July 31, 1999

This Week's Finds in Mathematical Physics (Week 135)

John Baez

Well, darn it, now I'm too busy running around to conferences to write This Week's Finds! First I went to Vancouver, then to Santa Barbara, and for almost a month now I've been in Portugal, bouncing between Lisbon and Coimbra. But let me try to catch up....

From June 16th to 19th, Steve Savitt and Steve Weinstein of the University of British Columbia held a workshop designed to get philosophers and physicists talking about the conceptual problems of quantum gravity:

1) Toward a New Understanding of Space, Time and Matter, workshop home page at http://axion.physics.ubc.ca/Workshop/

After a day of lectures by Chris Isham, John Earman, Lee Smolin and myself, we spent the rest of the workshop sitting around in a big room with a beautiful view of Vancouver Bay, discussing various issues in a fairly organized way. For example, Chris Isham led a discussion on "What is a quantum theory?" in which he got people to question the assumptions underlying quantum physics, and Simon Saunders led one on "Quantum gravity: physics, metaphysics or mathematics?" in which we pondered the scientific and sociological implications of the fact that work on quantum gravity is motivated more by desire for consistency, clarity and mathematical elegance than the need to fit new experimental data.

It's pretty clear that understanding quantum gravity will make us rethink some fundamental concepts - the question is, which ones? By the end of the conference, almost every basic belief or concept relevant to physics had been held up for careful scrutiny and found questionable. Space, time, causality, the real numbers, set theory - you name it! It was a bit unnerving - but it's good to do this sort of thing now and then, to prevent hardening of the mental arteries, and it's especially fun to do it with a big bunch of physicists and philosophers. However, I must admit that I left wanting nothing more than to do lots of grungy calculations in order to bring myself back down to earth - relatively speaking, of course.

I particularly enjoyed Chris Isham's talk about topos theory because it helped me understand one way that topos theory could be applied to quantum theory. I've tended to regard topoi as "too classical" for quantum theory, because while the internal logic of a topos is intuitionistic (the principle of exclude middle may fail), it's still not very quantum. For example, in a topos the operation "and" still distributes over "or", and vice versa, while failure of this sort of distributivity is a hallmark of quantum logic. If you don't know what I mean, try these books, in rough order of increasing difficulty:

2) David W. Cohen, An Introduction to Hilbert Space and Quantum Logic, Springer-Verlag, New York, 1989.

3) C. Piron, Foundations of Quantum Physics, W. A. Benjamin, Reading, Massachusetts, 1976.

4) C. A. Hooker, editor, The Logico-algebraic Approach to Quantum Mechanics, two volumes, D. Reidel, Boston, 1975-1979.

Perhaps even more importantly, topoi are Cartesian! What does this mean? Well, it means that we can define a "product" of any two objects in a topos. That is, given objects a and b, there's an object a x b equipped with morphisms

p: a x b → a


q: a x b → b

called "projections", satisfying the following property: given morphisms from some object c into a and b, say

f: c → a


g: c → b

there's a unique morphism f x g: c → a x b such that if we follow it by p we get f, and if we follow it by q we get g. This is just an abstraction of the properties of the usual Cartesian product of sets, which is why we call a category "Cartesian" if any pair of objects has a product.

Now, it's a fun exercise to check that in a Cartesian category, every object has a morphism

Δ: a → a x a

called the "diagonal", which when composed with either of the two projections from a x a to a gives the identity. For example, in the topos of sets, the diagonal morphism is given by

Δ(x) = (x,x)

We can think of the diagonal morphism as allowing "duplication of information". This is not generally possible in quantum mechanics:

5) William Wooters and Wocjciech Zurek, A single quantum cannot be cloned, Nature 299 (1982), 802-803.

The reason is that in the category of Hilbert spaces, the tensor product is not a product in the above sense! In particular, given a Hilbert space H, there isn't a natural diagonal operator

Δ: H → H tensor H

and there aren't even natural projection operators from H tensor H to H. As pointed out to me by James Dolan, this non-Cartesianness of the tensor product gives quantum theory much of its special flavor. Besides making it impossible to "clone a quantum", it's closely related to how quantum theory violates Bell's inequality, because it means we can't think of an arbitrary state of a two-part quantum system as built by tensoring states of both parts.

