In physical systems one is often interested in knowing if the ground state is unique or not. If one wants to prove such results for all values of some coupling constant then one needs to use Perron-Frobenius type arguments and positive cones. However in non relativistic quantum field theory, there are no obvious choice of cone. One choice however is intimately related to Malliavin Calculus, and gives a very useful representation of the problem in terms of normally distributed variables.

In this talk I will present the basic setup for non relativistic quantum field theory, and argue why it is not too different from the basic setup of Malliavin Calculus. I will then show how this can be used to analyse certain translation invariant models. If time permits I will also spend some time on open problems, where deeper probabilistic arguments are needed.