The Schwinger Approximation Scheme for Quantum Systems

Trond Digernes, Veeravalli S. Varadarajan, Srinivasa R. S. Varadhan

Joao C. Pinto Barros, Pierpaolo Fontana, Pasquale Sodano +1 more

In this chapter we review results on the lattice Schwinger model. In par-ticular, we show how the effect of the anomaly is reproduced on the lattice. We connect these results to recent developments in the field of quantum simulation of interacting field theories. Schemes for the quantum simulation of (approximations of) Schwinger models are discussed.

Krastan B. Blagoev, Fred Cooper, John F. Dawson +1 more

In this paper we discuss a Schwinger-Dyson (SD) approach for determining the time evolution of the unequal time correlation functions of a nonequilibrium classical field theory, where the classical system is described by an initial density matrix at time $t=0.$ We focus on $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ field theory in $1+1$ space-time dimensions where we can perform exact numerical simulations by sampling an ensemble of initial conditions specified by the initial density matrix. We discuss two approaches. The first, the bare vertex approximation (BVA), is based on ignoring vertex corrections to the SD equations in the auxiliary field formalism relevant for $1/N$ expansions. The second approximation is a related approximation made to the SD equations of the original formulation in terms of $\ensuremath{\varphi}$ alone. We compare these SD approximations as well as a Hartree approximation with exact numerical simulations. We find that both approximations based on the SD equations yield good agreement with exact numerical simulations and cure the late time oscillation problem of the Hartree approximation. We also discuss the relationship between the quantum and classical SD equations.

Paul Bracken

Trond Digernes, Erik Makino Bakken

Dmitry N. Voskresensky

Functional methods: Generating Φ {ϕ, G } functional of Baym on Schwinger-Keldysh contour; symmetries and conservation laws; thermodynamic consistency. (See [1]. Renormalization scheme can be found in [2].) Generalized kinetic approach: gradient approximation. (See [3]. Gradient expansion diagram technique see in [4].) Some helpful examples. (See [5].)