Friday, March 23, 2012

p-Adic homology and finite measurement resolution

Discretization in dimension D in terms of pinary cutoff means division of the manifold to cube-like objects. What suggests itself is homology theory defined by the measurement resolution and by the fluxes assigned to the induced Kähler form.

  1. One can introduce the decomposition of n-D sub-manifold of the imbedding space to n-cubes by n-1-planes for which one of the coordinates equals to its pinary cutoff. The construction works in both real and p-adic context. The hyperplanes in turn can be decomposed to n-1-cubes by n-2-planes assuming that an additional coordinate equals to its pinary cutoff. One can continue this decomposition until one obtains only points as those points for which all coordinates are their own pinary cutoffs. In the case of partonic 2-surfaces these points define in a natural manner the ends of braid strands. Braid strands themselves could correspond to the curves for which two coordinates of a light-like 3-surface are their own pinary cutoffs.

  2. The analogy of homology theory defined by the decomposition of the space-time surface to cells of various dimensions is suggestive. In the p-adic context the identification of the boundaries of the regions corresponding to given pinary digits is not possible in purely topological sense since p-adic numbers do not allow well-ordering. One could however identify the boundaries sub-manifolds for which some number of coordinates are equal to their pinary cutoffs or as inverse images of real boundaries. This might allow to formulate homology theory to the p-adic context.

  3. The construction is especially interesting for the partonic 2-surfaces. There is hierarchy in the sense that a square like region with given first values of pinary digits decompose to p square like regions labelled by the value 0,...,p-1 of the next pinary digit. The lines defining the boundaries of the 2-D square like regions with fixed pinary digits in a given resolution correspond to the situation in which either coordinate equals to its pinary cutoff. These lines define naturally edges of a graph having as its nodes the points for which pinary cutoff for both coordinates equals to the actual point.

  4. I have proposed earlier kenociteallb/categorynew what I have called symplectic QFT involving a triangulation of the partonic 2-surface. The fluxes of the induced Kähler form over the triangles of the triangulation and the areas of these triangles define symplectic invariants, which are zero modes in the sense that they do not contribute to the line element of WCW although the WCW metric depends on these zero modes as parameters. The physical interpretation is as non-quantum fluctuating classical variables. The triangulation generalizes in an obvious manner to quadrangulation defined by the pinary digits. This quadrangulation is fixed once internal coordinates and measurement accuracy are fixed. If one can identify physically preferred coordinates - say by requiring that coordinates transform in simple manner under isometries - the quadrangulation is highly unique.

  5. For 3-surfaces one obtains a decomposition to cube like regions bounded by regions consisting of square like regions and Kähler magnetic fluxes over the squares define symplectic invariants. Also Kähler Chern-Simons invariant for the 3-cube defines an interesting almost symplectic invariant. 4-surface decomposes in a similar manner to 4-cube like regions and now instanton density for the 4-cube reducing to Chern-Simons term at the boundaries of the 4-cube defines symplectic invariant. For 4-surfaces symplectic invariants reduce to Chern-Simons terms over 3-cubes so that in this sense one would have holography. The resulting structure brings in mind lattice gauge theory and effective 2-dimensionality suggests that partonic 2-surfaces are enough.

The simplest realization of this homology theory in p-adic context could be induced by canonical identification from real homology. The homology of p-adic object would the homology of its canonical image.

  1. Ordering of the points is essential in homology theory. In p-adic context canonical identification x=∑ xnpn→ ∑ xnp-n map to reals induces this ordering and also boundary operation for p-adic homology can be induced. The points of p-adic space would be represented by n-tuples of sequences of pinary digits for n coordinates. p-Adic numbers decompose to disconnected sets characterized by the norm p-n of points in given set. Canonical identification allows to glue these sets together by inducing real topology. The points pn and (p-1)(1+p+p2+...)pn+1 having p-adic norms p-n and p-n-1 are mapped to the same real point p-n under canonical identification and therefore the points pn and (p-1)(1+p+p2+...)pn+1 can be said to define the endpoints of a continuous interval in the induced topology although they have different p-adic norms. Canonical identification induces real homology to the p-adic realm. This suggests that one should include canonical identification to the boundary operation so that boundary operation would be map from p-adicity to reality.

  2. Interior points of p-adic simplices would be p-adic points not equal to their pinary cutoffs defined by the dropping of the pinary digits corresponding pn, n>N. At the boundaries of simplices at least one coordinate would have vanishing pinary digits for pn, n>N. The analogs of n-1 simplices would be the p-adic points sets for which one of the coordinates would have vanishing pinary digits for pn, n>N. n-k-simplices would correspond to points sets for which k coordinates satisfy this condition. The formal sums and differences of these sets are assumed to make sense and there is natural grading.

  3. Could one identify the end points of braid strands in some natural manner in this cohomology? Points with n≤ N pinary digits are closed elements of the cohomology and homologically equivalent with each other if the canonical image of the p-adic geometric object is connected so that there is no manner to identify the ends of braid strands as some special points unless the zeroth homology is non-trivial. In kenociteallb/agg it was proposed that strand ends correspond to singular points for a covering of sphere or more general Riemann surface. At the singular point the branches of the covering would co-incide.

