August 20, 1996

This Week's Finds in Mathematical Physics (Week 87)

John Baez

Let me continue summarizing what happened during July at the Mathematical Problems of Quantum Gravity workshop in Vienna. The first two weeks concentrated on the foundations of the loop representation of quantum gravity; the next week was all about black holes!

Tuesday, July 16th - Ted Jacobson gave an overview of "Issues of Black Hole Thermodynamics". There is a lot to say about this subject and I won't try to repeat his marvelous talk here. Let me just mention a very interesting technical point he made. The original Bekenstein-Hawking formula for the entropy of a black hole is

S = A/(4 ħ G)

where A is the area of the event horizon, ħ is Planck's constant, and G is Newton's constant. One way to try to derive this is from the partition function of a quantum field theory involving gravity and other fields. Jacobson sketched a heuristic calculation along these lines. When you do this calculation it's natural to worry why the other fields, representing various forms of matter, don't seem to contribute to the answer above. Also, when we do quantum field theory, there is often a difference between the "bare" coupling constants we put into the theory and the "renormalized" coupling constants that are what the theory predicts we'll observe experimentally. So it's natural to worry about whether it's the bare or renormalized Newton's constant G that enters the above formula --- even though quantum gravity is so unlike most other quantum field theories that it's unclear that this worry makes sense, ultimately.

Anyway, the nice thing is that these two worries cancel each other out. In other words: yes, it's the renormalized Newton's constant G --- the physically measured one --- that enters the above formula. But at least to first order in ħ, the difference between the bare G and the renormalized G is precisely due to the interactions between gravity and the matter fields (including the self-interaction of the gravitational field). In other words, the matter fields really do contribute to the black hole entropy, but this contribution is absorbed into the definition of the renormalized G.

In the most extreme case, the bare 1/G is zero, and the renormalized 1/G is entirely due to interactions between matter and gravity. This is Andrei Sakharov's theory of "induced gravity". According to Jacobson, in this case all of the black hole entropy is "entanglement entropy" --- this being standard jargon for the way that two parts of a quantum system can each have entropy due to correlations, even though the whole system has zero entropy. Unfortunately my notes do not allow me to reconstruct the wonderful argument whereby he showed this. (See "week27" for a more detailed explanation of entanglement entropy.)

Wednesday July 17th - There was a talk on "Colombeau theory" by a mathematician whose name I unfortunately failed to catch. Colombeau theory is a theory that allows you to multiply distributions, just like they said in school that you weren't allowed to do. So if for example you want to square the Dirac delta function, you can do it in the context of Colombeau theory. There has been a certain amount of debate, however, on whether Colombeau theory allows you to this multiplication in a useful way. There were a lot of physicists at this talk who would be willing and eager to master Colombeau theory if it let one solve the physics problems they wanted to solve. However, after much discussion, it appears that they didn't buy it. I believe that at best Colombeau theory provides a useful framework for understanding the ambiguities one encounters when multiplying distributions.

I say "ambiguities" rather than "disasters" because while the square of the Dirac delta function has no sensible interpretation as a distribution, there are many cases, such as when you try to multiply the Dirac delta function and the Heaviside function, where you can interpret the result as a distribution in a variety of ways. These ambiguous cases are the ones of greatest interest in physics. A nice place to see this in quantum field theory is in

1) G. Scharf, Finite quantum electrodynamics: the causal approach, Springer-Verlag, Berlin, 1995.

If you want to learn about Colombeau theory you can try:

2) J. F. Colombeau, "Multiplication of Distributions: a Tool in Mathematics, Numerical Engineering, and Theoretical Physics," Lecture Notes in Mathematics 1532, Springer, Berlin, 1992.

Later that day I had nice conversation with Jerzy Lewandowski on the approach to the loop representation where one uses smooth, rather than analytic, loops. (See "week55" for more on this issue.)

Thursday, July 18th - Carlo Rovelli spoke on "Black Hole Entropy", reporting some work he did with Kirill Krasnov. They have a nice approach to computing the black hole entropy using the loop representation of quantum gravity. A common goal among quantum gravity folks is to recover the Bekenstein-Hawking formula from some full-fledged theory of quantum gravity --- the original derivation being a curious "semiclassical" hybrid of quantum and classical reasoning. In a statistical mechanical approach, entropy should be the logarithm of the number of microstates some system can have in a given macrostate. So one wants to count states somehow. Basically what Rovelli and Krasnov do is count the number of ways a surface can be pierced by a spin network so as to give it a certain area. (This uses the formula for the area operator I descrbed in "week86".) They get an entropy proportional to the area, but not with the same constant as in the Bekenstein-Hawking formula.

