I gave a talk at Brown this week, so this post may be shorter than usual. On the topic of Brown I don’t have much original to say: the people were friendly, the buildings were brownish-colored, and bringing a car there was definitely a bad idea. Don’t park at Brown. Not even then.
There’s a quote from Werner Heisenberg that has been making the rounds of the internet. It comes out of a 1976 article by Felix Bloch where he describes taking a walk with Heisenberg, when the discussion turned to the subject of space and time:
I had just read Weyl’s book Space, Time and Matter, and under its influence was proud to declare that space was simply the field of linear operations.
“Nonsense,” said Heisenberg, “space is blue and birds fly through it.”
Heisenberg’s point is that sometimes in physics you need to ask what your abstractions are really describing. You need to make sure that you haven’t stretched your definitions too badly away from their original inspiration.
When people first hear that string theory requires eleven dimensions, many wonder if this point applies. In mathematics, it’s well known that a problem can be described in many dimensions more than the physical dimensions of space. There’s a lovely example in the book Flatterland (a sequel to Flatland, a book which any math-y person should read at least once) of the dimensions of a bike. The bike’s motion through space gives three dimensions: up/down, backward/forward, and left/right. However, the bike can move in other ways: its gears can each be in a different position, as can its handlebars, as can the wheels…in the end, a bike can be envisioned as having many more “dimensions” than our normal three-dimensional space, each corresponding to some internal position.
Is string theory like this? No.
The first hint of the answer comes from something called F theory. String theory is part of something larger called M theory, and since M theory has eleven dimensions this is usually the number of dimensions given. But F theory contains string theory in a certain sense as well, only F theory contains twelve dimensions.
So why don’t string theorists say that the world has twelve dimensions?
As it turns out, the extra dimension added by F theory isn’t “really” a dimension. It’s much more like the mathematical dimensions of a bike’s gears and wheels than it is like the other eleven dimensions of M theory.
What’s the difference? What, according to a string theorist, is the definition of a dimension of space?
It’s simple: Space is “blue” (or colorless, I suppose). Birds (and particles, and strings, and membranes) fly in it.
We’re using the same age-old distinction that Heisenberg was, in a way. What is space? Space is just a place where things can move, in the same way they move in our usual three dimensions. Space is where you have momentum, where that momentum can change your position. Space is where forces act, the set of directions in which something can be pulled or pushed in a symmetric way. Space can’t be reduced, at least not without a lot of tricks: a bird flying isn’t just another description of a lizard crawling, not in the way a bicycle’s gears moving can be thought of as turning through our normal three dimensions without any extra ones. And while F theory doesn’t fit this criterion, M theory really does. The membranes of M theory fly around in eleven dimensional space-time, just like a bird moves through three space and one time dimensions.
Space for a string theorist isn’t any crazier or more abstract than it is for you. It’s just a place where things can move.