From: baez@math.ucr.edu (John Baez) Newsgroups: sci.physics.research,sci.math,sci.physics Subject: This Week's Finds in Mathematical Physics (Week 67) Date: 24 Oct 1995 07:00:24 GMT 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. 2) Stephen W. Hawking, Virtual black holes, preprint available as hep-th/510029. 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 S^2 x S^2, CP^2, and K3. The latter two have orientation reversed versions, -CP^2 and -K3." S^2 x S^2 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 S^4 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 "theta 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. ISBN 9971-950-03-0. 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 "theta 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 CP^2 and one of -CP^2, 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 CP^2 and -CP^2. 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 S^2 x S^2's. Here I cannot resist adding that K3 is a 4-manifold whose intersection is E_8 + E_8 + H + H + H, where H is the 2x2 matrix 0 1 1 0 and E_8 is the nondegenerate symmetric 8x8 matrix describing the inner products of vectors in the wonderful lattice, also called E_8, which I discussed in "week65"! So E_8 raises its ugly head yet again.... By the way, H is just the intersection form of S^2 x S^2, while the intersection form of CP^2 is just the 1x1 matrix 1. 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 S^2 x S^2. 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." [deletia - djr] ============================================================================== From: drm@math.duke.edu (David R. Morrison) Newsgroups: sci.math.research,sci.physics.research Subject: Re: 4-manifolds / Hawking's new paper Date: 16 Oct 1995 15:15:37 GMT In article <45phtr$iup@galaxy.ucr.edu>, john baez wrote: > >In Hawking's new paper, "Virtual black holes," he writes: > >Barring some pure mathematical details, it seems that the topology of >simply connected four manifolds can be essentially represented by >glueing together three elementary units, which I shall call bubbles. >The three elementary units are S^2 x S^2, CP^2 and K3. The latter two >have orientation reversed versions [...]. Thus there are five building >blocks for simply connected four manifolds. [stuff deleted] >Can someone tell me what "pure mathematical details" he is sweeping >under the rug here, and what the precise result is? > > From Freedman's theorem (topological Poincare conjecture in dimension 4) and properties of the Rohlin invariant, it follows that the topology of a compact simply-connected smooth 4-manifold is determined by the "parity" of the bilinear form on H^2 (i.e., whether (h \cup h) [X] represents only even integers, or both even and odd integers), and by the Euler characteristic \chi and the signature \sigma. The known restrictions on these are that if the bilinear form is even, then \sigma is congruent to 0 mod 16. (For a space which is not necessarily a smooth manifold, \sigma only has to be congruent to 0 mod 8.) There is a conjecture known as the 11/8 conjecture which asserts that the ratio |\chi| / |\sigma| must be at least 11/8 in the even case (if \sigma is not 0). If it is true, then all topological types of smooth 4-manifolds can indeed be built up out of these simple pieces as Hawking asserts. However, to the best of my knowledge no one has proved this conjecture, although the people tried hard with the powerful new tools for studying 4-manfolds (the ``Seiberg-Witten invariants'') which were introduced last fall. One other warning about this result: this classifies the *topological* isomorphism class of a smooth manifold, but does not classify the *diffeomorphism* (or `smooth') class. As Donaldson showed, there can be many distinct differentiable manifolds of the same topological type. -- Dave Morrison ============================================================================== From: "Zhenghan Wang" Date: Fri, 27 Oct 1995 10:16:29 -0400 To: rusin@math.niu.edu, zwang@math.lsa.umich.edu Subject: Re: differentiable structures. Hi, sorry for replying late. I was out of town for two days. I think you can find this in R. Friedman and J. Morgan's paper in 1988 JDG, or better their book. Basically CP^{2}# 9(-CP^{2}) is a minimal elliptic surface E(1), then any p,q logarithm trnasforma gives different smoont mfds but same top mfd. regards zhenghan wang