# Systems of classical particles in the grand canonical ensemble, scaling limits and quantum field theory

January 11, 2006

Euclidean quantum fields obtained as solutions of stochastic partial pseudo
differential equations driven by a Poisson white noise have paths given by
locally integrable functions. This makes it possible to define a class of
ultra-violet finite local interactions for these models (in any space-time
dimension). The corresponding interacting Euclidean quantum fields can be
identified with systems of classical "charged" particles in the grand canonical
ensemble with an interaction given by a nonlinear energy density of the "static
field" generated by the particles' charges via a "generalized Poisson
equation". The infinite volume limit of such systems is discussed for models
with trigonometric interactions using a representation of such models as
Widom-Rowlinson models associated with a (formal) Potts models at imaginary
temperature. The continuum limit of the particle systems under consideration is
also investigated and the formal analogy with the scaling limit of
renormalization group theory is pointed out. In some simple cases the question
of (non-) triviality of the continuum limits is clarified.

Keywords:

Euclidean quantum field theories, scaling limits, stochastic constructions