Vedran Sohinger University of Warwick
The Euclidean Φ42 theory as the limit of an interacting Bose gas
Gibbs measures of nonlinear Schrödinger equations are a fundamental object used
to study low-regularity solutions with random initial data. In the dispersive
PDE community, this point of view was pioneered by Bourgain in the 1990s. On the
other hand, the nonlinear Schrödinger equation can be viewed a classical limit
of many-body quantum theory. We are interested in the problem of the derivation
of Gibbs measures as mean-field limits of Gibbs states in many-body quantum
The particular case we consider is when the dimension d=2 and when the
interaction potential is the delta function, which corresponds to the Euclidean
Φ42 theory. The limit that we consider corresponds to taking the density
to be large and the range of the interaction to be small in a controlled way.
Our proof is based on two main ingredients.
This is joint work with J.Fröhlich, A. Knowles, and B. Schlein.
- An infinite-dimensional stationary phase argument, based on a
functional integral representation.
- A Nelson-type estimate for a nonlocal field theory in two dimensions.