Quantization of classical strings - a beginner's question

[Qturtle]

Hey all. I've started to read and watch lectures on string theory. usually everyone starts with a classical relativistic particle action, and then goes to a classical relativistic string action. after they finish with the classical string they start the quantization process.

my question is somewhere before the quantization itself, but i don't see anyone mention this in the lectures or books i am reading so i thought to ask it here. I remember that when we where deriving the wave equation for a classical string, we had to use the assumption that the string have only small vibrations, and that it has to be tense. if the string have no tension then the equation of motion is not very wavelike.. and it is also very intuitive that if i just take a loose string, its motion is not only described by a simple wave equation. the same goes for a closed string - since if it is closed then it is usually loose and have less tension.
when looking at the derivations in the books, i don't see any mentions to the restrictions of the strings. it looks like the relativistic action that they provide is used to completely describe a a loose or open strings that are not tense, but can loosely move through space and everything. they provide some function that is used to provide the location of every point of the string in space, and there is no restriction on the shape that the string can have.
my question is regarding the nature of these strings - what are the restriction used to derive the relativistic (soon to be quantized) action? can these strings move however they want? can they wobble around themselves? can they have knots? because from what i see, they can ONLY have small vibrations propagating through them, provided they have tension, and that's it. at least that is what you get if you start from a classical string that is described by a wave equation. am i wrong?

Thank's in advance

 

[Demystifier]

Strings can wobble, have knots, and move however they want, as long as the total energy and momentum are conserved. The vibrations do not need to be small. But due to tension, larger vibrations and stretches take more energy, so such larger deformations occur less frequently.

 

[Qturtle]

thanks for the comment.
how is this possible if the string has tension? if you need to assume small vibrations to derive the wave equation of stings, and then you build a lagrangian that gives you the wave equation, shouldn't the restrictions still be valid?
please explain why do you say that there are no restrictions and how is this implied from the way we derived the action for the classical string.

 

[Ben Niehoff]

A string in string theory has intrinsic tension just like a particle has intrinsic mass. In fact, tension is the same as mass per unit length.

This means that yes, strings prefer to have zero length, and stretching them out has some energy cost. This could be stretching an open string by pulling on its endpoints, or stretching a closed string by giving it some angular momentum. The string also couples to the B field, which can cause it to stretch in different ways.

 

[Ben Niehoff]

thanks for the comment.
how is this possible if the string has tension? if you need to assume small vibrations to derive the wave equation of stings, and then you build a lagrangian that gives you the wave equation, shouldn't the restrictions still be valid?
please explain why do you say that there are no restrictions and how is this implied from the way we derived the action for the classical string.

Well, consider that fundamental strings are relativistic objects, so some of your assumptions here are not valid. In particular, disturbances in the string profile propagate at the speed of light. If you look at your classical, non-relativistic stretched string, and take the limit of parameters where the speed of wave propagation approaches the speed of light, you should arrive at the fundamental string.

 

[Qturtle]

so if i understand you right, the example of a nonrelativistic string which its equation of motion is derived using the assumptions of small oscillation and that can not do much other then oscillate with constant tension, is just a privet case of the more general relativistic string (that can do almost anything and get any shape) with its endpoints constrained to stay in one place and with its propagation speed taken to a value smaller enough then c?
if so, then i guess that what got me confused is that textbooks introduce you to this constrained non-relativistic string, and then move on to the non-constrained relativistic string (instead of starting with a non-constrained non-relativistic string)
the nonrelativistic string action is actually derived from simple Newtonian mechanics (which i guess is more intuitive and less abstract), where the nonrelativistic one is derived simply by assuming that the action is proportional to the string's surface are in spacetime.

so now i guess that my questions are:
1. how can i find an equivalent desription of a classical nonrelativistic string that can do more stuff and get in more complicated shapes like the relativistic one (and not the like the common textbook example)?
2. is there a more "mechanical" derivation of the relativistic string action, that uses relativistic dynamics and not the functional analysis of string surface area in spacetime?
3. are the solutions for the relativistic string that is not constrained in its end points (or have angular momentum) always leads for a string shrinking into a dot?
4. how can i get the relativistic string by taking the limit of the speed to c? i would expect to get the nonrelativistic string by letting the propagation be smaller then c, but not the other way around.