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Okay, now the time has come to speak of many things: of topoi, glueballs, communication between branches in the many-worlds interpretation of quantum theory, knots, and quantum gravity.
1) Robert Goldblatt, Topoi, the Categorial Analysis of Logic, Studies in logic and the foundations of mathematics vol. 98, North-Holland, New York, 1984.
If you've ever been interested in logic, you've got to read this book. Unless you learn a bit about topoi, you are really missing lots of the fun. The basic idea is simple and profound: abstract the basic concepts of set theory, so as to define the notion of a "topos", a kind of universe like the world of classical logic and set theory, but far more general!
For example, there are "intuitionistic" topoi in which Brouwer reigns supreme --- that is, you can't do proof by contradiction, you can't use the axiom of choice, etc.. There is also the "effective topos" of Hyland in which Turing reigns supreme --- for example, the only functions are the effectively computable ones. There is also a "finitary" topos in which all sets are finite. So there are topoi to satisfy various sorts of ascetic mathematicians who want a stripped-down, minimal form of mathematics.
However, there are also topoi for the folks who want a mathematical universe with lots of horsepower and all the options! There are topoi in which everything is a function of time: the membership of sets, the truth-values of propositions, and so on all depend on time. There are topoi in which everything has a particular group of symmetries. Then there are really high-powered things like topoi of sheaves on a category equipped with a Grothendieck topology....
And so on: not an attempt to pick out "the" universe of logic and mathematics, but instead, an effort to systematically examine a bunch of them and how they relate to each other. The details can be intimidating, but Goldblatt explains them very nicely. A glance at the subject headings reveal some of the delights in store: "elementary truth", "local truth", "geometric logic", etc..
What is a topos, precisely? Well, most people would need to limber up a little bit before getting the precise definition... so let me just start you off with some mental stretching exercises. In classical logic we are used to working with two truth values, True and False. Let's call the set of truth values Ω, just to make it sound impressive --- and because it's traditional in topos theory. So, we are used to doing all our logic with
Ω = {True, False}.
In set theory, one of the things we do with Ω is describe subsets of a given set X. In other words, to describe a subset Y of X, we can say for each member of X, whether it is True or False that it is a member of Y. Thus we can describe the subset Y by giving a function
f: X → Ω.
We say x is in Y if f(x) = True, but x is not in Y if f(x) = False.
Now say we wanted to describe a topos of "time-dependent sets". But instead of "time-dependent sets", let's act like topos theorists and call them simply "objects", and instead of talking about one being a "subset" of another, let's say one is a "subobject" of another. To keep life simple, let's consider only two times: today and tomorrow. So we can think of an object X in this topos as a pair (X1, X2) of sets: one set X1 today, and another set X2 tomorrow. We say that an object Y is a "subobject" of X if Y1 is a subset of X1 and Y2 is a subset of X2. The idea is that we want Y to be contained in X both today and tomorrow.
Now, to describe a subobject Y of X, we can what's in Y today, and also what's in Y tomorrow. We would like to do so using some kind of function, or what topos theorists call a "morphism",
f: X → Ω.
Clearly we can't do this with our old truth values set {True, False}. Instead, we should use a truth values object Ω that keeps track of what's true or false today and what's true or false tomorrow. In other words, Ω should now be the pair of sets
({True, False}, {True, False})
Now what is that "morphism" f exactly? Well, it's like one function today and another function tomorrow, or in other words, a pair of functions! In general, a morphism f: S → T between objects in this topos is a pair of functions (f1,f2), with f1: S1 → T1 and f2: S2 → T2. Then in our particular case, the morphism f: X → Ω will say which elements of X1 are in Y1, and which elements of X2 are in Y2.
This is a pretty simple example of what the objects and morphisms in a topos can be like. They can be a lot weirder. The key thing is that you can do a lot of the same things with them that you can do with sets and functions. Also, you can do a lot of the same things with Ω that you can with {True, False}. Note that in our example, like in the classical example where Ω = {True, False}, the subobject classifier has a bunch of logical operations on it: morphisms like
Not: Ω → Ω
And: Ω x Ω → Ω
Or: Ω x Ω → Ω
and so on. In our example, and in the classical example, these make Ω into what folks call a boolean algebra. Basically, all the usual rule of logic apply. In general, though, Ω only needs to be a Heyting algebra. This is more general than a boolean algebra, and it can be sort of intuitionistic in flavor; for example, Not Not doesn't need to equal the identity morphism 1: Ω → Ω, so proof by contradiction doesn't necessarily work. A typical example of a Heyting algebra Ω is the set of open sets in a topological space, with And and Or being intersection and union, and with Not being the interior of the complement. This gives a little hint as to what topoi have to do with topology.
