Quantum calculus talk of 2013

Just uploaded a larger version of my 2013 Pecha-Kucha talk “If Archimedes knew functions…”. The Pecha-Kucha format of presenting 20 slides with 20 seconds time each is fantastic to keep talks concise and to the point. The video has been produced by Diane Andonica from the Bok Center for teaching and learning. The event had been organized by Curt Mc Mullen and Sarah Koch. Was a lot of fun as in one hour , there were about 8 talks on various topics in math. Not the usual experience, where after 10 minutes, you have no idea any more what the speaker is even talking about. But one also must say that preparing such a talk is heavy.

There are some more annotations to the slides on this page and a write up on this ArXiv paper.

On Bowen-Lanford Zeta Functions

Zeta functions are ubiquitous in mathematics. One of the many zeta functions, the Bowen-Lanford Zeta function was introduced by my Phd dad Oscar Lanford and Rufus Bowen. I am in the process to wrap up a proof of a theorem which is so short that its statement can be done in 140 characters: A finite simple graph G=(V,E) defines H=(W,F) where W={simplices in G}, F={(a,b)| a intersect b}. If A=adjacency of H then 1+A is unimodular. See the actual titter announcement and the math table handout [PDF]. The proof is still quite complicated. The result was found in February 2016 when doing experiments related to intersection calculus. As we will see, one can reformulate the theorem in terms of the Bowen-Lanford zeta function of an intersection graph at z=-1 is either -1 or one. Actually, we know even that it is 1 if and only if the number of odd dimensional simplices in the original graph is even. Before we go into into the mathematics, a short explanation why this belongs to quantum calculus: we think of a graph as a geometric object on which one can do calculus. While the Fredholm determinant of the intersection graph of a graph G is not invariant under Barycentric refinements, it appears that the values of the inverse matrix (1+A(G’)) are invariant under Barycentric refinements of the graph G. If this is true then can again use this discrete intersection calculus to attach invariants to continuum geometries and distinguish spaces topologically.

Bowen Lanford Zeta functions

Given a geometric space there are two major ways to attach a zeta function to it. Given a Laplacian L=D2 one can look at the positive eigenvalues of D and form the sum \zeta(s)=\sum_n \lambda_n^{-s}. An example is D=i d/dx on the circle, where the positive eigenvalues are n with eigenvectors \exp(-inx). This works for any compact Riemannian manifold or for any finite simple graph. An other possibility is to look at a dynamical system, a time evolution on the space like a map T on the space for which {\rm Fix}(T^n) is finite for every n>0. The Artin-Mazur \zeta function is then defined as \zeta(z)=\exp(\sum_n |{\rm Fix}(T^n)| z^n/n). In the case of a finite simple graph, there is a natural time evolution, the subshift of finite type defined by the graph. In that case, the Artin-Mazur zeta function simplifies to 1/det(1-zA), where A is the adjacency matrix. It is called the Bowen-Lanford \zeta function. One can rewrite the function as a product \prod_p (1-p^{-s})^{-1}, where the product is over all prime orbits of T and p=\exp(|P|) where P is the orbit, |P| is the length and s=\log(z). It was just written in such a way that it looks like the Riemann zeta function again which is by Euler \prod_p (1-p^{-s})^{-1}, but where p runs over all the primes.

bowenlanford1 Oscar Lanford Rufus Bowen

Fredholm Determinant

Given a trace class operator A, one can define \det(1+A). It is 1/\zeta(-1) with the above notion of \zeta function and called Fredholm determinant. In the case when A is the adjacency matrix of a graph, then one can interpret \det(1+A) as a sum over all oriented paths (not necessarily connected) over the vertex set. The identity matrix 1 in 1+A has added the possibility that a path also stays at the same spot. A similar change happens when looking at the Laplacian L rather than the adjacency matrix, where the pseudo determinant det(L) gives the number of rooted spanning trees in the graph and det(1+L) the number of rooted spanning forests.

The Kustaanheimo prime

Paul Kustaanheimo
Paul Kustaanheimo (1924-1997) was a Finnish astronomer and mathematician. In celestial mechanics, his name is associated with the Kustaanheimo-Stiefel transform or shortly KS transform which allows to regularize the Kepler problem using Clifford algebras. In this elegant picture, the motion of the two bodies becomes a rotation in three dimensions rendering therefore the Kepler motion into a harmonic oscillator. The KS transform generalizes the Levi-Civita regularization, which was found already in 1920 by Tullio Levi-Civita. The Kustaaanheimo Stiefel transform is still studied today. See this article for example.

