Abstract
A common method to prepare states in AdS/CFT is to perform the Euclidean path integral with sources turned on for single-trace operators. These states can be interpreted as coherent states of the bulk quantum theory associated to Lorentzian initial data on a Cauchy slice. In this paper, we discuss the extent to which arbitrary initial data can be obtained in this way. We show that the initial data must be analytic and define the subset of it that can be prepared by imposing bulk regularity. Turning this around, we show that for generic analytic initial data the corresponding Euclidean section contains singularities coming from delta function sources in the bulk. We propose an interpretation of these singularities as non-perturbative objects in the microscopic theory.
Highlights
On the gravitational side, this statement is equivalent to saying that all that is needed at the semi-classical level to describe a state is Lorentzian initial data, that is data of the gravitational fields on a Cauchy slice Σ, see figure 1
We will call the states that describe semi-classical geometries |φ, Π since they are specified by the initial data, and they should be interpreted as coherent states of the full quantum theory
We have discussed the relation between CFT states prepared by a Euclidean path integral and coherent states of the dual bulk theory which are parametrized by a choice of initial data
Summary
To demonstrate our results in a simple and concrete setting we consider Einstein gravity in the bulk, minimally coupled to a scalar field,. The proof can be found in textbooks on PDEs, see for example section 2.2 theorem 10 (analyticity) in [39].4 To apply it to our Cauchy problem, consider a ball that includes a portion of the τ = 0 surface, e.g. as in figure 4. Given analytic initial data f, fτ it follows from the Cauchy-Kovalevskaya theorem that a unique analytic solution can be constructed in the neighbourhood of any point on the τ = 0, z > 0 surface. We will see in our examples (section 3) that this breakdown occurs at singularities which indicate locations in the bulk where delta-function sources have been turned on This is not an obstruction to extracting the corresponding λ, since unique analytic solutions can be obtained on Rd+1 minus these singular points.
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