String theory relativistic equations

[lkwarren01]

I'm an engineer trying to get a reasonable laymen's/conceptual understanding of string theory. I've finally gotten a general understanding of developing classical equations of motion, but I'm a little stuck on relativistic equations.

As I understand it, relativistic equations of motion are developed from classical equations by applying light cone coordinates then Fourier expansion. It's my understanding that there's a Fourier term for each spatial dimension of the theory, including compactified dimensions, which describes how the string vibrates in that dimension. Is that correct?

If yes, how is the geometry of the compactified dimensions--eg, C-Y, orbifold, torus, etc--taken into account? Thru the sigma variable in the Fourier term?

Thanks
Larry Warren, IL

 

[Ben Niehoff]

The Fourier expansion only works in flat space, strictly speaking. The compact directions can be flat tori.

In curved spaces, the normal modes will be something other than sines and cosines. However, the Fourier expansion will be good enough for very small oscillations.

 

[haushofer]

Have you read Zwiebach's text? That's, I think, the best text that could meet your prerequisites :)

Non-relativistically, we say that a free particle (no forces!) traces out a straight line in space. Straight lines are the shortest paths if the space is flat, which is the case in Newtonian physics. Relativistically, we extend this idea to say that particles trace out geodesics; the shortest paths in spacetime. However, usually one develops string theory in flat spacetime, where geodesics are again straight lines.

So: to determine how particles move in spacetime, we extremize the length they trace out in spacetime.

A string doesn't trace out a line (which is 1-dimensional) in spacetime, but a sheet (which is 2-dimensional). Now the idea is that the surface of this sheet is extremized. This is done by writing down the so-called Nambu-Goto action. In chapter 6 of Zwiebach you can read the details :)

 

[haushofer]

Compactifications are taken into account by imposing periodic conditions on the spacetime coordinates which describe the compactifications.

So, if you have a 26-dimensional string theory (describing only bosons), the spacetime points are described by coordinates

<br /> x^{\mu}, \ \ \ \ \mu = 0,1,2,\ldots,25<br />

If you now want to compactify this theory on a torus, you can do this as follows:

<br /> x^{25} = x^{25} + 2\pi R_1 n, \ \ \ \ \ x^{24} = x^{24} + 2\pi R_2 m, \ \ \ \ n,m \in Z<br />

where the R's describe the two radii of the torus (we call these parameters moduli). Of course, this choice is arbitrary; we are free to choose the coordinate parametrization! But in my example I chose x^24 and x^25.

Now, an open string can wrap around this torus w times. In that case, the sigma coordinate wraps around one of the cycli of the torus (say, described by R_1), and gets the same periodic identification (in this case, the same periodic identification as x^25).

For Calabi-Yau's this becomes a bit more technical, but the idea is the same. Again: Zwiebach describes this in full detail :)