What are strings "made of" in string theory?
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As of today, there is reason to believe that string theory itself (or rather the five different consistent versions of it) arises as the limit of a more fundamental theory that is still poorly understood and goes by the name of M-theory.
However, this still doesn't mean that the strings of string theory are made of something, and to make sense of the fact that the string description can arise as a limit to something else without there having to be 'stuff' that strings are made up of, I will talk about a toy model whose roots go back to a seminal paper of 't Hooft.
Consider a scalar theory in which the dynamical field is an [math]N\times N[/math] matrix. This means we have a Lagrangian of the form
[math]\mathcal L = N\,\mathrm{tr}(\partial_\mu\Psi\partial^\mu\Psi - V(\Psi)),[/math]
where [math]\Psi[/math] is a matrix and [math]V(\Psi)[/math] is an analytic expression in [math]\Psi[/math] consisting of [math]\Psi^2[/math] and higher order terms. The reason for including an overall factor of [math]N[/math] shall soon become clear.
What would the Feynman diagrams of such a theory look like? To get an idea of this, let's think of the components [math](\Psi)^a_b[/math] of the matrix [math]\Psi[/math] as independent fields. In general, (the trace of) a matrix product [math]\Psi^k[/math] when expressed in terms of the components is given by
[math]\mathrm{tr}(\Psi^k)=\sum_{a_1,\ldots,a_k}(\Psi)^{a_1}_{a_2}(\Psi)^{a_2}_{a_3}\cdots(\Psi)^{a_k}_{a_1},[/math]
where all the indices run from [math]1[/math] to [math]N[/math] in the sum. Since, these are the only combinations in which the component fields occur, the [math]k[/math]-point function [math]\langle (\Psi)^{a_1}_{b_1}\cdots (\Psi)^{a_k}_{b_k}\rangle[/math] is necessarily proportional to [math]\delta^{a_2}_{b_1}\delta^{a_3}_{b_2}\cdots \delta^{a_k}_{b_1}[/math] plus terms obtained by permuting indices.
This suggests that the Feynman diagrams for the matrix theory involve the following modifications to the propagators and vertices in the diagrams for the usual scalar theory, the double-lines having to do with the two indices that come attached to the components of the matrix-valued fields.
![]()
(Terms corresponding to permuted indices have been omitted because my proficiency with MS Paint is rather limited. Also missing, for the same reason, are tiny antiparallel arrows for distinguishing between upstairs and downstairs indices. More accurate renditions may be found in McGreevy's notes.)
Here is what a typical Feynman diagram for the matrix theory would look like.
![]()
In addition to integrating over all momenta available to every internal double-line, we also sum over all matrix indices that come attached to every (connected) internal single-line. In other words, for every 'face' in the Feynman diagram, we have a factor of [math]N[/math].
Now, as the usual Feynman rules will tell you, since I had included an overall factor of [math]N[/math] in the Lagrangian, there is a factor of [math]N[/math] for every vertex and a factor of [math]N^{-1}[/math] for every propagator (i.e. edge). So, all in all, if we denote the number of vertices, edges, faces in the diagram as [math]V,E,F[/math] respectively, the net factor overall carried by the diagram is [math]N^{V-E+F}[/math]. The combination [math]V-E+F[/math] is one you have probably seen in Euler's formula for polyhedra. What it tells you is that if a polyhedron is topologically a sphere with [math]h[/math] handles attached, then [math]V-E+F = 2-2h[/math]. This means that if we a embed a Feynman diagram in a surface so that all its faces are topologically equivalent to a disc (no handles within a face), then it carries a factor of [math]N^{2-2h}[/math], where [math]h[/math] is the number of handles on that surface. If you feel that I have pulled a fast one at this point, have a look at Matthew von Hippel's post, which is a much clearer exposition than I can manage.
Anyway, once you're convinced of this, we'll begin to take certain limits of the theory. Firstly, we go over to the strong coupling limit in which the coefficients of [math]\Psi^k[/math] in [math]V(\Psi)[/math] for [math]k>2[/math] become of the order of [math]1[/math] (or greater). What this tells you is that we can no longer ignore higher order Feynman diagrams with lots of vertices when we wish to compute scattering amplitudes. In other words, Feynman diagrams that fill out the surface they are embedded in become important. The situation looks somewhat like this picture which I have taken from the 't Hooft paper I mentioned at the beginning.
![]()
You'll notice that our Feynman diagrams start looking like the worldsheets traced out by strings as they move through time. Now, in addition, we also take the limit of [math]N[/math] becoming large, so that the contribution of surface-filling Feynman diagrams with more handles is suppressed by additional factors of [math]N^{-1}[/math], we get something like the genus expansion of perturbative string theory which says that the contribution of worldsheet diagrams in which closed loops of string are emitted and reabsorbed, are suppressed by a factor of the (closed) string coupling [math]g_{\mathrm{cs}} [/math].
Of course, [math]N[/math] is dimensionless while [math]g_{\mathrm{cs}} [/math] is not, but we can always introduce a fixed dimensionful parameter [math]\lambda[/math] (called the 't Hooft parameter) and make the identification [math]g_{\mathrm{cs}} =\lambda/N[/math]. So, the strong coupling, large [math]N[/math] limit of a matrix theory gives you Feynman diagrams that look and behave like string worldsheets at weak coupling! And if under certain (highly nontrivial) identifications, you find the scattering amplitudes of a matrix model in a certain limit and those of a stringy model agree, you have every right to say that the stringy model arises as a limit of the matrix model in question.
