December 12, 1994

This Week's Finds in Mathematical Physics (Week 46)

John Baez

I will be on sabbatical during the first half of 1995. I'll be roaming hither and thither, and also trying to get some work done on n-categories, quantum gravity and such, so this will be the last "This Week's Finds" for a while. I have also taken a break from being a co-moderator of sci.physics.research.

So, let me sign off with a roundup of diverse and sundry things! I'm afraid I'll be pretty terse about describing some of them. First for some news of general interest, then a little update on Seiberg-Witten theory, then some neat stuff on TQFTs, n-categories, quantum gravity and all that, and then various other goodies....

1) The speed of write, by Gary Stix, Scientific American, Dec. 1994, 106-111.

Goodbye, Gutenberg, by Jacques Leslie, WiReD 2.10, Oct. 1994, available via WWW as /electrosphere/ejournals.html

Among other things, the above articles show that Paul Ginsparg is starting to get the popular recognition he deserves for starting up hep-th. In case anyone out there doesn't know yet, hep-th is the "high-energy physics - theoretical" preprint archive, which revolutionized communications within this field by making preprints easily available world-wide, thus rendering many (but not all) aspects of traditional journals obsolete. The idea was so good it quickly spread to other subjects. Within physics it went like this:

   High Energy Physics - Theory (hep-th), started 8/91 
   High Energy Physics - Lattice (hep-lat), started 2/92 
   High Energy Physics - Phenomenology (hep-ph), started 3/92 
   Astrophysics (astro-ph), started 4/92 
   Condensed Matter Theory (cond-mat), started 4/92 
   General Relativity & Quantum Cosmology (gr-qc), started 7/92 
   Nuclear Theory (nucl-th), started 10/92 
   Chemical Physics (chem-ph), started 3/94 
   High Energy Physics - Experiment (hep-ex), started 4/94 
   Accelerator Physics (acc-phys), started 11/94 
   Nuclear Experiment (nucl-ex), started 11/94 
   Materials Theory (mtrl-th), started 11/94 
   Superconductivity (supr-con), started 11/94 

Similar archives are sprouting up in mathematics (see below - but also note the existence of the American Mathematical Society preprint server, described later in this week's finds).

There are many ways to access these preprint archives, since Ginsparg has kept up very well with the times - indeed, so much better than I that I'm afraid to go into any details for fear of making a fool of myself. The dernier cri, I suppose, is to access the archives using the World-Wide Web, which is conveniently done by opening the document

If this makes no sense to you, my first and very urgent piece of advice is to learn about the World-Wide Web (WWW), Mosaic, and the like, since they are wonderful and very simple to use! In the meantime, however, you can simply send mail to various addresses with subject header


and no message body, in order to get information. Some addresses are:               (accelerator physics)               (astrophysics)                (chemical physics)               (condensed matter)               (functional analysis)                  (general relativity / quantum cosmology)            (computational and lattice physics)                 (high energy physics phenomenological)                 (high energy physics formal)                 (high energy physics experimental)                (nuclear theory)                (nuclear experiment)                (materials theory)               (superconductivity) (algebraic geometry)                   (automorphic forms)                      (complex dynamics & hyperbolic geometry)                      (differential geometry & global analysis)               (non-linear systems)                 (computation and language)                 (e-mail address database)

One might also want to check out the:

Directory of Electronic Journals, Newsletters, and Academic Discussion Lists: Send e-mail to at the Association of Research Libraries, +1 (202) 296-2296, fax +1 (202) 872 0884.

2) Monopoles and four-manifolds, by Edward Witten, preprint available as hep-th/9411102.

The genus of embedded surfaces in the projective plane, by P. B. Kronheimer and T. S. Mrowka, preprint number #19941128001, available from the AMS preprint server under subject 57 in the Mathematical Reviews Subject Classification Scheme.

I don't have anything interesting to say about these papers that wasn't in "week44" and "week45", but anyone interested in following the revolution in Donaldson theory initiated by Seiberg and Witten will have to read them.

Let me say a bit about how the AMS preprint server works. Assuming you are hip to the WWW, just go to

You will then see a menu, and you can click on "Mathematical Preprints", and then "AMS Preprint Server", where preprints are classified by subject. Alternatively, click on "New Items This Month (all Subjects)".

On a related note, you can also get some AMS stuff using telnet by doing


and using


as login and password. This doesn't seem to get you to the preprints, though. For gopher fans,


has roughly similar effects.

