标题： 从庞加莱-卡坦形式到场论中泊松括号的葛斯坦哈伯代数 显示英文标题

标题： From the Poincaré-Cartan form to a Gerstenhaber algebra of Poisson brackets in field theory

摘要： 我们回顾了经典分析力学基本结构在德唐-魏尔(De Donder-Weyl, DW)协变哈密顿理论框架内向场论的最新推广。我们从庞加莱-卡坦(Poincaré-Cartan)形式出发，构造与辛形式相对应的多辛形式，其度数为n+1，其中n是时空的维数。动力学变量由微分形式表示，并且多辛形式引导出形式上的泊松括号的定义。泊松括号为动力学变量的外代数赋予了“高阶”格尔斯滕哈伯代数的结构。我们还简要概述了一种可能的场量子化方法，该方法基于DW哈密顿形式和形式的泊松括号。 显示英文摘要

摘要： We review the recent generalization of the basic structures of classical analytical mechanics to field theory within the framework of the De Donder-Weyl (DW) covariant canonical theory. We start from the Poincar\'e-Cartan form and construct the analogue of the symplectic form -- the polysymplectic form of degree n+1, where n is the dimension of the space-time. The dynamical variables are represented by differential forms and the polysymplectic form leads to the definition of the Poisson brackets on forms. The Poisson brackets equip the exterior algebra of dynamical variables with a structure of a "higher-order" Gerstenhaber algebra. We also briefly outline a possible approach to field quantization which proceeds from the DW Hamiltonian formalism and the Poisson brackets of forms.