God is no geometer, rather an unpredictable poet.In mathematics today, the spirit of set theory has been subsumed by the hierarchy of weak $n$-categories and their generalisations. Instead of an infinity of set elements, assuming an axiom of infinity, there is an infinite tower of increasingly complex equivalences and relations between relations. A mere set is but a $0$-category, sitting at the bottom of the ladder, and yet many constructions to date take their intuition from the land of sets.

Looking at the path table of monomials, in the spirit of Cantor many would see a partition of all possible words on an alphabet $\{ X, Y, Z, \cdots, \}$ grading the free monoid functor on the category of sets. The table stretches off to infinity without question.

But as physicists, we stare in amazement at the first few entries of the table and wonder how many infinities can be built with just these few words, the qubits and qutrits. What are these information sets? They live not in any ordinary category of sets, but in some other world. With qubits alone, dualities reside. As words grow longer, so do the possibilities of categorical composition. What was once a Hilbert space of dimension $n^d$, is now a small collection of words in the symbols of measurement.

The quantum information paradigm seems to hint at a topological topos underlying our observable universe, in the spirit of Brouwer. Such topological toposes include the toposes of sheaves on any topological space T, for which excluded middle, double negation, and axiom of choice fail. In such toposes, a truth value, is an open set U in T.

ReplyDeleteSo, you speak of a topological space as a lattice of subsets, with a $0$ for empty and a $1$ for the whole space. This categorical ladder aims to expand the notion of space by discussing not only $(0,1)$ type spaces, but also general dit spaces. Since the axiom of choice fails for simple sheaf toposes, we certainly don't want to impose it on the higher dimensional hierarchy!

ReplyDeleteIt is a while since we have discussed the rationale for higher categorical dimensions, but one way to see it is that quantum counting is about dimension (of Hilbert spaces, say) rather than about cardinality.

What I mentioned is concretely realized in n-dimensional projective space P^n, the space of n+1 state qudits. An atlas for P^n gives n+1 coordinate charts, yielding n+1 possible measurement values for a qudit, described by n+1 orthogonal projectors.

ReplyDeleteGates for such qudits are typically given by unitary transformations (isometries) on P^n, but more recently SLOCC transformations allow one to relax length preservation and merely preserve rank (collineations).

Given any collineation, on can reduce it to transformations on projective lines in P^n, hence any general qudit SLOCC gate can be decomposed into elementary qubit SLOCC gates. Hence, n-ary logical operations inevitably reduce to compositions of binary logical operations. This is just the quantum gate version of the combinatorial result that any permutation can be expressed as a product of elementary transpositions.

OK, so you bring up the reduction of qudit operations (as they are usually viewed in terms of classical groups) to binary ones, which would take place in the usual land of topological spaces ... but we could generalise the concept of qubit at the fundamental level of duality (like in my thesis) and then the projective geometry would be the binary shadow of a much richer world. This was what I had in mind. I don't see why we should stick to ordinary quantum mechanics.

ReplyDeleteA particular case of a Klein group is a Schottky group. This is a particular case where a system of closed curves (or Jordan curves) cuts a sphere into regions according to a Mobius or modular transformation. The nesting of regions inside bounded curves then has a self-similar structure, and a region defined as A is bounded by curves which define regions that are “out.” This nesting structure will divide the sphere into in and out regions that are Cantor sets. The drawings by Escher of tessellated disks, where the tilings were fish or angels and devils, is and example of this sort of structure. If one defines regions and in and out and removes them the result is a type of Cantor set. Here the set is a defined by a modular form that constructs the Klein group on the sphere. This can be extended to other spaces as well, such as hyperbolic spaces and the AdS spacetime.

ReplyDeleteOn a hyperbolic space the Laplace Beltrami operator describes a quantum system with group SL(2,R). Hyperbolic geometry is a wellspring of sorts for conformal quantum groups. The Schottky group (or space) is a manifestation of conformal quantum mechanics, and it further defines the conformal completion of the AdS spacetime. This gets into some deep issues I see in how general relativity and quantum mechanics are in some ways equivalent to each other.

You seem to get into a lot of set theory, which is a bit removed from my experience. In fact in graduate school I was advised to stay away from the subject. I do know a few things about it, but I am not much of an expert on the subject.

From the perspective of noncommutative geometry, qudit projective spaces take on a very simple form. Indeed, in this context, a projective space reduces to a finite point space. A priori, there is no ordering or labeling of such points. Choosing two points and assigning truth values to them is an arbitrary process.

ReplyDeleteWith such finite point spaces, it is better to just consider the points as objects, and consider morphisms over the objects. Ordinary quantum mechanics would then amount to the condition that morphism composition is associative. Relaxing associativity gives us a precategory (directed graph) useful for applications in quantum gravity.

Higher weak n-categories can then be defined inductively from such precategories a la Batanin and Baez.

Yes, I have been wondering a bit about the n-category definitions, and this is certainly motivated by Batanin's work, but I suspect that this physics will not follow the current routes to a recursive definition. For a start, we are thinking of multicategories and polycategories and not just ordinary categories. We have the possibility of generalising the use of source/targets to cyclic structures, and so on. The complexity of the objects means a very structured 'category of sets', rather than the old category of sets.

ReplyDeleteLawrence, we use a categorical language as a means of avoiding these kind of set theory problems.

I mentioned the Cantor set with Schottky groups, since you mentioned Cantor sets above.

ReplyDeleteI hope to get back to this later tonight, but I do need to ask a question. Is there a prospect that category theory might demonstrate an equivalency between and E_{8(8)} at the UV range, then various decompositions by reducing SL(2)'s say with E_{7(7)} and so forth down to the IR limit? If so this would then illustrate something about the mass gaps in the Zamolodchikov conformal theory.

I merely mentioned Cantor's development of Set Theory, rather than any particular example of an interesting set. By the way, you can use dollar signs for inline latex in the comments.

ReplyDeleteRe your question on $E_{8(8)}$. Certainly I believe that the 'operad combinatorics' we study is responsible for such a symmetry sequence, mathematically, but once again please note that I am always thinking of this equivalence in terms of a new cosmology where the UV/IR link is a question of principle and thus the CFT issue is not so interesting.

This sort of got behind me. The categorical theory and operands might be a tool for demonstrating that a low energy theory with conformal groups peeled off are equivalent to high energy theory. So in that way it might aid in a UV/IR correspondence.

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