August 10, 1994

This Week's Finds in Mathematical Physics (Week 37)

John Baez

Mainly this week I have various bits of news to report from the 7th Marcel Grossman Meeting on general relativity. It was big and had lots of talks. Bekenstein gave a nice review talk on entropy/area relations for black holes, and Strominger gave a talk in which he proposed a solution to the information loss puzzle for black holes. (Recall that if one believes, as most people seem to believe, that black holes radiate away all their mass in the form of completely random Hawking radiation, then there's a question about where the information has gone that you threw into the black hole in the form of, say, old issues of Phys. Rev. Lett.. Some people think the information goes into a new "baby universe" formed at the heart of the black hole - see "week31" for more. The information would still, of course, be gone from our point of view in this picture. Strominger proposed a set up in which one had a quantum theory of gravity with annihilation and creation operators for baby universes, and proposed that the universe (the "metauniverse"?) was in a coherent state, that is, an eigenstate of the annihilation operator for baby universes. This would apparently allow handle the problem, though right now I can't remember the details.) There were also lots of talks on the interferometric detection of gravitational radiation, other general relativity experiments, cosmology, etc.. But I'll just try to describe two talks in some detail here.

1) L. Lindblom, Superfluid hydrodynamics and the stability of rotating neutron stars, talk at MG7 meeting, Monday July 5, 1994, Stanford University.

Being fond of knots, tangles, and such, I have always liked knowing that in superfluids, vorticity (the curl of the velocity vector field) tends to be confined in "flux tubes", each containing an angular momentum that's an integral multiple of Planck's constant, and that similarly, in type II superconductors, magnetic fields are confined to magnetic flux tubes. And I was even more happy to find out that the cores of neutron stars are expected to be made of neutronium that is both superfluid and superconductive, and contain lots of flux tubes of both types. In this talk, which was really about a derivation of detailed equations of state for neutron stars, Lindblom began by saying that the maximum rotation rate of a rotating neutron star is due to some sort of "gravitational radiation instability due to internal fluid dissipation". I didn't quite understand the details of that, which weren't explained, but it motivated him to study the viscosity in neutron star cores, which are superfluid if they are cool enough (less than a billion degrees Kelvin). There are some protons and electrons mixed in with the neutrons in the core, and both the protons and neutrons go superfluid, but the electrons form a normal fluid. That means that there are actually two kinds of superfluid vortices - proton and neutron - in addition to the magnetic vortices. These vortices mainly line up along the axis of rotation, and their density is about 10^6 per square centimeter. Rather curiously, since the proton, neutron, and electron fluids are coupled due to β decay (and the reverse process), even the neutron vortices have electric currents associated to them and generate magnetic fields. This means that the electrons scatter off the neutron vortex cores as well as the proton vortex cores, which is one of the mechanisms that yields viscosity.

2) Abhay Ashtekar, Mathematical developments in quantum general relativity, a sampler, talk at MG7 meeting, Tuesday July 6, 1994, Stanford University.

This talk, in addition to reviewing what's been done so far on the "loop representation" of quantum gravity, presented two new developments that I found quite exciting, so I'd like to sketch what they are. The details will appear in future papers by Ashtekar and collaborators.

The two developments Ashtekar presented concerned mathematically rigorous treatments of the "reality conditions" in his approach to quantum gravity, and the "loop states" used by Rovelli and Smolin. First let me try to describe the issue of "reality conditions". As I described in "week7", one trick that's important in the loop representation is to use the "new variables" for general relativity introduced by Ashtekar (though Sen and Plebanski already had worked with similar ideas). In the older Palatini approach to general relativity, the idea was to view general relativity as something like a gauge theory with gauge group given by the Lorentz group, SO(3,1). But to do this one actually uses two different fields: a "frame field", also called a "tetrad", "vierbein" or "soldering form" depending on who you're talking to, and the gauge field itself, usually called a "Lorentz connection" or "SO(3,1) connection". Technically, the frame field is an isomorphism between the tangent bundle of spacetime and some other bundle having a fixed metric of signature +---, usually called the "internal space", and the Lorentz connection is a metric-preserving connection on the internal space.

