September 19, 1998

This Week's Finds in Mathematical Physics (Week 123)

John Baez

It all started out as a joke. Argument for argument's sake. Alison and her infuriating heresies.

"A mathematical theorem," she'd proclaimed, "only becomes true when a physical system tests it out: when the system's behaviour depends in some way on the theorem being true or false.

It was June 1994. We were sitting in a small paved courtyard, having just emerged from the final lecture in a one-semester course on the philosophy of mathematics - a bit of light relief from the hard grind of the real stuff. We had fifteen minutes to to kill before meeting some friends for lunch. It was a social conversation - verging on mild flirtation - nothing more. Maybe there were demented academics, lurking in dark crypts somewhere, who held views on the nature of mathematical truth which they were willing to die for. But were were twenty years old, and we knew it was all angels on the head of a pin.

I said, "Physical systems don't create mathematics. Nothing creates mathematics - it's timeless. All of number theory would still be exactly the same, even if the universe contained nothing but a single electron."

Alison snorted. "Yes, because even one electron, plus a space-time to put it in, needs all of quantum mechanics and all of general relativity - and all the mathematical infrastructure they entail. One particle floating in a quantum vacuum needs half the major results of group theory, functional analysis, differential geometry - "

"OK, OK! I get the point. But if that's the case... the events in the first picosecond after the Big Bang would have `constructed' every last mathematical truth required by any physical system, all the way to the Big Cruch. Once you've got the mathematics which underpins the Theory of Everything... that's it, that's all you ever need. End of story."

"But it's not. To apply the Theory of Everything to a particular system, you still need all the mathematics for dealing with that system - which could include results far beyond the mathematics the TOE itself requires. I mean, fifteen billion years after the Big Bang, someone can still come along and prove, say... Fermat's Last Theorem." Andrew Wiles at Princeton had recently announced a proof of the famous conjecture, although his work was still being scrutinised by his colleagues, and the final verdict wasn't yet in. "Physics never needed that before."

I protested, "What do you mean, `before'? Fermat's Last Theorem never has - and never will - have anything to do with any branch of physics."

Alison smiled sneakily. "No branch, no. But only because the class of physical systems whose behaviour depend on it is so ludicrously specific: the brains of mathematicians who are trying to validate the Wiles proof."

"Think about it. Once you start trying to prove a theorem, then even if the mathematics is so `pure' that it has no relevance to any other object in the universe... you've just made it relevant to yourself. You have to choose some physical process to test the theorem - whether you use a computer, or a pen and paper... or just close your eyes and shuffle neurotransmitters. There's no such thing as a proof which doesn't rely on physical events, and whether they're inside or outside your skull doesn't make them any less real."

And this is just the beginning... the beginning of Greg Egan's tale of an inconsistency in the axioms of arithmetic - a "topological defect" left over in the fabric of mathematics, much like the cosmic strings or monopoles hypothesized by certain physicists thinking about the early universe - and the mathematicians who discover it and struggle to prevent a large corporation from exploiting it for their own nefarious purposes. This is the title story of his new collection, "Luminous".

I should also mention his earlier collection of stories, named after a sophisticated class of mind-altering nanotechnologies, the "axiomatics", that affect specific beliefs of anyone who uses them:

1) Greg Egan, Axiomatic, Orion Books, 1995.

Greg Egan, Luminous, Orion Books, 1998.

Some of the stories in these volumes concern math and physics, such as "The Planck Dive", about some far-future explorers who send copies of themselves into a black hole to study quantum gravity firsthand. One nice thing about this story, from a pedant's perspective, is that Egan actually works out a plausible scenario for meeting the technical challenges involved - with the help of a little 23rd-century technology. Another nice thing is the further exploration of a world in which everyone has long been uploaded to virtual "scapes" and can easily modify and copy themselves - a world familiar to readers of his novel "Diaspora" (see "week115"). But what I really like is that it's not just a hard-science extravaganza; it's a meditation on mortality. You can never really know what it's like to cross an event horizon unless you do it....

Other stories focus on biotechnology and philosophical problems of identity. The latter sort will especially appeal to everyone who liked this book:

2) Daniel C. Dennett and Douglas R. Hofstadter, The Mind's I: Fantasies and Reflections on Self and Soul, Bantam Books, 1982.

