The String Coffee Table

Jim Graber wrote:

Are similar visualizations of the electron possible?

Depending on how much you can stretch your imagination, yes.

States of the string which look like spin-1/2 particles (in general) are in their bosonic oscillator ground state but have ‘spin field excitations’ turned on.

This first of all means that in terms of its spatial extension the string in a spin-1/2 particle state is in its gound state and hence oscillates as little as allowed by quantum mechanics. Imagine a Gaussian-shaped blob of string.

But there are also the fermionic degrees of freedom on the superstring, which I had mentioned before. For spin-1/2 particle states of string these are in what I propose here to call for pedagogical reasons a ‘rotor configuration’.

Ever heard of Hestenes’ school of thought called ‘Geometric Algebra’, where people are all excited about how intuitively accessible spinors become when you make proper use of Clifford algebraic notation?

The key idea of is that it is fun to think of Clifford algebra elements $\gamma^\mu$ (‘gamma matrices’) as vectors, think of products $\gamma^{[\mu}\gamma^{\nu]}$ of them as little planes (‘bivectors’) and realize that then the object

which is in the GA context sometimes called a ‘rotor’ is nothing but a prescription to rotate things by an angle alpha in the plane spanned by $\gamma^\mu$ and $\gamma^\nu$ - and is at the same time essentially a spinor!

In order to understand how this works see this post (and maybe related messages).

If what I describe in the posts at the links given above works for you and gives you a way to visualize spinors then I can provide you with a visualization of the spin-1/2 particle state of an open superstring! In that case, continue reading… :-)

With a basic understanding of Clifford algebra and spinors first note that the superstring is distinguished from the bosonic string by the fact that at every point of the superstring there sits a little Clifford generator (‘gamma-matrix’), which, as proposed here can maybe roughly (very roughly, actually) be visualized by thinking of little arrows at each point of the string.

So in contrast to the spinning particle there is not just a single copy $\gamma^\mu$ of the Clifford algebra (which contains in it all the information about rotations and spinors) but a continuum of such representations $\gamma^\mu(\sigma)$ at every $\sigma$, where $\sigma$ is a parameter running along the string. Which one of these copies should we insert into the formula for a rotor given above in order to get spinors?

The true answer involves concepts from superconformal field theory which are not terribly easy to visualize. But what essentially happens in the construction of ‘worldsheet spin fields’ is that you construct something like

This is supposed to indicate that you take bispinors as before, at seperate points on the worldsheet, and then integrate them from some reference point to the point $\sigma$ on the string. The resulting ‘integrated bispinor’ is conventionally called $H^a$, with $a \in \{0,1,2,3,4\}$.

(This procedure of integrating pairs of fermions is known as ‘bosonization’, essentially because the result is an even graded object. I have omitted a conventional factor of the imaginary unit from the definition of $H^a$ as well as lots of mathematical fine print.)

My point here is that the $H^a$ are in a way ‘averaged’ bivectors (in the Clifford algebra sense) over the worldsheet. Please note very well the quotation marks around the above five formulas are supposed to indicate that they only transport (hopefully) the general idea, while the true definition of these objects involves some technical subtleties which are however not important at all for our current goal of trying to get a heuristic picture of the spin-1/2 particle state of the string.

So my point is that if you understand intuitively why for the ordinary spin-1/2 particle the rotor

describes the spin degree of freedom of the particle (and the above links should enable you to do just that) then you can now understand how the ‘worldsheet spin field’

describes the analogous rotor for the superstring.

More precisely, if $|p,0\rangle$ is the state of the superstring in which it has center-of-mass momentum $p$ and all its oscillations are in its ground state $|0\rangle$, then the state of the string which describes a spin-1/2 particle looks like

Very roughly and on a very heuristic level, what I am trying to say here is that you can think of the superstring as a continuous chain of spin-1/2 points whose ‘averaged spin degrees of freedom produce the spin degree of freedom of the string as a whole which makes it look like a spin-1/2 particle’.

Finally note that all this is true for the open superstring. On the closed superstring there is yet another copy of the Clifford algebra at every point of the string and hence two different copies of the rotors/spin fields above.

As soon as you understand how two rotors define a differential form (see this for a simple introduction to this idea) you understand how acting with two copies of such spin fields on $|p,0\rangle$ you create a state of the closed superstring which is the quantum of a $p$-form field (a so-called Ramon-Ramond $p$-form field).

These $p$-form fields are essentially nothing but higher-$p$ generalizations of ordinary electromagnetism. More on the latter can be found here.

On the closed superstring you can also apply one rotor/spin field and instead of another such rotor turn on some of the oscillations of the string. The result is partly a spinor (due to the rotor) and partly a vector particle (due to the oscillations). This gives the gravitiono and dilatino states of the string!

Finally, if anyone reading this is feeling the desire to understand the subtleties and technical fine print which I have swept under the carpet, let me point out the references:

The canonical textbook reference is section 10.3 of Polchinski’s book. In equation (10.3.13) the true definition of the ‘string averaged bivectors’ H is given, while equation (10.3.29) gives the true definition of the ‘string rotor’ which is usually called a ‘spin field’.

Unfortunately crucial information about these spin fields is scattered all over the second volume of Polchinski’s book. For instance the crucial formula which shows that the spin fields actually do behave as spinors under multiplication with the Clifford elements y^m(s) is found in chapter 12, equation (12.4.7).

The fact that a spinor is ‘half a vector’ is nicely expressed in the absolutely crucial equation (12.4.18) of Polchinski’s book. This formula also describes the spacetime supersymmetry of the superstring, but in order to understand that one must know about the meaning of ‘superconformal ghost picture changing’. In my humble opinion all this is best learned not from Polchinski (where it is section 10.4 and the beginning of section 12.5) but from the textbook-like seminal paper

D. Friedan & E. Martinec & S. Shenker, Conformal invariance, Supersymmetry and String Theory, Nucl. Phys. B271 (1986), 93-165 .