Euclidean Preparation and Lorentzian State Dictionaries

A Euclidean path integral does more than compute a partition function: when it ends on a spatial slice, it prepares a state on that slice. In holography, regular bulk data on a Euclidean cap determine the position and momentum data of a Lorentzian bulk solution. The continuation contour also determines whether later insertions are time ordered, anti-time ordered, or placed on a real-time contour. This page works at leading large $N$ for a scalar in asymptotically AdS$_{d+1}$, first in the linearized approximation and then at the level of the variational principle.

Required background. Timelike Boundary, Causality, and Boundary-Value Problems supplies the admissible AdS boundary data and symplectic-flux condition. The GKPW Generating-Functional Relation supplies the source convention and Euclidean saddle prescription. States, Geometries, and Radial Quantization supplies the state–operator and energy dictionaries.

Helpful background. Initial Density Matrices and Contour Boundary Conditions explains how a general initial density matrix is encoded on a Schwinger–Keldysh contour.

A Euclidean cap prepares a wavefunctional

Take Euclidean global AdS with boundary cylinder time $\tau$,

$$\mathrm ds_E^2=L^2\left( \cosh^2\rho\,\mathrm d\tau^2+\mathrm d\rho^2 +\sinh^2\rho\,\mathrm d\Omega_{d-1}^2\right), \qquad \tau<0,$$

and a scalar with $m^2L^2=\Delta(\Delta-d)$ on the standard branch $\Delta=\Delta_+>d/2$. With a defining function $z\sim2e^{-\rho}$, prescribe the non-normalizable coefficient

$$\phi_E(z,\tau,\Omega) =z^{d-\Delta}J_-(\tau,\Omega)+z^\Delta A_-(\tau,\Omega)+\cdots$$

and demand vacuum regularity as $\tau\to-\infty$. The cap path integral with endpoint value $\varphi(\rho,\Omega)=\phi_E|_{\tau=0}$ is the wavefunctional $\Psi_{J_-}[\varphi]$. At a classical saddle, regularity and $J_-$ fix both $\phi_E$ and its Euclidean canonical momentum at the join. This is why a source on the cap prepares a state rather than merely imposing a Lorentzian forcing term. The saddle construction and its real-time gluing were developed systematically by Skenderis and van Rees 2008, §§2–3.

The ket cap is only one part of an expectation value. A bra is prepared by a second, oppositely oriented cap, commonly with the reflected source $J_+(\tau,\mathbf{x})=J_-^*(-\tau,\mathbf{x})$ for a pure normalized state. Independent cap sources instead prepare a transition amplitude or, with further contour segments, matrix elements of a density operator.

Gluing to Lorentzian AdS

Continue near the joining slice by $\tau=i t$. Lorentzian global AdS then has the site convention

$$\mathrm ds_L^2=L^2\left( \cosh^2\rho\,\mathrm dt^2-\mathrm d\rho^2 -\sinh^2\rho\,\mathrm d\Omega_{d-1}^2\right).$$

The field and canonical momentum obey

$$\phi_L(0,\rho,\Omega)=\phi_E(0,\rho,\Omega), \qquad \Pi_L(0,\rho,\Omega)=i\Pi_E(0,\rho,\Omega),$$

where the sign assumes a ket cap running from $\tau=-\infty$ to $0$ and $\tau=i t$ on the outgoing Lorentzian branch. Reversing the cap orientation reverses the corresponding momentum relation. The variational boundary terms at the join cancel precisely with these matching conditions; continuity of the field alone is not sufficient. The full contour prescription, including corners and operator insertions, is given in Skenderis and van Rees 2009, §§3–4.