(Image Source: originally astro.helsinki.fi

I learned about his attempt to do physics on finite fields in a linear algebra lecture of Ernst Specker. By the way, the absorption of mathematical material was a treat, but seeing lectures and side remarks from teachers like for example the above remark about Kustaanheimo is something which I consider now the most priceless part of my own college years. Knowledge is important, but it is cheep as it can again be looked up. Insight, excitement and suggestions from masters in the field is much more valuable. By the way, the picture on the Wikipedia entry of Specker from 1982 was taken just at the time, when I took his linear algebra course for us freshmen mathematicians and physicists (who took then both the same identical lectures in the first two years).

The fact that Specker mentioned Kustaanheimo is not an accident. Stiefel (1909-1978) (the same Stiefel from the Stiefel-Whitney classes of vector bundle), was an academic brother of Ernst Specker as they both were students of Heinz Hopf. Eduard Stiefel also founded the institute for applied mathematics at ETH. Stiefel was from 1948 on professor at ETH. Specker got his PHD in 1949 and was professor at ETH since 1955 so that Stiefel and Specker certainly knew each other. Kustaanheimo stayed in Zuerich in the summer of 1964 to work (as acknowledged in their paper) with Stiefel on the “perturbation theory of the Kepler problem” so that also Kustaanheimo and Specker must have known each other (even so Specker did not tell us (the linear algebra class) when mentioning the Kustaanheimo work).

Kustaanheimo muses about doing physics over a finite field. The starting point is to approximate a geometric space by a finite geometry. He gives an upper bound of the order of a googoplex but estimates the actual prime to be much smaller. The Kustaanheimo paper is of a similar scope then my “particles and prime” paper as it deals with a rather clear cut elementary number theoretical problem, which it is motivated by a speculation that our physical space could be modeled by a finite field. Here is the article (thanks to the Harvard librarians for digging the paper up):








Particles and Primes, Counting and Cohomology

I recently posted a “Particles and Primes” as well as the “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.

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

Counting and Cohomology

To this article:

There are various cohomologies for finite simplicial complexes. If the complex is the Whitney complex of a finite simple graph then many major results from Riemannian manifolds have discrete analogues. Simplicial cohomology has been constructed by Poincaré already for simplicial complexes. Since the Barycentric refinement of any abstract finite simplicial complex is always the Whitney complex of a finite simple graph, there is no loss of generality to study graphs instead of abstract simplicial complexes. This has many advantages, one of them is that graphs are intuitive, an other is that the data structure of graphs exists already in all higher order programming languages. A few lines of computer algebra system allow so to compute all cohomology groups. The matrices involved can however become large, so that alternative cohomologies are desired.

A second cohomology is Cech cohomology. It is defined once one has defined a notion of topology and so of a nerve graph of an open cover. Cech cohomology leads in general to much smaller matrices which makes the computations easier. But the construction of the nerve graph is not canonical. About the notion of a topological structure on the graph: Homeomorphisms defined in traditional topological graph theory assume the graph to be a one-dimensional simplicial complex. Graphs however, when equipped with the Whitney complex, are full grown-up geometric spaces for which many results from the continuum generalize. A notion of topology for such general graphs allows to define what a homeomorphism is. It also allows to define a Cech cover. See this article.

A third cohomology is de Rham cohomology. For de Rham cohomology, one first has to have a good notion of graph product, otherwise, it is not defined. There is indeed a natural topology which makes things work nicely. There is then a discrete de Rham theorem showing that the cohomologies are the same. Also de Rham cohomology leads in general to much smaller matrices, but it assumes that the graph is a product of other graphs or then patched together with different charts which are made of products. See the Kuenneth paper.

We recently stumbled upon a fourth cohomology for graphs: Morse cohomology. It has become a fancy cohomology in dynamical systems theory, especially after it got enhanced by Floer to prove the Arnold conjectures. The story in the continuum is quite well established with Thom, Milnor, Smale, Witten etc. We have not yet explored it in full generality, but here is what has been shown already (see this paper, [January 25, now on ArXiv]).

Theorem: given a finite simple graph G=(V,E). There exists a Morse function f:V1 -> R on the vertex set V1 of the Barycentric refinement G1 = (V1,E1) of G for which the Morse cohomology is defined and equivalent to simplicial cohomology.