There were some hopes that taking matter fields into account might give the right constant. But since everyone had been to Ted Jacobson's talk, this led to much interesting wrangling over whether Rovelli and Krasnov were using the bare or renormalized Newton's constant G, and whether the concept of bare and renormalized G even makes sense, ultimately! Also, there are some extremely important puzzles about what the right way to count states is, in these loop representation computations.

For more, try:

3) Carlo Rovelli, Loop quantum gravity and black hole physics, preprint available as gr-qc/9608032.

Kirill Krasnov, The Bekenstein bound and non-perturbative quantum gravity, preprint available as gr-qc/9603025.

Kirill Krasnov, On statistical mechanics of gravitational systems, preprint available as gr-qc/9605047.

Friday, July 19th - Don Marolf spoke on "Black hole entropy in string theory". He attempted valiantly to describe the string-theoretic approach to computing black hole entropy to an audience only generally familiar with string theory. I will not try to summarize his talk, except to note that he mainly discussed the case of a black hole in 5 dimensions, which was really a "black string" in 6 dimensions --- a solution with translational symmetry in the 6th dimension, but where the extra 6th dimension is so tiny that ordinary 5-dimensional folks think they've just got a black hole. (By the way, even the 6-dimensional approach is really just a way of talking about a string theory that fundamentally lives in 10 dimensions. This stuff is not for the faint-hearted.)

Here are a few papers on this subject by Marolf and Horowitz:

4) Gary Horowitz, The origin of black hole entropy in string theory, preprint available as gr-qc/9604051.

Gary T. Horowitz and Donald Marolf, Counting states of black strings with traveling waves, preprint available as hep-th/9605224.

Gary T. Horowitz and Donald Marolf, Counting states of black strings with traveling waves II, preprint available as hep-th/9606113.

Monday, July 22nd - Kirill Krasnov spoke on "The Einstein-Maxwell Theory of Black Hole Entropy". This was a report on attempts to see how his calculations of the black entropy in the loop representation changed when he took the electromagnetic field into account. The calculations were very tentative, for certain technical reasons I won't go into here, but they made even clearer the importance of the issue of how one counts states when computing entropy in this approach.

Later, I had a nice conversation with Carlo Rovelli about my hopes for thinking of fermions (e.g., electrons) as the ends of wormholes in the loop representation of quantum gravity. We came up with a nice heuristic argument to get the right Fermi statistics for these wormhole ends. Hopefully we can make this all more precise at some later date.

Tuesday, July 23rd - Ted Jacobson gave informal talks on two subjects, the first of which was "Transplanckian puzzle: origin of outgoing black hole modes." This dealt with the puzzling fact that in the standard computation of Hawking radiation, the rather low-frequency radiation which leaves the hole is the incredibly redshifted offspring of high-frequency modes which swung past the horizon shortly after the hole's formation --- modes whose wavelength is far smaller than the Planck length!

What if spacetime is "grainy" in some way at the Planck scale? Jacobson studied this using an analogy introduced by Unruh. If you have fluid flowing down a narrowing pipe, and at some point the velocity of the fluid flow exceeds the speed of sound in the fluid, there will be a "sonic horizon". In other words, there is a line where the fluid flow exceeds the speed of sound, and no sound can work its way upstream across that line. Now if you quantize the theory of sound in a simple-minded way you get "phonons" --- which have indeed been observed in solid-state physics. Unruh showed that in the case at hand you would get "Hawking radiation" of phonons from the sonic horizon, going upstream and getting shifted to lower frequencies as they go.

Jacobson considered what would happen if you actually took into account the graininess of the fluid. (He considered the theory of liquid helium, to be specific.) The graininess at the molecular scale means that the group velocity of waves drops at very high frequencies. So what happens instead of "Hawking radiation" is something rather different. Start with a high-frequency wave attempting to go upstream, starting from upstream of the sonic horizon. Its group velocity is very slow so it fails miserably and gets swept toward the sonic horizon, like a hapless poor swimmer getting pulled to the edge of a waterfall despite trying to swim upstream. But as it gets pulled near the horizon its wavelength increases, and thus group velocity increases, thus allowing it to shoot upstream at the last minute! (An analogous process is apparently familiar in plasma physics under the name of "mode conversion".) In this scenario, the Hawking radiation winds up resulting from incoming modes through this process of mode conversion --- modes that have short wavelength, but not as short as the intermolecular spacing (or Planck length, in the gravitational case.)

Ted Jacobson's second talk was even more interesting to me, but I'll postpone that for next Week.

Here, by the way, is a paper related to the talk by Pullin discussed in "week86":

5) Hugo Fort, Rodolfo Gambini and Jorge Pullin, Lattice knot theory and quantum gravity in the loop representation, preprint available as gr-qc/9608033.

© 1996 John Baez