Among these, one of my favorite is called "Closer". How close can you be to someone without actually being them? Would temporarily merging identities with someone you loved help you understand them better? Luckily for you penny-pinchers out there, this particular story is available free at the following website:

3) Greg Egan, Closer, http://www.eidolon.net/old_site/issue_09/09_closr.htm

Whoops! I'm drifting pretty far from mathematical physics, aren't I? Self-reference has a lot to do with mathematical logic, but.... To gently drift back, let me point out that Egan has a website in which he explains special and general relativity in a nice, nontechnical way:

4) Greg Egan, Foundations, http://www.netspace.net.au/~gregegan/FOUNDATIONS/index.html

Also, here are some interesting papers:

5) Gordon L. Kane, Experimental evidence for more dimensions reported, Physics Today, May 1998, 13-16.

Paul M. Grant, Researchers find extraordinarily high temperature superconductivity in bio-inspired nanopolymer, Physics Today, May 1998, 17-19.

Jack Watrous, Ribosomal robotics approaches critical experiments; government agencies watch with mixed interest, Physics Today, May 1998, 21-23.

What these papers have in common is that they are all works of science fiction, not science. They read superficially like straight science reporting, but they are actually the winners of Physics Today's "Physics Tomorrow" essay contest!

For example, Grant writes:

"Little's concept involved replacing the phonons - characterized by the Debye temperature - with excitons, whose much higher characteristic energies are on the order of 2 eV, or 23,000 K. If excitons were to become the electron-pairing `glue', superconductors with T_c's as high as 500 K might be possible, even under weak coupling conditions. Little even proposed a possible realization of the idea: a structure composed of a conjugated polymer chain (polyene) dressed with highly polarizable molecule (aromatics) as side groups. Simply stated, the polyene chain would be a normal metal with a single mobile electron per C-H molecular unit; electrons on separate units would be paired by interacting with the exciton field on the polarizable side groups."

Actually, I think this part is perfectly true - William A. Little suggested this way to achieve high-temperature superconductivity back in the 1960s. The science fiction part is just the description, later on in Grant's article, of how Little's dream is actually achieved.

Okay, enough science fiction! Time for some real science! Quantum gravity, that is. (Stop snickering, you skeptics....)

6) Laurent Freidel and Kirill Krasnov, Spin foam models and the classical action principle, preprint available as hep-th/9807092.

I described the spin foam approach to quantum gravity in "week113". But let me remind you how the basic idea goes. A good way to get a handle on this idea is by analogy with Feynman diagrams. In ordinary quantum field theory there is a Hilbert space of states called "Fock space". This space has a basis of states in which there are a specific number of particles at specific positions. We can visualize such a state simply by imagining a bunch of points in space, with labels to say which particles are which kinds: electrons, quarks, and so on. One of the main jobs of quantum field theory is to let us compute the amplitude for one such state to evolve into another as time passes. Feynman showed that we can do it by computing a sum over graphs in spacetime. These graphs are called Feynman diagrams, and they represent "histories". For example,

\u       e/
 \       /
  \__W__/
  /     \
 /       \
/d      ν\

Now, where do these rules for computing Feynman diagram amplitudes come from? They are not simply postulated. They come from perturbation theory. There is a general abstract formula for computing amplitudes in quantum field theory, but it's not so easy to use this formula in concrete calculations, except for certain very simple field theories called "free theories". These theories describe particles that don't interact at all. They are mathematically tractable but physically uninteresting. Uninteresting, that is, except as a starting-point for studying the theories we are interested in - the so-called "interacting theories".

The trick is to think of an interacting theory as containing parameters, called "coupling constants", which when set to zero make it reduce to a free theory. Then we can try to expand the transition amplitudes we want to know as a Taylor series in these parameters. As usual, computing the coefficients of the Taylor series only requires us to to compute a bunch of derivatives. And we can compute these derivatives using the free theory! Typically, computing the nth derivative of some transition amplitude gives us a bunch of integrals which correspond to Feynman diagrams with n vertices.

By the way, this means you have to take the particles you see in Feynman diagrams with a grain of salt. They don't arise purely from the mathematics of the interacting theory. They arise when we approximate that theory by a free theory. This is not an idle point, because we can take the same interacting theory and approximate it by different free theories. Depending on what free theory we use, we may say different things about which particles our interacting theory describes! In condensed matter physics, people sometimes use the term "quasiparticle" to describe a particle that appears in a free theory that happens to be handy for some problem or other. For example, it can be helpful to describe vibrations in a crystal using "phonons", or waves of tilted electron spins using "spinons". Condensed matter theorists rarely worry about whether these particles "really exist". The question of whether they "really exist" is less interesting than the question of whether the particular free theory they inhabit provides a good approximation for dealing with a certain problem. Particle physicists, too, have increasingly come to recognize that we shouldn't worry too much about which elementary particles "really exist".