After studying this sort of thing a while, it's rather hard to go on pretending that the Zermelo-Fraenkel axioms of set theory, which were cooked up in the early 20th century to escape the logical paradoxes of Russell and others, are the last word on "foundations". One can develop topos theory within set theory if one wishes, but one can also set up topos theory from scratch, as a kind of pluralistic foundation of mathematics.
For a deeper but still friendly and expository introduction to topoi, try
2) Saunders Mac Lane and Ieke Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Springer-Verlag, New York, 1992.
Here you can learn about "Brouwer's theorem: all functions are continuous" (in a suitably intuitionistic topos, of course). You can also learn topos-theoretic versions of Cohen's proofs of the independence of the continuum hypothesis and the axiom of choice.
Goldblatt's book teaches you all the category theory you need to learn about topoi... but for people who already know some category theory, let me give the precise definition of a topos (or more precisely, an elementary topos, to distinguish it from a "Grothendieck topos"): it's a category with finite limits and power objects. This automatically implies a lot of things, such as the existence of the subobject classifier Ω that I was talking about.
To get deeper into topos theory, try:
3) Michael Barr and Charles Wells, Toposes, Triples and Theories, Springer-Verlag, New York, 1983. Available for free electronically at http://www.cwru.edu/artsci/math/wells/pub/ttt.html
Now let me catch up on some things more directly related to physics:
4) Frank Close, Are glueballs and hybrids found?, available as hep-ph/9509245. To appear in Proceedings of Hadron95.
J. Sexton, A. Vaccarino, D. Weingarten, Numerical evidence for the observation of a scalar glueball, available as hep-lat/9510022.
Thanks go to Greg Kilcup for bringing these to my attention. Have they found a glueball??? That would be really exciting. What's a glueball, you ask? Well, quantum chromodynamics, our best theory of the strong force, says that that the strong force is carried by particles called "gluons". Like electromagnetism, the strong force is a gauge field, but it's a nonabelian gauge field, so the gluons themselves have charge, or "color". Thus they interact in a nonlinear way. This is what lets them bind together quarks in such a tight way. But perhaps, in addition to pairs of quarks and antiquarks held together by gluons --- i.e., mesons --- and triples of quarks held together by gluons --- i.e., baryons --- there could be short-lived assemblages consisting entirely of gluons, held together by their self-interactions. These are called glueballs, but we don't know if these exist.
However, to my surprise, it turns out that there are now some candidates out there! The first paper suggests that the f_0(1500), a neutral spin-zero particle with mass around 1500 MeV, is a glueball. The second paper argues instead that this is basically a quark-antiquark pair (made of a strange quark and a strange antiquark... where "strange" is the technical name for one of the 6 quarks!). It presents evidence from a numerical simulation and argues that the "θ" or f_J(1710), a neutral particle with even spin and mass 1710 MeV, is a glueball. Part of the subtlety here is that, thanks to the superposition principle, there is not a perfectly sharp distinction between a glueball with some virtual quark-antiquark pairs in it, and a quark-antiquark pair with a bunch of virtual gluons in it. There can be "hybrids" that are a bit of both a linear combination of a meson and a glueball! (This phenomenon of "hybridization" is also familiar in chemistry.)
It's tough to do nonperturbative computations in nonlinear gauge field theories --- basically one needs to approximately compute a path integral, using Monte Carlo technique, approximating spacetime by a lattice (in this case, a 16 x 16 x 16 x 24 lattice). Computing the properties of a glueball and matching it with an experimentally observed particle would be a marvelous confirmation of quantum chromodynamics. In addition, I find there to be something charming about the idea that in a nonabelian gauge theory we could have a particle made simply of the gauge field itself.
5) R. Plaga, Proposal for an experimental test of the many-worlds interpretation of quantum mechanics, preprint available as quant-ph/9510007.
