Real-Time Holographic Correlators

An ingoing condition at a future horizon computes retarded response because it is the solution regular in ingoing Eddington–Finkelstein coordinates: disturbances may fall through the future horizon but do not emerge from it. Combined with a prescribed boundary source and holographic renormalization, that causal interior condition selects the retarded Green function. The sign depends on the site convention $e^{-i\omega t}$, for which stable poles lie below the real frequency axis.

Required background. Gauge Theory and String-Theory Preparation supplies operator/source couplings and large-N normalization used below.

Helpful background. Quantum Gravity frames the semiclassical expansion, Quantum Information distinguishes response from state reconstruction, Black-Hole Perturbation Theory supplies the spectral boundary-value problem, and Gravitational Radiation and Waveforms provides complementary retarded-propagation intuition.

Horizon causality selects the solution

Consider a minimally coupled neutral scalar on a static black brane in its rest frame, with diagonal exterior metric and horizon generator $\partial_t$:

$$S_2 =- \frac12\int d^{d+1}x\,\sqrt{-g} \left(g^{MN}\partial_M\phi\partial_N\phi+m^2\phi^2\right), \qquad \phi=e^{-i\omega t+i\mathbf k\cdot\mathbf x}\phi_{\omega,\mathbf k}(r).$$

Near a nonextremal horizon, the two local behaviors are

$$\phi_{\omega,\mathbf k} \sim (r-r_h)^{\mp i\omega/(4\pi T)}.$$

Introduce $v=t+r_*$, with $dr_*/dr=\sqrt{g_{rr}/(-g_{tt})}$; this is $1/f(r)$ when $g_{tt}=-f$ and $g_{rr}=1/f$. The upper-sign choice in $\mp$—the negative exponent—combines with $e^{-i\omega t}$ to give $e^{-i\omega v}$ and is regular at the future horizon. The positive-exponent branch is regular at the past horizon and represents outgoing flux. Thus “ingoing” is a causal and geometric statement, not an arbitrary absorption rule.

The Lorentzian prescription, including its analytic continuation from thermal contours, is derived in Son and Starinets 2002, §§2–4, pp. 4–14. It assumes a stationary thermal background and linear response. A different initial state requires the appropriate real-time contour and gluing conditions.

Canonical radial momentum gives response

On shell, the quadratic action reduces to boundary terms. Define radial canonical momentum

$$\Pi(r,k) =-\sqrt{-g}\,g^{rr}\partial_r\phi(r,k).$$

Normalize the ingoing solution by its boundary source $J(k)$. The response must be conjugate to $J$, not to the unrescaled cutoff field. Here $d$ is the boundary spacetime dimension and standard scalar quantization uses the larger root $\Delta=d/2+\sqrt{d^2/4+m^2L^2}$ of $\Delta(\Delta-d)=m^2L^2$. Writing the generic distinct large-radius powers in units $L=1$,

$$\phi(r,k)=r^{\Delta-d}J(k)+r^{-\Delta}A(k)+\cdots, \qquad \Pi_J(k)=\lim_{r\to\infty}r^{\Delta-d}\Pi_{\rm ren}(r,k).$$

The extra factor follows directly from $\delta\phi\sim r^{\Delta-d}\delta J$ in the boundary variation. For the convention $G_R(t)=-i\theta(t)\langle[\mathcal O(t),\mathcal O(0)]\rangle$, linear response then gives

$$G_R(k)=-\frac{\Pi_J(k)}{J(k)}.$$

For a massless scalar with $\Delta=d$, this reduces to the familiar $-\lim\Pi_{\rm ren}/\phi$. For a generic mass, the raw cutoff ratio misses the source scaling. Logarithmic falloffs and alternate quantization require the corresponding renormalized boundary variation. Overall factors also depend on whether a tensor, vector, or scalar gauge-invariant master field is used. Local polynomial terms in $\omega$ and $\mathbf k$ are contact terms whose finite part can be scheme-dependent. Poles and the dissipative spectral density away from contact ambiguities are the robust data.

