Discretized and covariant path integrals for stochastic processes

Thibaut Arnoulx de Pirey (1) , Leticia F. Cugliandolo (2, 3) , Vivien Lecomte (4) , Frédéric van Wijland (5)

1 Department of Physics, Technion
2 LPTHE - Laboratoire de Physique Théorique et Hautes Energies
3 IUF - Institut universitaire de France
4 LIPhy - Laboratoire Interdisciplinaire de Physique [Saint Martin d’Hères]
5 MSC - Matière et Systèmes Complexes

Path integrals are a ubiquitous tool in theoretical physics. However, their use is sometimes hindered by the lack of control on various manipulations - such as performing a change of the integration path - one would like to carry out in the light-hearted fashion that physicists enjoy. Similar issues arise in the field of stochastic calculus, which we review to prepare the ground for a proper construction of path integrals. At the level of path integration, and in arbitrary space dimension, we not only report on existing Riemannian geometry-based approaches that render path integrals amenable to the standard rules of calculus, but also bring forth new routes, based on a fully time-discretized approach, that achieve the same goal. We illustrate these various definitions of path integration on simple examples such as the diffusion of a particle on a sphere.

Mots clés: Theoretical Physics; Path Integrals; Stochastic Calculus

Domaines: Physique [physics]

Thibaut Arnoulx de Pirey, Leticia F. Cugliandolo, Vivien Lecomte, Frédéric van Wijland. Discretized and covariant path integrals for stochastic processes. 2022. ⟨hal-03865795⟩