Lower bounds for Wiener integrals

Techniques are developed to sharpen lower bounds for density matrices occurring in statistical mechanics. The Wiener integrals are treated by insertion of trial functionals and parametric representations of unity that involve functionals of the path. Jensen's inequality is applied to suitable parameter-dependent path measures. These yield stronger forms than the basic Feynman bound. We also introduce trajectory insertions, and use coupling constant integration and the hierarchy for correlation functions to improve the bounds.