Quantum theory of particles and fields as an extension of a probabilistic variational approach to classical mechanics and classical field theory: I

Abstract

A theoretical scheme, based on a probabilistic generalization of Hamilton's principle, is elaborated to obtain a unified description of more general dynamical behaviors determined both from a Lagrangian function and by mechanisms not contemplated by this function. Within this scheme, quantum mechanics, classical field theory and a quantum theory for scalar fields are discussed. As a by-product of the probabilistic scheme for classical field theory, the equations of the De Donder–Weyl theory for multi-dimensional variational problems are recovered.