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While studying the AdS/CFT correspondence, one encounters very early the example of a scalar field in AdS. The general solution to the Klein-Gordon equation in the limit $z\rightarrow 0$ may be expressed as a linear combination of two solutions:

\begin{equation} \phi(z,\textbf{x})\rightarrow z^{\Delta_+}[A(\textbf{x})+O(z^2)]+z^{\Delta_-}[B(\textbf{x})+O(z^2)] \end{equation}

with $\Delta_\pm=\frac{d}{2}\pm\sqrt{\frac{d^2}{4}+m^2l^2}$.

In order to quantize the field, one can show that for $m^2>0$ only the plus sign leads to normalizable modes. However, unitarity in AdS space allows a slightly tachyonic bulk field, $m^2>-\frac{d^2}{4}$, in which modes of both signs are normalizable and we are free to choose either one.

When both modes are normalizable, how should one decide which quantization to choose (i.e which coefficient to set to $0$)? Are these boundary conditions related to two different physics in the bulk?

I should point out that I know how these are related to the boundary CFT theory: I think my question could be formulated without referring to the correspondence at all and is related to my lack of understanding of how BCs affect a QFT.

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As you have mentioned, the Breitenlohner-Freedman bound implies that the mass of the bulk scalar must be bounded below $m^2 L^2 > - \frac{d^2}{4}$.

If the mass of the scalar field $m^2 L^2 > -\frac{d^2}{4}+1$, then only one boundary condition is allowed, namely the Dirichlet boundary conditions. Here, we take $B(x)$ to be the source and $A(x)$ to be the response. This is known as standard quantization.

If the scalar field has mass $-\frac{d^2}{4} < m^2 L^2 < - \frac{d^2}{4} + 1$, then we can also impose Neumann boundary conditions which allow us to choose $A(x)$ as the source and $B(x)$ as the response. This is known as alternate quantization.

Different choices of boundary conditions define different theories in the bulk. Note that both boundary conditions give the same classical theory (since that only depends on equations of motion), but their path integrals are different so the quantum theories differ (the actions have different boundary terms).

In the context of AdS/CFT, the two different bulk theories (which differ only by their boundary conditions) are dual to different boundary theories. A famous example where this happens is Vasiliev higher spin theory which is dual to critical $O(N)$ vector model (with standard quantization) or free $O(N)$ vector model (with alternative quantization).

This stuff is discussed in https://arxiv.org/abs/hep-th/9905104 (Klebanov and Witten)

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  • $\begingroup$ as a small correction to the last paragraph, Vasiliev theory was shown to suffer from internal inconsistencies, which can eventually be linked to the fact that not every nice CFT has a nice AdS dual and vector models do not. $\endgroup$
    – John
    Commented Jun 11 at 20:16
  • $\begingroup$ @John can you explain what these inconsistencies are? Also, please provide a reference for this. $\endgroup$
    – Prahar
    Commented Jun 11 at 20:23
  • $\begingroup$ it is not yet a theory, it is the procedure to generate the most general ansatz for interactions. Nobody knows how to make it produce meaningful interactions (in the sense of giving finite physical observables). Moreover, this type of theories does not seem to exist (not every CFT is dual to something nice), see inspirehep.net/literature/1596923 for general nonexistence and inspirehep.net/literature/1388388 for why Vasiliev's equations are incomplete $\endgroup$
    – John
    Commented Jun 18 at 18:57

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