Anyway, this has made me feel for a while that topos theory isn't sufficiently "quantum" to be useful in understanding the peculiar special features of quantum physics. However, after Isham and I gave our talks, someone pointed out to me that one can think of a topological quantum field theory as a presheaf of Hilbert spaces over the category nCob whose morphisms are n-dimensional cobordisms. Now, presheaves over any category form a topos, so this means we should be able to think of a topological quantum field theory as a "Hilbert space object" in the topos of presheaves over nCob. From this point of view, the peculiar "quantumness" of topological quantum field theory comes from it being a Hilbert space object, while its peculiar "variability" - i.e., the fact that it assigns a different Hilbert space to each (n-1)-dimensional manifold representing space - comes from the fact that it's an object in a topos. (Topoi are known for being very good at handling things like "variable sets".) I'm not sure how useful this is, but it's worth keeping in mind.

While I'm talking about quantum logic, let me raise a puzzle concerning the Kochen-Specker theorem. Remember what this says: if you have a Hilbert space H with dimension but more than 2, there's no map F from self-adjoint operators on H to real numbers with the following properties:

a) For any self-adjoint operator A, F(A) lies in the spectrum of A,


b) For any continuous f: R → R, f(F(A)) = F(f(A)).

This means there's no sensible consistent way of thinking of all observables as simultaneously having values in a quantum system!

Okay, the puzzle is: what happens if the dimension of H equals 2? I don't actually know the answer, so I'd be glad to hear it if someone can figure it out!

By the way, I once wanted to do an undergraduate research project on mathematical physics with Kochen. He asked me if I knew the spectral theorem, I said "no", and he said that in that case there was no point in me trying to work with him. I spent the next summer reading Reed and Simon's book on Functional Analysis and learning lots of different versions of the spectral theorem. I shudder to think that perhaps this is why I spent years studying analysis before eventually drifting towards topology and algebra. But no: now that I think about it, I was already interested in analysis at the time, since I'd had a wonderful real analysis class with Robin Graham.

Okay, now let me say a bit about the next conference I went to. From June 22nd to 26th there was a conference on "Strong Gravitational Fields" at the Institute for Theoretical Physics at U. C. Santa Barbara. This finished up a wonderful semester-long program by Abhay Ashtekar, Gary Horowitz and Jim Isenberg:

6) Classical and Quantum Physics of Strong Gravitational Fields, program homepage with transparencies and audio files of talks at http://www.itp.ucsb.edu/~patrick/gravity99/

Like the whole program, the conference covered a wide range of topics related to gravity: string theory and loop quantum gravity, observational and computational black hole physics, and γ ray bursters. I can't summarize all this stuff; since I usually spend a lot of talking about quantum gravity here, let me say a bit about other things instead.

John Friedman gave an interesting talk on gravitational waves from unstable neutron stars. When a pulsar is young, like about 5000 years old, it typically rotates about its axis once every 16 milliseconds or so. A good example is N157B, a pulsar in the Large Magellanic Cloud. Using the current spindown rate one can extrapolate and guess that pulsars have about a 5-millisecond period at birth. It's interesting to think about what makes a newly formed neutron star slow down. Theorists have recently come up with a new possible mechanism: namely, a new sort of gravitational-wave-driven instability of relativistic stars that could force newly formed slow down to a 10-millisecond period. It's very clever: the basic idea is that if a star is rotating very fast, a rotational mode that rotates slower than the star will gravitationally radiate positive angular momentum, but such modes carry negative angular momentum, since they rotate slower than the star. If you think about it carefully, you'll see this means that gravitational radiation should tend to amplify such modes! I asked for a lowbrow analog of this mechanism and it turns out that a similar sort of thing is at work in the formation of water waves by the wind - with linear momentum taking the place of angular momentum. Anyway, it's not clear that this process really ever has a chance to happen, because it only works when the neutron star is not too hot and not too cold, but it's pretty cool.

Richard Price gave a nice talk on computer simulation of black hole collisions. Quantitatively understanding the gravitational radiation emitted in black hole and neutron star collisions is a big business these days - it's one of the NSF's "grand challenge" problems. The reason is that folks are spending a lot of money building gravitational wave detectors like LIGO:

7) LIGO project home page, http://www.ligo.caltech.edu/

8) Other gravitational wave detection projects, http://www.ligo.caltech.edu/LIGO_web/other_gw/gw_projects.html

and they need to know exactly what to look for. Now, head-on collisions are the easiest to understand, since one can simplify the calculation using axial symmetry. Unfortunately, it's not very likely that two black holes are going to crash into each other head-on. One really wants to understand what happens when two black holes spiral into each other. There are two extreme cases: the case of black holes of equal mass, and the case of a very light black hole of mass falling into a heavy one.

The latter case is 95% understood, since we can think of the light black hole as a "test particle" - ignoring its effect on the heavy one. The light one slowly spirals into the heavy one until it reaches the innermost stable orbit, and then falls in. We can use the theory of a relativistic test particle falling into a black hole to understand the early stages of this process, and use black hole perturbation theory to study the "ringdown" of the resulting single black hole in the late stages of the process. (By "ringdown" I mean the process whereby an oscillating black hole settles down while emitting gravitational radiation.) Even the intermediate stages are manageable, because the radiation of the small black hole doesn't have much effect on the big one.