Operationally, compute two independent asymptotic coefficients only after fixing the horizon normalization. A rescaling of the entire solution cancels in the response-to-source ratio, whereas adding a finite boundary counterterm changes only an allowed local polynomial. This makes it possible to compare integrations performed in different radial coordinates: translate the canonical momentum and counterterms together, then verify that the nonlocal part of the Green function and every pole agree.

For a decoupled field with a real two-derivative action and real $\omega,\mathbf k$, the Wronskian flux

$$\mathcal F = i\left(\phi^*\Pi-\phi\Pi^*\right)$$

is radially conserved. With the outward radial orientation used here, the ingoing solution has $\Pi\simeq i\omega C_h\phi$, where $C_h\gt0$, and hence $\mathcal F=-2\omega C_h|\phi_h|^2\lt0$ for $\omega\gt0$. This gives $-\operatorname{Im}G_R\ge0$ after the source normalization is included. Complex QNM frequencies do not obey this real-frequency conjugate-flux argument.

Analyticity, poles, and thermal relations

Retarded response is analytic for $\operatorname{Im}\omega\gt0$. Quasinormal frequencies occur when an ingoing solution is also normalizable with zero source; they are therefore poles of $G_R$. Stable modes have $\operatorname{Im}\omega\lt0$. A pole in the upper half-plane instead signals a dynamical instability or an inconsistent sign choice.

In a thermal state, the spectral density $\rho=-2\operatorname{Im}G_R$ relates Wightman functions by the KMS condition. The ingoing prescription alone determines the retarded function; obtaining all Schwinger–Keldysh components also requires the thermal state and contour. The radial response flow and its horizon limit are developed for general transport channels in Iqbal and Liu 2009, §§II–III, pp. 2–9.

Renormalization and failure modes

Near-boundary divergences are canceled by local covariant counterterms. Logs can introduce a scale and encode anomalies. Gauge fields and metric perturbations also satisfy radial constraints, so differentiating an unconstrained component can produce a spurious correlator. One should solve in gauge-invariant channels or impose the constraint before evaluating the action.

Extremal horizons require separate analysis because the near-horizon exponent and order of limits change. At nonzero momentum, branch cuts or pole condensation can replace a simple isolated-pole picture. Numerically, integrating outward from an exactly singular horizon point loses precision; a Frobenius expansion at $r_h+\epsilon$ supplies controlled initial data.

A useful final check is the static limit. For an operator dual to a stable scalar, the zero-frequency response computed from the Lorentzian solution should agree with the second derivative of the renormalized Euclidean free energy, after contact terms and ensembles are matched. Disagreement usually indicates a missing counterterm, a source normalization error, or a noncommuting frequency and momentum limit rather than new dissipative physics.

The next page takes the low-frequency limit of these correlators to obtain Kubo transport coefficients.

Exercises

Show which near-horizon branch is regular in the ingoing coordinate and determine the sign of its radial flux.

Solution

Near the horizon, $r_*\simeq(4\pi T)^{-1}\ln(r-r_h)$. The branch $(r-r_h)^{-i\omega/(4\pi T)}$ multiplied by $e^{-i\omega t}$ becomes $e^{-i\omega(t+r_*)}=e^{-i\omega v}$, so it is regular at the future horizon. Substitution into the conserved Wronskian shows flux directed toward decreasing exterior radius. Because absorption is nonnegative, the resulting spectral density has the retarded positivity sign. Reversing the exponent instead produces outgoing flux and the advanced prescription.

References

• Iqbal, Nabil, and Hong Liu. “Universality of the Hydrodynamic Limit in AdS/CFT and the Membrane Paradigm.” Physical Review D 79 (2009): 025023. DOI.
• Son, Dam T., and Andrei O. Starinets. “Minkowski-Space Correlators in AdS/CFT Correspondence: Recipe and Applications.” Journal of High Energy Physics 09 (2002): 042. DOI.