Lorentzian Witten Diagrams and Real-Time Orderings

Lorentzian Witten diagrams are defined by a contour, an initial state, and causal propagators. A Euclidean diagram has several inequivalent analytic continuations corresponding to time-ordered, Wightman, retarded, and out-of-time-order correlators. Their singularities and imaginary parts differ, so an ordering cannot be inferred from the same real cross ratios alone.

Required background. Euclidean preparation and Lorentzian states supplies the state map. AdS propagators supplies the causal Green functions.

Helpful background. Closed-time-path generating functionals supplies contour bookkeeping. Lorentzian kinematics and causal orderings supplies the boundary domains.

Contours define observables

On a Schwinger–Keldysh contour, every vertex receives a branch label and each internal line is a matrix $G_{ab}$ of time-ordered, anti-time-ordered, and Wightman functions. Euclidean caps may prepare the initial and final wavefunctionals. For a thermal state, the contour also implements the KMS displacement. The bulk boundary conditions at a horizon must match the desired response rather than being chosen solely for regularity.

For a two-point response,

$$G_R(x_1,x_2)=-i\,\theta(t_1-t_2) \langle[O(x_1),O(x_2)]\rangle,$$

and the bulk retarded propagator has support only when its first point lies in the causal future of the second. The real-time holographic prescription derives these objects by filling the contour with glued Lorentzian and Euclidean segments Skenderis and van Rees 2009.

First application: a retarded three-point function

Start from a Euclidean cubic contact diagram and assign boundary times $t_1,t_2,t_3$. The second-order response of $O_1$ to sources for $O_2,O_3$ is a sum of nested commutators, schematically

$$G_{R;1\mid23}=(-i)^2\theta(t_1-t_2)\theta(t_2-t_3) \langle[[O_1,O_2],O_3]\rangle+(2\leftrightarrow3).$$

In the bulk it is built from one retarded line directed toward the measured insertion and branch-summed Wightman lines. The answer vanishes unless the sources can influence the response through the boundary causal domain. Continuing the Euclidean insertion times with the ordered $i\epsilon_i$ prescription reproduces the same nested-commutator discontinuity, while the thermal retarded construction selects ingoing horizon conditions van Rees 2009.

Adversarial control: one continuation for two orderings

Assign the same $i\epsilon$ ordering to $\langle O_1O_2O_3\rangle$ and $\langle O_2O_1O_3\rangle$. When $O_1$ and $O_2$ cross a light cone, the two correlators lie on different sheets and their discontinuity gives the commutator. A shared continuation incorrectly sets this difference to zero or gives the wrong imaginary part. The failure is exposed by causal support and the spectral sign.

The evidence ceiling is a perturbative real-time boundary correlator for a specified state and contour. It is not state independent, and near-horizon secular growth or pinch singularities may require resummation. AdS integrals supplies a reusable Euclidean basis whose continuation must retain these sheet choices.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Skenderis, K., and van Rees, B. C. (2008), “Real-Time Gauge/Gravity Duality,” Physical Review Letters 101, 081601. arXiv:0805.0150.
• Skenderis, K., and van Rees, B. C. (2009), “Real-Time Gauge/Gravity Duality: Prescription, Renormalization and Examples,” Journal of High Energy Physics 2009(05), 085. arXiv:0812.2909.
• van Rees, B. C. (2009), “Real-Time Gauge/Gravity Duality and Ingoing Boundary Conditions,” Nuclear Physics B Proceedings Supplements 192–193, 193–196. arXiv:0902.4010.