Hi, If elementary particles are nothing but a mode of vibration of a string, and if strings are on the order of Planck's length, then how did we end up with particles that are as big as 10^20 times Planck's length?
In the Standard Model the particles start out as massless and dimensionless. They acquire mass through the Higgs mechanism. The proton and neutron have size because they are multiparticle systems with lots of space between the three quarks and an "ocean" of massless virtual gluons flying around holding it all together. As quantum particles they are defined by their wave functions. So the quarks and gluons could well be vibration modes of tiny strings which BTW are not the size of the PLanck length but somewhat bigger.
Unfortunately i) there is no link between particles and strings. Except methaphysical ones. ii) string theory is a misnomer, since current theory work with branes and would be called brane theory. In fact, is callled brane theory in research papers but "string theory" in popular talks and books. Emphasizing the myth around string theory. iii) The only "well-known" vibration modes are masless supersimmetric modes. There is a belief that would explain hypotetical particles like the graviton, but no prooof. iv) Nobody has found nonsupersimmetric nonzero mass modes of vibrations that coincide with known particles of the standard model. v) Nobody know if strings (or branes) are of the order of Planck's length. vi) Etc
(i) through (iv) are absolutely correct and I didn't deny them. On (v), string (and brane and M-) theorists assume they have a spacetime continuum available to them and that motion, implied in the worldsheet, is unproblematical. Furthermore they (mostly) treat gravity as negligible. This implies their dimensions are large compared to the Planck length where the Compton radius equals the Schwartzschild radius.
Thanks! Regarding (v) I mean that in proper formulation of string or brane theory there is nothing fixing really the scale of the string or branes. Whereas in other alternatives a Planck scale arise naturally, string or brane theorist choose the tension parameter which in last instance fix really the size of the string. The situation is similar to the status of 'hidden' dimensions, each author chooses a size: Planck scale (or several orders of magnitude larger, as you correctly note), mm range, cosmological, etc. I also was expresing idea that in cosmology is supposed that strings (branes) are of cosmological scale. For example in brane cosmology is assumed that 4D spacetime is a brane of size more large that radius of observed universe. This is 'explained' from dualities. I simply desired to express with my previous points that is said in popular books, talks, and magazines is very, very different is said in research articles on the topic.