October 23, 1995

This Week's Finds in Mathematical Physics (Week 67)

John Baez

I'm pretty darn busy now, so the forthcoming Weeks will probably be pretty hastily written. This time I'll mainly write about quantum gravity.

1) Margaret Wertheim, Pythagoras' Trousers: God, Physics, and the Gender Wars, Times Books/Random House, New York, 1995.

I enjoyed this book, despite or perhaps because of the fact that it may annoy lots of physicists. It notes that, starting with Pythagoras, theoretical physics has always had a crypto-religious aspect. With Pythagoras it was obvious; he seems to have been the leader of a special sort of religious cult. With people like Kepler, Newton and Einstein it is only slightly less obvious. The operative mythology in every case is that of the mage. Think of Einstein, stereotypically with long white hair (though most of best work was actually done before his hair got white), a powerful but benign figure devoted to finding out "the thoughts of God". The mage, of course, is typically male, and Wertheim argues that this makes it harder for women to become physicists. (A lot of the same comments would apply to mathematics.) It is not a very scholarly book, but I wouldn't dismiss it.

2) Stephen W. Hawking, Virtual black holes, available as hep-th/9510029.

Hawking likes the "Euclidean path-integral approach" to quantum gravity. The word "Euclidean" is a horrible misnomer here, but it seems to have stuck. It should really read "Riemannian", the idea being to replace the Lorentzian metric on spacetime by one in which time is on the same footing as space. One thus attempts to compute answers to quantum gravity problems by integrating over all Riemannian metrics on some 4-manifold, possibly with some boundary conditions. Of course, this is tough - impossible so far - to make rigorous. But Hawking isn't scared; he also wants to sum over all 4-manifolds (possibly having a fixed boundary). Of course, to do this one needs to have some idea of what "all 4-manifolds" are. Lots of people like to consider wormholes, which means considering 4-manifolds that aren't simply connected. Here, however, Hawking argues against wormholes, and concentrates on simply-connected 4-manifolds. He writes: "Barring some pure mathematical details, it seems that the topology of simply connected four-manifolds can be essentially represented by gluing together three elementary units, which I call bubbles. The three elementary units are S2 × S2, CP2, and K3. The latter two have orientation reversed versions, -CP2 and -K3." S2 × S2 is just the product of the 2-dimensional sphere with itself, and he argues that this sort of bubble corresponds to a virtual black hole pair. He considers the effect on the Euclidean path integral when you have lots of these around (i.e., when you take the connected sum of S4 with lots of these). He argues that particles scattering off these lose quantum coherence, i.e., pure states turn to mixed states. And he argues that this effect is very small at low energies except for scalar fields, leading him to predict that we may never observe the Higgs particle! Yes, a real honest particle physics prediction from quantum gravity! As he notes, "unless quantum gravity can make contact with observation, it will become as academic as arguments about how many angels can dance on the head of a pin". I suspect he also realizes that he'll never get a Nobel prize unless he goes out on a limb like this. He also gives an argument for why the "θ angle" measuring CP violation by the strong force may be zero. This parameter sits in front of a term in the Standard Model Lagrangian; there seems to be no good reason for it to be zero, but measurements of the neutron electric dipole moment show that it has to be less than 10-9, according to the following book...

3) Kerson Huang, Quarks, Leptons, and Gauge Fields, World Scientific Publishing Co., Singapore, 1982.

Perhaps there are better bounds now, but this book is certainly one of the nicest introductions to the Standard Model, and if you want to learn about this "θ angle" stuff, it's a good place to start.

Anyway, rather than going further into the physics, let me say a bit about the "pure mathematical details". Here I got some help from Greg Kuperberg, Misha Verbitsky, and Zhenghan Wang (via Xiao-Song Lin, a topologist who is now here at Riverside). Needless to say, the mistakes are mine alone, and corrections and comments are welcome!

