Boundary Conditions

The boundary condition admits independent periodic functions of ${\sigma }^{+}$, ${\sigma }^{-}$ for left-moving and right-moving modes respectively. It also admits linear solutions where the coefficients of ${\sigma }^{+}$, ${\sigma }^{-}$ are equal. The most general solution admits an expansion

$X_L^{\mu }({\sigma }^{+})=\frac{1}{2}(x^{\mu }-c^{\mu })+\frac{1}{2}{l}^2{p}^{\mu }{\sigma }^{+}+\frac{i}{2}l{\sum }_{n\ne 0}\frac{1}{n}{\tilde{\alpha }}_n^{\mu }{e}^{-2in{\sigma }^{+}}$

$X_R^{\mu }({\sigma }^{-})=\frac{1}{2}(x^{\mu }+c^{\mu })+\frac{1}{2}{l}^2{p}^{\mu }{\sigma }^{-}+\frac{i}{2}l{\sum }_{n\ne 0}\frac{1}{n}{\alpha }_n^{\mu }{e}^{-2in{\sigma }^{-}}$.

• It is convenient to set $c^{\mu }=0$.
• The constant $x^{\mu }$ can be interpreted as the position of the centre of mass of a string in spacetime ${\mathbb{R}}^{1,D-1}$.
• The coefficient of the linear terms, $p^{\mu }$, has the interpretation of spacetime momentum.
• The coefficients ${\tilde{\alpha }}_n^{\mu }$ and ${\alpha }_n^{\mu }$ correspond to the excitations of the left-moving and right-moving modes of the string. We require that the coordinate functions $X^{\mu }$ are real, i.e., $({\alpha }_n^{\mu }{)}^{\ast }={\alpha }_{-n}^{\mu }\text{ and }({\tilde{\alpha }}_n^{\mu }{)}^{\ast }={\tilde{\alpha }}_{-n}^{\mu }$.
• Finally, $l$ is a parameter with dimensions of length. By convention, we set $l:=1/\sqrt{\pi T}$, and often introduce another related parameter ${\alpha }^{\prime }:=l^2/2=1/2\pi T$.

Open Strings

Open strings admit a similar mode expansion. Since the string now has a spatial boundary, we must re-examine the action principle and ensure that the boundary terms cancel out. Variation of the gauge-fixed action principle produces a boundary condition

$\delta S=-T\int d\tau {\left[{\partial }_{\sigma }{X}_{\mu }\delta X^{\mu }\right]}_{\sigma =0}^{\sigma =\pi }$

which must vanish to have a consistent action principle.

There are two ways to cancel boundary variations:

1. Dirichlet boundary conditions: $\delta X^{\mu }{|}_{\sigma =0,\pi }=0$
2. Neumann boundary conditions: ${\partial }_{\sigma }{X}_{\mu }{|}_{\sigma =0,\pi }=0$.

In principle, we could choose either of the two boundary conditions independently for each index $\mu =0,1,...,D-1$ and boundary $\sigma =0,\pi$. Dirichlet boundary conditions break Poincaré symmetry though.

Here, we consider Neumann boundary conditions for all coordinates at both boundaries. This corresponds to a freely propagating open string. This forces the left-moving and right-moving modes to combine into trigonometric functions. The general solution takes the form

$X^{\mu }(\tau ,\sigma )=x^{\mu }+l^2{p}^{\mu }\tau +il{\sum }_{n\ne 0}\frac{1}{n}{\alpha }_n^{\mu }{e}^{-in\tau }\cos (n\sigma )$,

where the coefficients have a similar interpretation as before.

Reminder: $T_{\pm }={\partial }_{\pm }{X}^{\mu }{\partial }_{\pm }{X}_{\mu }=0$.