We extend our study of the large-
N expansion of general nonequilibrium many-body systems with matrix degrees of freedom
M, and its dual description as a sum over surface topologies in a dual string theory, to the Keldysh-rotated version of the Schwinger-Keldysh formalism. The Keldysh rotation trades the original fields ${M}_{\ifmmode\pm\else\textpm\fi{}}$---defined as the values of
M on the forward and backward segments of the closed time contour---for their linear combinations
Mcl and
Mqu, known as the ``classical'' and ``quantum'' fields. First we develop a novel ``signpost'' notation for nonequilibrium Feynman diagrams in the Keldysh-rotated form, which simplifies the analysis considerably. Before the Keldysh rotation, each world-sheet surface $\mathrm{\ensuremath{\Sigma}}$ in the dual string theory expansion was found to exhibit a triple decomposition into the parts ${\mathrm{\ensuremath{\Sigma}}}^{\ifmmode\pm\else\textpm\fi{}}$ corresponding to the forward and backward segments of the closed time contour, and ${\mathrm{\ensuremath{\Sigma}}}^{\ensuremath{\wedge}}$ which corresponds to the instant in time where the two segments meet. After the Keldysh rotation, we find that the world-sheet surface $\mathrm{\ensuremath{\Sigma}}$ of the dual string theory undergoes a very different natural decomposition: $\mathrm{\ensuremath{\Sigma}}$ consists of a ``classical'' part ${\mathrm{\ensuremath{\Sigma}}}^{\mathrm{cl}}$ and a ``quantum embellishment'' part ${\mathrm{\ensuremath{\Sigma}}}^{\mathrm{qu}}$. We show that both parts of $\mathrm{\ensuremath{\Sigma}}$ carry their own independent genus expansion. The nonequilibrium sum over world-sheet topologies is naturally refined into a sum over the double decomposition of each $\mathrm{\ensuremath{\Sigma}}$ into its classical and quantum part. We apply this picture to the classical limits of the quantum nonequilibrium system (with or without interactions with a thermal bath), and find that in these limits, the dual string perturbation theory expansion reduces to its appropriately defined classical limit.
We extend our study of the large-
N expansion of general non-equilibrium many-body systems with matrix degrees of freedom
M, and its dual description as a sum over surface topologies in a dual string theory, to the Keldysh-rotated version of the Schwinger-Keldysh formalism. The Keldysh rotation trades the original fields
M± -- defined as the values of
M on the forward and backward segments of the closed time contour -- for their linear combinations
Mcl and
Mqu, known as the "classical" and "quantum" fields. First we develop a novel "signpost" notation for non-equilibrium Feynman diagrams in the Keldysh-rotated form, which simplifies the analysis considerably. Before the Keldysh rotation, each worldsheet surface
Σ in the dual string theory expansion was found to exhibit a triple decomposition into the parts
Σ± corresponding to the forward and backward segments of the closed time contour, and
Σ∧ which corresponds to the instant in time where the two segments meet. After the Keldysh rotation, we find that the worldsheet surface
Σ of the dual string theory undergoes a very different natural decomposition:
Σ consists of a "classical" part
Σcl, and a "quantum embellishment" part
Σqu. We show that both parts of
Σ carry their own independent genus expansion. The non-equilibrium sum over worldsheet topologies is naturally refined into a sum over the double decomposition of each
Σ into its classical and quantum part. We apply this picture to the classical limits of the quantum non-equilibrium system (with or without interactions with a thermal bath), and find that in these limits, the dual string perturbation theory expansion reduces to its appropriately defined classical limit.
We present a unified framework for equilibrium and nonequilibrium many-body perturbation theory. The most general nonequilibrium many-body theory valid for general initial states is based on a time-contour originally introduced by Konstantinov and Perel'. The various other well-known formalisms of Keldysh, Matsubara and the zero-temperature formalism are then derived as special cases that arise under different assumptions. We further present a single simple proof of Wick's theorem that is at the same time valid in all these flavors of many-body theory. It arises simply as a solution of the equations of the Martin-Schwinger hierarchy for the noninteracting many-particle Green's function with appropriate boundary conditions. We further discuss a generalized Wick theorem for general initial states on the Keldysh contour and derive how the formalisms based on the Keldysh and Konstantinov-Perel'-contours are related for the case of general initial states.
This chapter explores to what extent the closed-contour Schwinger–Keldysh approach and the Lindblad Master equation can be connected with each other. Here, the connection with the Schwinger–Keldysh closed-contour approach does not involve the full machinery of the Green’s functions method, but rather refers directly to the time evolution of the many-body density matrix, which contains a forward evolution operator from the reference time t ₀ to the measuring time t and a backward evolution operator from t back to t ₀. The key approximations to derive the Lindblad Master equation are specified in detail. As an example, a two-level system coupled to a phonon bath is explicitly considered.