What still has to be done is to identify a general class of Morse-Smale functions for which the Morse cohomology works and is equivalent to the simplicial cohomology. Here is proof: there is a natural Morse function on the Barycentric refinement: it is the function f(x) = dim(x), the dimension of the point x, when it was a simplex in the original graph (remember that in the Barycentric refinement G1 the vertices were the simplices in G). It is locally injective (aka a coloring) by definition and a Morse function in the sense that the graph Sf(x) generated by all vertices y in the unit sphere S(x) of a vertex x for which f(y)Evako sphere (see the Brouwer paper.

For the function f=dim, every point is a critical point. The Morse index of a critical point is defined to be equal to m, if the sphere Sf(x) has dimension m-1. Actually, when adding the point x to the graph f(y)< f(x), a m-dimensional handle (m-ball) has been added. Its boundary is the sphere. The chain complex X for Morse cohomology are functions on critical points, graded by the Morse index. We have X= X1 + …. + Xm. So, 0-forms are functions on vertices of Morse index 0, 1-forms are functions on vertices of Morse index 1 etc For Morse cohomology, one defines an exterior derivative d: Xm -> Xm+1 given by df(x) = sumy n(x,y) f(y), where x is a point of Morse index m+1 and the sum is over critical points of Morse index m. The number n(x,y) is the intersection number of the stable manifold of x and the unstable manifold of y. The difficulty to make this general is that one has to be able to define stable and unstable manifolds as well as to have a reasonable intersection number.

Where does counting come in? We can for each integer n define a graph G(n) for which the vertices are the square free integers in the interval [2,n] and where two integers are connected, if one is contained in the other. Here is the graph G(30). What just happened when adding the number 30, is that a two dimensional handle (disc) has been added to the graph. It destroyed a one-dimensional circle and changed the first Betti number of the graph. It turns out that during counting, the topology of the graph G(n) does not change when adding an integer for which the Moebius function is zero. Actually, the Moebius function of a vertex x is a minus the Poincaré-Hopf index of x. And The Euler characteristic X(G(n)) is equal to 1-M(n), where M(n) is the Mertens function. In the case n=30 for example, we have mu(30)=mu(3*5*2) = -1 so that the index is 1. Indeed, when we added the vertex 30, the unit sphere of 30 is a 1-sphere of Euler characteristic 0 so that the index is i(30) = 1-X(S(30)) = 1. The Betti number has increased by 1 as a one dimensional hole has been filled. As described in the text (PDF), counting is a rather dramatic process, telling the story of birth and death of spheres. The example is interesting also as it is a case, where one can almost immediately see the relation between Morse cohomology and simplicial cohomology.

the graph G(30)

Quaternions and Particles

The standard model of particle physics is not so pretty, but it is successful. Many lose ends and major big questions remain: is there a grand unified gauge group? Why are there three generations of particles? Why do neutrini oscillate? How is general relativity included? (See for example page 540 in Woit’s online monograph).

When experimenting with quaternion primes, especially in connection with Goldbach, I recently stumbled over a combinatorial structure within quaternion integers which resembles some structures seen in the standard model. This structure is purely combinatorial and could suggest that some relations seen in elementary particles appear unavoidable. Maybe there is more behind it, maybe it is naive, maybe it is just amusing like this story. For now, I see it as a caricature showing that the structures seen in the Standard model are not so arbitrary. Nice would be to show that the structure is unavoidable. Here is a recent new write-up. It might score a bit high both on the crackpot index for primes as well as the crackpot index for physics. I leave it to you to judge. For me it has just been fun and will also in future be a bit motivation to learn more about the physics of particles.

The starting point is the mathematical structure of division algebras. The requirement of being a division algebra restricts the structure severely: according to a theorem of Hurwitz, (Frobenius 1878, Hurwitz 1922, Mazur 1938), there are only 4 normed division algebras and 3 associative normed division algebras. In a physical frame work with quantum evolution and representability of observables as operators, where we want to have both associativity and completeness, the list of normed division algebras which are associative and complete drops down to two: they are the complex numbers C and the quaternions H. It is rare in mathematics that a categorical structure has so few constituents. There are other ways to distinguish C and H: these are the only linear spaces for which the unit sphere is a continuous Lie group. As symmetries are of enormous important in physics as stressed by Emmy Noether, this is a relevant feature also distinguishing the complex numbers and the quaternions.

