kvantti said:
If you don't want to think of elementary particles as 0-dimensional "dots", you have to go with strings or with some other more fundamental structure of nature, e.g. branes or spinfoams.
Strings were found handy because they get rid of infinities in our current theories, such as the singularity in general relativity. Also, one vibrational mode of a string was realized to be the graviton, so it was the first glimpse of hope to include gravity into the quantum theory.
The 2d worldsheet of strings is the only manifold which gives local conformal invariance. To copy and paste a post I made on another forum (and pulled mostly from GS&W - Superstrings) :
In string theory the Lagrangian can take a form roughly (up to some factors etc) of
\mathcal{L} = \int d^{n+1}\sigma \sqrt{h}h^{\alpha \beta}(\sigma) g_{\mu\nu}(X) \partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}
Not particularly pleasant, but it's very general and this form is used in standard quantum field theory where n=0, the objects are zero dimensional points with g_{\mu\nu} = \eta_{\mu\nu}. This is just the mathematical extension to higher dimensions to see if anything is interesting. The integral is over n+1 dimensions because you're including time, so it's over an n+1 dimensional 'manifold' (just think surface or volume)
g_{\mu\nu} is the space-time metric, the one which you'd see in general relativity. h_{\alpha \beta} is the metric which describes the geometry of the n+1 dimensional manifold, basically the shape the quantum object 'sweeps out' as it moves through time and space, h^{\alpha \beta} is it's inverse and \sqrt{h} is the square root of the determinant of that metric. X^{\mu} is the position function of the object. Just like \mathbf{x}=(x,y,z) says where an object is in 3d space, X^{\mu} does it in 10.
Local conformal invariance is sort of like a position dependent rescaling of the n+1 dimensional manifold (ie a rescaling of the surface where the rescaling changes with position) which leaves the integral unchanged. Specifically, it's the transformation h_{\alpha \beta} \to \Lambda(\sigma)h_{\alpha \beta}. Crunching the numbers, this turns the integral into
\mathcal{L} = \int d^{n+1}\sigma \Lambda^{\frac{1}{2}(n+1)-1}\sqrt{h}h^{\alpha \beta}(\sigma) g_{\mu\nu}(X) \partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}
But we want invariance, so \Lambda^{\frac{1}{2}(n+1)-1} = 1 so \frac{1}{2}(n+1)-1 = 0, so n=1. Therefore the
ONLY theory which has local conformal invariance is that which works with 2 dimensional manifolds, one dimension of which is time, so the object is extended through 1 spatial dimension, a string.
