This is a follow-up to an intriguing question last year about tension in string theory.
What are the strings in string theory composed of?
I am serious. Strings made of matter are complex objects that require a highly specific form of long-chain inter-atomic bonding (mostly carbon based) that would be difficult to implement if the physics parameters of our universe were tweaked even a tiny bit. That bonding gets even more complicated when you add in elasticity. The vibration modes of a real string are the non-obvious emergent outcome of a complex interplay of mass, angular momentum, various conservation laws, and convenient linearities inherent in of our form of spacetime.
In short, a matter-based vibrating real string is the outcome of the interplay of most of the more important physics rules of our universe. Its composition — what is is made of — is particularly complex. Real strings are composed out of a statistically unlikely form of long-chain bonding, which in turn depend on the rather unlikely properties that emerge from highly complex multiparticle entities called atoms.
So how does string theory handle all of this? What are the strings in string theory made of, and what is it about this substance that makes string-theories simple in comparison to the emergent and non-obvious complexities required to produce string-like vibrations in real, matter-based strings?
Addendum 2012-12-28 (all new as of 2012-12-29):
OK, I'm trying to go back to my original question after some apt complaints that my addendum yesterday had morphed it into an entirely new question. But I don't want to trash the great responses that addendum produced, so I'm trying to walk the razor's edge by creating an entirely new addendum that I hope expands on the intent of my question without changing it in any fundamental way. Here goes:
The simplest answer to my question is that strings are pure mathematical abstractions, and so need no further explanation. All of the initial answers were variants of that answer. I truly did not expect that to happen!
While such answers are sincere and certainly well-intended, I suspect that most people reading my original question will find them a bit disappointing and almost certainly not terribly insightful. They will be hoping for more, and here's why.
While most of modern mathematical physics arguably is derived from materials analogies, early wave analogies tended towards placing waves within homogeneous and isotropic "water like" or "air like" media, e.g. the aether of the late 1800s.
Over time and with no small amount of insight, these early analogies were transformed into sets of equations that increasingly removed the need for physical media analogies. The history of Maxwell's equations and then SR is a gorgeous example. That one nicely demonstrates the remarkable progress of the associated physics theories away from using physical media, and towards more universal mathematical constructs. In those cases I understand immediately why the outcomes are considered "fundamental." After all, they started out with clunky material-science analogies, and then managed over time to strip away the encumbering analogies, leaving for us shiny little nuggets of pure math that to this day are gorgeous to behold.
Now in the more recent case of string theory, here's where I think the rub is for most of us who are not immersed in it on a daily basis: The very word "string" invokes the image of a vibrating entity that is a good deal more complicated and specific than some isotropic wave medium. For one thing the word string invokes (perhaps incorrectly) an image of an object localized in space. That is, the vibrations are taking place not within some isotropic field located throughout space, but within some entity located in some very specific region of space. Strings in string theory also seem to possess a rather complicated and certainly non-trivial suite materials-like properties such as length, rigidity, tension, and I'm sure others (e.g. some analog of angular momentum?).
So, again trying to keep to my original question:
Can someone explain what a string in string theory is made of in a way that provides some insight into why such an unusually object-like "medium of vibration" was selected as the basis for building all of the surrounding mathematics of string theory?
From one excellent comment (you know who you are!), I can even give an example of the kind of answer I was hoping for. Paraphrasing, the comment was this:
"Strings vibrate in ways that are immediately reminiscent of the harmonic oscillators that have proven so useful analytically in wave and quantum theory."
Now I like that style of answer a lot! For one thing, anyone who has read Feynman's section on such oscillators in his lectures will immediately get the idea. Based on that, my own understanding of the origins of strings has now shifted to something far more specific and "connectable" to historical physics, which is this:
Making tuning forks smaller and smaller has been been shown repeatedly in the history of physics to provide an exceptionally powerful analytical method for analyzing how various types of vibrations propagate and interact. So, why not take this idea to the logical limit and make space itself into what amounts to a huge field of very small, tuning-fork-like harmonic oscillators?
