Building a path-integral calculus: a covariant discretization approach
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 important. 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.
Many different definitions are used to define path-integral weight. In statistical mechanics, time-discretization is the standard approach; it implies that, 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. In such an approach, no path-integral based calculus is possible. We explain which are the 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 problem, through a direct time-discretization procedure. We also compare our time-discretized approach to other definitions of path-integral weights that were used in field theories of quantum problems.
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