The higher arithmetic in C and H are both quite well established: one has prime factorization and fundamental theorem of arithmetic in each case. While the arithmetic in C is quite old as it has been done by Gauss, the prime factorization in H is more complex: The work from Hurwitz to Conway and Smith have established the fundamental theorem of arithmetic in H: the factorization is unique there only up to unit migration, recombination and meta commutation. (The word higher arithmetic is due to Davenport and more adequate than “elementary number theory” as we know that elementary number theory is one of the most difficult topics in mathematics overall.)

Lets start with C, the associative commutative normed division algebra. The number theory in C deals with the structure of the Gaussian integers. There are three type of primes, the inert ones, the split ones and the ramified ones. The inert ones are the rational primes of the form 4k+3 together with the negative ones. So, the inert Gaussian primes are { …-11,-7,-3,3,7,11…,-11i,-7i,-3i,3i,7i,11i,…}. Then there are the split primes which come in groups of 8. They are of the form a+i b with p = a2 + b2 being prime. The 8 primes belonging to p=5 are 1+2i,1-2i,-1+2i,-1-2i, 2+i,2-i, -2+i,-2-i. Then there are the ramified primes: they belong to the rational prime p and there are exactly 4 elements: {1+i,1-i,-1+i,-1-i}. Now there are two symmetries we can let work on these primes. The first is multiplication with the units U = {1,i,-1,-i}. The second is the dihedral group generated by U and conjugation a+ib -> a-ib. The equivalence classes modulo V correspond one to one to the rational primes. For the equivalence classes modulo U, it depends on the type. For every rational prime p=4k+1 there are two equivalence classes a+ib, b+ia for which the arithmetic norm a2 = b2=p. For ramified primes, there is only one equivalence class 1+i with arithmetic norm 2. For inert primes p=4k+3, there is also only one U-equivalence class, but now, not the norm but the square root of the norm is equal to the prime p.Now, what does this have to do with particle physics? The starting point is quadratic reciprocity. Given two primes p,q, there is a number (p|q) called the Jacobi symbol. Now, primes of the form 4k+1 behave like Bosons and primes of the form 4k+3 behave like Fermions: the reason is the quadratic reciprocity theorem of Gauss: (p|q) = (q|p) if and only if one of the primes p,q is a Boson and (p|q) = -(q|p) if both are Fermions. The fact that primes of the form 4k+1 behave like Bosons is not surprising as they are actually the product of two Gaussian primes a+ib, a-ib by the Fermat two square theorem. We can now look at these two primes as a Fermion pair. An other input comes from the double cover V of U, which allows to define a quantity called charge. Double covers are everywhere in physics like Spin(n) double covering SO(n). A fancy way to describe this through a short exact sequence 1->Z/(2Z) -> V -> U -> 1 telling that dihedral groups covers the cyclic group. Since a+ib and a-ib are not U-equivalent if N(a+ib) is an odd prime, we look at them as Fermions of different charge. The most natural choice is to call them electron and positron. The neutral 4k+3 Fermions are then the neutrini. We still have to place the primes with arithmetic norm 2. Since their V and U equivalence classes are the same, they are neutral. They are also light. Why not associate them with the Higgs particle?

Lets look at the primes in the quaternion algebra H. The Lagrange four square theorem prevents the existence of neutrini type primes on the coordinate axes. All primes (a,b,c,d) have at least 2 entries which are nonzero. Now, primes in H come in two types, there are the Lipschitz primes (a,b,c,d), where all a,b,c,d are integers or then the Hurwitz primes (a+1/2,b+1/2,c+1/2,d+1/2), where a,b,c,d are integers. In both cases, a quaternion integer is a prime if N(a,b,c,d) = a2 + b2 + c2 + d2 is prime. Lets again look at symmetries. The first one is the group U of units in the algebra. It is a finite group, the binary tetrahedral group. The first group $V$ is the group generated by the permutations of coordinates and the conjugation (a,b,c,d) -> (a,-b,-c,-d). We call U the weak group of symmetries because it is generated by a discrete subgroup of SU(2). V is the strong group of symmetries because it is generated by a discrete subgroup of U(3). There is a third group W which is a subgroup of V and which only take even permutations. It is a subgroup of SU(3). As V is a double cover of W, giving an equivalence class of V and a sign called charge determines an equivalence class in W. The equivalence classes in V are easy to describe. Every integer is modulo V equivalent to an integer (a,b,c,d) with 0 ≤ a &leq b ≤ c ≤ d. Such an integer together with a charge determines an equivalence class in W. Here comes the interesting part: what are the weak equivalence classes within the strong ones? It is a bit surprising that the answer is very simple. Except for the integers with arithmetic norm 2, the equivalence classes consist either of two or three elements. When seeing this there is almost no other reflex possible as seeing the odd primes as “hadrons” and the equivalence classes either as baryons or mesons. The individual elements in the equivalence classes are then the quarks.

