Real time quantum field theory on a computer: The Hartree ensemble approximation

The main motivation for this thesis is the physics of the very early universe
and of heavy ion collisions. These two fields are described using quantum field theory. In order to do calculations in (quantum) field theory one often makes use of perturbation theory and/or the imaginary time formalism. However, if one wants to describe nonequilibrium processes such as phase transitions or thermalization, in which nonperturbative effects play a role, new approximation methods are needed.

Commonly used nonperturbative real-time approximation schemes include the
large-$n$, the Hartree and the classical approximations. They all have their
limitations: the related large-$n$ and Hartree include quantum corrections, but fail to properly describe thermalization, due to the lack of direct scattering.
The classical approximation does include scattering and leads to (classical)
thermalization, but has problems with the Rayleigh Jeans divergencies and cannot describe quantum thermalization.
More recently, approximations based on an expansion of the $2PI$ effective
action or on a truncation of the Schwinger-Dyson equations, both including
direct scattering, have been shown to lead to thermalization. These methods give good results, but the treatment of inhomogeneous systems seems numerically too demanding.

In this thesis an approximation is introduced and studied, which tries to
improve on both the classical and the Hartree approximation. In the Hartree
approximation the initial density matrix is taken to be Gaussian, leading to the absence of direct scattering terms. In the classical approximation one usually writes the initial density matrix as an ensemble average over initial
conditions. By expanding a general density matrix on a (over complete) basis of Gaussian density matrices, one can use the Hartree or Gaussian approximation to calculate the dynamics in each of the `realizations' and take an ensemble average to allow for more general initial density matrices. At the same time these `realizations' can be inhomogeneous (i.e. space dependent), leading to indirect scattering and thus removing the principal obstruction for thermalization.

The thesis examines this approximation in a simple scalar $\phi^4$ theory in 1+1 dimensions. In the first chapters it studies if the model thermalizes starting from some out of equilibrium initial condition. Starting at low momenta the system indeed thermalizes. However this process only slowly proceeds to higher momenta, and a full quantum thermal distribution is never found. Deviations start to emerge and the system evolves towards a classical distribution, caused by the fact that the Hartree equations of motion can be derived from a Hamiltonian. Timescales for both processes, thermalization and the flow to classical behaviour are derived and studied in both phases of the theory.
Because of the numerical task involved, further approximations are also
investigated, especially to allow for the extension to 3+1 dimensions.
The final chapter studies the topological defects present in the theory, kinks
and antikinks. It makes a comparison between the classical theory and the
Hartree approximation. As initial conditions both semi-stationary and colliding
configurations are studied. Some important differences between the classical and quantum theory are found.