Peierls brackets in non-Lagrangian field theory

TL;DR

This paper introduces a generalized covariant Poisson bracket, extending Peierls' bracket, for non-Lagrangian field theories using Lagrange structures, facilitating their quantization within the path-integral framework.

Contribution

It establishes a systematic way to define covariant Poisson brackets for non-Lagrangian dynamics, bridging path-integral and deformation quantization methods.

Findings

Defines covariant Poisson brackets from Lagrange structures.

Generalizes Peierls' bracket to non-Lagrangian theories.

Connects path-integral and deformation quantization approaches.

Abstract

The concept of Lagrange structure allows one to systematically quantize the Lagrangian and non-Lagrangian dynamics within the path-integral approach. In this paper, I show that any Lagrange structure gives rise to a covariant Poisson brackets on the space of solutions to the classical equations of motion, be they Lagrangian or not. The brackets generalize the well-known Peierls' bracket construction and make a bridge between the path-integral and the deformation quantization of non-Lagrangian dynamics.