Here is the situation from a number theory perspective: the fundamental theorem of algebra for quaternions (see the book of Conway and Smith) can be restated that any quaternion integer z of arithmetic norm p can be written for any factorization p=p1 …pn be written modulo V as a product [z1] … [zn], where [zi] is an equivalence class of primes with arithmetic norm pi. In the case when the primes are all odd and adjacent primes have different norm, the factorization is given as an ordered product of mesons or baryons. We hope to be able to assign charge to quarks in a natural way by looking at all possible factorizations and assign charge in a simple combinatorial way. Its clear that this leads to charges of individual members which are multiple of 1/6 and as we believe actually to a multiple of 1/3. We have not done this combinatorics yet but instead assigned charge ad hoc to individual elements of the equivalence classes. Our assignment is simple and unique but it has the disadvantage yet that it rules out baryons of charge 2 which have been observed in nature. I hope that looking at all possible factorization of a quaternion integer modulo the symmetry groups U,V,W allows to define charge in a more natural way. As for now, this is a rather concrete combinatorial problem.

Here is a picture of all the hadrons in the case of the prime p=107. There are two baryons and two mesons. The first baryon has charge 1, the second has charge 0. Then there are two mesons of charge 1.


Bosonic and Fermionic Calculus

Traditional calculus often mixes up different spaces, mostly due to pedagogical reasons. Its a bit like function overload in programming but there is a prize to be payed and this includes confusions when doing things in the discrete.

Here are some examples: while in linear algebra we consider row and column vectors, in multivariable calculus, we only look at one type of vectors. We even throw affine and linear vectors into one pot. This is perfectly fine as we would produce unnecessary complications like also to consider the cross product of two vectors as a vector and not a covector. We also treat differential forms as vector fields and define the divergence as a type of exterior derivative rather than an adjoint. This is all fine. But there are places, where pedagogy does the wrong thing. An example is when defining line integrals or flux integrals. Some textbooks produce an intermediate integral which is the scalar line or scalar surface integral. It not only complicates things as more integrations are built in, these integrals are also completely different beasts. One only gets fully aware of such things, when working in a discrete calculus like when looking at calculus on graphs.

Here is a first riddle, which many students ponder when learning calculus. How come that if we compute an arc length or surface area that the result is independent of how we integrate. You see, if we take the function |r'(t)|, the speed, then this is a direction independent notion, a non-negative scalar. If we travel from A to B and parametrize this with a curve r(t), then ∫ab |r'(t)| dt is a non-negative number. Also if we integrate backwards. Arc length is independent of the parametrization. If we look at a line integral ∫ab F(r(t)) r'(t) dt however, then the orientation matters. Similarly, when we compute a surface area ∫ ∫ |ru x rv| du dv then this does not depend on the surface orientation, while the flux integral ∫ ∫ F(r(u,v)) . ru x rv du dv does depend on the orientation. You might bend your mind and somehow blame the dt or du dv or then just think you don’t understand well enough. What actually happens is that the objects are of completely different nature. In reality, we have an integration which is orientation dependent, and then there is an integration which does not depend on orientation. Putting these two type of integrals so close together like many calculus books do, not only confuses, it also complicates things.

When looking at integrals ∫ f(t) |r'(t)| dt , then one should see them as modifications of the arc length. They can model average along a curve or mass of an inhomogeneous wire. Mathematically they are one dimensional valuations with respect to an inhomogeneous background measure. The line integral of a vector field however integrates a 1-form which is intrinsically a completely different objects, as differential forms are asymmetric tensors. Its like confusing Fermions with Bosons or symmetric tensors with asymmetric tensors. Mixing such things up is extremely poor taste, even if it is done with the best pedagogical intentions.
Now there are notions in the continuum, which are close to the discrete and are intuitive: this is done with differentials which are quite handy. Still, like non-standard calculus, the overhead appears too big. In order to define differentials properly and appreciate them, it needs quite a bit of sophistication. Similarly, to understand non-standard analysis, it requires some logic and real analysis background in order not to get lost. I personally use differentials and non-standard analysis only on an intuitive level, where it can be quite power ful but not teach it as it can become a source for serious errors.