For an orthonormal normal-mode basis $u_n$ of global AdS, the linearized problem reduces mode by mode to a harmonic oscillator of frequency $\omega_n>0$. If $J_n(\tau)$ is the projection of the cap source onto mode $n$, regularity gives, up to the fixed bulk-to-boundary normalization,

$$a_n=\int_{-\infty}^{0}d\tau\,e^{\omega_n\tau}J_n(\tau).$$

With a reflected bra source, the resulting real classical field is

$$\phi_L(t)=\sum_n\frac{1}{2\omega_n} \left(a_ne^{-i\omega_nt}u_n+a_n^*e^{i\omega_nt}u_n^*\right).$$

The decaying Euclidean solution therefore selects positive frequency on the ket branch. The omitted proportionality factor is not arbitrary: it is fixed by the scalar action prefactor, the normalization of $u_n$, and the GKPW source normalization.

First application: a small coherent excitation

Choose a smooth source supported in $\tau_1<\tau<\tau_2<0$ and concentrated in one global harmonic $n$. For example, $J_n(\tau)=j e^{-(\tau-\tau_0)^2/(2\sigma^2)}$ gives

$$a_n=j\sqrt{2\pi}\,\sigma \exp\!\left(\omega_n\tau_0+\frac{\omega_n^2\sigma^2}{2}\right)$$

when the Gaussian tails are negligible at the join and at $\tau=-\infty$. At leading order, the CFT state is a coherent excitation of the operator mode and the bulk field oscillates as $a_ne^{-i\omega_nt}+a_n^*e^{i\omega_nt}$. Moving the source deeper down the cap suppresses high-energy modes by $e^{\omega_n\tau_0}$; the complete Laplace transform, not its central time alone, controls the high-frequency tail. This is an explicit, calculable map from a Euclidean preparation to Lorentzian normalizable initial data.

Adversarial check: contour ordering and source placement

Operator ordering is contour ordering. A Lorentzian insertion displaced as $t\mapsto t-i0$ relative to another lies later on the standard ket contour and produces the usual time-ordered boundary value. Retarded correlators require a Schwinger–Keldysh contour and the corresponding causal combination; they are not obtained merely by replacing Euclidean frequency with a real number. In momentum space the time-ordered continuation carries the Feynman prescription $p^2\mapsto p^2+i0$, while an ingoing condition at a horizon belongs to the retarded prescription.

As a decisive check, move a source insertion across the Euclidean–Lorentzian join. It is then a Lorentzian source acting on an already prepared state, not part of the state wavefunctional. Alternatively reverse the $i0$ displacement: the pole side and operator ordering reverse, and the resulting amplitude is generally the complex conjugate or a different contour-ordered correlator. A calculation unchanged by either operation has erased contour data and cannot specify a Lorentzian state dictionary.

Controlled limits and handoff

The mode formula is linearized and assumes a pure state prepared by smooth caps. Interactions couple modes, large sources backreact on the metric, and a thermal or otherwise mixed state requires an enlarged contour. Those changes preserve the gluing principle but replace the Gaussian wavefunctional by a nonlinear saddle, possibly with several competing saddles. The present construction supplies the state data needed by Heavy States, Coherent States, and Semiclassical Geometries; complete real-time black-hole contours belong to the later thermal and nonequilibrium treatment.

Exercises

For a single oscillator, verify that a delta-function cap source $J(\tau)=j\,\delta(\tau-\tau_0)$ with $\tau_0<0$ prepares $a=j e^{\omega\tau_0}$, and determine the effect of sending $\tau_0\to-\infty$.

Solution

Substitution in the cap transform gives $a=j e^{\omega\tau_0}$. Since $\omega>0$, the amplitude vanishes as $\tau_0\to-\infty$: Euclidean evolution projects onto the vacuum unless the source strength is scaled exponentially.

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, Kostas, and Balt C. van Rees. “Real-Time Gauge/Gravity Duality.” Physical Review Letters 101 (2008): 081601. doi:10.1103/PhysRevLett.101.081601. arXiv:0805.0150.
• Skenderis, Kostas, and Balt C. van Rees. “Real-Time Gauge/Gravity Duality: Prescription, Renormalization and Examples.” Journal of High Energy Physics 05 (2009): 085. doi:10.1088/1126-6708/2009/05/085. arXiv:0812.2909.