# Fourier Expansions for Closed Strings and Parity

I'm revising some string basics, and have come across the following problem. For closed strings one introduces the worldsheet parity operator

$$\Pi : \sigma \mapsto \ell-\sigma$$

where $\ell$ is the string length. In order to relate the left-movers and right-movers by parity in the Fourier expansion one assumes that $X$ and $P$ are parity even.

But why is this reasonable? I don't understand what forces $X$ and $P$ to have a specific behaviour under parity. Indeed surely a generic closed string will not have $X(\sigma)=X(\ell-\sigma)$!

Is there something I'm missing here? Thanks in advance for any helpful thoughts.

Edit

I wonder whether the notes I'm reading are only dealing with unoriented string theory. Perhaps that would be the reason for assuming the evenness of $X$ and $P$. Could someone possibly confirm that I've got the correct meaning of unoriented?

• Yes, it concerns unoriented theories. Without orientation, "increasing" $\sigma$ should be equivalent to "decreasing" $\sigma$, so a new symmetry $\sigma \to l-\sigma, \tau'=\tau$ is needed, generated by the worldsheet parity operator. In consistent string theories, only parity-even states are kept (Ref : Polchinski, Vol $1$, page $29$). – Trimok Dec 11 '13 at 19:58

What you're missing is that the condition $$\Pi |\psi\rangle = |\psi\rangle$$ does not imply $X(\sigma)=X(\ell-\sigma)$ – a condition which would force the closed string to go back and forth along the same path and effective become an open string.
Instead, the condition above implies (is equivalent to) a much weaker condition that the complex amplitude $A_{\rm forward}$ for the closed string to be in a particular generic profile $X(\sigma)$ is the same as the complex amplitude $A_{\rm backward}$ that the string goes along the same path in the opposite direction, $X(\sigma)\to X(\ell-\sigma)$.
It means that in the theory constrained by $\Pi=1$ above, we only allow states composed of $$|\psi\rangle_{\rm forward} + |\psi\rangle_{\rm backward}$$ where the two pieces only differ by the direction of the string (whether the $\sigma$ is increasing or decreasing along a particular part of the string). The relative sign between the two terms is the eigenvalue of $\Pi$ when the ket vectors are properly normalized.