Right now I'm in a country estate called Les Treilles in southern France, at a conference organized by Alexei Grinbaum and Michel Bitbol:
1) Philosophical and Formal Foundations of Modern Physics, http://www-drecam.cea.fr/Phocea/Vie_des_labos/Ast/ast_visu.php?id_ast=762
It's very beautiful here, but about 20 philosophers, physicists and mathematicians have agreed to spend six days indoors discussing quantum gravity, the history of relativity, quantum information theory and the like. And guess what? Now it's our afternoon off, and I'm spending my time writing This Week's Finds! Some people just don't know how to enjoy life.
In fact, I want to continue telling you The Tale of Groupoidification. But before I do, here's a puzzle that Jeffrey Bub raised the other night at dinner. It's not hard, but it's still a bit surprising.
You and your friend each flip a fair coin and then look at it. You can't look at your friend's coin; they can't look at yours. You can't exchange any information while the game is being played, though you can choose a strategy beforehand. Each of you must guess whether the other's coin lands heads up or tails up. Your goal, as a team, is to maximize the chance that you're both correct.
What's the best strategy, and what's the probability that you both guess correctly?
Here's an obvious line of thought.
Since you don't have any information about your friend's coin flip, it doesn't really matter what you guess. So, you might as well guess "heads". You'll then have a 1/2 chance of being right. Similarly, your friend might as well guess "heads" - or for that matter, "tails". They'll also have a 1/2 chance of being right. So, the chance that you're both right is 1/2 × 1/2 = 1/4.
I hope that sounds persuasive - but you can actually do much better!
How? I'll give away the answer at the end.
Jeffrey Bub is famous for his work on the philosophy of quantum mechanics, and in his talk today he mentioned a similar but more sophisticated game, the Popescu-Rohrlich game. Here you and your friend each flip coins as before. But now, after looking at your coin, you each write either "yes" or "no" on a pad of paper. Your goal, as a team, is to give the same response when at least one coin lands heads up, but different responses otherwise.
Classically the best you can do is both say "yes" - or, if you prefer, both say "no". Then you'll have a 3/4 chance of winning. But, if before playing the game you and your friend prepare a pair of spin-1/2 particles in the Bell state, and you each keep one, you can use these to boost your chance of winning to about 85%!
I think the underlying idea first appeared here:
1) S. Popescu and D. Rohrlich, Nonlocality as an axiom, Found. Phys. 24 (1994), 379-385.
For the "game" version, try this:
2) Nicolas Gisin, Can relativity be considered complete? From Newtonian nonlocality to quantum nonlocality and beyond, available as quant-ph/0512168.
There's a lot more to say about this - especially about the "Popescu-Rohrlich box", a mythical device which would let you win all the time at this game, but still not allow signalling. The existence of such a box is logically possible, but forbidden by quantum mechanics. It can only exist in certain "supra-quantum theories" which allow even weirder correlations than quantum mechanics.
But, I don't understand this stuff, so you should just read this:
3) Valerio Scarani, Feats, features and failures of the PR-box, available as quant-ph/0603017.
Okay - now for our Tale. I want to explain double cosets as spans of groupoids... but it's best if I start with some special relativity.
Though Newton seems to have believed in some form of "absolute space", the idea that motion is relative predates Einstein by a long time. In 1632, in his Dialogue Concerning the Two Chief World Systems, Galileo wrote:
Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need throw it no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction.As a result, the coordinate transformation we use in Newtonian mechanics to switch from one reference frame to another moving at a constant velocity relative to the first is called a "Galilei transformation". For example:
When you have observed all these things carefully (though doubtless when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still.
(t, x, y, z) |→ (t, x + vt, y, z)
By the time Maxwell came up with his equations describing light, the idea of relativity of motion was well established. In 1876, he wrote:
Our whole progress up to this point may be described as a gradual development of the doctrine of relativity of all physical phenomena. Position we must evidently acknowledge to be relative, for we cannot describe the position of a body in any terms which do not express relation. The ordinary language about motion and rest does not so completely exclude the notion of their being measured absolutely, but the reason of this is, that in our ordinary language we tacitly assume that the earth is at rest.... There are no landmarks in space; one portion of space is exactly like every other portion, so that we cannot tell where we are. We are, as it were, on an unruffled sea, without stars, compass, sounding, wind or tide, and we cannot tell in what direction we are going. We have no log which we can case out to take a dead reckoning by; we may compute our rate of motion with respect to the neighboring bodies, but we do not know how these bodies may be moving in space.So, the big deal about special relativity is not that motion is relative. It's that this is possible while keeping the speed of light the same for everyone - as Maxwell's equations insist, and as we indeed see! This is what forced people to replace Galilei transformations by "Lorentz transformations", which have the new feature that two coordinate systems moving relative to each other will disagree not just on where things are, but when they are.
