The multi-time propagators and the consistency condition

Abstract: The time evolution of a wave function with $N$ time variables through the Feynman picture of quantum mechanics is derived. However, these evolutions will be compatible if and only if the $N$ Lagrangians satisfy a certain relation called the consistency condition or integrability condition which could be expressed in terms of the Wilson line. This consistency condition violates if there presents the interaction. As a consequence of this consistency condition, the evolution of the wave function gives rise to a key feature called the "path-independent" property on the space of time variables. This would suggest that one must consider all possible paths not only on the space of dependent variables but also on the space of independent variables. In the view of the geometry, this consistency condition can be considered as a zero curvature condition and the multi-time evolution can be treated as a compatible parallel transport on flat space of time variables.