We review the present status of gauge theories built on various quantum space–times described by noncommutative space–times. The mathematical tools and notions underlying their construction are given. Different formulations of gauge theory models on Moyal spaces as well as on quantum spaces whose coordinates form a Lie algebra are covered, with particular emphasis on some explored quantum properties. Recent attempts aiming to include gravity dynamics within a noncommutative framework are also considered.

中文翻译：

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量子空间的规范理论

我们回顾了建立在非对易时空所描述的各种量子时空之上的规范理论的现状。给出了构建它们的基础的数学工具和概念。涵盖了 Moyal 空间以及其坐标形成李代数的量子空间的规范理论模型的不同表述，特别强调了一些已探索的量子特性。还考虑了最近旨在将引力动力学包含在非交换框架内的尝试。