As Einstein wrote in 1905:
Examples of this sort, together with the unsuccessful attempts to discover any motion of the earth relative to the "light medium", suggest that the phenomena of electrodynamics as well as mechanics possess no properties corresponding to the idea of absolute rest. They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics are valid. We will elevate this conjecture (whose content will be called the "principle of relativity") to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with it, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body. These two postulates suffice for attaining a simple and consistent theory of the electrodynamics of moving bodies based on Maxwell's theory for stationary bodies.So, what really changed with the advent of special relativity? First, our understanding of precisely which transformations count as symmetries of spacetime. These transformations form a group. Before special relativity, it seemed the relevant group was a 10-dimensional gadget consisting of:
Nowadays this is called the "Galilei group":
With special relativity, the relevant group became the "Poincare group":
It's still 10-dimensional, not any bigger. But, it acts differently as transformations of the spacetime coordinates (t,x,y,z).
Another thing that changed was our appreciation of the importance of symmetry! Before the 20th century, group theory was not in the toolkit of most theoretical physicists. Now it is.
Okay. Now suppose you're the only thing in the universe, floating in empty space, not rotating. To make your stay in this thought experiment a pleasant one, I'll give you a space suit. And for simplicity, suppose special relativity holds true exactly, with no gravitational fields to warp the geometry of spacetime.
Would the universe be any different if you were moving at constant velocity? Or translated 2 feet to the left or right? Or turned around? Or if it were one day later?
No! Not in any observable way, at least! It would seem exactly the same.
So in this situation, it doesn't really make much sense to say "where you are", or "which way you're facing", or "what time it is". There are no "invariant propositions" to make about your location or motion. In other words, there's nothing to say whose truth value remains unchanged after you apply a symmetry.
Well, almost nothing to say! The logicians in the crowd will note that you can say "T": the tautologously true statement. You can also say "F": the tautologously false statement. But, these aren't terribly interesting.
Next, suppose you have a friend also floating through space. Now there are more interesting invariant propositions. There's nothing much invariant to say about just you, and nothing to say about just your friend, but there are invariant relations. For example, you can measure your friend's speed relative to you, or your distance of closest approach.
Mathematicians study invariant relations using a tool called "double cosets". I want to explain these today, since we'll need them soon in the Tale of Groupoidification.
"Double cosets" sound technical, but that's just to keep timid people from understanding the subject. A double coset is secretly just an "atomic" invariant relation: one that can't be expressed as "P or Q" where P and Q are themselves invariant relations - unless precisely one of P or Q is tautologously false.
So, atomic invariant relations are like prime numbers: they can't be broken down into simpler bits. And, as we'll see, every invariant relation can be built out of atomic ones!
Here's an example in the case we're considering:
"My friend's speed relative to me is 50 meters/second, and our distance of closest approach is 10 meters."
This is clearly an invariant relation. It's atomic if we idealize the situation and assume you and your friends are points - so we can't ask which way you're facing, whether you're waving at each other, etc.
To see why it's atomic, note that we can always find a frame of reference where you're at rest and your friend is moving by like this:
If you and your friend are points, the situation is completely described (up to symmetries) by the relative speed and distance of closest approach. So, the invariant relation quoted above can't be written as "P or Q" for other invariant relations.
The same analysis shows that in this example, every atomic invariant relation is of this form:
"My friend's speed relative to me is s, and our distance of closest approach is d."for some nonnegative numbers s and d.
(Quiz: why don't we need to let s be negative if your friend is moving to the left?)
From this example, it's clear there are often infinitely many double cosets. But there are some wonderful examples with just finitely many double cosets - and these are what I'll focus on in our Tale.
Here's the simplest one. Suppose we're doing projective plane geometry. This is a bit like Euclidean plane geometry, but there are more symmetries: every transformation that preserves lines is allowed. So, in addition to translations and rotations, we also have other symmetries.
For example, imagine taking a blackboard with some points and lines on it:
\ / ------------x-----------x----------- \ / \ / \ / \ / \ / x / \ / \ / \We can translate it and rotate it. But, we can also view it from an angle: that's another symmetry in projective geometry! This hints at how projective geometry arose from the study of perspective in painting.
We get even more symmetries if we use a clever trick. Suppose we're standing on the blackboard, and it extends infinitely like an endless plain. Points on the horizon aren't really points on the blackboard. They're called "points at infinity". But, it's nice to include them as part of the so-called "projective plane". They make things simpler: now every pair of lines intersects in a unique point, just as every pair of points lies on a unique line. You've probably seen how parallel railroad tracks seem to meet at the horizon - that's what I'm talking about here. And, by including these extra points at infinity, we get extra symmetries that map points at infinity to ordinary points, and vice versa.