John Gribbin brought this one to my attention and asked me what I thought about it. Basically, the idea here is to isolate an ion from its environment in an "ion trap", and then perform a measurement on with two possible outcomes on another quantum system, and to excite the ion only if the first outcome occurs, before the ion has had time to "decohere" or get "entangled" with the environment. Then one checks to see if the ion is excited. The idea is that even if we didn't see the outcome that made us excite the ion, we might see the ion excited, because it was excited in the other "world" or "branch" --- the one in which we did see the outcome that made us excite the ion. The author gets fairly excited himself, suggesting that "outside physics, interworld communication would lead to truly mind-boggling possibilities".
Does this idea really make sense? First of all, I don't think of this sort of thing as a test of the many-worlds interpretation; I think that all sufficiently sensible interpretations of quantum mechanics (not necessarily very sensible, either!) give the same concrete predictions for all experiments, when it comes to what we actually observe. They may make us tell very different stories about what is happening behind the scenes, but not of any testable sort. As soon as one comes up with something that makes different predictions, I think it is (more or less by definition) not a new "interpretation" of quantum theory but an actual new theory. And I don't think the many-worlds interpretation is that.
So the question as I see it is simply, will this experiment work? Will we sometimes see the ion excited even when we didn't excite it? It seems hard; usually the decoherence between the two "branches" prevents this kind of "inter-world communication" (not that I'm particularly fond of this way of talking about it). What exactly is supposed to make this case different? The problem is that the paper is quite sketchy at the crucial point... just when the rabbit being pulled from the hat, as it were. I haven't put much time into analyzing it, but some people interested in this sort of thing might enjoy having a go at it.
6) Nicholas Landsman, Against the Wheeler-DeWitt equation, preprint available as gr-qc/9510033.
I haven't read this one yet, but I had some nice talks with Landsman about his ideas on quantization of constrained systems (see "week60") back when I was in Cambridge England. Quantizing constrained systems is the main problem with the so-called "canonical" approach to quantum gravity (see "week43"), so I was eager to see it applied to gravity, and I guess that's what he's done. The title of the paper is deliberately provocative... hmmm, I guess I'd better read it soon! Here's the abstract:
7) Pavel Etingof and David Kazhdan, Quantization of Lie bialgebras, I, preprint available in AMSTeX form as q-alg/9506005.
Quantization of Poisson algebraic groups and Poisson homogeneous spaces, preprint available in AMSTeX form as q-alg/9510020.
It sounds like Etinghof and Kazhdan are making serious progress on some questions of Drinfeld about when you can quantize Lie bialgebras and their kin. More stuff I need to read! I need to invent a time machine and keep running it backwards to make my weekends longer and read this stuff!
8) Steve Carlip, Statistical mechanics and black hole entropy, preprint available as gr-qc/9509024.
Claudio Teitelboim, Statistical thermodynamics of a black hole in terms of surface fields, preprint available as hep-th/9510180.
Steve Carlip's paper is a nice introduction to recent ideas, many of them his, on deriving black hole area/entropy relations by thinking of the entropy as associated to degrees of freedom of a field living on the event horizon. I haven't read Teitelboim's paper, but it sounds related.
9) Jorge Griego, Is the third coefficient of the Jones knot polynomial a quantum state of gravity?, preprint available as gr-qc/9510051.
The Kauffman bracket and the Jones polynomial in quantum gravity, preprint available as gr-qc/9510050.
In the loop representation of quantum gravity, states of quantum gravity give rise to link invariants. Which link invariants, though? The Kauffman bracket comes from a state of quantum gravity with cosmological constant... that is something I understand pretty well by now. But Gambini and Pullin also have an argument suggesting that the second coefficient of the Jones polynomial (also known as the Arf invariant) is a state of quantum gravity without cosmological constant. I've tried to make this argument more rigorous and never succeeded. They also floated a conjecture that all the coefficients of the Jones polynomial are states of quantum gravity. This confuses me a lot, because the Jones polynomial depends on the orientations of the components of a link, while states of quantum gravity should give link invariants that are independent of orientations. I guess all the odd coefficients of the Jones polynomial are orientation dependent. Thus I'm not shocked that Griego has done calculations indicating that the third coefficient does not come from a state of quantum gravity.
© 1995 John Baez
baez@math.removethis.ucr.andthis.edu
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