Now that I can at least understand as an argument for why strings "resonated" well with a lot of physicists as an interesting approach to unifying physics.
Addendum 2018-03-28: The Answer (no kidding!)
This year for the first time I submitted an essay, Fundamental as Fewer Bits, to the annual FQXi foundational questions essay contest. In the essay I propose that Kolmogorov complexity provides a more automated way, less human-biased way to apply Occam's Razor to physics theories, literally by trying to find the least-bits representation of the Kolmogorov sense of program-like data compression. (My thanks to Garrett Lisi for noticing the connection to Occam's Razor; I had not thought of my essay that way.)
The contest, which this year goes on until May 1, 2018, proved to be much more interesting and interactive than I had anticipated. In the course of looking at other essays I dove into the details of how string theory originated. I was amazed to find out that the concept has some very solid experimental data behind it… at a scale about $10^{20}$ times larger than the one at which it is now described!
As it turns out, string theory originated in some extremely interesting 1960s and 1970s experimental research on hadrons. A hadron is any particle composed of quarks, and includes both two-quark bosonic mesons and three-quark fermionic baryons such as protons, neutrons, and the more exotic $\Lambda$ particles. Being composed of quarks, all hadrons are of course bound together by the strong force, and therein likes the real, experimentally meaningful Answer to what the questions of what strings are composed of:
All real, experimentally meaningful strings are composed of the strong force.
It seems that most (perhaps all) hadrons have excited states in which their spins are augmented in increments of 2. For example, both the proton and neutron normally has a spin of $\frac{1}{2}$, but both also have higher spins states of e.g. $\frac{5}{2}=\frac{1}{2}+2$ and $\frac{9}{2}=\frac{1}{2}+4$.
These higher spins states also have higher masses. Amazingly, when all the possible states are plotted in a spin-versus-mass-squared graph, the result is beautiful set of straight lines with even spacing between the 2-spin additions. These lovely and highly unexpected lines are called Regge trajectories, and they are the true origins of string theory.
Theoretical analyses of these remarkable regularities could be explained by assuming them to be the stationary vibration modes of a string. In fact, if you think in terms of how a skip rope can have one, two, or even more loops in it when handled by an expert, you are not too far off the mark. At the time there was hope that these remarkable string-like vibration models might lead to a deeper understanding of both fundamental and composite particles. However, quantum chromodynamics (QCD) instead began to dominate, while Regge trajectories continued to pose theoretical problems. It looked like the end for hadron-level strings and string vibrations, despite the truly remarkable and still unexplained regularities seen in Regge trajectories.
Then something very strange happened, an event that to my way of thinking was one of the least rational and most bizarre events in the entire history of physics. I call it the Deep Dive. It has features that I would more typically associate with the ancient and fascinating history of religious revelation and the founding of new religions than I would with scientific analysis.
While they were not the only people involved, in 1974 physicists Scherk and Schwarz wrote a conventional looking paper, Dual Models for Non-Hadrons, with an extremely unconventional conclusion tucked away inside. The conclusion was this: Because the two-spin increments of hadron strings bore several mathematical resemblances to the proposed properties of spin-2 gravitons (the still-hypothetical quantized particles of gravity), they were in some way one and the same thing, and the concept of string vibrations should therefore be moved from out of hadrons and into the domain of quantum gravity.
This enormous leap of faith was the origin of what we now call "string theory".
There was of course a "tiny" problem, in the most literal since of the word: This abrupt leap from very real, experimentally meaningful string-like vibration modes in hadrons to gravitons plunged the needed size scales down to the experimentally inaccessible Planck foam level. This was a drop of about 20 orders of magnitude, with a comparable increase in the energy levels needed to access the proposed vibrations. Even worse, all of the severe constraints on vibration modes imposed by hadron "architectures" were instantly removed by this proposed drop, allowing the number of vibration modes of the now-abstract strings composed of a now-abstract substance (maybe mass-energy?) to explode into at least $10^{500}$ possibile vacuum states.