I gave a more formal introduction to projective geometry in "week106" and "week145", and "week178". If you read these, you'll know that points in the projective plane correspond to lines through the origin in a 3d space. And, you'll know a bit about the group of symmetries in projective geometry: it's the group G = PGL(3), consisting of 3×3 invertible matrices, modulo scalars.
(I actually said SL(3), but I was being sloppy - this is another group with the same Lie algebra.)
For some great examples of double cosets, let F be the space of "flags". A "flag" is a very general concept, but in projective plane geometry a flag is just a point x on a line y:
-----------------x---------------- yAn amazing fact is that there are precisely 6 atomic invariant relations between a pair of flags. You can see them all in this picture:
\ / ------------x-----------x'---------- \ / y \ / \ / \ / \ / x" / \ / \ y'/ \y"There are six flags here, and each exemplifies a different atomic invariant relation to our favorite flag, say (x,y).
For example, the flag (x',y') has the following relation to (x,y):
"The point of (x',y') lies on the line of (x,y), and no more."By "no more" I mean that no further incidence relations hold.
There's a lot more to say about this, and we'll need to delve into it much deeper soon... but not yet. For now, I just want to mention that all this stuff generalizes from G = PGL(3) to any other simple Lie group! And, the picture above is an example of a general concept, called an "apartment". Apartments are a great way to visualize atomic invariant relations between flags.
This "apartment" business is part of a wonderful theory due to Jacques Tits, called the theory of "buildings". The space of all flags is a building; a building has lots of apartments in it. Buildings have a reputation for being scary, because in his final polished treatment, Tits started with a few rather unintuitive axioms and derived everything from these. But, they're actually lots of fun if you draw enough pictures!
Next, let me explain why people call atomic invariant relations "double cosets".
First of all, what's a relation between two sets X and Y? We can think of it as a subset S of X × Y: we say a pair (x,y) is in S if the relation holds.
Next, suppose some group G acts on both X and Y. What's an "invariant" relation? It's a subset S of X × Y such that whenever (x,y) is in S, so is (gx,gy). In other words, the relation is preserved by the symmetries.
Now let's take these simple ideas and make them sound more complicated, to prove we're mathematicians. Some of you may want to take a little nap right around now - I'm just trying to make contact with the usual way experts talk about this stuff.
First, let's use an equivalent but more technical way to think of an invariant relation: it's a subset of the quotient space G\(X × Y).
Note: often I'd call this quotient space (X × Y)/G. But now I'm writing it with the G on the left side, since we had a left action of G on X and Y, hence on X × Y - and in a minute we're gonna need all the sides we can get!
Second, recall from last Week that if G acts transitively on both X and Y, we have isomorphisms
X ≅ G/H
Y ≅ G/K
for certain subgroups H and K of G. Note: here we're really modding out by the right action of H or K on G.
Combining these facts, we see that when G acts transitively on both X and Y, an invariant relation is just a subset of
G\(X × Y) ≅ G\(G/H x G/K)
Finally, if you lock yourself in a cellar and think about this for a few minutes (or months), you'll realize that this weird-looking set is isomorphic to
This notation may freak you out at first - I know it scared me! The point is that we can take G, mod out by the right action of K to get G/K, and then mod out by the left action of H on G/K, obtaining
Or we can take G, mod out by the left action of H to get H\G, and then mod out by the right action of K on H\G, obtaining
And, these two things are isomorphic! So, we relax and write
A point in here is called a "double coset": it's an equivalence class consisting of all guys in G of the form
for some fixed g, where h ranges over H and k ranges over K.
Since subsets of H\G/K are invariant relations, we can think of a point in H\G/K as an "atomic" invariant relation. Every invariant relation is the union - the logical "or" - of a bunch of these.
So, just as any hunk of ordinary matter can be broken down into atoms, every invariant statement you can make about an entity of type X and an entity of type Y can broken down into "atomic" invariant relations - also known as double cosets!
So, double cosets are cool. But, it's good to fit them into the "spans of groupoids" perspective. When we do this, we'll see:
This relies on the simpler slogan I mentioned last time:
Let's see how it goes. Suppose we have two sets on which G acts transitively, say X and Y. Pick a figure x of type X, and a figure y of type Y. Let H be the stabilizer of x, and let K be the stabilizer of y. Then we get isomorphisms
X ≅ G/H
Y ≅ G/K
The subgroup H ∩ K stabilizes both x and y, and
Z = G/(H ∩ K)
is another set on which G acts transitively. How can we think of this set? It's the set of all pairs of figures, one of type X and one of type Y, which are obtained by taking the pair (x,y) and applying an element of G. So, it's a subset of X × Y that's invariant under the action of G. In other words, it's an invariant relation between X and Y!
