I could understand the derivation of the "bulk-to-boundary" propagators ($K$) for scalar fields in $AdS$ but the iterative definition of the "bulk-to-bulk" propagators is not clear to me.

On is using the notation that $K^{\Delta_i}(z,x;x')$ is the bulk-to-boundary propagator i.e it solves $(\Box -m^2)K^{\Delta_i}(z,x;x') = \delta (x-x')$ and it decays as $cz^{-\Delta _i}$ (for some constant $c$) for $z \rightarrow 0$. Specifically one has the expression, $K^{\Delta_i}(z,x;x') = c \frac {z^{\Delta _i}}{(z^2 + (x-x')^2)^{\Delta_i}}$

Given that this $K$ is integrated with boundary fields at $x'$ to get a bulk field at $(z,x)$, I don't understand why this is called a bulk-to-boundary propagator. I would have thought that this is the "boundary-to-bulk" propagator! I would be glad if someone can explain this terminology.

*Though the following equation is very intuitive, I am unable to find a derivation for this and I want to know the derivation for this more generalized expression which is written as*,

$\phi_i(z,x) = \int d^Dx'K^{\Delta_i}(z,x;x')\phi^0_i(x') + b\int d^Dx' dz' \sqrt{-g}G^{\Delta_i}(z,x;z',x') \times$
$\int d^D x_1 \int d^D x_2 K^{\Delta_j}(z,x;x_1)K^{\Delta_k}(z,x;x_2)\phi^0_j(x_1) \phi^)_k(x_2) + ...$

where the "b" is as defined below in the action $S_{bulk}$, the fields with superscript of $^0$ are possibly the values of the fields at the boundary and $G^{\Delta_i}(z,x;z',x')$ - the "bulk-to-bulk" propagator is defined as the function such that,

$(\Box - m_i^2)G^{\Delta_i}(z,x;z',x') = \frac{1}{\sqrt{-g}} \delta(z-z')\delta^D(x-x')$

- Here what is the limiting value of this $G^{\Delta_i}(z,x;z',x')$ that justifies the subscript of $\Delta_i$.

Also in this context one redefined $K(z,x;x')$ as,

$K(z,x;x') = lim _ {z' \rightarrow 0} \frac{1}{\sqrt{\gamma}} \vec{n}.\partial G(z,x;z',x')$
where $\gamma$ is the metric $g$ restricted to the boundary.

How does one show that this definition of $K$ and the one given before are the same? (..though its very intuitive..)

I would also like to know if the above generalized expression is somehow tied to the following specific form of the Lagrangian,

$S_{bulk} = \frac{1}{2} \int d^{D+1}x \sqrt{-g} \left [ \sum _{i=1}^3 \left\{ (\partial \phi)^2 + m^2 \phi_i^2 \right\} + b \phi_1\phi_2 \phi_3 \right ]$

Is it necessary that for the above expression to be true one needs multiple fields/species? Isn't the equation below the italicized question a general expression for any scalar field theory in any space-time?

- Is there a general way to derive such propagator equations for lagrangians of fields which keep track of the behaviour at the boundary?

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