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arXiv:1608.04580 · EPJ Web of Conferences

Applicability of the Wigner functional approach to evolution of quantum fields

Andrey Leonidov, Anna Radovskaya

Published Jan 1, 2016• 5 citations
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Abstract

Evolution of highly excited quantum field is considered in the framework of Keldysh formalism . It is demonstrated that leading order (LO) term of semiclassical approximation appears as well-known Classical Statistical Approximation (CSA). In simple case of spatially homogeneous scalar field analytical expressions for leading and next-to-leading (NLO) order are presented. It is shown that the range of applicability of CSA strongly depends on the properties of the initial state of the system.

DOI
10.1051/epjconf/201612505013

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Evolution of highly excited quantum field is considered in the framework of Keldysh formalism . It is demonstrated that leading order (LO) term of semiclassical approximation appears as well-known Classical Statistical Approximation (CSA). In simple case of spatially homogeneous scalar field analytical expressions for leading and next-to-leading (NLO) order are presented. It is shown that the range of applicability of CSA strongly depends on the properties of the initial state of the system.
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.
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 develop a consistent perturbation theory in quantum fluctuations around the classical evolution of a system of interacting bosons. The zero-order approximation gives the classical Gross-Pitaevskii equations. In the next order we recover the truncated Wigner approximation, where the evolution is still classical but the initial conditions are distributed according to the Wigner transform of the initial density matrix. Further corrections can be characterized as quantum scattering events, which appear in the form of a nonlinear response of the observable to an infinitesimal displacement of the field along its classical evolution. At the end of the paper we give a few numerical examples to test the formalism.
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.