# Poincare invariance of Dirichlet and Neumann boundary conditions

The action which describes a string propagating in a $D$ dimensional spacetime, with given metric $g_{\mu\nu}$, is given by the Polyakov action $$S_{\text{p}}=-\frac{T}{2}\int \mathrm{d}\sigma\mathrm{d}\tau\sqrt{-h}\eta^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}g_{\mu\nu}\tag{1}$$ where the symbols have their usual meaning. It is not hard to check that action is invariant under Poincare transformations $$\delta X^{\mu}(\sigma,\tau)=a^{\mu}_{~~~\nu}X^{\nu}(\sigma,\tau)+b^{\mu}\tag{2}.$$ When all the dust is settled (i.e. after gauge fixing and Weyl transformations) the Polyakov action becomes $$S_{\text{P}}=\frac{T}{2}\int \mathrm{d}\sigma\mathrm{d}\tau \left((\dot{X})^{2}-(X')^{2}\right)\tag{3}$$ where $\dot{X}=\partial_{\tau}X^{\mu}$ and $X'=\partial_{\sigma}X^{\mu}$. Variation with respect to $X^{\mu}$ yields the equation of motion $$(-\partial^{2}_{\tau}+\partial^{2}_{\sigma})X^{\mu}-T\int\mathrm{d}\tau\left[X'\delta X^{\mu}|_{\sigma=\pi}+X'\delta X^{\mu}|_{\sigma=0}\right]=0.\tag{4}$$ The $\sigma$ boundary terms tell us what type of strings we have, either closed or open strings.

• For open string equation (4) becomes $(-\partial^{2}_{\tau}+\partial^{2}_{\sigma})X^{\mu}=0$ where we assume that the end points of the string follow the Neumann boundary conditions $$\partial_{\sigma}X^{\mu}(\tau,\sigma)=\partial_{\sigma}X^{\mu}(\tau,\sigma+n).\tag{5}$$ One interesting feature is that the Neumann boundary conditions remains invariant under global Poincare transformation since \begin{eqnarray} \partial_{\sigma}X'^{\mu}|_{\sigma=0,n} & = & \partial_{\sigma}\left(a^{\mu}_{~~~\nu}X^{\nu}(\sigma,\tau)+b^{\mu}\right)|_{\sigma=0,n} \\ & = & a^{\mu}_{~~~\nu}~\partial_{\sigma}X^{\nu}|_{\sigma=0,n}\\ & = & 0 \\ \end{eqnarray}

• Whereas the Dirichlet boundary conditions $$X^{\mu}(\tau,\sigma=0)=X^{\mu}_{0}\qquad\qquad X^{\mu}(\tau,\sigma=n)=X^{\mu}_{n}$$ break the Poincare invariance, as $$X'^{\mu}|_{\sigma=0,n}=\left(a^{\mu}_{~~~\nu}X^{\nu}(\sigma,\tau)+b^{\mu}\right)|_{\sigma=0,n}\neq X^{\mu}_{0,n}$$ which simply means that under a Poincare transformation the ends of the string actually change.

Does the spectrum of string excitations keep any signature of this (non) invariance under Poincare transformations? If so, how can that result be interpreted?

• v5: How did $\pi$ become $n$ in various places? – Qmechanic Nov 19 '16 at 18:33