Lagrange structure and quantization

TL;DR

This paper introduces a novel path-integral quantization method for dynamical systems that do not necessarily derive from an action principle, using the concept of Lagrange structures and topological sigma-models.

Contribution

It develops a general quantization framework based on Lagrange structures, extending traditional methods to non-Lagrangian systems with a BRST and AKSZ formulation.

Findings

Quantization of non-Lagrangian systems demonstrated

Lagrange structures generalize the action principle

Equivalent boundary dynamics via topological sigma-models

Abstract

A path-integral quantization method is proposed for dynamical systems whose classical equations of motion do \textit{not} necessarily follow from the action principle. The key new notion behind this quantization scheme is the Lagrange structure which is more general than the Lagrangian formalism in the same sense as Poisson geometry is more general than the symplectic one. The Lagrange structure is shown to admit a natural BRST description which is used to construct an AKSZ-type topological sigma-model. The dynamics of this sigma-model in $d+1$ dimensions, being localized on the boundary, are proved to be equivalent to the original theory in $d$ dimensions. As the topological sigma-model has a well defined action, it is path-integral quantized in the usual way that results in quantization of the original (not necessarily Lagrangian) theory. When the original equations of motion come from the action principle, the standard BV path-integral is explicitly deduced from the proposed quantization scheme. The general quantization scheme is exemplified by several models including the ones whose classical dynamics are not variational.