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Friday, November 25, 2005

Categories in physics and the Trivium

I mentioned last time that our understanding of even simple mathematical notions such as the natural numbers or integers is far from complete ("The sphere spectrum is the true integers"). More evidence for this comes in this paper, 'The Dirichlet Hopf algebra of arithmetics', ArXiv: math-ph/0511079, which studies the combinatorial intricacy of co-addition and co-multiplication operations, dual to addition and multiplication, in the natural numbers. Dual operations correspond to ways of decomposing entities. Amongst other goals, Fauser and Jarvis are trying to understand why number theoretic terms, such as values of the Riemann zeta function, appear in work on renormalization in quantum field theory, such as those found by Kreimer and Connes. There appears to be a shared combinatorial structure between these two fields, which was treated in this workshop.

As with the work of Baez and Coecke (25 October), the idea is afoot to strip quantum theory down to its combinatorial bones. Fauser and Jarvis argue

The appearance of complex numbers, an algebraically closed field, is often argued to be a key feature, e.g. for producing interference effects, but we doubt this. A ‘phase’ may be modelled by a finite cyclic group also. The expectation values can be obtained in a topos theoretic setting using more general truth objects and therewith related subobject classifiers. This will be explored elsewhere. We think that the present work shows at least, that for the identification of the algebraic structures involved in quantum mechanics and quantum field theory, a characteristic free approach to quantum mechanics would be of great help. Especially the interpretation of combinatorial factors, normalizations etc. would benefit from such a view, even if the complex number field is finally adopted. The appearance of number theoretic functions in renormalization supports this point of view. (p. 41)
The authors also mention that: "The present structure is more compatible with a 2-category picture, but we refrained here from exploring this in a first exposition." (p. 3)

The Hopf structures the authors are studying are closely related to tensor categories, which seem to be cropping up all over mathematical physics. Perhaps Levin and Wen know why. In their paper, 'A unification of light and electrons through string-net condensation in spin models', they claim:
In a crystal, atoms organize themselves into a very regular pattern - a lattice. Since different lattice structures are distinguished by their symmetries, we can use group theory to classify all the 230 crystals in three dimensions. In much the same way, string-net condensed states are highly structured. The different possible structures are described by solutions to (3). Tensor category theory provides a classification of the solutions of (3), which leads to a classification of string-net condensates. Thus tensor category theory is the underlying mathematical framework for understanding string-net condensed phases, just as group theory is for symmetry breaking phases.
Someone who has done interesting things with tensor categories is Michael Müger. You can read about why one of his papers was a fast breaking paper. He conjectures that "all existing (and future) applications of subfactor theory to low-dimensional topology ‘factor through category theory’—as is by now well known for the knot invariants of Jones and HOMFLY" (p. 154). (Many of the interviews for fast breaking papers across a wide range of disciplines are interesting.)

On a different subject, I came across this interesting advocacy of a return to a mediaeval syllabus by Dorothy Sayers. The word 'trivial' derives from the term 'Trivium', the first part of the syllabus, covering Grammar, Dialectic, and Rhetoric, but the Trivium was anything but trivial. Is this grounding what is needed to lay the foundations for the life of learning Alasdair MacIntyre wishes us to enjoy:
...we have to stop thinking of teaching and learning as activities restricted to specialized, compartmentalized area of life within schools, colleges and universities. Of course, schools, colleges and universities have their own highly specific tasks, but these tasks need to be defined in terms of their contribution to lifelong learning and teaching, most of it carried out in nonscholastic and nonacademic contexts. We need, that is, to think of formal academic education not primarily as a preparation for something else, a life of work, which terminates when that life of work begins, but rather as itself the beginning of, and the providing of skills, virtues and resources for, a lifelong education directed toward and informed by the achievement of the good. We need, for example, to teach our students to read, so that they go on reading throughout their lives. We need to make such reading a way of illuminating their social relationships, so that their familial and communal lives continue to be enriched by a stock of common reading. We need to rethink the time-scale of education so that we make one of the tests of the adequacy of what we teach now the answer to the question: "What will our students be reading when they are forty, sixty, seventy-five?" and to accept that if they are not then returning to the Republic and the Confessions, to Don Quixote and Dostoievski and Borges, we will have failed as teachers. (The Privatization of Good: An Inaugural Lecture 359, The Review of Politic 52(3), 344-377, 1990, available on JSTOR)
Finally, a good 'Bibliography for Philosophical Materials Pertaining to Mathematics and Proof'.


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