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 develops the general closed-time-path formalism, while Mathematical QFT develops exact KMS analyticity.

The chapter overview compares these assumptions and limitations.

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.