Higher dimensions in physics?

Don't be shocked by the idea that the universe has more spacte-time dimensions than four. When someone once told me that string theory only works in 26 dimensions (it's now 11, btw), the common response among my undergrad fellows was "that's absurd - the universe is 4 dimensional, and any theory in 26 dimensions is so far removed that it must be nonsense".

Well, we must think about it more carefully. Is our common experience really four dimensional? (or 3 dimensional, seen as a movie). Lets use Heisenberg's and Einstein's "Positivist" method : don't assume anything unless you can actually measure it. So, why how can we assume that the universe is 4-d ? Well, what does it really mean for something to live in higher d? There are a couple of ways to think about it. One is, if it is left alone in empty space, it doesn't spontaneously change (except for uniform translation along a geodesic). We can see that this is violated if we choose the wrong dimensionality for our guess : if we look at 2-d space, in which we really have a 3-d ball, then the uniform translation of that 3-d ball will make it appear to grow and shrink in our 2-d guess. Thus it is not simply translating in our 2-d space, so that must be the wrong dimensionality. Right? This is a nice physical way to say what dimension we are in.

Another cool example is orientation on a mobius strip. Imagine we live in a 2-d space. We take two little arrows and line them up with eachother, then we throw one of them off. It will eventually come back to us from the other side (there's no problem with this, it doesn't change the dimensionality by our definition), but now something is amiss. If we catch it, we find that it has flipped orientation (relative to its brother which we kept with us). What has happened? Well, our arrow has violated our definition of "rest" : it was moving through space with no interactions and it was changed - this means that the arrow must really be a higher-dimensional object. We are defining "normal space" to be the 2-d space that we can see locally (because any local experiment we do will show that space is 2-d : for example, if we carry an arrow around, it will never flip), so we are not allowed to think of our space as actually being a mobius strip. It must be a cylinder, with an extra dimension extending from it. Whenever our object moves around the cylinder, it also rotates in this other dimension. What we see in "normal space" is really just the 2-d slice of the larger 3-d space, projected on our cylinder. Imagine the hand of a clock, projected on a 1-d line : it appears to oscillate between pointing in opposite directions. This is the same thing that our arrow does as it moves along : it rotates in this extra dimension, so it appears to flip orientation (& back) as it moves around our 2-d space. Of course, being as wise as we are, we know this is analogous to leaving the 3-d space fixed, and making our 2-d space have an explicit loop (the mobius strip), but while this may seem pretty, it leaves us ill-equipped for extending this idea to our own 4-d spacetime.

Now, isospin causes us a problem. Isospin is the idea that a particle is "degenerate" with another particle : the two are indistinguishable in all ways, except their name. (isospin is not actually realized in nature, except in perturbation. The up & down quark are an iso-couplet if the electroweak force is turned off, the commonly known kaons are nearly an isocouplet, and we can thing of any set of degenerate energy levels as an isomultiplet). So, if our particle, call it an apple, is sitting in space, it could turn into its iso-partner, call it a pear, without us knowing. Thus an 'apple' sitting in space could turn into a 'pear' at any time. So, our space must not be 4-d space-time, because our definition has been violated : an apple does not just translate, it can also turn into a pear!

So, now we go back to our analogy of a 3-sphere moving through a plane. In the plane we saw a circle grow and shrink. So, we extend our space from 2-d by adding another dimension; this dimension is a label for how big the "circle" is. (this is a funny way of thinking of 3-d space : two axes are spacial, and the other labels how big a slice of a sphere is in that plane). This is analogous to an apple turning into a pear. Thus, we should be able to extend our space to a higher dimension, in which 4 axes are space, and the other labels whether we have an apple or a pear. Thus, instead of our apple turning into a pear and back, we are sliding along this 5th axis !!

So, this is a cute idea and convinces us that our idea of plain space-time dimensionality may be naive. It is pointless to consider a 4-d background if objects can spontaneously change under only uniform translation.


Charles Bloom / cb at my domain
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