Taking the path integral more seriously

Leticia F. Cugliandolo, Vivien Lecomte, Frédéric van Wijland

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.

Mark S. Swanson

Mark S. Swanson

Leticia F. Cugliandolo, Vivien Lecomte, Frédéric van Wijland

Abstract 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.

Lawrence S. Schulman

Karthik Rajeev

Abstract We consider the backreaction of a quantum system q on an effectively classical degree of freedom C that is interacting with it. The backreaction equation based on the standard path integral formalism gives the so-called ‘in-out’ backreaction equation, which has several serious pathologies. One can use a different backreaction prescription, referred to as the ‘in-in’ approach, which resolves all the issues of ‘in-out’ backreaction equation. However, this procedure is usually invoked in a rather ad hoc manner. Here we provide a rigorous path integral derivation of the ‘in-in’ approach by exploiting the concept of quantum evolution along complex time contours. It turns out that, this approach can also be used to study both the ‘in-in’ and ‘in-out’ backreaction equations in a unified manner.