In classical mathematics, these distinctions only start to matter when looking at geometric measure theory or integral geometry, where unlike in measure theory, the objects under consideration have more structure. They can be duals of differential forms or then have internal simplicial structure. What is going on is that integrals like length, area or volume are valuations which integrates over non-oriented simplicial complexes, while line, flux integrals or any integral entering a fundamental theorem of calculus integrates over oriented simplicial complexes.

It starts already in one dimensions. When looking at the fundamental theorem of calculus, the derivative of a function is a 1-form. Integration then depends on the orientation and ∫ab f'(x) dx = – ∫ba f'(x) dx When computing an area, we add up positive quantities and whether we go from left to right or right to left should not matter. Its like adding up numbers on a spread sheet, where it does not matter whether we add from the left to the right or from the right to the left. That we are dealing with two type of integrals becomes only visible in the discrete. In the first case the function f'(x) is a actually a function on the edges of the graph, while in the second case, |r'(x)| is a scalar given as the square root of r'(x) . r'(x) which is a function on vertices. Not that I believe these things should be pointed out in a calculus course, it is the teachers and especially the textbook writers who have to be aware of it. When explaining calculus, we take great care to hide these difficulties: in the following 15 minute review for example, the integral is defined as Archimedes did it and hide the fact that for the fundamental theorem of calculus, the integrand is a 1-form. It would be a big no-no to explain the difficulty.

We can even hide the difficulty when dealing with calculus on a one dimensional graph like in this Pecha-Kucha (20 x 20 seconds) talk. Its just when we work on general graphs and especially when looking at PDEs on graphs, things start to matter.

Lets look at an other example: the case of surface integrals. Then in the discrete, we deal with valuations counting triangles, possibly with weights. In the flux integral case, however, we have differential forms, functions on oriented triangles and the answer is now orientation dependent. Multivariable calculus is already tough enough, why complicate things? Knowing the
background actually gives more arguments to consider the simplified version, where we only have one type of integral in each dimension: the line integral of a vector field along one dimensional objects, the flux integral of a vector field along a two dimensional surface and the three dimensional integral along a three dimensional solid. Identifying 1 forms with 2 forms and identifying 0 forms with 3 forms in three dimensions is of course ok as well as not mentioning differential forms at all and just look at scalar functions as well as vector fields.

To summarize, its good to be aware that there is a symmetric calculus which features an integration without fundamental theorem but which belongs to valuations in integral geometry. Then there is an anti-symmetric calculus which enters the fundamental theorem of calculus, which is Stokes in higher dimensions. The first is a Bosonic calculus, the second is a Fermionic calculus. As in complexity theory (Permanents and Determinants, NP or P) or physics (stability of matter), the Fermionic calculus is often more pleasant. In calculus, it is the fundamental theorem of calculus which allows us to take shortcuts. In Bosonic calculus, there is often no short cut (similarly as can not compute permanents easily), nor can we count the number of triangles in a large graph easily without just counting them.

What does this have to do with quantum calculus? It is there that we can no more hide our ignorance and have to distinguish between summing up function values attached to oriented simplices or then non-oriented simplices. And this distinction also is important when building interaction calculus which is a new type of calculus, which does not seem to exist in the continuum. There is a Bosonic version which for example is used when defining Wu characteristic (the analogue of Euler characteristic in interaction calculus), and then there a Fermionic version which leads like in traditional calculus to a simplicial cohomology. Now, cohomology is always a very Fermionic construct as we need d2 = 0. Interaction cohomology is exciting, as it allows to attach in an entirely algebraic way numbers to topologies. It allows to distinguish the Moebius strip from the cylinder for example: See the computation.

