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 KΔi(z,x;x′) is the bulk-to-boundary propagator i.e it solves (◻−m2)KΔi(z,x;x′)=δ(x−x′) and it decays as cz−Δi (for some constant c) for z→0. Specifically one has the expression, KΔi(z,x;x′)=czΔi(z2+(x−x′)2)Δi
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,
ϕi(z,x)=∫dDx′KΔi(z,x;x′)ϕ0i(x′)+b∫dDx′dz′√−gGΔi(z,x;z′,x′)×
∫dDx1∫dDx2KΔj(z,x;x1)KΔk(z,x;x2)ϕ0j(x1)ϕ)k(x2)+...
where the "b" is as defined below in the action Sbulk, the fields with superscript of 0 are possibly the values of the fields at the boundary and GΔi(z,x;z′,x′) - the "bulk-to-bulk" propagator is defined as the function such that,
(◻−m2i)GΔi(z,x;z′,x′)=1√−gδ(z−z′)δD(x−x′)
- Here what is the limiting value of this GΔi(z,x;z′,x′) that justifies the subscript of Δi.
Also in this context one redefined K(z,x;x′) as,
K(z,x;x′)=limz′→01√γ→n.∂G(z,x;z′,x′)
where γ 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,
Sbulk=12∫dD+1x√−g[∑3i=1{(∂ϕ)2+m2ϕ2i}+bϕ1ϕ2ϕ3]
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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