# How are string vibration modes related to particle identity?

I understand that the vibration modes of an elementary string determines the identity of its particle. When I first heard this, I visualized a stiff circular steel band-like entity vibrating with 2, 3, 4, 5,... nodes or antinodes, the number determining the particle. This was further supported when I asked a well-known physicist how one gets from equations to physical strings, and he said he recognized the harmonic nature of the solutions.

Bu then I noticed that physicists seem to accept the image of a soft spaghetti-like string flopping about with no well-defined vibrational modes. I asked another physicist about this, and his answer was simply that a string has multiple modes, not necessarily in a harmonic progression.

So my questions are 1) How do you visualize the vibrations of a string, and 2) how are they related to particle identity?

One needs to carefully distinguish between the classical and the quantum mechanical treatment here. For simplicitly, I will talk about the bosonic string, but the pertinent points also carry over to the superstring.

# The classical string

The classical string is literally a one-dimensional object floating in some n-dimensional Lorentzian spacetime. In analogy to a relativistic particle, it traces out a worldsheet, and its motion is governed by a rule to extremize the Lorentzian area of this worldsheet, just like particles extremize the length of their worldlines between two fixed event (i.e. they follow geodesics). The action for this motion is the famous Nambu-Goto action. It is classically equivalent to the computationally much mroe tractable Polyakov action, whose equation of motion for the string coordinates $X^\mu(\sigma,\tau)$, where $\sigma$ is the spatial position along the string and $\tau$ the time along the worldsheet is just a simple wave equation: $$\left(\partial_\tau^2 - \partial_\sigma^2\right)X^\mu = 0$$ On a closed string, we have $X^\mu(l,\tau) = X^\mu(0,\tau)$ as a boundary condition. We can then write down the general form of a solution to this wave equation: $$X_\pm^\mu(\tau\pm\sigma) = \underbrace{\frac{x^\mu + c^\mu}{2}}_{\text{initial position}} + \underbrace{\frac{\pi\alpha'}{l}p^\mu(\tau\pm\sigma)}_{\text{center-of-mass motion}} + \underbrace{\sum_{n\in\mathbb{Z}-\{0\}} \frac{\alpha^{\pm\mu}_n}{n}\mathrm{e}^{-\frac{2\pi\mathrm{i}}{l}n(\tau\pm\sigma)}}_{\text{vibrational modes}}$$ Here, the $x$ and $c$ are for our purposes irrelevant constants, and likewise the $p$ and $\alpha'$. The relevant term is the last one: The $\frac{1}{n}$ is a normalization factor, the $\alpha_n$ a Fourier coefficient (i.e. one should think of a function $\alpha^{\pm\mu} : \mathbb{Z}\to\mathbb{R}, n\mapsto \alpha^{\pm\mu}_n$ as being the Fourier transform of $X^\mu_\pm$), and the exponential simply the $n$-th harmonic vibration on the string. Neither the derivation of this solution nor its exact mathematical form is relevant for the basic thrust of this answer.

As you might recall, waves can be split into a right-moving and a left-moving part, the $X_-$ is "right"-moving and the $X_+$ is "left"-moving in our convention. This is nothing but the decomposition of the string motion into the different parts of its motion. Most notably, the last term is just a sum over the Fourier modes of the string - it represents the actual, classical vibration of the string. Each $\alpha^{\pm\mu}_n$ is a unique, well-defined harmonic mode; the entirety of the modes gives a vibration which of course need not correspond to any nice harmonic vibration.

Visualizing these vibrations is as easy as imagining a (non-streching) thin wristband flying through a room. Once in a lifetime you might get a pure harmonic vibration on it, but generally, it will just wobble about in a superposition of the many possible modes.

# The quantum string

Now it gets non-visual. After the success of quantum mechanics and in particular the process of quantization of classical theories, the physicist of course asks what the quantum theory corresponding to the classical string is. In this quantum theory, the former state variables of the string - the $X^\mu$ - become operators that act upon the quantum state of space. To each string, there is an associated space of state, but this space does not contain "vibrations". Together with the $X^\mu$, every $\alpha^{\pm\mu}_n$ has become an operator, too. There is a lowest energy state for the string, called $\lvert 0\rangle$, and we call states like $\alpha^{+\mu}_n\alpha^{-\nu}_m\lvert 0\rangle$ "excitations" of the string.

The former vibrational modes now each generate such states. In analogy to quantum field theory, we anticipate that these states will turn out to have some sort of particle nature (whatever that means), and in the end, they do - they correspond to usual QFT particles in the low-energy approximation of string theory.

The somewhat baffling thing is that not all such excitations are actual states. It is here that the quantum theory begins to strongly diverge from the classical picture of superposing harmonic waves on the string: Only very specific combinations of the different $\alpha^{\pm\mu}_n$ (determined by "level-matching", their ghost number, or other equivalent criteria) acting on the ground state produce physically meaningful quantum states. The others either don't exist to begin with or are equivalent to at first glance completely different combinations, for technical reasons related to the fact that the theory of the string is a gauge theory.

It is a gauge theory because I tricked you - string theory not only considers the $X^\mu$ as dynamical variables but also the worldsheet metric. The equations of motion for the string only look like the classical wave equation in the so-called "flat gauge", but the quantization procedure must respect the original character of the theory as a gauge theory.

It is therefore unreasonable to expect a direct map between the physical idea of vibrations of the string and the states that the fully quantized theory will exhibit. Sorry.