Boundary Condition and the Auxiliary Phase in Feynman Path Integral

When employing Feynman path integrals to compute propagators in quantum physics, the concept of summing over the set of all paths is not always naive. In fact, an auxiliary phase often has to be included as a weight for each summand. In this article we discuss the nature of those phase factors for the various types of boundary conditions including all three of the Dirichlet, Neumann and Robin types, as well as their mixtures. We verify that for a free particle confined on a line segment, the resulting formula on the propagator matches those arising from the Schrodinger equation, with a trivial normalization factor.