Breitenlohner-Freedman bound

The Breitenlohner-Freedman bound permits a scalar field in Anti-de Sitter spacetime to have a moderately negative squared mass without the instability that the same parameter would imply in flat spacetime. In ${\mathrm{AdS}}_{d+1}$ of curvature radius $L$, the bound is $m^2{L}^2\ge -d^2/4$ for the standard scalar dynamical stability problem with appropriate boundary conditions.

The near-boundary exponents satisfy $\Delta (\Delta -d)=m^2{L}^2$, hence ${\Delta }_{\pm }=d/2\pm \sqrt{d^2/4+m^2{L}^2}$. The bound keeps these exponents real. The spacetime geometry and boundary conditions are part of the dynamical stability statement; a negative squared mass cannot be classified without them.^[ref]