But I digress! My point was simply to say that Feynman diagrams arise from approximating interacting theories by free theories. The details are complicated and in most cases nobody has ever succeeded in making them mathematically rigorous, but I don't want to go into that here. Instead, I want to turn to spin foams.

Everything I said about Feynman diagrams has an analogy in this approach to quantum gravity. The big difference is that ordinary "free theories" are formulated on a spacetime with a fixed metric - usually Minkowski spacetime, with its usual flat metric. Attempts to approximate quantum gravity by this sort of free theory failed dismally. Perhaps the fundamental reason is that general relativity doesn't presume that spacetime has a fixed metric - au contraire, it's a theory in which the metric is the main variable!

So the idea of Freidel and Krasnov is to approximate quantum graivty with a very different sort of "free theory", one in which the metric is a variable. The theory they use is called "BF theory". I said a lot about BF theory in "week36", but here the main point is simply that it's a topological quantum field theory, or TQFT. A TQFT is a quantum field theory that does not presume a fixed metric, but of a very simple sort, because it has no local degrees of freedom. I very much like the idea that a TQFT might serve as a novel sort of "free theory" for the purposes of studying quantum gravity.

Everything that Freidel and Krasnov do is reminscent of familiar quantum field theory, but also very different, because their starting-point is BF theory rather than a free theory of a traditional sort. For example, just as ordinary quantum field theory starts out with Fock space, in the spin network approach to quantum gravity we start with a nice simple Hilbert space of states. But this space has a basis consisting, not of collections of 0-dimensional particles sitting in space at specified positions, but of 1-dimensional "spin networks" sitting in space. (For more on spin networks, see "week55" and "week110".) And instead of using 1-dimensional Feynman diagrams to compute transition amplitudes, the idea is now to use 2-dimensional gadgets called "spin foams". The amplitudes for spin foams are easy to compute in BF theory, because there are a lot of explicit formulas using the so-called "Kauffman bracket", which is an easily computable invariant of spin networks. So then the trick is to use this technology to compute spin foam amplitudes for quantum gravity.

Now, I shouldn't give you the wrong impression here. There are lots of serious problems and really basic open questions in this work, and the whole thing could turn out to be fatally flawed somehow. Nonetheless, something seems right about it, so I find it very interesting.

Anyway, on to some other papers. I'm afraid I don't have enough energy for detailed descriptions, because I'm busy moving into a new house, so I'll basically just point you at them....

7) Abhay Ashtekar, Alejandro Corichi and Jose A. Zapata, Quantum theory of geometry III: Non-commutativity of Riemannian structures, preprint available as gr-qc/9806041.

This is the long-awaited third part of a series giving a mathematically rigorous formalism for interpreting spin network states as "quantum 3-geometries", that is, quantum states describing the metric on 3-dimensional space together with its extrinsic curvature (as it sits inside 4-dimensional spacetime). Here's the abstract:

"The basic framework for a systematic construction of a quantum theory of Riemannian geometry was introduced recently. The quantum versions of Riemannian structures - such as triad and area operators - exhibit a non-commutativity. At first sight, this feature is surprising because it implies that the framework does not admit a triad representation. To better understand this property and to reconcile it with intuition, we analyze its origin in detail. In particular, a careful study of the underlying phase space is made and the feature is traced back to the classical theory; there is no anomaly associated with quantization. We also indicate why the uncertainties associated with this non-commutativity become negligible in the semi-classical regime."

In case you're wondering, the "triad" field is more or less what mathematicians would call a "frame field" or "soldering form" - and it's the same as the "B" field in BF theory. It encodes the information about the metric in Ashtekar's formulation to general relativity.

Moving on to matters n-categorical, we have:

8) Andre Hirschowitz, Carlos Simpson, Descente pour les n-champs (Descent for n-stacks), approximately 240 pages, in French, preprint available as math.AG/9807049. math.AG/9807049.

Apparently this provides a theory of "n-stacks", which are the n-categorical generalization of sheaves. Ever since Grothendieck's 600-page letter to Quillen (see "week35"), this has been the holy grail of n-category theory. Unfortunately I haven't mustered sufficient courage to force my way through 240 pages of French, so I don't really know the details!