Interaction cohomology

Classical calculus we teach in single and multi variable calculus courses has an elegant analogue on finite simple graphs. The discrete theory is completely analogue, Stokes theorem is almost tautological and simplicial cohomology is easy to compute just by finding the nullity of some Laplacians. Taylor expansions on graphs are just wave equation solutions on graphs. Any partial differential equation which is based on the Laplacian can be considered on an arbitrary network. While multi-variable calculus in 2 or 3 dimensions is quite old (Stokes, Ampere, Green, Gauss), the higher dimensional calculus came surprisingly late. The reason is that first, linear algebra, tensor calculus and especially multi-linear algebra had to be developed which was necessary because we are rather clumsy in accessing lower dimensional parts of space without sheaf theoretical constructs like differential forms. In higher dimensions, the frame work of differential forms in arbitrary dimensions had been formalized first by Élie Cartan around the same time than Poincaré created algebraic topology while simultaneously developing then also discrete calculus. Both in the continuum as well as in the discrete, the calculus is tied to the notion of Euler characteristic. This is visible when looking at theorems like Gauss-Bonnet-Chern (in differential geometry), Poincareé-Hopf (in differential topology), Brouwer-Lefschetz-Hopf (fixed point theorem) or Riemann-Roch (in algebraic geometry) which tap into various parts of calculus: curvature, indices of critical points, or Brouwer indices or the dimension of linear systems to divisors need calculus for their definition) Euler characteristic is the only quantity which is invariant under refinements of the geometry (Barycentric refinements) and is a linear valuation in the sense of integral geometry. But this uniqueness result (which has been proven and re-proven again and again by various mathematicians) does not exclude that nonlinear quantities exist which play the analogue role. Such quantities exist in the form of Wu characteristics. They all satisfy natural counting principles like that w(G x H) = w(G) w(H) and w(G + H) = w(G) + w(H), if x is the Graph product and + the disjoint union of graphs. Now, since Euler characteristic can be computed using calculus too. The reason is given by Euler-Poincaré relations. There is the natural question now, whether there is a calculus which belongs to Wu characteristic. Such a calculus indeed exists and so far, I don’t know whether it is related to something analogue in the continuum. What is interesting about this “interaction calculus” is that it leads to a cohomology which can “see” things which traditional cohomology (Simplicial, Cech or de Rham cohomologies) can not see. When doing computations with this cohomology it appeared that it can distinguish the cylinder and Möbius strip. Simplicial cohomology (which is by de Rham equivalent to the cohomology of traditional calculus) can not do this without taking Z2 target group (Stiefel-Whitney classes). A case study [PDF] gives a detailed computation.

A nonorientable two dimensional graph with the topology of the Moebius strip

A nonorientable two dimensional graph with the topology of the Moebius strip

A two dimensional orientable graph with the topology of the cylinder

A two dimensional orientable graph with the topology of the cylinder

Wu Characteristic

Update: March 8, 2016: Handout for a mathtable talk on Wu characteristic.

Gauss-Bonnet for multi-linear valuations
deals with a number in discrete geometry. But since the number satisfies formulas which in the continuum need differential calculus, like curvature, the results can be seen in the light of quantum calculus. Here are some slides:

So, why is the Wu characteristic an object of quantum calculus? It is a combinatorial invariant, a quantity which does not change under Barycentric subdivision. An other reason is Gauss-Bonnet which holds for this quantity and for discrete structures. When Barycentrically refined again and again, the graph converges to a smooth manifold, where the corresponding curvatures involve curvature tensors, which are objects from classical calculus using classical partial derivatives. One must therefore see the discrete curvature values as quantities in a discrete quantum calculus setting. It is a bit mysterious still, how the complicated expressions of the Riemann curvature tensor in the continuum become such simple expressions in the discrete. But in the discrete we have magical abilities to see into the lower dimensional building blocks in space, something which is more difficult in the continuum. In classical calculus, we need to probe these quantities using a tomographic methods called integral geometry or sheaf theoretical methods. Some quantities like length of a curve, or area of a surface or volume are quite intuitive. There are other quantities which are not, like “length of a surface”. One can measure the length of a surface; it is just something we usually do not do, or are not interested in. For discrete surfaces given by a graph, the length of the surface is the number of edges it contains. For an icosahedron it has 30. The area is the number of triangles, 20, which gives the Icosahedron the name. Some combination of the quantities can be seen topologically. One important one is the Euler characteristic, which is area minus length plus number of points. In the case of the Icosahedron, this is 12-30+20=2, a relation which has been found by Rene Descartes already. The story is described nicely in the book “Descartes’s Secret Notebook: A True Tale of Mathematics, Mysticism, and the Quest to Understand the Universe“, by Amir Aczel. It was Euler who proved the formula first. But the story continues to be interesting then as the formula turned out to be wrong in general. There are surfaces, like the Kepler polyhedra, for which the Euler characteristic is different from 2. Imre Lakatos analyzes the evolution of a theorem brilliantly in his book “Proofs and Refutations”. Now, the Wu characteristic is even more exciting than the Euler characteristic, because it allows to probe continuum spaces like varieties in a different manner. The number is a quadratic valuation: it does not only probe the cardinalities of the pieces of the fabric of space, but also how these pieces interact. For a lemniscate for example, the Wu characteristic is 7. It is different from the Euler characteristic, which is minus 1.

