Holographic Initial States and Euclidean Caps

A Euclidean cap prepares a Lorentzian holographic state only after its topology, sources, regularity, and junction data are specified. A smooth source on the cap produces normalizable Lorentzian initial data after the source is turned off; a boundary source that persists at the join instead changes the Lorentzian Hamiltonian. Regularity of the cap and finite norm are independent tests.

Required background. Euclidean Preparation and Lorentzian State Dictionaries supplies the basic state map, and Schwinger–Keldysh Contours and Real-Time Bulk Geometries supplies the gluing rules.

Helpful background. Initial Density Matrices and Contour Boundary Conditions fixes the QFT data, while Initial Density Matrices and Boundary EFT explains boundary-localized state deformations.

Euclidean wavefunctionals and normalizable data

For a boundary field configuration $\varphi_0$ at $\tau=0$, a Euclidean half-space path integral defines

$$\Psi_{\lambda_E}[\varphi_0] =\int_{\phi(0)=\varphi_0}^{\mathrm{regular}} \mathcal D\phi\, \exp\!\left[ -S_E[\phi]-\int_{\tau<0}\lambda_E\,\mathcal O \right].$$

At large $N$, the integral is approximated by a regular Euclidean bulk saddle. The source $\lambda_E(\tau,\mathbf x)$ fixes the nonnormalizable asymptotic coefficient on the cap. The value and oriented canonical momentum at $\tau=0$ become Lorentzian Cauchy data:

$$\phi_L(0,\mathbf x)=\phi_E(0,\mathbf x), \qquad \pi_L(0,\mathbf x)=-i\,\pi_E(0,\mathbf x),$$

with the sign determined by the chosen Wick rotation and outward normals. The full contour prescription of Skenderis and van Rees 2009 makes this continuation part of the variational problem rather than an after-the-fact rule.

A small excited state

Let $\lambda_E=\epsilon f(\tau,\mathbf x)$ have compact support away from the gluing surface. To first order,

$$\delta\phi_E(X) =-\epsilon\int_{\partial M_E}d^dx\, K_E(X;x)f(x),$$

where $K_E$ is the regular Euclidean bulk-to-boundary propagator. Continue $\delta\phi_E$ and its momentum to $t=0$. Because the Lorentzian boundary source vanishes, the subsequent solution has only normalizable asymptotic data,

$$\delta\phi_L(t,z,\mathbf x) \sim z^\Delta\,\delta\langle\mathcal O(t,\mathbf x)\rangle .$$

This is the first application: the profile $f$ prepares a controlled small excitation, and the Lorentzian one-point function is found by evolving the matched data. At quadratic order, reflection of the cap and positivity of the Euclidean norm provide an additional state check. Euclidean-source constructions of excited holographic states are developed explicitly by Botta-Cantcheff, Martínez, and Silva 2016.

Regularity and the adversarial cap

Two failures should be separated.

Geometric failure. If a thermal cap has Euclidean time period different from the value required by horizon smoothness, the tip has a conical singularity. The path integral then prepares a geometry with an inserted defect, not the claimed smooth thermal state.

State failure. If the matched Lorentzian coefficient contains a nonnormalizable mode while the claimed Hamiltonian has no source, the cap has prepared a driven problem or a nonnormalizable state. Smoothness in the interior does not repair the boundary mismatch.

An admissible cap can also fail to exist for the requested topology or charges. In that case the correct result is absence of this preparation saddle, not permission to impose arbitrary Lorentzian data.

Scope, limit order, and handoff

The protocol fixes one state in a specified semiclassical code sector. Backreaction is controlled when the Euclidean source and inserted energy remain within the saddle expansion. Taking $N\to\infty$ first suppresses fluctuations of the prepared geometry; it does not prove that the exact state is uniquely specified by one classical cap.

Thermal and Nonequilibrium QFT owns general initial density matrices; curved-spacetime QFT owns state admissibility. This page supplies only the holographic cap and junction calculation.

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

• Botta-Cantcheff, Marcelo; Martínez, Pedro J.; and Silva, Guillermo A. “On Excited States in Real-Time AdS/CFT.” Journal of High Energy Physics 2016, 171 (2016). doi:10.1007/JHEP02(2016)171.
• Skenderis, Kostas, and Balt C. van Rees. “Real-Time Gauge/Gravity Duality: Prescription, Renormalization and Examples.” Journal of High Energy Physics 2009, 085 (2009). doi:10.1088/1126-6708/2009/05/085.