Quantum Statistical Mechanics by Perturbation Resummation Methods

S. Nakajima

Carl S. Helrich

C. F. Hart

The Laplace-transformed Heisenberg equations of motion are resummed in such a way that the resulting iterative series expansion of any operator is given in terms of exact perturbed operator self-energies. We show how to incorporate this result into a recently introduced Liouvillian Green's-function and self-energy approach to nonequilibrium statistical mechanics. The spontaneous emission of a two-level atom is discussed as a simple example of this new resummation technique.

Michael J. Brown, Ian B. Whittingham

Approximations based on two-particle irreducible (2PI) effective actions (also known as Φ-derivable, Cornwall–Jackiw–Tomboulis or Luttinger–Ward functionals depending on context) have been widely used in condensed matter and non-equilibrium quantum/statistical field theory because this formalism gives a robust, self-consistent, non-perturbative and systematically improvable approach which avoids problems with secular time evolution. The strengths of 2PI approximations are often described in terms of a selective resummation of Feynman diagrams to infinite order. However, the Feynman diagram series is asymptotic and summation is at best a dangerous procedure. Here we show that, at least in the context of a toy model where exact results are available, the true strength of 2PI approximations derives from their self-consistency rather than any resummation. This self-consistency allows truncated 2PI approximations to capture the branch points of physical amplitudes where adjustments of coupling constants can trigger an instability of the vacuum. This, in effect, turns Dyson's argument for the failure of perturbation theory on its head. As a result we find that 2PI approximations perform better than Padé approximation and are competitive with Borel–Padé resummation. Finally, we introduce a hybrid 2PI–Padé method.

Ivo Sachs, Siddhartha Sen, James Sexton

Satoshi Iso, Hikaru Ohta, Takao Suyama

In classical and quantum systems, perturbation of an evolution equation is often invalidated by secular terms which diverge at late times. The diverging behavior of evolution can be remedied by various techniques of resummation such as the renormalization group or multi-scale analysis. In this paper, we prove that, in a generic quantum-mechanical system, secular terms can be systematically removed to all orders in the Dyson series by the method of improved (renormalized) perturbation. A recurrence relation to provide an explicit method to remove the secular terms is given. As a by-product, we give a simple method to obtain energy eigenvalues and decay rates to all orders of perturbation.