Regge Functional - Quantum Calculus

A finite abstract simplicial complex G, a finite set of non-empty sets closed under the operation of taking non-empty subsets, has not only a wonderful topology in which the stars $U(x) = \{ y \in G, x \subset y \}$ form a basis and the cores $C(x) = \{ y \in G, y \subset x\}$ are closed, but also a hyperbolic structure as the unit sphere $S(x) = \overline{U(x)} \setminus U(x)$ is the join of the stable sphere $S^-(x) = C(x) \setminus U(x)$ and unstable sphere $S^+(x) = U(x) \setminus C(x)$ . The closed sets in G are the sub simplicial complexes and the open sets are complements of simplicial sub-complexes. Open sets naturally form a delta set and so also have cohomology and compatabiility as expressed in the fusion inequality for the Betti vectors. Every finite simple graph of course comes with natural simplicial complex, the Whitney complex in which the complete subgraphs define the elements of the complex. There is a natural dimension functional R(x)=dim(x)=|x|-1 on G and also a natural Dirac operator. This motivated to look at a delta-set triple (G,R,D) in more generality as it includes the topos of delta sets as well as open sets in simplicial complexes. Delta set triples and simplicial complexes in in particular form one of the most attractive geometric structures possible in which one has everything we dream about: arithmetic, topology, algebra, analysis and physics can be done in it.

While this is a wonderful general frame work for geometry which generalizes zero dimensional geometry which is the topos of finite sets, it is – like the topos of finite sets- too rich for physics. While the task of the mathematician is to build structures and formulate and prove theorems, the task of physics is to select out the right structure that fits observations we see in nature. This is done partly through no-go theorems like for example Bertant’s theorem [a PDF from a multivariable class taught in 2002 when we still taught Kepler’s laws to first year students] which classifies Euclidean spaces in which the Newtonian dynamics produces stable structures. This is only possible in dimension 1 (the harmonic oscillator x”=-x and dimension 3 (the Kepler problem $x''=-x/|x|^3$ ) . An other example is the Frobeninus theorem classifying the structure of associative real normed division algebras. An other example is the Frobenius theorem classifying the structure of associative real normed division algebras.Restrictions can also be done by using functionals. Among all curves in a geometry the geodesics form an important role, the Einstein equations are extrema of the Hilbert functionals. Any probability distribution that is relevant can be described by entropy maximum. Laws in thermodynamics are given by extrema of functionals like Free energy. In geometry, manifolds can be characterized as having particularly simple small spheres, manifolds with the smallest Lusternik Schnirelman category possible so that the structure can not be collapsed. This brings us to the third reduction of generality: simplicity. This can be tricky because things need to be simple but not too simple. For geometric structures, this is already great. Take the smallest subset of abstract simplicial complexes which contains 1 and is closed under the join operation. This produces the arithmetic of the integers. Take the smallest subset of simplicial complexes which contains the zero dimensional simplices and is closed under the join operation. This produces the ring of partitions. Take the set of L-S-category 2 simplicial complexes which contains $S^0=\{ \{1\},\{2\} \}$ and is closed under both the operation of join and taking unit spheres. This is the set of spheres. Manfolds are the smallest class which have the property that all unit spheres are spheres. An example to justify why this is a good definition is the theorem that any level surface in a manifold is again a manifold (see i.e this blog entry from January 14, 2024, ).

If one is using functionals to get structures that have a chance of being a model of a physical theory, one also there should look at simplicity. The Einstein equations R-Sg/2=T look simple but actually are quite complicated even in simple cases. The theory is still ugly in some sense: particles with mass are described by paths = one dimensional distributions in space time. The tensor T on the other hand is assumed to be smooth where masses are “smeared out”. The real theory should be an integro differential equation like int he simplest case of Newtonian dynamics, where the Vlasov equations generalize Newtonian dynamics in that the point measure of the particles is replaced by a smooth function. The Vlasov equations on a manifold $f''=\int_M \nabla V(f(x)-f(y)) \; d \mu(y)$ look nice but are already quite a complicated integro-differential equations. See part 3 in “Probability Theory and Stochastic Processes with Applications”. In an honest theory for gravity, one should have a tensor T, which describes the density of particles moving micro-canonically along geodesics in the metric g satisfying R-Sg2=T. I’m not aware than anybody has build such a thing, except for numerical schemes. These frame works are however extremely ugly in that different aspects are covered in different ways, like Post Newtonian approximations. One tries to find a metric which is singular along two geodesic paths which are geodesics in the metric. It does not make sense for a mathematician at all: geodesic equations can only be defined in places where coordinates are smooth but if we look at the motion of a black hole binary, then each has an event horizon meaning that the event horizon consists of two tubes in a Lorentzian manifold. The centers of the tubes now have to be geodesics in that singular manifold. One can not even begin to say how bad and ugly this all looks. We would really like to have a theory in which the “Three Kepler Problems” can be solved nicely. The Newtonian Kepler problem works fine in a system like the earth and moon, the Quantum mechanical Kepler problem describes well the periodic system of elements, the gravitational Kepler problem is still a big mess.

There was a nice suggestion for a discrete Hilbert action in a discrete 4-manifold. Regge looked at a geometric realization and then looked at the excess angle of the dual sphere to a triangle, which is nice as 2-tensors like R, g,T should live on triangles (like differential 2-forms which are also (0,2) tensors, both in the complete symmetric or complete asymmetric case, one can just look at functions on simplices, where in the anti-symmetric case, a given choice of orientation (coordinate system) gives concrete values. This has in the discrete already been realized by Whitney. The choice of orientation on the simplices (the do not have to be compatible at all with each other) is irrelevant for anything interesting. It is like the choice of a basis in linear algebra, a choice of frames in differential geometry, a choice of gauge in field theories, a choice of difeomorphism in a frame work of general covariance. For me personally, geometric realizations are a big no-no. You see that in Chapter 42 of “Gravitation”. While we always can embed a smooth or discrete manifold in a higher dimensional Euclidean space, the question is what dimension we chose. The discrete analog of excess angle is of course the length. We see that when combining Descartes theorem about excess angles and the Pusieux formula for curvature in 2-manifolds, or when looking at discrete versions of curvature.

The simplest discrete Regge functional in a 4-manifold involves the stable and unstable spheres of a triangle. In a 4-manifold, the unit sphere is a 3-sphere. It is the join of the stable and unstable sphere. It is all defined by the length of the unstable circle as the length of the stable circle is always 6 (3 points and 3 edges are strictly contained in a triangle). The unstable circle consists of hyper chambers ( $K_5$ complexes) and tetrahedra ( $K_4$ complexes) containing the triangular “bone” x. Taking the sum or average of these lengths is a natural thing to do. I myself can experiment with it for example by taking a random 4 manifold in a higher dimensional manifold like a 6 manifold and compute these lengths. As a mathematician, one can now ask for 4 manifold with a given number of vertices for example which minimize this Regge functional. Then hope for the best that these 4-manifold say anything about physics. This is computationally challenging as even in small 4-manifolds there are maybe tens of thousands of triangles. We have for each then compute the dual sphere which (with the code I have myself) can take hours. In order to be able to find minima one would have to compute the functional in a fraction of a second. And these computations are not even yet using the softened integral geometric frame work. I had hoped to be able to say a bit more about this during the presentation. I mostly talked about “Alte Sachen”:

Author: oliverknill