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In string theory, it is assumed that all particles can be described as quanta corresponding to the excitations of only one kind of fundamental string.

How can in principle the different motion patterns of one kind of string give rise to the whole particle zoo in the standard model? How are the different properrties, that characterize an elementary particle such as their

  • mass
  • spin
  • charges

encoded in the allowed motion pattterns of the string?

(I know that the vacuum corresponding to our standard model can not be uniquely identified at present, but I am interested in the general concepts and ideas that should in principle give rise to the different characteristics of the particles we observe.)


This is just an attempt to extract and reask the useful part of this now a bit too overloaded original question.

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1 Answer 1

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A string is a "particle with a complicated internal structure". To see the rough emergence of particle species, you may start with a hydrogen-string analogy. The hydrogen atom is a bound state of a proton and an electron. It may be in various energy eigenstates described by the quantum numbers $(n,l,m)$. They have a different angular momentum and its third polarization and different energies that mostly depend on $n$.

It's similar for a string. A string may be found in various states. The exact "spectrum" i.e. composition of these states depends on the background where the string propagates and the type of string theory (more precisely, the type of the string theory vacuum). But for the rough picture, consider string theory in the flat space, e.g. in the 26-dimensional spacetime. Take an open string. Its positions $X^\mu(\sigma)$ may be Fourier decomposed and each of the Fourier modes, labeled by a positive integer $n$, produces coordinates of a 24-dimensional harmonic oscillator.

So an open string is equivalent to a $24\infty$-dimensional harmonic oscillator (yes, it's twenty-four times infinity). Each of the directions in this oscillator contributes $Nn/ \alpha'$ to the squared mass $m^2$ of the resulting particle where $N$ is the total excitation level of the harmonic oscillators that arise from the $n$-th Fourier mode. At any rate, the possible values of the squared mass $m^2$ of the particle are some integer multiples of $1/\alpha'$. This dimensionful parameter $1/\alpha'$ is also called $1/l_{string}^2=m_{string}^2$.

The ground state of the string, $|0\rangle$ of the harmonic oscillator, is a tachyonic particle with $m^2=-1/\alpha'$ in the case of bosonic strings. These tachyons are filtered away in the superstring. The first excited state of an open string is $\alpha^\mu_{-1}|0\rangle$ which carries one spacetime Lorentz vector index so all these states behave as a vector with $m^2=0$. They give you a gauge boson. And then there are massive modes with $m^2\gt 0$. Closed strings of similar masses have twice larger number of indices, so for example, the massless closed string states inevitably produce a graviton.

So different masses of the resulting particles arise from different values of $Nn$ – and the very fact that the values may be different for different excitations is analogous to the same feature of the hydrogen atom or any other composite particle in the world. In string theory, however, one may also produce states with different values of the angular momentum – also somewhat analogous to the hydrogen atom which is a sufficient model – or different values of the electric charge and other charges.

For example, in some Kaluza-Klein-like vacua, the number of excitations of $X^5_{n}$, the Fourier modes of the (circular) fifth dimension $X^5$, will be interpreted as the electric charge and it will behave as the electric charge in all physical situations, too. There are other ways how $U(1)$ electric-like charges and other charges arise in string theory. See e.g. this popular review


of ways how Yang-Mills gauge groups and charges may emerge from different formulations and vacua of string theory. If even this review is too technical, you will have to be satisfied with the popular Brian-Greene-like description stating that particles of different mass, spin, or charges emerge from strings vibrating in different ways. I am sort of puzzled about your question – and afraid that my answer will be either too simple or too off-topic given your real question – because you must have heard and read these basic insights about string theory about hundreds of times already.

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Hi Lumo, thanks for this nice answer to this originally not my question :-). Please dont be disappointed or annoyed about it, the puzzle has a simple solution. My question here is just an attempt to reask the useful enough for this site content if this question from another user called Neo. In the corresponding metadiscussion, I asked the mods to associate this question with Neo's account since it is not my own. –  Dilaton Nov 19 '12 at 13:42
Many people here on physics SE were interested in the answer to this only reasonable question buried in the unsuccessful post. They asked for a popular anwer and they should perfectly like what you explain here. Cheers –  Dilaton Nov 19 '12 at 13:45

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