Path integrals and lower bounds for density matrices

Feynman has established a variational principle for the coordinate space representation of the canonical density matrix. It uses real trial actions in place of the actual real action. This principle is extended by dividing the original temperature interval, using matrix multiplication, and trial actions that depend on the end points. The result is a series of better lower bounds. A detailed analysis is made of the soluble harmonic oscillator case using free particle and mean path trial actions.Work supported by a grant from the National Science Foundation.