Physicists think of a dimension as an independent (perpendicular) direction in which objects could move. There are other definitions of dimension that are interesting to mathematicians (like a fractal dimension), but those types are beyond the scope of this course.

After a good deal of work, physicists discovered that quantum mechanics doesn’t make sense with two or more time dimensions, because you can’t define energy unambiguously. On the other hand, extra space dimensions are fine. There is no theoretical reason to believe that the idea of extra space dimensions is inherently evil. Theoretical physicists have mightily enjoyed thinking about whether extra dimensions of space might exist and -- if they do -- how we would measure their existence. I am one of the people guilty of having a LOT of fun doing this sort of thing. :-)

Trying to think in extra dimensions makes the brain hurt. What is easy is imagining a lower-dimensional world than ours by projecting our world onto a surface.

Imagine that all people and objects in our lives are two-dimensional (2D), living in a 2D world -- say on a tabletop. What’s different about their lives than for our lives? Take some time while you’re reading this to try to draw on pieces of paper what life would be like in only 2D. Try to also think about how would you explain the idea of a 3rd dimension to a flatlander. How would you explain a soccer ball? How would you communicate with a flatlander on another tabletop if their photons were stuck living on their 2D world? Can flatlanders eat, excrete, & have sex, or only some of those things?

4D to us is like 3D is to the flatlanders. It’s hard for our brains to comprehend naturally. The really neat thing is that theoretical physicists have developed ways of making extra-dimensional physics easier to grasp. Essentially, what we do is work with projections, which is like looking at a 3D object from different 2D perspectives and then knitting the 2D perspectives together into a 3D whole. This is a little bit like assembling a CT scan: what the CT machine does is takes lots of 2D pictures of you and then uses software to join the pictures together into a 3D image that your doctor can interpret.

Theories incorporating extra dimensions go back over 80 years, but string theory is new in requiring more space dimensions. How would we visualize having extra dimensions that are too small for us to see?

Consider an ant on a telephone wire:-

￼A big ant can walk only in the big (extended) dimension. It is entirely too large and clumsy to feel the second dimension of the telephone wire -- the small circular dimension. A tiny wee ant, on the other hand, knows that the wire surface is two-dimensional. Which it is! What this example shows you is that if you want to discover small curled-up extra dimensions you need to be capable of resolving distances that are very small.

Extra dimensions have not yet been seen. Physicists are looking hard for evidence of extra dimensions in a variety of very ingenious tabletop & collider experiments. The experimental constraints differ for gravity-only extra dimensions (GOEDs) versus universal extra dimensions (UEDs). State-of-the-art experiment says that universal extra dimensions must be smaller than 0.01fm, which is one percent of a millionth of a billionth of a metre. Pretty small, eh! For gravity-only extra dimensions, the constraints are much weaker because gravity is a weak force, making precise experiments difficult. The limit that experimentalists have obtained so far is that any gravity-only extra dimensions -- which are dimensions into which we cannot move and into which we cannot shine light (photons) -- must be smaller than 0.15mm. It’s pretty mindblowing to realize that we could have an extra dimension curled up at every point in our four-dimensional world and it could be as big as the thickness of a human hair.

Exciting a physical field like an electron wavefunction in an extra dimension does not work for arbitrary configurations. Quantum mechanics insists that you must fit an integer number of De Broglie wavelengths around the circumference of the circular extra dimension, or else the wavefunction would cancel itself out. This holds true for both particles and strings.

One ingredient we need to work out the energy cost of fitting $n$ De Broglie wavelengths around the extra dimension of radius $R$ is the quantization condition. Written in equations, it is $$ n\lambda= 2\pi R \,. $$ ￼Another ingredient we need is the De Broglie wavelength formula $$ \lambda = {\frac{h}{p}} $$ ￼Combining these two pieces of information, we can find the energy for a KK mode with quantum number $n$ $$ {\frac{E_{\rm KK}^{(n)}}{E_s}}= (n\hbar c){\frac{\ell_s}{R}} $$ ￼The derivation of this mathematical equation is completely unimportant for our purposes here. What matters physically is that the energy cost of a KK mode is expensive at small radius.

Here is a picture of an n=4 KK mode. The dotted line is intended to get across the idea that there are four De Broglie wavelengths fit around the circle. The solid line represents the circular extra dimension. ￼

The following example is a qualitatively different mode. It is a winding mode. The physically important thing is that the body of the string is wrapped around the circular extra dimension. ￼

Particles can’t do this! They can move around the extra dimension, but they can’t wrap it.

What is the energy cost of winding a string w times around a circular extra dimension? $$ {\rm{Energy}} = {\frac{{\rm (Energy)}}{\rm (length)}} \cdot {\rm (length)} = {\frac{\hbar c}{2\pi\ell_s^2}} \cdot (w \cdot 2\pi R) $$ ￼￼So for winding modes ￼$$ {\frac{E^{(w)}_{\rm winding}}{E_s}} = (w\hbar c) {\frac{R}{\ell_s}} $$ Again, how I derived this formula is not important. What matters is that the energy cost of winding modes is proportional to the radius $R$ -- winding modes are energetically expensive at large radius. This is the opposite behaviour to what we saw for KK modes.

T-duality is a symmetry that relates (a) small radius to large radius and (b) momentum modes to winding modes. Pictorially,

￼￼ ￼Duality is when two different-looking physical systems turn out (usually in a huge surprise to physicists of the day) to have the same underlying physics. Having duality is not generic. To explain this, consider a language analogy for KK modes and winding modes. In the analogy, KK modes are like English and winding modes are like French. Anyone who has ever tried to learn another language seriously will know that there are typically concepts in one language (in our case, English) that are inefficient to convey in the other language (in our case, French). Having a duality is like being in the overlap region of the Venn Diagram where English and French words correspond one-to-one. T-duality is such a situation. For every thing you can do with KK modes in string theory, you can do it with winding modes if you just T-dualize.