Finsler Geometrical Path Integral

TL;DR

This paper introduces a Finsler geometric approach to defining the path integral, providing a coordinate-free and covariant quantisation scheme that overcomes limitations of traditional methods.

Contribution

It proposes a novel Finsler geometric formulation of the path integral, enabling a purely Lagrangian-based, coordinate-independent quantisation framework.

Findings

Finsler structure provides a natural geometric setting for path integrals.

The scheme is coordinate-free and covariant, independent of time coordinate choices.

Potential to unify and justify path integral quantisation in a geometric manner.

Abstract

A new definition for the path integral is proposed in terms of Finsler geometry. The conventional Feynman's scheme for quantisation by Lagrangian formalism suffers problems due to the lack of geometrical structure of the configuration space where the path integral is defined. We propose that, by implementing the Feynman's path integral on an extended configuration space endowed with a Finsler structure, the formalism could be justified as a proper scheme for quantisation from Lagrangian only, that is, independent from Hamiltonian formalism. The scheme is coordinate free, and also a covariant framework which does not depend on the choice of time coordinate.