kepler icosahedron

Barycentric refinement

When passing from the discrete to the continuum, topologists have invented Barycentric subdivision. This is usually done in a Euclidean setting. When thinking about topology in a quantum way, we don’t want to use Euclidean space but start with discrete structures. Combinatorial topology has come up with the concept of Barycentric subdivision for abstract simplicial complexes. These are structures imposed on a set which similar than topological, order, measure or algebraic structures are given by a specifying a set of subsets, in this case a set of non-empty subsets which are closed under the operation of taking non-empty subsets. Barycentric subdivision takes this set of subsets as basic point set and builds a new simplicial complex structure on this, given by the order structure. If you think this set-up is hard to understand, I agree as simplicial complexes usually are introduced later in a mathematics curriculum. Much more intuitive and approachable is the language of graphs. A finite simple graph G=(V,E) comes naturally equipped with a simplicial complex: It is the Whitney complex which consists of the set of all subsets which form complete graphs. The Barycentric subdivision of a graph is a new graph, where the complete subgraphs are the new vertices and where two such new points are connected, if one was originally contained in the other. This immediately produces a holographic picture as once the subdivision is done, one can repeat it. The basic idea is to look at complete sub graphs of a graph as points. If one takes this seriously then already a Barycentric limit is intrinsic in the finite structure. If the original graph has no triangles or -if one looks at graphs equipped with the 1-skeleton simplicial complex structure as many graph theorists do – then the Barycentric limiting space is the topological group of dyadic integers. This is a quantum analogue of the circle but unlike the circle, which has a continuum of translations, there is a smallest translation in the group of dyadic integers. It is called the adding machine. In ergodic theory (that’s how I got into it), the dynamical system is known as the von Neumann Kakutani system, an interval exchange transformation which is the unique dynamical system which is a fixed point of the 2:1 integral extension. Unlike the usual integers, the dyadic integers form a compact space and space is quantized in the sense that there is a smallest translation. What is interesting is when one looks at the Laplacian of the limiting structure. The Laplacian is a fundamental object in any geometry as it allows to define what “light evolution” is and since light can be used to measure distances, it allows to recover all the geometry. It turns out that the Laplacian of the Barycentric limiting geometry is universal. (See The graph spectrum of barycentric refinements and Universality for Barycentric subdivision.) In the one dimensional case, its spectral density of states is the equilibrium measure of a Julia set of the quadratic map belonging to the “tail” -2 of the Mandelbrot set. Unlike most Julia sets, it is a smooth measure and still self similar. The inverse of the integrated density of states satisfies F(2x) = T F(x) where T is a quadratic map. If one looks at the situation in 2 or more dimensions, the limiting density of states is universal too but it is not identified yet. The spectral behaviour is no more smooth as in the one dimensional case. There appear gaps in the spectrum: barycentric limit in dimension 2
There is an other story appearing when looking at Barycentric subdivision. If one counts how the number of points grows under subdivision, one can encode this with a single universal matrix A whose eigenvalues are k!. The eigen vectors of its transpose are produce Barycentric characteristic numbers, where the first one with eigenvalue 1 is Euler characteristic. The picture immediately shows that Euler characteristic is unique in the sense that it is the only valuation which is a homotopy invariant assigning the value 1 to points. The other invariants scale under Barycentric refinement. But there is a surprise: for geometric graphs, half of them vanish! This is related to Dehn-Sommerville-Klee invariants which have been known for a while. One can prove this elegantly using ideas from differential geometry: there is a Gauss-Bonnet, Poincare-Hopf result as in differential geometry or differential topology (its just much easier to understand). It turns out that Gauss-Bonnet is the expectation over Poincare-Hopf if one averages over all possible functions (waves). This is very nice. And now the quantum calculus ideas discussed before in the context of the Sard theorem kicks in. One can write the Poincare-Hopf indices (integers!) as invariants of smaller dimensional spaces allowing induction. This is outlined here kept as a 2 page document so that can be tweeted.

Some video: