Particles and Primes, Counting and Cohomology

Source: Pride and Prejudice, 2005
Judy Dench plays the role of Lady Catherine de Bourgh.

I recently posted a “Particles and Primes” as well as a
“Counting and Cohomology” article on the ArXiv, because, as it is a truth universally acknowledged, that an article in possession of a good result, must be in want of a place to be read. But as it happens, there appeared to be the danger that the shades of Pemberley would be thus polluted; the articles were put on hold. Certainly, the Jane Austen quip in the title did not help. Fortunately “Counting and Cohomology” was not chosen as a title in the second article and indeed, that paper got cleared earlier. Having the articles flagged did not come as a surprise: The topics of “primes” and “particles” in one document appears forbidding, especially as it is written by somebody without connections, fortune nor good breeding. Indeed, there is a crackpot index for primes as well as in fundamental physics.

Well, the articles appear both a bit off the grid or even appear like a hoax and I don’t blame the moderators (or bots) of the ArXiv to have a closer look. [Update September 25: the audit trigger could also have been a text overlap between my “experiments paper” and the new write up. Indeed, the later expands a small paragraph in the former 72 page write-up.] Obviously, the two texts were written hastily due to self-imposed deadlines at the end of vacations in order not having the papers rot and eventually be forgotten somewhere in the hard drive like so many others before. Lets look at their core results:

Particles and Primes

“Particles and Primes” is about an affinity between the structure of primes in division algebras and the particle structure in the standard model. The whole thing might sound first like a Sokal type hoax, but it is not. Heaven forbid however trying to publish this. It barely got into the ArXiv, where only screening for abuse happens. The document got bumped into physics although. The paper was intended to be combinatorial (it is the structure of two different equivalence relations on the class of primes in division algebras) and lead to more questions in mathematics. It even mentions that it is hardly of any value for physics as physics by definition is a theory which makes qualitative predictions or verifications of experimental observations.

The starting point are the theorems of Frobenius or Hurwitz. Any of these two results assure that the complex numbers C and the quaternions H are the only algebraically complete associative division algebras. (Algebraic completeness must be first defined in non-Abelian cases; but nobody lesser than Eilenberg and Niven have done that and proven a fundamental theorem of algebra for H). As vector spaces, C and H can also be characterized as the only linear spaces in which the unit spheres are continuous Lie groups: it is really remarkable that only the d-spheres for d=1 or d=3 carry a Lie group structure. The allegory is that primes in C are like Leptons and primes in H behave like Hadrons and that the units or the symmetries of the gauge groups produce the gauge Bosons, photons and gluons. Since the prime 2 is not yet accounted for, and because it is neutral and Boson like, adding norm = mass to an integer, why not associate it with the Higgs? The text cautions at various places about the danger of blind associations but still, this part ranks high on the crackpot index, no doubt. But lets look at the math:

The primes in C are the Gaussian primes, the primes in H are either Lipschitz or Hurwitz primes. In order to study the structure of these primes, I had looked at additive features related to Goldbach (which is a topic dangerously close on the crackpot abyss given the immense number of nonsense appearing even in print (proofs of Goldbach for example)) but also at various equivalence relations among the primes. There is a weak equivalence relation in which two primes p,q are equivalent if p = u q or p = q u for some unit u. As the units in C are a discrete subgroup of U(1) and the units in H are a discrete subgroup of SU(2), the name “weak” seems adequate. Then there is a strong equivalence, where two primes p,q are equivalent if one can find a permutation combined with a sign change of its coordinates to make two equivalent. This group can be realized and generated by discrete subgroups of three complex sub-planes as well as a discrete subgroup of the unitary group U(3) permuting the spacial coordinates. (When using SU(3), one can see the sign of the charges). The name “strong” can therefore be justified. The primes 2+5i or 5-2i for example are strongly equivalent in C and the primes (39,33,27,17)/2 and (-33,-17,27,39)/2 are strongly equivalent Hurwitz primes in H. While the strong equivalence classes in C are in 1-1 correspondence with rational primes, one can look at the weak equivalence classes among the strong equivalence classes of primes in H. By Hurwitz, the H-primes with norm p have p+1 weak equivalence classes. The result tells that if we look at the weak equivalence classes among strong equivalence classes, then there are only two cases for odd primes!

Particles and Primes theorem: the strong equivalence classes of odd primes in H can be partitioned into equivalence classes of two or three. We can assign in a consistent way a fractional charge to each of the elements so that the total charge of an equivalence class is an integer. The charges are multiplies of 1/3.