    The obvious guess is that the singular points are associated with the covering characterized by the value of Planck constant. As a matter fact, the original assumption was that all points of the partonic 2-surface are singular in this sense. It would be however enough to make this assumption for the ends of braid strands only. The orbits of braid strands and string world sheet having braid strands as its boundaries would be the singular loci of the covering.

For background see the chapter Quantum Adeles of "Physics as Generalized Number Theory".


At 10:33 PM, Anonymous ◘Fractality◘ said...



At 2:53 AM, Blogger Ulla said...

The mystery now is, where the
hell did these Riemann zeros come from?
- they might be vibrations of something
- a mysterious, suggestive mathematical ‘coincidence’ involving something called the Selberg Trace Formula — and that ties in with certain unexpected connections

So if we’ve got vibrations of a mysterious ‘something’ underlying the number system, it’s a
mystery about the system of positive integers, about ‘order’, and arguably even about time.

The set of primes and the set of Riemann zeros are in some sense ‘dual’ structures. There’s a variant of what’s called a ‘Fourier duality’ between them. you can use the zeros to generate the set of primes and vice versa.
The Riemann zeros are very different – they’re
not integers, they’re what we call ‘transcendental’, irrational
numbers. the zeros are the problem.

One suspects that if a mathematical structure underlying or ‘explaining’ the Riemann zeros were to emerge – that is, if in fifty or a hundred years someone comes up with something new which ‘explains’ the zeros in the way the zeros ‘explain’ the primes – then that new structure is just going to open up another even deeper mystery. Paul Erdös said, it’s going to be at least a
million years before we understand the primes, and even then we won’t really understand them.

Without the set of positive integers, those other mathematical problems couldn’t exist. So the problem of the primes is the problem in a sense. all science is entirely built on measurement, and you can’t measure anything until you can count. All our rational scientific thought relies on these very basic ideas of order and counting.

So, in probing the mystery of the prime numbers we’re effectively on a sort of journey to the centre of the mind, or of the collective human psyche, and ultimately to the point where that interfaces with the physical world which it finds itself inhabiting.

One of few texts actually in favour of TGD:) A long text. Interview with Matthew Watkins.

At 3:44 AM, Blogger Ulla said...

Watkins page at the Essex University

At 4:24 AM, Blogger Ulla said...

a Jungian perspective on the use of 'emotional' language in descriptions of number theoretical phenomena

I also learned they try primes as some smart or intellingent meeter system or evolution.

At 12:39 AM, Anonymous said...


thank you for an excellent link. I was in contact with Matthew Watkins for years ago. "Strategy for proving Riemann hypothesis" might be still found at the pages of MW.

The interview is very interesting. In particular, the comments about Connes work were interesting: Connes is bringing in p-adic numbers fields and adeles in his attempts to understand RH.

A very interesting notion is the formal thermodynamical interpretation of RH. While reading the interview I realized that zero energy ontology replacing thermodynamics with its square root could resolve the conceptual difficulties posed by the necessity to assume complex temperature which simply fails to make sense physically.

I will write a separate posting about the topic.

At 10:47 PM, Anonymous said...

Matthew Watkins seems to have very similar views about what science is to him.

Exploration, adventure rather than opportunistic career building based on endless application of some method to produce items to curriculum vitae.

Also for me science of this kind looks absolutely boring and I have never been able to even imagine spending my life in such activities although the academic social pressures forced me to try hard hard before the final choice! I was expected to be young brisk academic male devoting himself to intriguing and striking up acquaintances with correct persons. I was not! For long time I was ashamed of this until I was mature to follow my own star.

MW seems to have even adopted similar working habits. Never-ending books. Updates, additions, corrections. Also to me seeing a printed article with my name appearing as an author is like looking my own grave stone.

At 11:35 PM, Blogger Ulla said...

Well, that's what is left after you are gone :)

To me those early texts would be very valueable.

You meeting people seem like a brilliant idea :) God beware :)

I thought how is it possible that he has a university job at all, it would not be here in Finland.

BTW your Kari Enquist was on finnish Tv talking about teasing in university world. He did not know of anything such. He was proud of being a skeptic head even. I have read something about that and if it is not teasing then what is? Apparently lying is favoured in the university world too? I was upset long after that dam program. Everyone must have known he sat there lying and nobody said anything. That is the way the phenomenon is living on well. The conflict has to be taken.

At 4:06 AM, Anonymous said...

People at the top of power hegemony rarely know about bullying! If KE would admit that bullying is everyday reality, he would lose his position in academy immediately.

Finnish academic world is very much like Franco's Spain. The old fellows continue to hold power long after the official age for retirement. This leads to a corruption extending from top to bottom and everyone pretends that everything is ok. Fear combined with opportunism has impressive corruptive power.

The incredible farce played by the so called "top leaders" of Finnair is nothing exceptional: same happens in the academic world too. Peer review ("vertaisarvionti") is a powerful method to filter out all but mediocrits.

I do not believe that much can be done for this by any control mechanisms. There is only one cure: do not participate the academic theatre and choose freedom instead of academic rewards - such as those enjoyed by KE for preaching materialistic world view as something forced by fundamental science. This medicine cures but tastes rather bitter!;-)


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