For the following two n-category papers, exploring some themes close to my heart, I'll just quote the abstracts:

9) Michael Batanin, Computads for finitary monads on globular sets, preprint available at http://www.ics.mq.edu.au/~mbatanin/papers.html

"This work arose as a reflection on the foundation of higher dimensional category theory. One of the main ingredients of any proposed definition of weak n-category is the shape of diagrams (pasting scheme) we accept to be composable. In a globular approach [due to Batanin] each k-cell has a source and target (k-1)-cell. In the opetopic approach of Baez and Dolan and the multitopic approach of Hermida, Makkai and Power each k-cell has a unique (k-1)-cell as target and a whole (k-1)-dimensional pasting diagram as source. In the theory of strict n-categories both source and target may be a general pasting diagram.

The globular approach being the simplest one seems too restrictive to describe the combinatorics of higher dimensional compositions. Yet, we argue that this is a false impression. Moreover, we prove that this approach is a basic one from which the other type of composable diagrams may be derived. One theorem proved here asserts that the category of algebras of a finitary monad on the category of n-globular sets is equivalent to the category of algebras of an appropriate monad on the special category (of computads) constructed from the data of the original monad. In the case of the monad derived from the universal contractible operad this result may be interpreted as the equivalence of the definitions of weak n-categories (in the sense of Batanin) based on the `globular' and general pasting diagrams. It may be also considered as the first step toward the proof of equivalence of the different definitions of weak n-category.

We also develop a general theory of computads and investigate some properties of the category of generalized computads. It turned out, that in a good situation this category is a topos (and even a presheaf topos under some not very restrictive conditions, the property firstly observed by S. Schanuel and reproved by A. Carboni and P. Johnstone for 2-computads in the sense of Street)."

10) Tom Leinster, Structures in higher-dimensional category theory, preprint available at http://www.dpmms.cam.ac.uk/~leinster

"This is an exposition of some of the constructions which have arisen in higher-dimensional category theory. We start with a review of the general theory of operads and multicategories. Using this we give an account of Batanin's definition of n-category; we also give an informal definition in pictures. Next we discuss Gray-categories and their place in coherence problems. Finally, we present various constructions relevant to the opetopic definitions of n-category.

New material includes a suggestion for a definition of lax cubical n-category; a characterization of small Gray-categories as the small substructures of 2-Cat; a conjecture on coherence theorems in higher dimensions; a construction of the category of trees and, more generally, of n-pasting diagrams; and an analogue of the Baez-Dolan slicing process in the general theory of operads."

Okay - now for something completely different. In "week122" I said how Kreimer and Connes have teamed up to write a paper relating Hopf algebras, renormalization, and noncommutative geometry. Now it's out:

11) Alain Connes and Dirk Kreimer, Hopf algebras, renormalization and noncommutative geometry, preprint available as hep-th/9808042.

Also, here's an introduction to Kreimer's work:

12) Dirk Kreimer, How useful can knot and number theory be for loop calculations?, Talk given at the workshop "Loops and Legs in Gauge Theories", preprint available as hep-th/9807125.

Switching over to homotopy theory and its offshoots... when I visited Dan Christensen at Johns Hopkins this spring, he introduced me to all the homotopy theorists there, and Jack Morava gave me a paper which really indicates the extent to which new-fangled "quantum topology" has interbred with good old- fashioned homotopy theory:

12) Jack Morava, Quantum generalized cohomology, preprint available as math.QA/9807058 and http://hopf.math.purdue.edu/

Again, I'll just quote the abstract rather than venturing my own summary:

"We construct a ring structure on complex cobordism tensored with the rationals, which is related to the usual ring structure as quantum cohomology is related to ordinary cohomology. The resulting object defines a generalized two- dimensional topological field theory taking values in a category of spectra."

Finally, Morava has a student who gave me an interesting paper on operads and moduli spaces:

13) Satyan L. Devadoss, Tessellations of moduli spaces and the mosaic operad, preprint available as math.QA/9807010.

"We construct a new (cyclic) operad of `mosaics' defined by polygons with marked diagonals. Its underlying (aspherical) spaces are the sets M_0,n(R) of real points of the moduli space of punctured Riemann spheres, which are naturally tiled by Stasheff associahedra. We (combinatorially) describe them as iterated blow-ups and show that their fundamental groups form an operad with similarities to the operad of braid groups."

© 1998 John Baez
baez@math.removethis.ucr.andthis.edu