The multi-time propagators and the consistency condition

For a nonrelativistic quantum system of $N$ particles, the wave function is a function of $3N$ spatial coordinates and one temporal coordinate. The relativistic generalization of this wave function is a function of $N$ time variables known as the multitime wave function, and its evolution is described by $N$ Schrödinger equations, one for each time variable. To guarantee the existence of a nontrivial common solution of these $N$ equations, the $N$ Hamiltonians must satisfy a compatibility condition known as the integrability condition. In this work, the integrability condition is expressed in terms of Lagrangians. The time evolution of a wave function with $N$ time variables is derived in Feynman's picture of quantum mechanics. However, these evolutions are compatible if and only if the $N$ Lagrangians satisfy a certain relation called the consistency condition, which can be expressed in terms of Wilson line. As a consequence of this consistency condition, the evolution of the wave function gives rise to a key feature called the “path-independence” property on the space of time variables. This suggests that one must consider all possible paths not only on the space of dependent variables (spatial variables) but also on the space of independent variables (temporal variables). Geometrically, this consistency condition can be regarded as a zero-curvature condition and the multitime evolutions can be treated as compatible parallel transport processes on the flat space of time variables.