# What makes the strings (or $p$-branes) that occur in string theory vibrate?

A normal spring (for example in my ballpoint) is made of particles that interact with each other wich makes the string vibrate with a frequency that depends on the material (the particles).

Now a string in string theory doesn´t consist out of particles between wich a force is working. No parts of the string exchanges a force carrier particle with other parts of the string, like particles that constitute a normal string do with other particles of the string in, say again, my ballpoint.

Then how can a string vibrate if there aren´t forces at work in the string? Or is this vibration just an analogy taken from real springs, without explaining, as can be done in real springs, why the string can vibrate? Can string theory account for the vibrational motions of it´s elementary entities whose different vibrational modes can explain the different elementary particles and their charges and thus the interactions of elementary particles? I can recall reading two of Brian Greens books; in one book he writes that the masses of elementary particles are due to vibrational modes, and in another book he writes it´s the Higgs-mechanism that does the job, but this aside.

So the main question is simple: what makes a string vibrate?

The point is that there is no "part of the string" for a fundamental string. So there is no need for a "carrier particle" or something like that.

Given the action for a free string, the Nambu-Goto action, you find the equation of motion that the string must satisfy and you discover that the solution contains a translational part associated to the center of mass of the string plus an infinity of transverse vibration modes (no longitudinal modes).

After quantization of the system, by promoting the string coordinates to operators and imposing the canonical commutation relations, you find that the spectrum of mass of the string depends on the modes of vibration, in particular for the bosonic string:

$$\alpha' m^2 = N-1$$

where N is the number operator eigenvalue. In this way you see that the fundamental level of the bosonic theory, $$N=0$$, is tachyonic. The first excited level is massless (graviton in closed bosonic theory), and so on.

Admittedly I've never heard about the possibility of the string standing still. But I think that in a quantum theory it's not possible to be perfectly still, otherwise your position is determined with arbitrary precision. So the string is forced to oscillate in one of the possible modes.

Moreover the string has a tension. So in some sense the string will try to squeeze itself, and is it only by vibrating that the string can be stable.

• Why has a string no parts? How can it vibrate? The parts are inextractibly bound up with each other, but that doesn´t mean there are no parts. It´s the same as saying that quarks don´t exist, and don´t forget that string theory has it´s origins in quark physics. The fact that the the strings coordinates are promoted to operators is for me already a proof that a string consists of parts. I think the uncertainty-relations can influence only the translational part of the string (vibrations are to good to be true), and maybe the tension (?) is zero at minimum size, so the string can be stable. Apr 11, 2016 at 12:49
• Is the fact that a string has a tension not contradictory to the fact that is has no parts? The difference with a matter made string that the stringmatter is continu, so no forces between costitutive particles can develop, like in a macroscopic string. Where there´s tension there´s force, so what force makes the string vibrate? Or can´t string theory account for that vibration? Apr 11, 2016 at 13:01
• That the string has no part is a postulate. You want it to be a fundamental entity. Otherwise you must study the behaviour of single string pieces (actually there are some people that use the so called string bit approach), that at the end will be point-like and you end up with a quantum field theory. Apr 11, 2016 at 14:33