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For a free scalar field of mass $m$ in Anti-de Sitter spacetime of dimension $d+1$ and radius $L$, use coordinates withThe conformal boundary is at $z=0$. The Klein-Gordon equation gives the radial leading balance $z^2{\varphi }_{zz}-(d-1)z{\varphi }_z-m^2{L}^2\varphi \simeq 0$. A power law $z^{\Delta }$ therefore obeys the AdS scalar mass--dimension relationFor distinct roots and generic nonresonant parameters,In standard quantization, $J$ is the source of the dual local field operator; $A$ determines its expectation value after holographic renormalization, with a normalization fixed by the bulk action. ^[ref]

The Breitenlohner-Freedman bound $m^2{L}^2\ge -d^2/4$ makes $\nu$ real. At $\nu =0$ the roots coincide and a second solution involves $z^{d/2}\log z$. Resonant expansions can also contain logarithms. In the appropriate mass window, Alternative quantization of an AdS scalar field changes which coefficient acts as source. ^[ref]

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]

Daniel Z. Freedman is a theoretical physicist whose work includes supergravity and the stability of fields in Anti-de Sitter spacetime. With Peter Breitenlohner, he derived the condition known as the Breitenlohner-Freedman bound.

Under the appropriate boundary conditions, a scalar field in Anti-de Sitter spacetime can be stable even when its mass-squared parameter is negative, provided it remains above the bound. Curved background geometry changes the stability criterion familiar from flat spacetime.^[ref] His MIT profile also identifies his 1976 collaborative construction of supergravity.^[ref]

Peter Breitenlohner was a theoretical physicist whose work with Daniel Z. Freedman established the Breitenlohner-Freedman bound for fields in Anti-de Sitter spacetime.

For a scalar field in $(d+1)$-dimensional Anti-de Sitter spacetime with curvature radius $L$, the familiar form is $m^2{L}^2\ge -d^2/4$, with stability also requiring suitable boundary conditions. The negative right-hand side explains the surprising feature: a negative mass-squared parameter alone does not establish instability in this curved background. Their result links field dynamics to the geometry and boundary conditions of spacetime.^[ref]