Is the one-dimensional string actually a real string, with some diameter to its cross section -- just as a rope has some thickness?

Question

Would not a one-dimensional string be just a Platonic idea, and not actually physical? Because if physical, the one dimensional string would comprise a line of point particles, each point particle being subject to the Heisenberg uncertainty principle along the other two axes where the point particle is located on the string (the other two axes being the ones orthogonal to the string at the location where the point particle exists).

Or, is the one-dimensional string actually a real string, with some diameter to its cross section -- just as a rope has some thickness?

Answer 1

Within string theory the string is a chord that has vibrations that are quantum mechanical. From the perspective of string theory alone there is nothing that composes the string. The string is described by an equation of the form below, which is somewhat simplified: $$ X^\mu~=~x^\mu~+~p^\mu\tau~+~\sum_{n=-\infty}^\infty \alpha^\mu_ne^{in(\sigma~-~\tau)}. $$ Here $\sigma$ parameterizes the spatial dimension of a string, and $\tau$ gives the time, which then defines the string surface. For a string in a loop this is a cylinder, and for an open string this is a ribbon. The operator $\alpha^\mu_n$ with $\alpha^\mu_{-n}~=~\bar\alpha^\mu_n$ is the string mode operator. These are essentially harmonic oscillator operators that raise and lower states. What is important for physics is the target map from the string world sheet to spacetime. The field theory on the string world sheet, which is a deep subject in its own right, is mapped to spacetime as a conformal field theory.

It is a gadget whereby quantum fields are parameterized along. This permits structure which one is not able to find easily in ordinary quantum field theory. As such from the perspective of string theory one should regard this largely as a sort of machinery.

Having said that however, we could well enough think of a string as a flux tube. In fact this is in part where the stringy idea originated. An open string at the string scale $\ell_s~\simeq~\sqrt{8\pi}\ell_p$, $\ell_p~=~\sqrt{G\hbar/c^3}~=~1.6\times 10^{-33}cm$ is similar to a meson with two quarks at the ends connected by a flux tube of QCD gauge bosons or gluons. In fact within M-theory a form of D-brane is the $D0$-brane, and these are the endpoints of $D1$-branes. The $D1$-brane is an STU-dual of the string. An open string correspondingly connected to a $Dp$-brane has Chan-Paton factors at their endpoints connected to the brane. These also are in a sense analogous to quarks. So while we have had this great theoretical advance with strings, it has elements taking us back to particles!

Answer 2

While I agree with Lawrence B. Crowell's answer, let me formulate answers to the OP in a more elementary language.

String theory is really a theory of strings, and can not be reduced to the study of its "constituent points", simply because these would-be constituent points can not be identified in an unambiguous manner. In other words, you can choose a parametrization of your string (say we take an open string and label the endpoints by $x=0$ and $x=1$) and then consider the point $x=0.4$. But this point has no physical reality, as another observer could choose a different parametrization. And it is fundamental in string theory that all parametrizations are equivalent.

You then mention the Heisenberg uncertainty principle. And yes, this principle indeed applies: the string, when quantized, has some fuzzyness that is taken into account by the path integral formulation of string theory (this is similar to what is done with quantum fields).

For your last question, yes the string is a physical string, which has a characteristic length (usually called the string length $l_s = \sqrt{\alpha '}$ ; but this is not the absolute length of the string, as this length can vary) and a cross section, which is vanishing in the classical description.