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

TL;DR

This paper develops a probabilistic variational framework unifying classical mechanics, classical field theory, and quantum mechanics, deriving equations for scalar fields and connecting to De Donder-Weyl theory.

Contribution

It introduces a probabilistic extension of Hamilton's principle that unifies various dynamical theories and derives multi-dimensional variational equations.

Findings

Unified description of classical and quantum dynamics

Derivation of De Donder-Weyl equations from probabilistic principles

Extension of variational methods to quantum and field theories

Abstract

A theoretical scheme, based on a probabilistic generalization of the Hamilton's principle, is elaborated to obtain an 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.