By contrast, the case of two black holes of equal mass is less well understood. We can treat the early stages, where relativistic effects are small, using a post-Newtonian approximation, and again we can treat the late stages using black hole perturbation theory. But things get complicated in the intermediate stage, because the radiation of each hole greatly effects the other, and there is no real concept of "innermost stable orbit" in this case. To make matters worse, the intermediate stage of the process is exactly the one we really want to understand, because this is probably when most of the gravitational waves are emitted!

People have spent a lot of work trying to understand black hole collisions through number-crunching computer calculation, but it's not easy: when you get down to brass tacks, general relativity consists of some truly scary nonlinear partial differential equations. Current work is bedeviled by numerical instability and also the problem of simulating enough of a region of spacetime to understand the gravitational radiation being emitted. Fans of mathematical physics will also realize that gauge-fixing is a major problem. There is a lot of interest in simplifying the calculations through "black hole excision": anything going on inside the event horizon can't affect what happens outside, so if one can get the computer to find the horizon, one can forget about simulating what's going on inside! But nobody is very good at doing this yet... even using the simpler concept of "apparent horizon", which can be defined locally. So there is some serious work left to be done!

(For more details on both these talks, go to the conference website and look at the transparencies.)

I also had some interesting talks with people about black hole entropy, some of which concerned a new paper by Steve Carlip. I'm not really able to do justice to the details, but it seems important....

9) Steve Carlip, Entropy from conformal field theory at Killing horizons, preprint available at gr-qc/9906126.

Let me just quote the abstract:

On a manifold with boundary, the constraint algebra of general relativity may acquire a central extension, which can be computed using covariant phase space techniques. When the boundary is a (local) Killing horizon, a natural set of boundary conditions leads to a Virasoro subalgebra with a calculable central charge. Conformal field theory methods may then be used to determine the density of states at the boundary. I consider a number of cases - black holes, Rindler space, de Sitter space, Taub-NUT and Taub-Bolt spaces, and dilaton gravity - and show that the resulting density of states yields the expected Bekenstein-Hawking entropy. The statistical mechanics of black hole entropy may thus be fixed by symmetry arguments, independent of details of quantum gravity.

There was also a lot of talk about "isolated horizons", a concept that plays a fundamental role in certain treatments of black holes in loop quantum gravity:

10) Abhay Ashtekar, Christopher Beetle, and Stephen Fairhurst, Mechanics of isolated horizons, preprint available as gr-qc/9907068.

11) Jerzy Lewandowski, Spacetimes admitting isolated horizons, preprint available as gr-qc/9907058.

For more on isolated horizons try the references in "week128".

Finally, on a completely different note, I've recently seen some new papers related to the McKay correspondence - see "week65" if you don't know what that is! I haven't read them yet, but I just want to remind myself that I should, so I'll list them here:

12) John McKay, Semi-affine Coxeter-Dynkin graphs and $G \subseteq SU_2(C)$, preprint available as math.QA/9907089.

13) Igor Frenkel, Naihuan Jing and Weiqiang Wang, Vertex representations via finite groups and the McKay correspondence, preprint available as math.QA/9907166.

Quantum vertex representations via finite groups and the McKay correspondence, preprint available as math.QA/9907175.

Next time I want to talk about the big category theory conference in honor of MacLane's 90th birthday! Then I'll be pretty much caught up on the conferences....

Robert Israel's answer to my puzzle about the Kochen-Specker theorem:

It's not true in dimension 2.  Note that for a self-adjoint 
2x2 matrix A, any f(A) is of the form a A + b I for some 
real scalars a and b (this is easy to see if you diagonalize 
A).  The self-adjoint matrices that are not multiples of I 
split into equivalence classes, where A and B are equivalent 
if B = a A + b I for some scalars a, b (a <> 0).  Pick a 
representative A from each equivalence class, choose F(A) 
as one of the eigenvalues of A, and then F(a A + b I) = 
a F(A) + b.  Of course, F(b I) = b.  Then F satisfies the 
two conditions.

The reason this doesn't work in higher dimensions is that 
in higher dimensions you can have two self-adjoint matrices 
A and B which don't commute,  F(A) = G(B) for some functions 
F and G, and F(A) is not a multiple of I.

Robert Israel                               israel@math.ubc.ca
Department of Mathematics       http://www.math.ubc.ca/~israel 
University of British Columbia            
Vancouver, BC, Canada V6T 1Z2

© 1999 John Baez