In Assa Auerbach's book (Ref. 1), he gave an argument saying that in the normal process of path integral, we lose information about ordering of operators by ignoring the discontinuous path.
What did he want to say? I don't think there is any problem related to the ordering of operators.
- Assa Auerbach, Interacting Electrons and Quantum Magnetism, p.102, just below eq. (10.6).