  In this paper, we analyze in a mathematically rigorous fashion the validity
of the Boltzmann transport equation within quantum field theory. We work within
the specific model of a hermitian, scalar field with polynomial
self-interaction in two-dimensional Minkowski space. Our main results are as
follows: Firstly, that one can obtain a non-perturbative, exact
integro-differential equation for the number densities, which we called the
pre-Boltzmann equation. We secondly take the long-time-dilute-medium limit of
this equation, to obtain a simpler equation. This limiting equation is
qualitatively similar to the Boltzmann equation, but it involves additional
re-scattering terms. These terms disappear if perform a perturbation expansion
in the coupling constant and ignore the loop corrections (Born approximation).
If loop corrections are included, then we argue that for consistency, one must
also keep corresponding rescattering terms which are normally ignored. Our
analysis is hence of potential relevance for physical applications of the
Boltzmann equation wherein loop effects are essential, such as in the standard
scenario of baryogensis in the Early Universe. Our analysis is performed in the
context of flat spacetime, but in such a way that the main ingredients can be
transferred, straightforwardly to the case of a curved spacetime of
Robertson-Walker-type. Our main technical tools are methods from constructive
quantum field theory, as well as a general method called "projection
technique". This turns out to give convergent expansions, and a rather elegant
way of organizing the combinatorics of the various quantum field theoretic
expansions in the analysis.
