Schwinger–Keldysh Contours and Real-Time Bulk Geometries

A Schwinger–Keldysh calculation requires two Lorentzian legs and a specified initial density matrix. Holographically, each contour segment has a bulk saddle; fields and oriented canonical momenta are matched where the segments join. The construction produces retarded and fluctuation correlators from one solution and enforces $Z[J,J]=1$. Omitting a leg or mismatching the junction data changes the observable rather than approximating it.

Required background. Lorentzian Holographic Correlators and Infalling Conditions supplies the horizon prescription, and Closed-Time-Path Generating Functionals in Practice supplies the doubled contour.

Helpful background. Unitarity, Normalization, and Largest-Time Identities supplies the decisive consistency tests; KMS Relations and Fluctuation–Dissipation supplies the thermal relation.

The doubled generating functional

For an initial density matrix $\rho_0$ at time $t_0$,

$$Z[J_1,J_2] =\operatorname{Tr}\!\left( U_{J_1}(t_f,t_0)\rho_0 U_{J_2}^{\dagger}(t_f,t_0) \right).$$

The forward and backward sources generate contour-ordered correlators. Unitarity gives

$$Z[J,J]=\operatorname{Tr}\rho_0=1,$$

and in the average/difference basis $J_r=(J_1+J_2)/2$, $J_a=J_1-J_2$, every term in the influence action contains at least one $a$ field.

For a thermal state, an Euclidean segment of length $\beta$ prepares $\rho_\beta$. Its endpoints attach to the two Lorentzian copies. The bulk saddle therefore consists of an Euclidean black-hole segment and two Lorentzian exteriors with the same induced data. The real-time gauge/gravity construction was formulated in this glued form by Skenderis and van Rees 2009.

Junction conditions and orientation

Vary the on-shell action on two neighboring contour segments. At their common hypersurface $\Sigma$, stationarity requires

$$\phi_L\big|_\Sigma=\phi_R\big|_\Sigma, \qquad \pi_L^{(\mathrm{out})}+\pi_R^{(\mathrm{out})}=0,$$

where $\pi=\sqrt{\lvert\gamma\rvert}\,n^M\partial_M\phi$ and each normal points outward from its own segment. The second equation is momentum continuity once orientations are included. Dropping the orientation sign creates a spurious source localized at the gluing surface.

For a free scalar, solve once with sources $(J_1,J_2)$ and evaluate the quadratic on-shell action. In the $r/a$ basis it has the form

$$W^{(2)} =\int_{\omega,\mathbf k} \left[ J_a(-k)G_R(k)J_r(k) +\frac{i}{2}J_a(-k)G_{\mathrm{sym}}(k)J_a(k) \right].$$

With $\rho=-2\operatorname{Im}G_R$ and $G_{\mathrm{sym}}=\tfrac12\langle\{\mathcal O,\mathcal O\}\rangle$, thermal KMS gives

$$G_{\mathrm{sym}}(\omega,\mathbf k) =\frac12\coth\!\left(\frac{\beta\omega}{2}\right)\rho(\omega,\mathbf k).$$

Thus the same glued solution yields dissipation and fluctuations rather than two unrelated prescriptions.

Thermal two-leg application

Choose equal Euclidean boundary sources to prepare the thermal state, then independently vary $J_1$ and $J_2$ on the Lorentzian legs. Infalling and outgoing components are not chosen independently: the cap periodicity and junction conditions fix their thermal linear combinations. Differentiating once with respect to $J_a$ and once with respect to $J_r$ gives $G_R$; differentiating twice with respect to $J_a$ gives the symmetrized kernel.

For an eternal thermal black brane, this construction reproduces the full $2\times2$ Schwinger–Keldysh propagator found by Herzog and Son 2003.

The calculation should reproduce three identities: $W[J_a=0]=0$, $G_A=G_R^*$ on the real axis for Hermitian operators, and the KMS formula above. These are stronger checks than agreement with one Euclidean continuation.

Adversarial gluing test

Set $\phi_L=\phi_R$ but deliberately take the same outward momentum on both sides. The uncancelled surface variation violates stationarity. Equivalently, omit the backward leg: then $Z[J,J]$ is no longer forced to unity and the largest-time identity fails. A numerical solution that is smooth on each segment can therefore still represent the wrong contour.

The contour, state, ordering, and final-time identification must be declared before a real-time quantity is named. The classical glued saddle controls the stated large-$N$ regime; it does not encode exact finite-$N$ late-time recurrences. Thermal and Nonequilibrium QFT owns the general closed-time-path formalism, while Mathematical QFT owns exact KMS analyticity.

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

• Herzog, Christopher P., and Dam T. Son. “Schwinger–Keldysh Propagators from AdS/CFT Correspondence.” Journal of High Energy Physics 2003, 046 (2003). doi:10.1088/1126-6708/2003/03/046.
• 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.