The Minor Complication with Open String Poisson Brackets

String Theory without the Baggage, Episode 30.

Hey Friends,

This one might sound similar to last time, but if anything is more important. The issue of open strings is tricky, and we again take on the task of filling in some of the gaps in string theory texts. This time we find an overlooked subtly that preempts a bit of the complication in the quantum theory.

We’re calling it String Theory without the Baggage, not String Theory without the gory details!

Best,

Sean

XC Ski Season officially opens today!!!

Lighting Summary of the Closed String

Last time we took a look at how the symplectic form for the string was adapted to closed string boundary conditions. That is, the periodic case where

\(X^{\mu}(\tau,\sigma) = X^{\mu}(\tau,\sigma + \pi).\)

Yes, just π. It was a little weird, but now that we know that’s implicitly part of what it takes to get us to

\(\delta(x) = \sum_{n\in\mathbb{Z}}e^{inx},\)

which is defined on the interval -π to π

Specifically, we saw that

\(\omega = \kappa_{0}\, dx^{\mu} \wedge dp_{\mu} + \sum_{n=1}^{\infty} \kappa_{n}\,d\alpha^{\mu}_{-n} \wedge d\alpha_{n \mu} +\bar{\kappa}_{m} \,d\bar{\alpha}^{\mu}_{-n} \wedge d\bar{\alpha}_{n \mu} \)

gave rise to the Poisson brackets

\(\left\{ A , B \right\} = \kappa_{0}\left(\frac{\partial A}{\partial x^{\lambda}}\frac{\partial B}{\partial p_{\lambda}} - \frac{\partial B}{\partial x^{\lambda}}\frac{\partial A}{\partial p_{\lambda}} \right) + \sum_{n=1}^{\infty}\kappa_{n}\left(\frac{\partial A}{\partial \alpha^{\lambda}_{-n}}\frac{\partial B}{\partial \alpha_{n\lambda}} - \frac{\partial B}{\partial \alpha^{\lambda}_{-n}}\frac{\partial A}{\partial \alpha_{n\lambda}}\right) + \sum_{n=1}^{\infty}\bar{\kappa}_{n}\left(\frac{\partial A}{\partial \bar{\alpha}^{\lambda}_{-n}}\frac{\partial B}{\partial \bar{\alpha}_{n\lambda}} - \frac{\partial B}{\partial \bar{\alpha}^{\lambda}_{-n}}\frac{\partial A}{\partial \bar{\alpha}_{n\lambda}}\right).\)

We verified by direct computation that

\(\left\{ X^{\mu}(\tau,\sigma) , \dot{X}_{\nu}(\tau,\sigma^{\prime}) \right\} = \ell^{2}\,\delta^{\mu}_{\nu}\,\delta(\sigma-\sigma^{\prime}).\)

consistent with the symplectic pairing:

\(X^{\mu}\leftrightarrow P_{\mu} = \dot{X}^{\mu}.\)

Today we’ll repeat this procedure - using essentially the same formulae - for the open string.

The Open String

The boundary conditions for the open string demand that no momenta should flow off the endpoints. That is, they are so-called Neumann conditions:

\(X^{\prime\mu} = \frac{\partial X^{\mu}}{\partial \sigma} \Big|_{\sigma = 0,\pi} = 0.\)

This gives a slightly different mode expansion,

\(X^{\mu} = x^{\mu} + \ell^{2}p^{\mu}\tau + \sum_{n\neq 0} \frac{i\ell}{n} \alpha^{\mu}_{n} e^{-in\tau}\cos n\sigma,\)

which basically amounts to identifying the left and right moving mode coefficients. The associated canonical string momentum is

\(\dot{X}^{\mu} = \ell^{2}p^{\mu} + \ell\sum_{n\neq 0} e^{-in\tau}\cos n\sigma.\)

Let us now compute the bracket { X , P }.

