Simple bulk reconstruction in anti-de Sitter/conformal field theory correspondence

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The free scalar commutator itself is not nonzero for a time-like separation; however, it is singular on

the light-cone, which dominates the commutator if we regularize the local operators by a smearing.

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where t = (x − π )/2. If we regard the integrand as a periodic function, it is singular at x = 0.

Note that it is regular if ( − d)/2 is a non-negative integer and eiπ = 1, but these two conditions are not consistent with d = odd. Thus, for large n the contribution from the region near x =

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or |x| − d for eiπ = 1, the contribution from the region near x = 0 for large n is O(1/n−d+1 ).

Combining these with ψnlCF T → n−d/2 , we find φ(ρ = 0, t = 0) = n O(nd/2−1 ) aˆ†n00 + h.c..

The summation n O(nd/2−1 ) diverges, and then the time integral of Eq. (A43) is localized

on an arbitrary small region containing t = ±π /2.

For d = even, the smearing function includes the log cos t factor and it is singular at t =

±π /2 as a periodic function. Then, for this case also, we can see that the discussions for d =

odd can be applied.

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