# Luis M. A. Bettencourt

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

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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.