Furthermore, it's the smallest invariant subset of X × Y that contains the pair (x,y). So, it's an atomic invariant relation - or in other words, a double coset!
Now, let's see how to get a span of groupoids out of this. We have a commutative diamond of group inclusions:
H∩K / \ / \ / \ v v H K \ / \ / \ / v v GThis gives a commutative diamond of spaces on which G acts transitively:
G/(H∩K) / \ / \ / \ v v G/H G/K \ / \ / \ / v v G/GWe already have names for three of these spaces - and G/G is just a single point, say *:
Z / \ / \ / \ v v X Y \ / \ / \ / v v *Now, in "week249" I explained how you could form the "action groupoid" X//G given a group G acting on a space X. If I were maniacally consistent, I would write it as G\\X, since G is acting on the left. But, I'm not. So, the above commutative diamond gives a commutative diamond of groupoids:
Z//G / \ / \ / \ v v X//G Y//G \ / \ / \ / v v *//GThe groupoid on the bottom has one object, and one morphism for each element of G. So, it's just G! So we have this:
Z//G / \ / \ / \ v v X//G Y//G \ / \ / \ / v v GSo - voila! - our double coset indeed gives a span of groupoids
Z//G / \ / \ / \ v v X//G Y//GX//G is the groupoid of figures just like x (up to symmetry), Y//G is the groupoid of figures just like y, and Z//G is the groupoid of pairs of figures satisfying the same atomic invariant relation as the pair (x,y). For example, point-line pairs, where the point lies on the line! For us, a pair of figures is just a more complicated sort of figure.
But, this span of groupoids is a span "over G", meaning it's part of a commutative diamond with G at the bottom:
Z//G / \ / \ / \ v v X//G Y//G \ / \ / \ / v v GIf you remember everything in "week249" - and I bet you don't - you'll notice that this commutative diamond is equivalent to diamond we started with:
H∩K / \ / \ / \ v v H K \ / \ / \ / v v GWe've just gone around in a loop! But that's okay, because we've learned something en route.
To tersely summarize what we've learned, let's use the fact that a groupoid is equivalent to a group precisely when it's "connected": that is, all its objects are isomorphic. Furthermore, a functor between connected groupoids is equivalent to an inclusion of groups precisely when it's "faithful": one-to-one on each homset. So, when I said that:
what I really meant was:
C / \ / \ / \ v v A B \ / \ / \ / v v Gwhere A,B,C are connected groupoids and the arrows are faithful functors.
This sounds complicated, but it's mainly because we're trying to toss in extra conditions to make our concepts match the old-fashioned "double coset" notion. Here's a simpler, more general fact:
where a "G-set" is a set on which G acts. This is the natural partner of the slogan I explained last Week, though not in this language:
Things get even simpler if we drop the "faithfulness" assumption, and simply work with groupoids over G, and spans of these. This takes us out of the traditional realm of group actions on sets, and into the 21st century! And that's where we want to go.
Indeed, for the last couple weeks I've just been trying to lay out the historical context for the Tale of Groupoidification, so experts can see how the stuff to come relates to stuff that's already known. In some ways things will get simpler when I stop doing this and march ahead. But, I'll often be tempted to talk about group actions on sets, and double cosets, and other traditional gadgets... so I feel obliged to set the stage.
Okay - here's the answer to the puzzle. Close your eyes if you want to think about it more.
An optimal strategy is for you and your friend to each look at your own coin, and then guess that the other coin landed the other way: heads if yours was tails, and tails if yours was heads. With this strategy, the chance you're both correct is 1/2.
Or, you can both guess that the other coin landed the same way. This works just as well.
The point is: you and your friend can do twice as well at this game if you each use the result of your own coin toss to guess the result of the other's coin toss!
It seems paradoxical that using this random and completely uncorrelated piece of information - the result of your own coin toss - helps you guess what your friend's coin will do, and vice versa.
But of course it doesn't. You each still have just a 1/2 chance of guessing the other's coin toss correctly. What the trick accomplishes is correlating your guesses, so you both guess right or both guess wrong together. This improves the chance of winning from 1/2 × 1/2 (the product of two independent probabilities) to 1/2.
By the way, the translation of the passage by Einstein is due to Michael Friedman, a philosopher at Stanford; he used it in his talk at this conference. There's a lot more to say about talks at this conference. Let's see if I get around to it.
Also by the way: if you fix a collection of n G-sets, there's always a Boolean algebra of n-ary invariant relations. Only the case n = 2 is related to double cosets, but everything else I said generalizes easily to higher n using "n-legged spans" of groupoids: an obvious generalization of the 2-legged spans I've been discussing so far. In Boolean algebra people often use the term "atom" to stand for an element that can't be written as "P or Q" unless exactly one of P or Q is tautologously false.
© 2007 John Baez