Yet, it doesn't mean that there are more elementary things that strings are comprised of. In fact, if the above toy model is indicative of how things will actually turn out in our pursuit of a fundamental theory describing the Universe as we know it, strings might not even be physical objects to begin with, but instead be effective descriptions of highly abstract entities in some limit.
However, this still doesn't mean that the strings of string theory are made of something, and to make sense of the fact that the string description can arise as a limit to something else without there having to be 'stuff' that strings are made up of, I will talk about a toy model whose roots go back to a seminal paper of 't Hooft.
Consider a scalar theory in which the dynamical field is an [math]N\times N[/math] matrix. This means we have a Lagrangian of the form
[math]\mathcal L = N\,\mathrm{tr}(\partial_\mu\Psi\partial^\mu\Psi - V(\Psi)),[/math]
where [math]\Psi[/math] is a matrix and [math]V(\Psi)[/math] is an analytic expression in [math]\Psi[/math] consisting of [math]\Psi^2[/math] and higher order terms. The reason for including an overall factor of [math]N[/math] shall soon become clear.
What would the Feynman diagrams of such a theory look like? To get an idea of this, let's think of the components [math](\Psi)^a_b[/math] of the matrix [math]\Psi[/math] as independent fields. In general, (the trace of) a matrix product [math]\Psi^k[/math] when expressed in terms of the components is given by
[math]\mathrm{tr}(\Psi^k)=\sum_{a_1,\ldots,a_k}(\Psi)^{a_1}_{a_2}(\Psi)^{a_2}_{a_3}\cdots(\Psi)^{a_k}_{a_1},[/math]
where all the indices run from [math]1[/math] to [math]N[/math] in the sum. Since, these are the only combinations in which the component fields occur, the [math]k[/math]-point function [math]\langle (\Psi)^{a_1}_{b_1}\cdots (\Psi)^{a_k}_{b_k}\rangle[/math] is necessarily proportional to [math]\delta^{a_2}_{b_1}\delta^{a_3}_{b_2}\cdots \delta^{a_k}_{b_1}[/math] plus terms obtained by permuting indices.
This suggests that the Feynman diagrams for the matrix theory involve the following modifications to the propagators and vertices in the diagrams for the usual scalar theory, the double-lines having to do with the two indices that come attached to the components of the matrix-valued fields.
(Terms corresponding to permuted indices have been omitted because my proficiency with MS Paint is rather limited. Also missing, for the same reason, are tiny antiparallel arrows for distinguishing between upstairs and downstairs indices. More accurate renditions may be found in McGreevy's notes.)
Here is what a typical Feynman diagram for the matrix theory would look like.
Now, as the usual Feynman rules will tell you, since I had included an overall factor of [math]N[/math] in the Lagrangian, there is a factor of [math]N[/math] for every vertex and a factor of [math]N^{-1}[/math] for every propagator (i.e. edge). So, all in all, if we denote the number of vertices, edges, faces in the diagram as [math]V,E,F[/math] respectively, the net factor overall carried by the diagram is [math]N^{V-E+F}[/math]. The combination [math]V-E+F[/math] is one you have probably seen in Euler's formula for polyhedra. What it tells you is that if a polyhedron is topologically a sphere with [math]h[/math] handles attached, then [math]V-E+F = 2-2h[/math]. This means that if we a embed a Feynman diagram in a surface so that all its faces are topologically equivalent to a disc (no handles within a face), then it carries a factor of [math]N^{2-2h}[/math], where [math]h[/math] is the number of handles on that surface. If you feel that I have pulled a fast one at this point, have a look at Matthew von Hippel's post, which is a much clearer exposition than I can manage.
Anyway, once you're convinced of this, we'll begin to take certain limits of the theory. Firstly, we go over to the strong coupling limit in which the coefficients of [math]\Psi^k[/math] in [math]V(\Psi)[/math] for [math]k>2[/math] become of the order of [math]1[/math] (or greater). What this tells you is that we can no longer ignore higher order Feynman diagrams with lots of vertices when we wish to compute scattering amplitudes. In other words, Feynman diagrams that fill out the surface they are embedded in become important. The situation looks somewhat like this picture which I have taken from the 't Hooft paper I mentioned at the beginning.
You'll notice that our Feynman diagrams start looking like the worldsheets traced out by strings as they move through time. Now, in addition, we also take the limit of [math]N[/math] becoming large, so that the contribution of surface-filling Feynman diagrams with more handles is suppressed by additional factors of [math]N^{-1}[/math], we get something like the genus expansion of perturbative string theory which says that the contribution of worldsheet diagrams in which closed loops of string are emitted and reabsorbed, are suppressed by a factor of the (closed) string coupling [math]g_{\mathrm{cs}} [/math].
Of course, [math]N[/math] is dimensionless while [math]g_{\mathrm{cs}} [/math] is not, but we can always introduce a fixed dimensionful parameter [math]\lambda[/math] (called the 't Hooft parameter) and make the identification [math]g_{\mathrm{cs}} =\lambda/N[/math]. So, the strong coupling, large [math]N[/math] limit of a matrix theory gives you Feynman diagrams that look and behave like string worldsheets at weak coupling! And if under certain (highly nontrivial) identifications, you find the scattering amplitudes of a matrix model in a certain limit and those of a stringy model agree, you have every right to say that the stringy model arises as a limit of the matrix model in question.
Yet, it doesn't mean that there are more elementary things that strings are comprised of. In fact, if the above toy model is indicative of how things will actually turn out in our pursuit of a fundamental theory describing the Universe as we know it, strings might not even be physical objects to begin with, but instead be effective descriptions of highly abstract entities in some limit.
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Auf Deutsch: Woraus “bestehen” Strings in der Stringtheorie?