Bulk-to-Boundary and Bulk-to-Bulk Propagators

AdS propagators are Green functions with specified boundary falloff and analytic prescription. Their normalization fixes every later OPE coefficient and exchange residue. This page constructs the Euclidean scalar bulk-to-boundary kernel in Poincaré AdS, then separates it from Feynman, Wightman, and retarded bulk-to-bulk functions.

Required background. Bulk fields and boundary operators supplies $m^2L^2=\Delta(\Delta-d)$. The GKPW generating functional supplies the source limit.

Helpful background. Causal Green functions supplies the Lorentzian distinctions. Dictionary normalization and global data supplies the coefficient checks.

The normalized scalar kernel

In Euclidean Poincaré AdS$_{d+1}$,

$$ds^2=L^2\frac{dz^2+d\mathbf x^2}{z^2}, \qquad z>0,$$

a scalar solution with standard quantization behaves as $\phi\sim z^{d-\Delta}\phi_{(0)}+z^\Delta A$. Translation, rotation, dilation, and inversion covariance fix the kernel up to normalization:

$$K_\Delta(z,\mathbf x;\mathbf x') =C_\Delta\left(\frac{z}{z^2+\lvert\mathbf x-\mathbf x'\rvert^2}\right)^\Delta, \qquad C_\Delta=\frac{\Gamma(\Delta)}{\pi^{d/2}\Gamma(\Delta-d/2)}.$$

For $\Delta>d/2$, distributional integration gives $z^{\Delta-d}K_\Delta\to\delta^{(d)}(\mathbf x-\mathbf x')$, so $\phi=\int d^dx'K_\Delta\phi_{(0)}$ has the required source. The normalization and the exceptional cases requiring analytic continuation follow from the original normalized correlator construction Freedman et al. 1999.

The Euclidean bulk Green function instead satisfies

$$(-\nabla^2+m^2)G_E(X,Y)=\frac{\delta^{d+1}(X-Y)}{\sqrt g}$$

with a declared boundary condition. It depends on the AdS chordal invariant and has a spectral representation. In the Breitenlohner–Freedman window both $\Delta_+$ and $\Delta_-$ may be admissible; choosing one changes the theory.

First application: covariance and delta normalization

Under $(z,\mathbf x)\mapsto(a z,a\mathbf x)$, $K_\Delta$ scales as $a^{-\Delta}$ in the boundary coordinate, exactly the transformation required for a source of an operator of dimension $\Delta$. To verify the delta limit, set $\mathbf x-\mathbf x'=z\mathbf y$; the integral becomes

$$z^{\Delta-d}\int d^dx'K_\Delta f(\mathbf x') =C_\Delta\int d^dy\frac{f(\mathbf x-z\mathbf y)}{(1+\mathbf y^2)^\Delta} \longrightarrow f(\mathbf x).$$

The stated $C_\Delta$ makes the last integral unity. This single fixture detects an omitted $L$ power, an incorrect $\sqrt g$, or a mismatch between source and normalizable falloffs.

Adversarial control: the wrong causal propagator

Replace $G_E$ in a Euclidean exchange diagram by a Lorentzian retarded propagator. The latter has causal support and is not an elliptic inverse on Euclidean AdS; its singular support and boundary values fail the Euclidean Green equation. Similarly, replacing a retarded propagator by a Feynman one changes a response function into a time-ordered correlator. Analytic continuation is part of the observable, not a cosmetic $i\epsilon$.

The evidence ceiling is an exactly normalized free Green function on a fixed AdS background. Interactions, loop boundary conditions, $1/N$, $\alpha'$, KK, and state dependence enter later. Contact diagrams are the first use of the kernel in perturbation theory.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Breitenlohner, P., and Freedman, D. Z. (1982), “Stability in Gauged Extended Supergravity,” Annals of Physics 144, 249–281. doi:10.1016/0003-4916(82)90116-6.
• Freedman, D. Z., Mathur, S. D., Matusis, A., and Rastelli, L. (1999), “Correlation Functions in the CFT$_d$/AdS$_{d+1}$ Correspondence,” Nuclear Physics B 546, 96–118. arXiv:hep-th/9804058.
• Witten, E. (1998), “Anti-de Sitter Space and Holography,” Advances in Theoretical and Mathematical Physics 2, 253–291. arXiv:hep-th/9802150.