There is therefore some affinity with the Hadron structure in elementary particle physics. It is most likely just an amusing structural coincidence but it motivates to look at quaternion-valued quantum mechanics which I actually consider very promising. It can be implemented quite easily, if one just looks at it on three complex planes hinged together on a real line, a picture which also allows to see a U(3) or SU(3) action in the model and triggers associations with neutrini oscillations (a strange phenomenon where particles flip between different flavors). There is a quaternion wave theory done by Rudolf Fueter (who was a student of Hilbert and worked in Zuerich) in the 30ies which shows that it is actually quite natural to look at quaternion-valued fields (=functions from the geometry to H). Fueter developed a function theory on quaternions.

Counting and Cohomology

“Counting and Cohomology” introduces a sequence of graphs related to counting for which all the cohomology groups can be computed explicitly in number theoretical terms. The seqeuence of Betti numbers is unbounded. The paper also notices that on any Barycentric refinement of an arbitrary finite simple graph, there is a Morse function for which the Morse cohomology is equivalent to the simplicial cohomology of the graph. It is a first step towards a Morse cohomology for finite simple graphs. Both papers contain mathematical results which could be formulated as theorems (elementary but nevertheless results which have proofs). The first one is an obvious relation between a topological notion, the Euler characteristic, and a number theoretical notion, the Mertens function, whose growth rate is obviously of importance in relation with the Riemann hypothesis.

Counting and Cohomology Theorem: Let G(n) be the graph with vertex set {2,,..n} for which two vertices are connected if one divides the
other. Then the Euler characteristic of G(n) is equal to 1-M(n), where M(n) is the Mertens function.

The proof is more interesting than the result as it shows that counting is a Morse theoretical process. The counting function f(n)=n has all the features, a Morse function in the continuum has. Either the intersection of a small sphere with { y | f(y) less than f(x) } is contractible, meaning that nothing interesting happens topologically when adding the point or then { y | f(y) less than f(x) } intersected with a small sphere is a m-sphere in a precise sense first defined by Evako (for the definition, see for example the abstract in this paper on Jordan-Brouwer). The process of adding a critical point is the discrete analogue of the formulation that a critical point has a m-dimensional stable manifold. Of course, for networks, we don’t have any notion of tangent spaces, Hessian etc, but we can manage very well with spheres. Graph theory is a theory of spheres, as spheres are everywhere: we have unit spheres but then also a very elegant inductively defined notion due to Evako of what a sphere is. The graphs G(n)={ f ≤ n } form a Morse filtration. The Poincare-Hopf index i(n) of an integer n is equal to -mu(n), where mu is the Moebius function of n. Adding a new square free n, this is a homotopy deformation G(n-1) -> G(n), otherwise a topological disc = handle of dimension m+1 is added. Its unit sphere is a graph theoretical sphere S of dimension m and Euler characteristic X(S) = 1+(-1)m satisfying 1-X(S)=i(n) = (-1)m. We also can keep track what happens with the cohomologies. If the sphere S is m dimensional then we either increase or decrease the m’th or (m+1)’th Betti number by 1. So, this is an example, where we can study high dimensional graphs, where many Betti numbers are nonzero. It is computationally hard for example to compute the 10’th Betti number of a huge network but in this number theoretical setup, we have everything tied to number theory which now can be computed very easily. I managed to get the cohomologies the traditional way up to n=250 by computing the kernels of Laplacians but things are getting harder.

Example 1: if n is a prime, then the step G(n-1) -> G(n) adds a single isolated vertex to the graph. Its unit sphere is the empty graph, a m=-1 dimensional sphere. The Euler characteristic grows by 1 since i(n)=-1. We have added a zero dimensional handle. We also increased the Betti number b0.

Example 2: if n=6, then the step G(5) -> G(6) adds a vertex 6 connected to the vertices 2 and 3. This adds a 1-dimensional handle whose unit sphere is m=0 dimensional. The Euler characteristic drops from 3 to 2. The zero’th Betti number b0 has decreased.

Example 3: if n=15, then the step G(14) -> G(15) adds a vertex 15 connected to the vertices 3 and 5. This adds again a 1-dimensional handle whose unit sphere is m-0 dimensional. The Euler characteristic again drops from 3 to 2. But now, the first Betti number b1 has increases as the first cycle {5,15,3,6,2,10} is born.

Example 4: if n=30, then the step G(29) -> G(30) adds a vertex 30 connected to the circle 2-6-3-15-5-10-2 killing that circle and increasing the Euler characteristic by 1. Indeed i(30) = -1 as a one dimensional circle has been the unit sphere. The first Betti number has decreased by 1, compatible with the increase of the Euler characteristic.