Pre-holography

We construct a symplectic isomorphism $\mathfrak{h}$ from classical Klein Gordon solutions of mass $m$ on ($d+1$)-dimensional Lorentzian anti–de Sitter space (equipped with the usual symplectic form) to a certain symplectic space of functions on its conformal boundary (only) for all integer and half-integer $\Delta$ ($=\frac{d}{2}+\frac{1}{2}(d^2+4m^2{)}^{1/2}$). $\mathfrak{h}$ induces a large family of new examples of Rehren’s algebraic holography in which the net of local quantum Klein Gordon algebras in AdS is seen to map to a suitably defined net of local algebras for the (generalized free) scalar conformal field with anomalous dimension $\Delta$ on $d$-dimensional Minkowski space (the AdS boundary). Relatedly, we show for these models that Bertola et al.’s boundary-limit holography becomes a quantum duality (only) if the test functions for boundary Wightman distributions are restricted in a particular way.

Figure 1

The support (dark shading) of a “typical” classical solution for the bulk massless scalar field on ${\mathrm{AdS}}_{1+1}$ and a choice (light shading) of chart $\mathcal{P}$ for Poincaré coordinates. (See endnote 11.)