String Perturbation Theory on the Schwinger-Keldysh Time Contour

We perform the large-$N$ expansion in the Schwinger-Keldysh formulation of nonequilibrium quantum systems with matrix degrees of freedom and study universal features of the anticipated dual string theory. We find a rich refinement of the topological genus expansion: In the original formulation, the future time instant where the forward and backward branches of the Schwinger-Keldysh time contour meet is associated with its own world sheet genus expansion. After the Keldysh rotation, the world sheets decompose into a classical and quantum part.

Figure 1

(a) The Schwinger-Keldysh contour $𝒞=C_{+}\cup C_{-}$. (b) Worldlines of particles corresponding to two of the four $G_{\pm \pm }$ propagators: $⟨{\varphi }_{+}(t_1){\varphi }_{+}(t_2){⟩}_0$ and $⟨{\varphi }_{-}(t_1){\varphi }_{+}(t_2){⟩}_0$.

Figure 2

A typical surface $\mathrm{\Sigma }$ and its triple decomposition into the forward region ${\mathrm{\Sigma }}^{+}$, the backward region ${\mathrm{\Sigma }}^{-}$, and the wedge region ${\mathrm{\Sigma }}^{\wedge }$ connecting them.

Figure 3

A typical surface $\mathrm{\Sigma }$ and its decomposition into the classical and quantum part ${\mathrm{\Sigma }}^{\mathrm{cl}}$ and ${\mathrm{\Sigma }}^{\mathrm{qu}}$.