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Recently, doubts have been cast on the validity of the continuous-timecoherent state path integral. This has led to controversies regarding thecorrect way of performing calculations with path integrals, and to severalalternative definitions of what should be their continuous limit. Furthermore,the issue of a supposedly proper ordering of the Hamiltonian operator,entangled with the continuous-time limit, has led to considerable confusion inthe literature. Since coherent state path integrals are at the basis of themodern formulation of many-body quantum theory, it should be laid on solidfoundations.  Here, we show that the issues raised above are coming from the illegitimateuse of the (standard) rules of calculus, which are not necessarily valid inpath integrals. This is well known in the context of stochastic equations, inparticular in their path integral formulation. This insight allows for solvingthese issues and addressing the correspondence between the various orderings atthe level of the path integral. We also use this opportunity to address theproper calculation of a functional determinant in the presence of aHubbard-Stratonovich field, which shares in the controversies.

Hubbard-Stratonovich transformation and consistent ordering in the coherent state path integral: insights from stochastic calculus