Monte Carlo approaches to effective field theories

In this paper, we explore the application of continuum Monte Carlo methods to effective field theory models. Effective field theories, in this context, are those in which a Fock space decomposition of the state is useful. These problems arise both in nuclear and condensed matter physics. In nuclear physics, much work has been done on effective field theories of mesons and baryons. While the theories are not fundamental, they should be able to describe nuclear properties at low energy and momentum scales. After describing the methods, we solve two simple scalar field theory problems; the polaron and two nucleons interacting through scalar meson exchange. The methods presented here are rather straightforward extensions of methods used to solve quantum mechanics problems. Monte Carlo methods are used to avoid the truncation inherent in a Tamm-Dancoff approach and its associated difficulties. Nevertheless, the methods will be most valuable when the Fock space decomposition of the states is useful. Hence, while they are not intended for ab initio studies of QCD, they may prove valuable in studies of light nuclei, or for systems of interacting electrons and phonons. In these problems, a Fock space decomposition can be used to reduce the number of degrees of freedom and to retain the rotational symmetries exactly. The problems we address here are comparatively simple, but offer useful initial tests of the method. We present results for the polaron and two non-relativistic nucleons interacting through scalar meson exchange. In each case, it is possible to integrate out the boson degrees of freedom exactly, and obtain a retarded form of the action that depends only upon the fermion paths. Here we keep the explicit bosons, though, since we would like to retain information about the boson components of the states and it will be necessary to keep these components in order to treat non-scalar of interacting bosonic fields.