3) Spin networks in quantum gravity, by Carlo Rovelli and Lee Smolin, to appear.

This paper is closely related to the earlier one in which Rovelli and Smolin argue that discreteness of area and volume arise naturally in the loop representation of quantum gravity, and also to my own paper on spin networks. (See "week43" for more on these, and a brief intro to spin networks.) Basically, while my paper shows that spin networks give a kind of basis of states for gauge theories with arbitrary (compact, connected) gauge group, in this paper Rovelli and Smolin concentrate on the gauge groups SL(2,C) and SU(2), which are relevant to quantum gravity, and work out a lot of aspects particular to this case, in more of a physicist's style. This makes spin networks into a practical computational tool in quantum gravity, used to great effect in the paper on the discreteness of area and volume.

4) Recent mathematical developments in quantum general relativity, by Abhay Ashtekar, 14 pages in TeX format available as gr-qc/9411055 (discussed in "week37").

Coherent state transforms for spaces of connections, by Abhay Ashtekar, Jerzy Lewandowski, Donald Marolf, Jose Mourao and Thomas Thiemann, 38 pages in LaTeX format, available as gr-qc/9412014 (discussed in "week43")

These are two papers on the loop representation of quantum gravity which I talked about in earlier "finds", and are out now. The former is a nice review of recent mathematically rigorous work; the latter takes a tremendous step towards handling the infamous "reality conditions" problem.

5) Differential geometry on the space of connections via graphs and projective limits, by Abhay Ashtekar and Jerzy Lewandowski, 54 pages in LaTeX format, available as hep-th/9412073

I've spoken quite a bit about doing rigorous functional integration in gauge theory using ideas from the loop representation; this paper treats functional derivatives and other things that are more differential than integral in nature. This is crucial in quantum gravity because the main remaining mystery, the Wheeler-DeWitt equation or Hamiltonian constraint, involves a differential operator on the space of connections. (For a wee bit more, try "week11" or "week43", where the Hamiltonian constraint is simply written as G_{00} = 0.)

Let me quote their abstract:

In a quantum mechanical treatment of gauge theories (including general relativity), one is led to consider a certain completion, A, of the space of gauge equivalent connections. This space serves as the quantum configuration space, or, as the space of all Euclidean histories over which one must integrate in the quantum theory. A is a very large space and serves as a ``universal home'' for measures in theories in which the Wilson loop observables are well-defined. In this paper, A is considered as the projective limit of a projective family of compact Hausdorff manifolds, labelled by graphs (which can be regarded as ``floating lattices'' in the physics terminology). Using this characterization, differential geometry is developed through algebraic methods. In particular, we are able to introduce the following notions on A: differential forms, exterior derivatives, volume forms, vector fields and Lie brackets between them, divergence of a vector field with respect to a volume form, Laplacians and associated heat kernels and heat kernel measures. Thus, although A is very large, it is small enough to be mathematically interesting and physically useful. A key feature of this approach is that it does not require a background metric. The geometrical framework is therefore well-suited for diffeomorphism invariant theories such as quantum general relativity.

6) Edge states in gravity and black hole physics, by A. P. Balachandran, L. Chandar, Arshad Momen, 22 pages in RevTeX format, available as gr-qc/9412019.

Ever since it started seeming that black holes have an entropy closely related to (and often proportional to) the area of their event horizons, many physicists have sought a better understanding of this entropy. In many ways, the nicest sort of explanation would say that the entropy was due to degrees of freedom living on the event horizon. A concrete calculation along these lines was recently made by Steve Carlip (see "week41") in the context of 2+1-dimensional gravity. The mechanism is mathematically very similar to what happens in (a widely popular theory of) the fractional quantum Hall effect! In both cases, Chern-Simons theory on a 3d manifold with boundary gives rise to an interesting field theory on the boundary, or "edge". The above paper clarifies this, especially for those of us who don't understand the fractional quantum Hall effect too well. Let me quote:

Abstract: We show in the context of Einstein gravity that the removal of a spatial region leads to the appearance of an infinite set of observables and their associated edge states localized at its boundary. Such a boundary occurs in certain approaches to the physics of black holes like the one based on the membrane paradigm. The edge states can contribute to black hole entropy in these models. A "complementarity principle" is also shown to emerge whereby certain "edge" observables are accessible only to certain observers. The physical significance of edge observables and their states is discussed using their similarities to the corresponding quantities in the quantum Hall effect. The coupling of the edge states to the bulk gravitational field is demonstrated in the context of (2+1) dimensional gravity.