First of all, Hawking must be talking about homeomorphism classes of compact oriented simply-connected smooth 4-manifolds, rather than diffeomorphism classes, because if we take the connected sum of 9 copies of CP2 and one of -CP2, that has infinitely many different smooth structures. Why the physics depends only on the homeomorphism class is beyond me... maybe he is being rather optimistic. But let's follow suit and talk about homeomorphism classes. Folks consider two cases, depending on whether the intersection form on the second cohomology is even or odd. If our 4-manifold has an odd intersection form, Donaldson showed that it is an connected sum of copies of CP2 and -CP2. If its intersection form is even, we don't know yet, but if the "11/8 conjecture" is true, it must be a connected sum of K3's and S2 × S2's. Here I cannot resist adding that K3 is a 4-manifold whose intersection is E8 ⊕ E8 ⊕ H ⊕ H ⊕ H, where H is the 2×2 matrix

 0 1
 1 0

and E8 is the nondegenerate symmetric 8×8 matrix describing the inner products of vectors in the wonderful lattice, also called E8, which I discussed in "week65"! So E8 raises its ugly head yet again.... By the way, H is just the intersection form of S2 × S2, while the intersection form of CP2 is just the 1×1 matrix


Even if the 11/8 conjecture is not true, we could if necessary resort to Wall's theorem, which implies that any 4-manifold becomes homeomorphic - even diffeomorphic - to a connected sum of the 5 basic types of "bubbles" after one takes its connected sum with sufficiently many copies of S2 × S2. This suggests that if Euclidean path integral is dominated by the case where there are lots of virtual black holes around, Hawking's arguments could be correct at the level of diffeomorphism, rather than merely homeomorphism. Indeed, he says that "in the wormhole picture, one considered metrics that were multiply connected by wormholes. Thus one concentrated on metrics [I'd say topologies!] with large values of the first Betti number[....] However, in the quantum bubbles picture, one concentrates on spaces with large values of the second Betti number."

4) Ted Jacobson, Thermodynamics of spacetime: the Einstein equation of state, available as gr-qc/9504004.

Well, here's another paper on quantum gravity, also very good, which seems at first to directly contradict Hawking's paper. Actually, however, I think it's another piece in the puzzle. The idea here is to derive Einstein's equation from thermodynamics! More precisely, "The key idea is to demand that this relation hold for all the local Rindler causal horizons through each spacetime point, with [the change in heat] and [the temperature] interpreted as the energy flux and Unruh temperature seen by an accelerated observer just inside the horizon. This requires that gravitational lensing by matter energy distorts the causal structure of spacetime in just such a way that the Einstein equation holds". It's a very clever mix of classical and quantum (or semiclassical) arguments. It suggests that all sorts of quantum theories on the microscale could wind up yielding Einstein's equation on the macroscale.

5) Lee Smolin, The Bekenstein bound, topological quantum field theory and pluralistic quantum field theory, available as gr-qc/9508064.

This is a continued exploration of the themes of Smolin's earlier paper, reviewed in "week56" and "week57". Particularly interesting is the general notion of "pluralistic quantum field theory", in which different observers have different Hilbert spaces. This falls out naturally in the n-categorical approach pursued by Crane (see "week56"), which I am also busily studying.

6) Rodolfo Gambini, Octavio Obregon and Jorge Pullin, Towards a loop representation for quantum canonical supergravity, available as hep-th/9508036.

Some knot theorists and quantum group theorists had better take a look at this! This paper considers the analog of SU(2) Chern-Simons theory where you use the supergroup GSU(2), and perturbatively work out the skein relations of the associated link invariant (up to a certain low order in ħ). If someone understood the quantum supergroup "quantum GSU(2)", they could do this stuff nonperturbatively, and maybe get an interesting invariant of links and 3-manifolds, and make some physicists happy in the process.

7) Roh Suan Tung and Ted Jacobson, Spinor one-forms as gravitational potentials, available as gr-qc/9502037.

This paper writes out a new Lagrangian for general relativity, closely related to the action that gives general relativity in the Ashtekar variables. It's incredibly simple and beautiful! I am hoping that if I work on it someday, it will explain to me the mysterious relation between quantum gravity and spinor fields (see the end of "week60").

8) Joseph Polchinski and Edward Witten, Evidence for heterotic - type I string duality, available as hep-th/9510169.

I'm no string theorist, so I've been lagging vastly behind the new work on "dualities" that has revived interest in the subject. Roughly speaking, though, it seems folks have discovered a host of secret symmetries relating superficially different string theories... making them, in some deeper sense, the same. The heterotic and type I strings are two of the most famous string theories, so if they were really equivalent as this paper suggests, it would be very interesting.

© 1995 John Baez