  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.
