Moyal implementation of flow equations—a non-perturbative approach to quantum many-body systems
Johannes N. Kriel, Frederik G. Scholtz, Jacobus Daniel Thom · Journal of Physics A Mathematical and Theoretical · 2007
We show how Wegner's flow equations can be reformulated as ordinary differential equations through the use of the Moyal bracket. In finite-dimensional Hilbert spaces the introduction of the Moyal bracket leads naturally to the identification of a small expansion parameter, namely the inverse of the dimensionality of the space. This expansion corresponds to a non-perturbative treatment of the coupling constant. In the case of infinite-dimensional spaces ℏ plays the role of the small parameter and the Moyal formulation then allows for a semi-classical treatment of the flow equation. We demonstrate these statements for the Lipkin and Dicke models as well as for the symmetric x 4 and double-well potentials.