I explore all of this a bit more — actually hmm, less than I just did here — in a mini-essay attached to my FQXi essay discussion. In that mini-essay I argue that It's Time to Get Back to Real String Theory. That is, there remains to this day a very real and extraordinarily interesting data puzzle in the very existence of Regge trajectories. This is a mystery that still needs to be resolved! This data is another example of how spin is a remarkably deep and fundamental concept, one for which physics still seems to be missing some kind of critical piece or pieces.
Regarding the question "What are the strings in string theory made of?", the answer could not be any clearer: In the real, experimentally meaningful strings found back in hadron research in the 1960s and 1970s, they are a function of the strong force, constrained in interesting and limiting ways by the quarks that enable a string-like topology to exist in the first place. This is all very real, very meaningful physics.
For the Planck-level strings that were proposed essentially by revelation, that is, by a leap of faith from experimentally meaningful physics down 20 orders of magnitude to the inaccessible level of the Planck foam, based on no more than a superficial mathematical resemblance, and with utter abandonment of any of the original tight constraints on both substance (the strong force) and vibration modes (the "topologies" of mesons and baryons), the epistemological nature of the Deep Dive now also allows me to provide a more logically precise and self-consistent answer: The substance of which Planck-level strings are composed is exactly the same as the substance that angels use to bind themselves to each other while dancing on the head of a pin.
If you think that's an unfair comparison for a scientific discussion, no problem: Just state exactly what scientific experiment should be performed to prove that Planck-scale strings are not composed of the same substance that angels use bind themselves to each other while dancing on the head of a pin.
If string theory is truly science, and if the half-billion dollars of critically important research funding that has been spent on it over four decades has not been a complete waste of money, then defining a simple test to prove that Planck-level string theory is more than just an untestable religious revelation gussied up with loads of equations should be no problem at all. Right?
Best Answer
OP wrote(v4):
Well, the relativistic string should not be confused with the non-relativistic material string, compare e.g. chapter 6 and 4 in Ref. 1, respectively. In contrast, the relativistic string is e.g. required to be world-sheet reparametrization-invariant, i.e. the world-sheet coordinates are no longer physical/material labels of the string, but merely unphysical gauge degree of freedom.
Moreover, in principle, all dimensionless continuous constants in string theory may be calculated from any stabilized string vacuum, see e.g. this Phys.SE answer by Lubos Motl.
OP wrote(v1):
What are strings made of?
One answer is that it is only meaningful to answer this question if the answer has physical consequences. Popularly speaking, string theory is supposed to be the innermost Russian doll of modern physics, and there are no more dolls inside that we can explain it in terms. However, we may be able to find equivalent formulations.
For instance, Thorn has proposed in Ref. 2 that strings are made of point-like objects that he calls string bits. More precisely, he has shown that this string bit formulation is mathematically equivalent to the light-cone formulation of string theory; first in the bosonic string and later in the superstring. The corresponding formulas are indeed quadratic a la harmonic oscillators (cf. a comment by anna v) with the twist that the "Newtonian mass" of the string bit oscillators are given by light-cone $P^+$ momentum. Thorn was inspired by fishnet Feynman diagrams (think triangularized world-sheets), which were discussed in Refs. 3 and 4. However, string bit formulation does not really answer the question What are strings made of?; it merely adds a dual description.
References:
B. Zwiebach, A first course in String Theory.
C.B. Thorn, Reformulating String Theory with the 1/N Expansion, in Sakharov Memorial Lectures in Physics, Ed. L. V. Keldysh and V. Ya. Fainberg, Nova Science Publishers Inc., Commack, New York, 1992; arXiv:hep-th/9405069.
H.B. Nielsen and P. Olesen, Phys. Lett. 32B (1970) 203.
B. Sakita and M.A. Virasoro, Phys. Rev. Lett. 24 (1970) 1146.