The "new variables" trick is to use the fact that SO(3,1) has as a double cover the group SL(2,C) of two-by-two complex matrices with determinant one. (For people who've read previous posts of mine, I should add that the Lie algebra of SL(2,C) is called sl(2,C) and is the same as the complexification of the Lie algebra so(3), which allows one to introduce the new variables in a different but equivalent way, as I did in "week7".) Ignoring topological niceties for now, this lets one reformulate complex general relativity (that is, general relativity where the metric can be complex-valued) in terms of a complex-valued frame field and an SL(2,C) connection that is just the Lorentz connection in disguise. The latter is called either the "Sen connection", the "Ashtekar connection", or the "chiral spin connection" depending on who you're talking to. The advantage of this shows up when one tries to canonically quantize the theory in terms of initial data. (For a bit on this, try "week11".) Here we assume our 4-dimensional spacetime can be split up into "space" and "time", so that space is a 3-dimensional manifold, and we take as our canonically conjugate fields the restriction of the chiral spin connection to space, call it A, and something like the restriction of the complex frame field to a complex frame field E on space. (Restricting the complex frame field to one on space is a wee bit subtle, especially because one doesn't really want a frame field or "triad field", but really a "densitized cotriad field" - but let's not worry about this here. I explain this in terms even a mathematician can understand in my paper "string.tex", available by ftp along with all my "week" files as described below.) The point is, first, that the A and E fields are mathematically very analogous to the vector potential and electric field in electromagnetism - or really in SL(2,C) Yang-Mills theory - and secondly, that if you compute their Poisson brackets, you really do see that they're canonically conjugate. Third and best of all, the constraint equations in general relativity can be written down very simply in terms of A and E. Recall that in general relativity, 6 of Einstein's 10 equations act as constraints that the metric and its time derivative must satisfy at t = 0 in order to get a solution at later times. In quantum gravity, these constraints are a big technical problem one has to deal with, and the point of Ashtekar's new variables is precisely that the constraints simplify in terms of these variables. (There's more on these constraints in "week11".)

The price one has paid, however, is that one now seems to be talking about complex-valued general relativity, which isn't what one had started out being interested in. One needs to get back to reality, as it were - and this is the problem of the so-called "reality conditions". One approach is to write down extra constraints on the E field that say that it comes from a real frame field. These are a little messy. Ashtekar, however, has proposed another approach especially suited to the quantum version of the theory, and in his talk he filled in some of the crucial details.

Here, to save time, I will allow myself to become a bit more technical. In the quantum version of the theory one expects the space of wavefunctions to be something like L^2 functions on the space of connections modulo gauge transformations - actually this is the "kinematical state space" one gets before writing the constraints as operators and looking for wavefunctions annihilated by these constraints. The problem had always been that this space of L^2 functions is ill-defined, since there is no "Lebesgue measure" on the space of connections. This problem is addressed (it's premature to say "solved") by developing a theory of generalized measures on the space of connections and proving the existence of a canonical generalized measure that deserves the name "Lebesgue measure" if anything does. One can then define L^2 functions and work with them. For compact gauge groups, like SU(2), this was done by Ashtekar, Lewandowski and myself; see e.g. the papers "state.tex" and "conn.tex" available by ftp. In the case of SU(2), Wilson loops act as self-adjoint multiplication operators on the resulting L^2 space. But in quantum gravity we really want to use gauge group SL(2,C), which is not compact, and we want the adjoints of Wilson loop operators to reflect that fact that the SL(2,C) connection A in quantum gravity is really equal to Γ + iK, where Γ is the Levi-Civita connection on space, and K is the extrinsic curvature. Both Γ and K are real in the classical theory, so the adjoint of the quantum version of A should be Γ - iK, and this should reflect itself in the adjoints of Wilson loop operators.

The trick, it turns out, is to use some work of Hall which appeared in the Journal of Functional Analysis in 1994 (I don't have a precise reference on me). The point is that SL(2,C) is the complexification of SU(2), and can also be viewed as the cotangent bundle of SU(2). This allows one to copy a trick people use for the quantum mechanics of a point particle on R^n - a trick called the Bargmann-Segal-Fock representation. Recall that in the ordinary Schrodinger representation of a quantum particle on R^n, one takes as the space of states L^2(R^n). However, the phase space for a particle in R^n, which is the cotangent bundle of R^n, can be identified with C^n, and in the Bargmann representation one takes as the space of states HL^2(C^n), by which I mean the holomorphic functions on C^n that are in L^2 with respect to a Gaussian measure on C^n. In the Bargmann representation for a particle on the line, for example, the creation operator is represented simply as multiplication by the complex coordinate z, while the annihilation operator is d/dz. Similarly, there is an isomorphism between L^2(SU(2)) and a certain space HL^2(SL(2,C)). Using this, one can obtain an isomorphism between the space of L^2 functions on the space of SU(2) connections modulo gauge transformations, and the space of holomorphic L^2 functions on the space of SL(2,C) connections modulo gauge transformations. Applying this to the loop representation, Ashtekar has found a very natural way to take into account the fact that the chiral spin connection A is really Γ + iK, basically analogous to the fact that in the Bargmann multiplication by z is really q + ip (well, up to various factors of sqrt(2), signs and the like).

Well, that was pretty sketchy and probably not especially comprehensible to anyone who hasn't already worried about this issue a lot! In any event, let me turn to the other good news Ashtekar reported: the constuction of "loop states". Briefly put (I'm getting worn out), he and some collaborators have figured out how to rigorously construct generalized measures on the space of connections modulo gauge transformations, starting from invariants of links. This begins to provide an inverse to the "loop transform" (which is a construction going the other way).

© 1994 John Baez