I can't resist adding that I have a personal stake in the notion that a lot of interesting things about quantum gravity will only show up when we consider it on manifolds with boundary, including the area-entropy relations. The loop representation of quantum gravity has a lot to do with knots and links, but on a manifold with boundary it has a lot to do with tangles, which can contain arcs that begin and end at the boundary. I wrote a paper on this a while back:

7) Quantum gravity and the algebra of tangles, by John Baez, Jour. Class. Quantum Grav. 10 (1993), 673 - 694.

and I'll be coming out with another in a while, co-authored with Javier Muniain and Dardo Piriz. The importance of manifolds with boundary for cutting-and-pasting constructions is also well-known in the theory of "extended" TQFTs (topological quantum field theories). These cutting and pasting operations should allow one to describe extended TQFTs in n dimensions purely algebraically using "higher-dimensional algebra". So part of the plan here is to understand better the relation between quantum gravity, TQFTs, and higher-dimensional algebra. Along these lines, a very interesting new paper has come out:

8) On algebraic structures implicit in topological quantum field theories, by Louis Crane and David Yetter, 13 pages in LaTeX format available as hep-th/9412025, figures available by request.

This makes more precise some of Louis Crane's ideas on "categorification". Nice TQFTs in 3 dimensions have a lot to do with Hopf algebras (like quantum groups), or alternatively, their categories of representations, which are certain braided monoidal categories. In this paper it's shown that nice TQFTs in 4 dimensions have a lot to do with Hopf categories, or alternatively, their categories of representations, which are certain braided monoidal 2-categories.

9) On the definition of 2-category of 2-knots, by V. M. Kharlamov and V. G. Turaev, preprint.

This preprint, which I obtained through my network of spies, seems to be implicitly claiming that the work of Fischer describing 2d surfaces in R^4 via on braided monoidal 2-categories (see "week12") is a bit wrong, but they don't come out and say quite what if anything is really wrong.

10) Non-involutory Hopf algebras and 3-manifold invariants, by Greg Kuperberg, preprint #19941128002, available from the AMS preprint server under subject 57 or 16 in the Mathematical Reviews Subject Classification Scheme.

I noted the existence of a draft of this paper, and related work, in "week38". Let me quote:

Abstract: We present a definition of an invariant #(M,H), defined for every finite-dimensional Hopf algebra (or Hopf superalgebra or Hopf object) $H$ and for every closed, framed 3-manifold M. When H is a quantized universal envloping algebra, #(M,H) is closely related to well-known quantum link invariants such as the HOMFLY polynomial, but it is not a topological quantum field theory.

Okay, now for some miscellaneous interesting things...

11) If Hamilton had prevailed: quaternions in physics, by J. Lambek, McGill University preprint, Nov. 1994.

Lambek is mainly known for work in category theory, but he has a strong side-interest in mathematical physics. This paper is, first of all, a "nostalgic account of how certain key results in modern theoretical physics (prior to World War II) can be expressed concisely in the labguage of quaternions, thus suggesting how they might have been discovered if Hamilton's views had prevailed." But there is a very interesting new thing, too: a way in which the group SU(3) X SU(2) X U(1) shows up naturally when considering Dirac's equation a la quaternions. This group is the gauge group of the Standard Model! Lambek modestly says that there does not appear to be any significance to this coincidence... but it would be nice, wouldn't it?

12) The life and times of Emmy Noether; contributions of E. Noether to particle physics, by Nina Byers, 32 pages in RevTeX format, available as hep-th/9411110.

Reminiscences about many pitfalls and some successes of QFT within the last three decades, by B. Schroer, 52 pages, 'shar'-shell-archiv, consisting of 5 files, available as hep-th/9410085.

My encounters - as a physicist - with mathematics, R. Jackiw, 13 pages in LaTeX format, available as hep-th/9410151.

These are some interesting historical/biographical pieces.

13) Speedup in quantum computation is associated with attenuation of processing probability, by Karl Svozil, available as hep-th/9412046.

The subject of quantum computation has become more lively recently. I haven't had time to look at this paper, but quoting the abstract:

Quantum coherence allows the computation of an arbitrary number of distinct computational paths in parallel. Based on quantum parallelism it has been conjectured that exponential or even larger speedups of computations are possible. Here it is shown that, although in principle correct, any speedup is accompanied by an associated attenuation of detection rates. Thus, on the average, no effective speedup is obtained relative to classical (nondeterministic) devices.

© 1994 John Baez