
One must find the algebraic points, the cognitive representation, for all these preferred extremals representing points of WCW (one must have preferred coordinates for H - the symmetries of imbedding space crucial for TGD and making it unique, provide the preferred coordinates)!
- What is so beautiful is that in given cognitive resolution defined by the extension of rationals inducing the discretization of space-time surface, the cognitive representation defines the coordinates of the space-time surfaces as a point of WCW. This huge infinite-dimensional space WCW discretizes and the situation can be handled using finite mathematics.
Connection with Kim's work
So: what is then the connection with the work and ideas of Kim. Also he is interested in the above problem of finding rational points of given surface. There has been a lot of progress in understanding the problem: here I an only refer to the popular article.
- One step of progress has been the realization that if one uses the fact that the solutions are common to both reals and various p-adic number fields helps a lot. The reason is that for rational points the rationality implies that the solution of equation representable as infinite power series of p contains only finite number powers of p. If one manages to prove the this happens for even single prime, a rational solution has been found.
The use of reals and all p-adic numbers fields is nothing but adelic physics. Real surfaces and all its p-adic variants form pages of a book like structure with infinite number of pages. The rational points or points in extension of rationals are the cognitive representation and are points common to all pages in the back of the book.
This generalizes also to algebraic extensions of rationals. Solving the number theoretic problem is in TGD framework nothing but finding the points of the cognitive representation. The surprise for me was that this viewpoint helps in the problem rather than making it more complex. There are however problematic situations in some cases the hypothesis about fintie set of algebraic points need not make sense. A good example is Fermat for x+y=1. All rational points and also algebraic points are solutions. For x2+ y2=1 the set of Pythagorean triangles characterizing the solutions is infinite. How to cope with these situations in which one has accidental symmetries as one might say.
- Kim however argues that one can make even further progress by considering the situation from even wider perspective by making the problem even bigger. Introduce what popular article calls the space of spaces. The space of string world sheets is what string models suggests. The "world of classical worlds", WCW is what TGD suggests. One can get a wider perspective of the problem of finding algebraic points of a surface by considering the problem in the space of surfaces and at this level it might be possible to gain much more understanding. The notion of WCW would not mean horrible complication of a horribly complex problem but possible manner to understand the problem!
- A further TGD based simplification would be M8-H (H=M4×CP2) duality in which space-time surfaces at the level of M8 are algebraic surfaces which are mapped to surfaces in H identified as preferred extremals of action principle by the M8-H duality. Algebraic surfaces satisfying algebraic equations are very simple as compared to preferred extremals satisfying partial differential equations but "preferred" is what makes possible the duality. This huge simplification of the solution space of field equations guarantees holography necessitated by general coordinate invariance implying that space-time surfaces are analogous to Bohr orbits. It would also guarantee the huge symmetries of WCW making it possible to have Kähler geometry.
This suggests in TGD framework that one finds the cognitive representation at the level of M8 using methods of algebraic geometry and maps the points to H by using the M8-H duality. TGD and octonionic variant of algebraic geometry would meet each other.
It must be made clear that now solutions are not points but 4-D surfaces and this probably means also that points in extension of rationals are replaced with surfaces with imbedding space coordinates defining function in extensions of rational functions rather than rationals. This would bring in algebraic functions. This might provide also a simplification by providing a more general perspective. Also octonionic analyticity is extremely powerful constraint that might help.
The crux of the Kim's idea is that somehow symmetries of space of spaces could come in rescue in the attempts to understand the rational points of surface. The notion of WCW suggest in TGD framework rather concrete realization of this idea that I have discussed from the point of view of construction of quantum states.
- A little bit more of zero energy ontology (ZEO) is needed to follow the argument. In ZEO causal diamonds (CDs) defined as intersections of future and past directed light-cones with points replaced with CP2 and forming a scale hierarchy are central. Space-time surfaces are preferred extremals with ends at the opposite boundaries of CD indeed looking like diamond. Symplectic group for the boundaries of causal diamond (CD) is the group of isometries of WCW. Its Lie algebra has structure of Kac-Moody algebra with respect to the light-like radial coordinate of the light-like boundary of CD, which is piece of light-cone boundary. This infinite-D group plays central role in quantum TGD: it acts as WCW isometries and zero energy states are invariant under its action at opposite boundaries.
- As one replaces space-time surface with a cognitive representation associated with an extension of rationals, WCW isometries are replaced with their infinite discrete subgroup acting in the number field define by the extension of rationals defining the adele. These discrete isometries do not leave the cognitive representation invariant but replace with it new one having the same number of points and one obtains entire orbit of cognitive representations. This is what the emergence of symmetries in wider conceptual framework would mean.
- One can in fact construct invariants of the symplectic group. Symplectic transformations leave invariance the Kaehler magnetic fluxes associated with geodesic polygons with edges identified as geodesic lines of H. The simplest polygons are geodesic triangles. The symplectic fluxes associated with the geodesic triangles define symplectic invariants characterizing the cognitive representation. For the twistor lift one must allow also M4 to have analog of Kähler form and it would be responsible for CP violation and matter antimatter asymmetry. Also this defines symplectic invariants so that one obtains them for both M4 and CP2 projections and can characterize the cognitive representations in terms of these invariants.
More complex cognitive representations in an extension containing the given extension are obtained by adding points with coordinates in the larger extension and this gives rise to new geodesic triangles and new invariants.
- Also in this framework one can have accidental symmetries. For instance, M4 with CP2 coordinates taken to be constant is a minimal surface, and all rational and algebraic points for given extension belong to the cognitive representation so that they ar infinite. Could this has something to do with the fact that we understand M4 so well and have even identified space-time with Minkowski space! Linear structure would be cognitively easy for the same reason and this could explain why we must linearize.
CP2 type extremals with light-like M4 geodesic as M4 projection is second example of accidental symmetries. The number of rational or algebraic points with rational M4 coordinates at light-like curve is infinite - the situation is very similar to x+y=1 for Fermat. Simplest cosmic strings are geodesic sub-manifolds, that is products of plane M2 ⊂ M4 and CP2 geodesic sphere. Also they have exceptional symmetries.
What is interesting from the point of view of proposed model of cognition is that these cognitively easy objects play a central role in TGD: their deformations represent more complex dynamical situations. For instance, replacing planar string with string world sheet replaces cognitive representation with a discrete or perhaps even finite one in M4 degrees of freedom.
For a summary of earlier postings see Latest progress in TGD.