  Path integrals are a central tool when it comes to describing quantum or
thermal fluctuations of particles or fields. Their success dates back to
Feynman who showed how to use them within the framework of quantum mechanics.
Since then, path integrals have pervaded all areas of physics where fluctuation
effects, quantum and/or thermal, are of paramount importance. Their appeal is
based on the fact that one converts a problem formulated in terms of operators
into one of sampling classical paths with a given weight. Path integrals are
the mirror image of our conventional Riemann integrals, with functions
replacing the real numbers one usually sums over. However, unlike conventional
integrals, path integration suffers a serious drawback: in general, one cannot
make non-linear changes of variables without committing an error of some sort.
Thus, no path-integral based calculus is possible. Here we identify which are
the deep mathematical reasons causing this important caveat, and we come up
with cures for systems described by one degree of freedom. Our main result is a
construction of path integration free of this longstanding problem, through a
direct time-discretization procedure.
