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I am reading String Theory and M-Theory: A Modern Introduction by Becker, Becker and Schwartz. I've tried to read this book before but not succeeded because I didn't know enough math or physics. This time, I feel like I am following it much better but I still don't understand what a string is.

All the concepts I've read so far either to fall into the category of stuff that could be described by a harmonic oscillator or the category of stuff that could be described by a point. But what does a string offer that these concepts don't? I still don't really understand what a string it is and I think that's partly because I don't see what the idea offers that's new.

My Question:

How is a string in string theory different from a harmonic oscillator or a point?

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A string is a one-dimensional object, and not a zero-dimensional like a point.

This means that the object traced out by it following its trajectory is a two-dimensional surface, the worldsheet, while particles - points - trace out one-dimensional objects, the worldlines.

Much of the difference between string theory and theories with particles as the fundamental objects arises from this conceptually simple difference - in particular, the worldsheets of two strings merging and unmerging again are also just smooth 2D surfaces, while the corresponding object in particle theories - a Feynman diagram - has these ugly vertices instead of being smooth.

Also, 2D "world-objects" are special in that the natural action functional on them - the Nambu-Goto action or the related Polyakov action - is conformally invariant (and only on them - the Polyakov action is Weyl invariant only in 2D), leading to a much more constrained theory compared to a generic QFT, because it forces the theory to have the Virasoro algebra as an infinite-dimensional symmetry.

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  • $\begingroup$ I think my confusion is around the dimensionality. I'm going to guess a particle's one dimensional trajectory corresponds to proper time. If so, what are the two dimensions we use for strings? I will guess again and suppose they are $\sigma$ and $\tau$. But what does that mean? I thought proper time was an affine parameterization of the curve distance traveled along a curve on on a spacetime manifold. What would the $\sigma$ be parameterizing? $\endgroup$ – Stan Shunpike Feb 18 '15 at 1:09
  • $\begingroup$ @StanShunpike: The $\sigma$ is the spatial coordinate on the string, the $\tau$ is the time variable, i.e. the $\sigma$ is where on the string you look and the $\tau$ is when. $\endgroup$ – ACuriousMind Feb 18 '15 at 1:12
  • $\begingroup$ But for strings does $\tau$ signify proper time or regular time? $\endgroup$ – Stan Shunpike Feb 18 '15 at 1:19
  • $\begingroup$ @StanShunpike: Whatever you like, it's just a reparametrization, anyway. $\endgroup$ – ACuriousMind Feb 18 '15 at 1:20

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