Secret Link Uncovered Between Pure Math and Physics

I learned about a possible existence of a very interesting link between pure mathematics and physics (see this). The article told about ideas of number theorist Minhyong Kim working at the University of Oxford. As I read the popular article, I realized it is something very familiar to me but from totally different view point.

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.

1. 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.

2. 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 coordinats of a point of the space-time surface.
3. 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.
4. 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)!

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.

1. 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.

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 simplication by providing a more general perspective. Also octonionic analyticity is extremely powerful constraint that might help.

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

Articles and other material related to TGD.