Quantum Calculus

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. The motion becomes a rotation in three dimension. The KS transform generalizes the Levi-Civita regularization, found in 1920.

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 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.

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. 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 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.

Lets look at this idea of 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, but motivated by a speculation that our physical space could be modeled by a finite field. Here is the article (thanks to the Harvard library for digging that up):

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 is a double jeopardy, especially by 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 of the ArXiv to have a closer look. [Update September 25: I think now that the trigger could also have been a text overlap between the Experiments paper and the new write up. Indeed, the later is an expansion of a paragraph in the former.] Obviously, the texts were written a bit 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 again at the 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 like a 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 (the structure of some equivalence relations) and lead to more questions in mathematics. It even mentions that it is hardly of any value for physics.

The starting point is the theorem of Frobenius or Hurwitz. It assures 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). As vector spaces they could also be characterized as the linear spaces in which the unit spheres are continuous Lie groups: only the d-spheres for d=1 or d=3 carry a Lie group structure. The allegory is that C harbors Fermions and H harbors hadrons and that the units or symmetries produce the gauge Bosons, photons and gluons. Since we did not know what to do with the prime 2, and because it is neutral and boson like, adding norm =mass to an integer, why not call it 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 (very dangerous on the crackpot abyss) 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 or sign change of its coordinate to make two equivalent. This group can be realized by discrete subgroups of three complex sub-planes as well as a discrete subgroup of the unitary group U(3). (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 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.

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. 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. 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 sqeuence 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 (very 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) 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 very high dimensional graphs where many Betti numbers are nonzero. It is computationally hard for example to compute the 100’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 and harder as the matrices become huge.

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.

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 S^–_f(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 S^–_f(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 + …. + X_m. 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: X_m -> X_m+1 given by df(x) = sum_y 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.

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.

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:
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:

In this recent paper the topic of level surfaces in a d-graph is studied. This is a core calculus topic and belongs to quantum calculus, as space is a discrete space which has properties shared by Euclidean space like that unit spheres are graph theoretical spheres (removing one vertex produces a contractible graph and recursively, every unit sphere is a sphere). One application is the visualization of Chladni surfaces as seen in the following video:

The definition of a level surface of a function f on the vertex set of the graph G is the following: consider the set of all complete subgraphs of G and connect two of these “points” if one is contained in the other. This is a larger graph G’ called Barycentric refinement of G. Now look at the subgraph of G’ which has as vertices all complete subgraphs of G on which the function f changes sign. The magic is that if G is a d-graph, then this level surface { f=c } is a (d-1) graph
as long as the value c is not taken by any vertex of G.

If we are given k functions, we can start with the first function, look at its level surface f1=c1, then extend the other functions to this new graph, then look at the level surface f2=c2 etc. For almost all values c1,…,ck, this produces a d-k graph. This is a discrete Sard result. However, there is a catch: the order of the functions matters (no surprise in a quantum setting) but the regularity is very helpful.

Closer to classical calculus is to look at k functions f1,…,fk and now look at all simplices in G’ on which all functions simultaneously change sign. Now the set of values c1,…,ck for which we get a (d-k) graph has positive measure in general. We are however closer to classical calculus or algebraic geometry.

Here is the simplest case: take a 2-graph (the discrete analog of a two dimensional surface) and two functions f,g. A Lagrange problem is to find the extrema of f under the constraint g=c. One can now do this in the same way as in classical
multivariable calculus: define the discrete gradient of f on a triangle (xyz) rooted at a vertex x as the two vector df = . Lets look at the level curve g=c. It consists of all edges in G on which f changes sign. If c is a value not taken by g, then g=c is a union of circular graphs. The reason is that on each triangle, either none or two of the edges have the property that g changes sign on them. Since every edge has two triangles as neighbors, this implies that every edge has two neighbors in G’. Similar, any level curve of f is a union of closed circular graphs.
Now, if the gradients df and dg are not parallel, then the simultaneous zero locus { f=c, g=d } is a discrete set. Only points for which the Lagrange equations df = L dg, g=c hold are candidates for extrema for f.

More generally, in higher dimensions, the discrete algebraic set {f1=c1,….,fk=ck} is a d-k graph, if at every point, the discrete gradients satisfy a maximal rank condition.

This topic of doing calculus without limits is not only a game. It can be used to investigate completely unexplored problems like classifying the nature of level surfaces of the ground state of the Laplacian in a graph. The ground state is the eigenvector to the smallest nonzero eigenvalue. It has one connected nodal surface (both in the graph (Fiedler) as well as in the Riemannian manifold case (Courant)) by the Courant-Fiedler theorem. What is the topology of this surface if G is a d-sphere? This question can be explored now in a combinatorial setting. If the answer would be that the nodal surface is a d-1 sphere, then this would be a strong indication that the same holds also in the continuum.

• It leads to a natural calculus on graphs. Here is an attempt to summarize calculus. One can explain some main ideas in multivariable calculus, differential geometry and topology in discrete settings. Calculus on graphs by nature is coordinate free. Yes, one needs to introduce orientations of edges for example to define the gradient but essential notions like the gradient which determines how light or heat propagates are coordinate independent.
• It allows experimentation with small worlds. One can look at the Maxwell equations, wave equations, heat flows etc on small graphs and experiment. Functional analysis reduces to linear algebra, there are no existence and uniqueness problems to solve. No unbounded operators, no regularity issues appear. Experimentation is esspecially interesting with discrete partial differential equations.
• We do not need to change notation from traditional calculus. Discrete calculus as a reinterpretation of the calculus we know and hardly new. Newton did it, Archimedes did it. The point is that we do not have to change the language. Indeed, language for discrete calculus, including numerical methods is usually very difficult to read, with lots of indices etc. But the translation does not need any new language. We can talk about gradients, curl and divergence for example using the same notation. We just have to reinterpret things. By defining polynomials, exponentials in a convenient way, we have the same results as in the continuum.