Showing posts with label dimensions. Show all posts
Showing posts with label dimensions. Show all posts

January 21, 2011

Real dimensions


Here's some more cool stuff about dimensions. It's about real dimensions that are not really real (it's funny how the word real has different meanings in English).

Usually when we think about dimensions, they are integer numbers, like 3 for our regular space, and 4 if we include time to make it space-time.

In physics we sometimes have to compute infinitely long sums (integrals if you know calculus). Sometimes these sums in 4-dimensional space are infinite. That's not what we want.

We can fix this by making the dimension slightly less than 4, for instance 3.99, which is a real number (and not an integer). The infinite sums now splits nicely in a finite term, which is the answer we seek, and an infinite term that we trash.

Then, in the end, we let the dimension go almost back to 4, say 3.999999999999.

It's magic, isn't it? And what does it mean? It's just a math trick that let us do computations, nothing more and nothing less. The dimension of our real space-time is still 4 >:)

(The trick outlined above is called dimensional regularization. It was invented by the physicists t'Hooft and Veltman who won the Nobel Prize in 1999. I have no idea how to picture a 3.99 dimensional space, and there's no need to do it as long as the math works. The picture above is a 4-dimensional hypercube from Wikipedia)

January 19, 2011

Higher dimensions


I'm working late today, preparing a lecture for Friday morning. It's on a cool subject, something called Hamilton's canonical equations. It's a funny game that takes place in a 6-dimensional space (so-called phase space).

The first 3 dimensions are the familiar world where we live.

Maybe we can use the other dimensions for something funny, like telepathy or getting in touch with the dead? Sorry, folks. You can't >:(

The first 3 dimensions describe where we are, and the last 3 dimensions describe how fast we're moving and in which direction. The 6-dimensional space is just a convenient way to represent our motion with mathematics. That's it.

Hamilton's equations are fundamental in both classical physics and quantum mechanics. In the quantum world of superstrings and such, there are other funny things coming into the theory as well. But still it's just a convenient way to describe physics in the language of mathematics.

Higher dimensions can't be used for anything exotic at all in our peaceful lives on Earth.

Sorry if you got disappointed, but I can tell you that theoretical physics is really cool >:)

(I found the picture of Sir William Rowan Hamilton (1805-1865) on Google Images. The photographer is probably dead long time ago, so I guess he doesn't mind if I use the photo. Sir William looks a little bit grumpy, doesn't he? Anyway, he was a really smart guy)
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