In quantum field theory, models are often constructed using the path integral formulation, treating spacetime as a very fine lattice so that the math is straightforward. To ensure that time evolution is unitary in the resulting model, time evolution in the euclidean path integral construction should be positive definite. This article explains how to show that it is positive definite in a few standard families of models, including models with only unconstrained scalar fields, nonlinear sigma models whose target space is a sphere, principal chiral models, and Yang-Mills models. This article's approach differs from the usual textbook approach by using a path integral that includes an explicit initial state and that includes only one time-step. This simplifies the analysis and clarifies its relationship to the general principles of quantum theory.
