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Theoretical and Mathematical Physics

Keldysh diagram technique with non-Gaussian initial correlations

A. G. Mikhaylenko, A. G. Semenov

Published Apr 1, 2026• 2 citations
Publisher

Abstract

Abstract We modify the nonequilibrium Keldysh diagram technique to systematically account for non-Gaussian initial correlations. We represent information about the initial state of the system as an additional term in the Keldysh action, which is expressed in terms of the cumulants of the initial Wigner functional. This additional term leads to the appearance of additional vertices in the diagram technique. We study the role of non-Gaussian initial correlations in the further evolution of correlation functions. In particular, we show that the presence of non-Gaussian correlations of odd order leads to the generation of a mean field. We obtain Dyson equations with partially resummed self-energies and prove that diagrams with a finite number of loops contain a finite number of non-Gaussian cumulants of the initial Wigner functional. We apply the modified diagram technique to the description of the evolution of a scalar field after a quench and show that in the limit of short interaction times this technique leads to exact results.

DOI
10.1134/s0040577926040094

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Abstract We modify the nonequilibrium Keldysh diagram technique to systematically account for non-Gaussian initial correlations. We represent information about the initial state of the system as an additional term in the Keldysh action, which is expressed in terms of the cumulants of the initial Wigner functional. This additional term leads to the appearance of additional vertices in the diagram technique. We study the role of non-Gaussian initial correlations in the further evolution of correlation functions. In particular, we show that the presence of non-Gaussian correlations of odd order leads to the generation of a mean field. We obtain Dyson equations with partially resummed self-energies and prove that diagrams with a finite number of loops contain a finite number of non-Gaussian cumulants of the initial Wigner functional. We apply the modified diagram technique to the description of the evolution of a scalar field after a quench and show that in the limit of short interaction times this technique leads to exact results.
The nonequilibrium dynamics of quantum fields is an initial-value problem, which can be described by Kadanoff-Baym equations. Typically, and, in particular, when numerical solutions are demanded, these Kadanoff-Baym equations are restricted to Gaussian initial states. However, physical initial states are non-Gaussian correlated initial states. In particular, renormalizability requires the initial state to feature nnn-point correlations that asymptotically agree with the vacuum correlations at short distances. In order to identify physical nonequilibrium initial states, it is therefore a precondition to describe the vacuum correlations of the interacting theory within the nonequilibrium framework. In this paper, Kadanoff-Baym equations for non-Gaussian correlated initial states describing vacuum and thermal equilibrium are derived from the 2PI effective action. A diagrammatic method for the explicit construction of vacuum and thermal initial correlations from the 2PI effective action is provided. We present numerical solutions of Kadanoff-Baym equations for a real scalar ${\ensuremath{\Phi}}^{4}$ quantum field theory, which take the thermal initial four-point correlation as the leading non-Gaussian correction into account. We find that this minimal non-Gaussian initial condition yields an approximation to the complete equilibrium initial state that is quantitatively and qualitatively significantly improved as compared to Gaussian initial states.
We present a nonequilibrium Green's-functional approach to study the dynamics following a quench in weakly interacting Bose-Hubbard model (BHM). The technique is based on the self-consistent solution of a set of equations which represents a particular case of the most general set of Hedin's equations for the interacting single-particle Green's function. We use the ladder approximation as a skeleton diagram for the two-particle scattering amplitude useful, through the self-energy in the Dyson equation, for finding the interacting single-particle Green's function. This scheme is then implemented numerically by a parallelized code. We exploit this approach to study the correlation propagation after a quench in the interaction parameter, for one and two dimensions. In particular, we show how our approach is able to recover the crossover from the ballistic to the diffusive regime by increasing the boson-boson interaction. Finally we also discuss the role of a thermal initial state on the dynamics both for one- and two-dimensional BHMs, finding that, surprisingly, at high temperature a ballistic evolution is restored.
We reformulate time evolution of systems in mixed states in terms of the classical observables of correlators using the Weyl correspondence rule. The resulting equation of motion for the Wigner functional of the density matrix is found to be of the Liouville type. To illustrate the methods developed, we explicitly consider a scalar theory with quartic self-interaction and derive the short time behaviour with the non-interacting thermal density matrix as initial condition. In the scalar case, the complete correlator hierarchy is studied and restrictions are derived for spatially homogeneous initial conditions and systems with unbroken symmetry.
The problem of quantum scalar field evolution after an instantaneous local perturbation (quench) is considered. A new approach to descriptions of a quench from an arbitrary initial state is developed in the framework of the Keldysh technique. This approach does not require the procedure of the analytical continuation, which can be ambiguous in some cases. The evolution of the energy density after local quench is calculated for a simple case, and its dependence on the interaction region width and the initial conditions is analyzed.
We present a method, based on the Keldysh formalism, for deriving stochastic master equations that describe the non-Markovian dynamics of a quantum system coupled to a Gaussian environment. This approach yields a compact expression for the system's propagator, which we show to be equivalent to existing formulations, such as the stochastic von Neumann equation (SVNE). A key advantage of our method is its generality: It can be extended to describe any open-system evolution defined on a suitable ordering contour. As a result, we adapt it to derive generalized versions of the SVNE that account for initial system-environment correlations, as well as stochastic equations that incorporate information about the statistics of energy flows in the environment. The insights offered by our technique further allow us to examine the nature of the noise processes appearing in the SVNE. We prove that its solution can be expressed in terms of a single physical noise, without any loss of information. Finally, we propose a semiclassical scenario in which this noise can be interpreted as arising from an initial measurement process on the environment.
We develop a perturbative framework for systems coupled quadratically to bosonic environments with quadratic Hamiltonians and linear Lindblad operators. The method applies to both single-mode and multimode environments whose initial states may be non-Gaussian and nonstationary. By combining linear evolution in a doubled operator space with a characteristic-function representation, we express ordered multitime environmental correlation functions in terms of derivatives of the initial characteristic function. These correlation functions enter a recursive construction of the time-convolutionless generator at arbitrary finite order. We illustrate the method with a qubit coupled to a damped mode via a two-photon interaction. Initial Fock and thermal states with identical first and second field moments produce different second-order generators through their fourth moments.