The first term is the same as with the closed string case,

\(\left\{ X^{\mu}(\tau,\sigma) , \dot{X}_{\nu}(\tau,\sigma^{\prime}) \right\}_{0} = \ell^{2}\delta^{\mu}_{\nu}.\)

The mode expansion terms:

\(\left\{ X^{\mu}(\tau,\sigma) , \dot{X}_{\nu}(\tau,\sigma^{\prime}) \right\}_{n} = \kappa_{n}\left( \frac{\partial X^{\mu}(\tau,\sigma)}{\partial \alpha_{-n}^{\lambda}}\frac{\partial \dot{X}_{\nu}(\tau,\sigma^{\prime})}{\partial \alpha_{n\lambda}} - \frac{\partial \dot{X}^{\mu}(\tau,\sigma^{\prime})}{\partial \alpha_{-n}^{\lambda}}\frac{\partial X_{\nu}(\tau,\sigma)}{\partial \alpha_{n\lambda}} \right).\)

are now given in terms of

\(\frac{\partial X^{\mu}(\tau,\sigma)}{\partial \alpha_{-n}^{\lambda}} = -\delta^{\mu}_{\lambda}\frac{i\ell}{n}e^{in\tau}\cos n\sigma,\)

and

\(\frac{\partial \dot{X}_{\nu}(\tau,\sigma^{\prime})}{\partial \alpha_{n\lambda}} =\delta_{\nu}^{\lambda} \ell e^{-in\tau}\cos n\sigma^{\prime}.\)

Putting these together we find, keeping in mind that cosine is an even function of its argument,

\(\left\{ X^{\mu}(\tau,\sigma) , \dot{X}_{\nu}(\tau,\sigma^{\prime}) \right\}_{n} = \frac{-2i\kappa_{n}}{n}\delta^{\mu}_{\nu}\ell^{2}\cos n\sigma\cos n\sigma^{\prime}.\)

In the closed string case, we found that the normalization factors were given by

\(\kappa_{n} = in.\)

Therefore,

\(\left\{ X^{\mu}(\tau,\sigma) , \dot{X}_{\nu}(\tau,\sigma^{\prime}) \right\} = \delta^{\mu}_{\nu}\ell^{2}\left( 1 + 2\sum_{n=1}^{\infty}\cos n\sigma\cos n\sigma^{\prime}\right).\)

Which is almost we we want, but not quite.

An Orientation Wrinkle

We were expecting to find a delta function, and hoping that the various cosine modes would conspire to give us one, as happened with the closed string. In that context

\(\delta(x) = \sum_{n\in\mathbb{Z}}e^{inx} = 1 + \sum_{n=1}^{\infty}(e^{inx} + e^{-inx}) = 1 + 2\sum_{n=1}^{\infty}\cos nx.\)

Here we have something different. Recall from elementary trigonometry that

\(\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta.\)

Therefore

\(\cos\alpha \cos\beta =\cos(\alpha+\beta) + \cos(\alpha-\beta).\)

This suggests that

\(1 + 2\sum_{n=1}^{\infty}\cos n\sigma \cos n\sigma^{\prime} = 1 + 2\sum_{n=1}^{\infty}\left(\cos n(\sigma + \sigma^{\prime}) + \cos n(\sigma - \sigma^{\prime})\right) \)

Finally, then we have

\(\left\{ X^{\mu}(\tau,\sigma) , \dot{X}_{\nu}(\tau,\sigma^{\prime}) \right\} = \frac{1}{2}\ell^{2}\delta^{\mu}_{\nu}\left( \delta( \sigma +\sigma^{\prime}) + \delta( \sigma - \sigma^{\prime} )\right).\)

Curiously, none of GSW, Polchinski or either of Kaku’s texts highlighted this fact. Of course, this fact is dealt with quite explicitly in the relevant mathematicians’ texts.

Remarks on the difference.

Now for the open string X is invariant under the map that flips the orientation of σ:

\(X^{\mu}(\tau,\sigma) = X^{\mu}(\tau,-\sigma).\)

This is notably not the case for closed strings, where this map also exchanges the modes

\(\sigma\rightarrow -\sigma \Rightarrow \alpha^{\mu}_{n}\leftrightarrow \bar{\alpha}^{\mu}_{n}.\)

This is not entirely unexpected, as open strings manifest in the quantum theory as unorientated strings. That is, we take this symmetry seriously. It’s fun to see hints of how that will express itself already.

This is done by twisting the theory so as to enforce that worldsheet reflection symmetry, locally.

In the language of Vertex Operator Algebras, we have the option to study the untwisted and twisted versions of the same theory.