We had an interesting FB discussion with Stephen King and others about discrete and continuum. As it usually occurs, people tended to talk about different things and misunderstand each other. Therefore I decided to write down my thoughts and emphasize those points that I see as important.
1. Is discretization fundamental or not?
The conversation inspired the question whether discreteness is something fundamental or not. If it is assumed to be fundamental, one encounters problems. The discrete structures are not unique. One has deep problem with the known space-time symmetries. Symmetries are reduced to discrete subgroup or totally lost. A further problem is the fact that in order to do physics, one must bring in topology and length measurements.
In discrete situation topology, in particular space-time dimension, must be put in via homology effectively already meaning use of imbedding to Euclidian space. Length measurement remains completely ad hoc. One feeds in information, which was not there by using hand weaving arguments like infrared limit. It is possible to approximate continuum by discretization but discrete to continuum won't go.
In hype physics these hand weaving arguments are general. For instance, the emergence of 3-space from discrete Hilbert space is one attempt to get continuum. One puts in what is factually a discretization of 3-space and then gets 3-space back at IR limit and shouts "Eureka!". It is regrettable that funding goes to the production of this kind of pseudophysics and pseudomathematics.
2. Can one make discretizations unique?
Then discussion went to numerics. Numerics is for mathematicians same as eating to poets. One cannot avoid it but luckily you can find people doing the necessary programming if you are a professor. Finite discretization is necessary in numerics as is highly unique.
I do not have anything personal against discretization as a numerical tool. On the contrary, I see finite discretization as absolutely essential element of physics when it includes also correlates of cognition. Cognition is discrete and finite: this is the basic clue.
3. Can discretization be performed without lattices?
- Cognitive representations are discretizations of for instance space-time and one should be able to do this in unique manner if one is talking about fundamental physics. In TGD p-adic physics as physics of cognition led to powerful calculational recipes. In p-adic thermodynamics the predictions come in power series of p-adic prime p and for the values of p assignable to elementary particles the two lowest terms give practically exact result. Corrections are of order 10^(-76) for electron characterized by Mersenne prime M_127= 2^127-1.
- Adelic physics provides the formulation of p-adic physics: it is assumed that cognition is universal. Adele is a book like structure having as pages reals and extensions of various p-adic number fields induced by given extension of rationals. Each extension of rationals defines its own extension of the rational adele by inducing extensions of p-adic number fields. Common points between pages consist of points in extension of rationals. The books associated with the adeles give rise to infinite library.
At space-time level the points with coordinates in extension define what I call cognitive representation. In the generic case it is discrete and has finite number of points. The loss of general coordinate invariance is the obvious objection. In TGD however the symmetries of imbedding space fix the coordinates used highly uniquely. M^8-H duality and octonionic interpretation implies that M^8 octonionic linear coordinates are highly unique. Note that M^8 must be complexified.
- Discretization by cognitive representation is unique for given extension defining the measurement resolution. At the limit of algebraic numbers algebraic points form a dense set of real space-time surface and p-adic space-time surfaces so that the measurement resolution is ideal.
Given polynomial defining space-time surfaces in M^8 defines via its roots extension of rationals. The hierarchy of extensions defines an evolutionary hierarchy. The dimension n of extension defines kind of IQ measuring algebraic complexity and n corresponds also to effective Planck constant labelling phases of dark matter in TGD sense so that a direct connection with physics emerges.
Imbedding space assigns to a discretization a natural metric. Distances between points of metric are geodesic distances computed at the level of imbedding space.
4. Simple extensions of rationals as codons of space-time genetic code
- One might think that discretization for partial differential equations involving derivatives forces to introduce lattice like structures. This is not needed in TGD. At the level of M^8 ordinary polynomials give rise to octonionic polynomials and space-time surfaces are algebraic surfaces for which imaginary or real part of octonionic polynomial in quaternionic sense vanishes. The equations are purely algebraic. This is essential!
For surfaces defined by polynomials the roots of polynomial are enough to fix the space-time surface: discretization is not an approximation but gives an exact result! This could be called number theoretical holography.
What about the field equations at the level of H=M^4xCP_2? M^8-H duality maps these surfaces to preferred extremals as 4-surfaces in H: they are minimal surfaces with singularities and determined in terms of polynomials too.
- The simplest assumption is that the polynomials have rational coefficients. One can consider also algebraic coefficients. In both cases also WCW is discretized and given point -space-time surface in QCD has coordinates given by the points of the number theoretically universal cognitive representation of the space-time surface. Even real coefficients are possible and this seems to be needed to obtain WCW as a continuum.
5. Are octonionic polynomials enough or are also analytic functions needed?
- The extensions of rationals define and infinite hierarchy. In particular, sequences of extensions of extensions of... define analog for the inclusions of hyper-finite actors my belief is that these two hierarchies are closely related. Galois group for a extension of extension contains Galois group of extension as normal subgroup and is therefore *not simple*. Extension hierarchies correspond to inclusion hierarchies for normal subgroups. Simple Galois groups are in very special position and associated with simple extensions, which are fundamental building bricks of inclusion hierarchies. They are like elementary particles and define fundamental space-time regions.
- One can therefore talk about elementary space-time surfaces in M^8 and their compositions by function composition of octonionic polynomials. Simple groups would label elementary space-time regions. They have been classified: https://en.wikipedia.org/wiki/List_of_finite_simple_groups .
- There is also an analogy with genes. Extensions with simple Galois groups could be seen as codons and sequences of extension obtained by functional composition as analogs of genes. I have even conjectured that the space-time surfaces associated with genes could quite concretely correspond to extensions of extensions of ...
See the chapter Does M8-H duality reduce classical TGD to octonionic algebraic geometry? or the article Are fundamental entities discrete or continuous and what discretization at fundamental level could mean?.
- What about analytic functions as limits of polynomials? Also their imaginary and real part can vanish in quaternionic sense and could define space-time surfaces - analogs of transcendentals as space-time surfaces. It is not clear whether these could be allowed or not.
- Could one have a universal polynomial like function giving algebraic numbers as the extension of rationals defined by its algebraic roots. Could Riemann zeta could code algebraic numbers as an extension via its roots. I have conjectured that roots are algebraic numbers: could they span all algebraic numbers? It is known that the real or imaginary part of Riemann zeta along s=1/2 critical line can approximate any function to arbitrary accuracy: also this would fit with universality. Could one think that the space-time surface defined as root of octonionic continuation of zeta could be universal entity analogous to a fixed point of iteration in the construction of fractals?
- Riemann zeta has counterpart in all extensions of rationals known as Dedekind zeta. Therefore that it seems that Riemann zeta is not unique. One can ask whether Dedekind zetas associated with simple Galois groups are special and whether Dedekind zetas associated with extensions of extensions of .... can be constructed by using the simple building brick zetas. One can also construct iterates of Riemann zeta having at least the same roots as zeta by the rule f(s)-->zeta(f(s))+zeta(0), zeta(0)=-1/2. zeta is not a fixed point of this iteration but one could ask whether the fixed point is obtained at the limit of infinite number of iterations.