Random fields, large deviations and triviality in quantum field theory. Part I

Adnan Aboulalaa

Published 2019-03-20Version 1

The issue of the existence and possible triviality of the Euclidean quantum scalar field in dimension 4 is investigated by using some large deviations techniques. As usual, the field $\phi_{d}^{4}$ is obtained as a limit of regularized fields $\phi_{d, k}^{4}$ associated with a probability measures $\mu_{k,V}$, where $k, V$ represent an ultraviolet and volume cutoffs. The result obtained is the following alternative: (1) either the renormalized field $\phi_{4}^{4}$ is the trivial free field, or (2) the almost sure limit of the density of $\mu_{k,V}$, with respect to the Gaussian free field measure, exists and is equal to $0$. This implies, in the second case, that $\mu_{k,V}$ can not have a strong limit as $k \longrightarrow \infty$. These assertions are valid in finite volume. They also hold for vector fields and can be extended to polynomial Lagrangians.