标题： 量子变分学在格点上 显示英文标题

标题： Quantum variational calculus on a lattice

摘要： 我们解决了在非交换空间或时空上变分法的长期问题，针对具有平凡喷射丛的一类重要模型。 我们的方法包括Anderson变分双重复形$\Omega(J^\infty)$的量子版本，并包含欧拉-拉格朗日方程和部分诺特定理。 我们详细展示了如何在作为离散非交换几何的$\Bbb Z^m$晶格上的自由场中应用此方法，得到标量场的克莱因-戈登方程，包括在一般度规和规范场背景下的情况，作为作用量的运动欧拉-拉格朗日方程。 在平坦度规的情况下，我们也得到了晶格上标量场的严格在壳守恒的能量-动量张量和诺特电荷。 显示英文摘要

摘要： We solve the long-standing problem of variational calculus on a noncommutative space or spacetime for a significant class of models with trivial jet bundle. Our approach entails a quantum version of the Anderson variational double complex $\Omega(J^\infty)$ and includes Euler-Lagrange equations and a partial Noether's theorem. We show in detail how this works for a free field on a $\Bbb Z^m$ lattice regarded as a discrete noncommutative geometry, obtaining the Klein-Gordon equation for a scalar field, including with a general metric and gauge field background, as the Euler-Lagrange equations of motion for an action. In the case of a flat metric we also obtain an exactly on-shell conserved stress-energy tensor and Noether charges for a scalar field on the lattice.