The Hartree approximation in quantum field theory

Yasushi Takahashi, Hiroomi Umezawa

Suk-Joon Lee

Starting from full quantum field theory, various mean field approaches are derived systematically. With a full consideration of external source dependence, the stationary phase approximation of an action gives a nuclear mean field theory which includes quantum correlation effects (such as particle-hole or ladder diagram) in a simpler way than the Brueckner-Hartree-Fock approach. Implementing further approximation, the result can be reduced to Hartree-Fock or Hartree approximation. The role of the source dependence in a mean field theory is examined.

C. Di Castro, Roberto Raimondi

Luís M. A. Bettencourt, Karen Pao, J. G. Sanderson

We study the dynamics of a spatially inhomogeneous quantum $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ field theory in $1+1$ dimensions in the Hartree approximation. In particular, we investigate the long-time behavior of this approximation in a variety of controlled situations, both at zero and finite temperature. The observed behavior is much richer than that in the spatially homogeneous case and, unlike the latter, shows some universality at long times. Nevertheless, we show that, within a family of Gaussian spatial profiles for the mean field, the evolution fails to thermalize in a quantum canonical sense, as expected from analogous results in closely related (mean field) transport theory. We argue that this dynamical approximation is best suited as a means to study the short-time decay of spatially inhomogeneous fields and in the dynamics of coherent quasiclassical inhomogeneous configurations (e.g. topological defects) in a background of dynamical self-consistent quantum fluctuations.

Alexander L. Fetter

Quantum-field theory is used to study noninteracting spinless particles in two dimensions obeying fractional statistics (anyons). In the self-consistent and conserving Hartree approximation, this description reproduces the previous random-phase approximation, demonstrating both the existence of a gapless undamped collective mode and a gauge-invariant Meissner effect, very similar to that found in the BCS theory.

M. Verschl, M. Sorg