Luis M. A. Bettencourt

Los Alamos National Laboratory

" Testing real time truncations of Dyson-Schwinger Equations in Quantum Mechanics "

Several attempts exist in the literature to construct truncations of Quantum Field theories capable of capturing their long time dynamics, and in particular the ingredients necessary to generate approach to thermal equilibrium.

The most direct approach to this problem starts with the Dyson-Schwinger equations for field correlators and truncates this hierarchy at a given order eg. through a 1/N expansion. Beyond mean fields such equations become non-local in time. This is a huge constraint on the possibility of their solution.

To circumvent this problem it was proposed that ab initio equal time field correlators could be used. The (Dyson-Schwinger) equation of motion for the the generating functional for equal time Green's functions can be written down for full, connected or 1PI field correlators. If these generating functionals have a series expansion an approximation can be achieved by truncating the series at a given order. Analogously another truncation can be achieved by using collecting terms of order 1/N to a certain order.

In order to study the properties of these aproximations and their validity against an exact quantum system I discussed the time evolution of the coupled hierarchy of equal time Green's functions for a system of N anharmonic oscillators.

This can be compared to the quantum roll for these oscillators. Under the assumption of radial symmetry, which is equivalent to a N dimensional single oscillator problem, the Schroedinger equation for the system is solved.

The truncated Green's functions beyond the second order truncation, have the property that they do not correspond to a positive definite probability. I discussed some of their general properties.

I also considered what possible variational truncations might be successful by projecting the exact answer on a particular set of basis wave functions.

A preprint is in preparation and the corresponding link will be added as soon as possible.

Luis M. A. Bettencourt T-6 Theoretical Astrophysics Los Alamos National Laboratory MS B288 Los Alamos, NM 87545 Tel. +1 505 665 1898