Title: Noncommutative spaces with twisted symmetries and second quantization Show Chinese title

Title: 具有扭转变换对称性的非交换空间与二次量子化

Abstract: In a minimalistic view, the use of noncommutative coordinates can be seen just as a way to better express non-local interactions of a special kind: 1-particle solutions (wavefunctions) of the equation of motion in the presence of an external field may look simpler as functions of noncommutative coordinates. It turns out that also the wave-mechanical description of a system of n such bosons/fermions and its second quantization is simplified if we translate them in terms of their deformed counterparts. The latter are obtained by a general twist-induced *-deformation procedure which deforms in a coordinated way not just the spacetime algebra, but the larger algebra generated by any number n of copies of the spacetime coordinates and by the particle creation and annihilation operators. On the deformed algebra the action of the original spacetime transformations looks twisted. In a non-conservative view, we thus obtain a twisted covariant framework for QFT on the corresponding noncommutative spacetime consistent with quantum mechanical axioms and Bose-Fermi statistics. One distinguishing feature is that the field commutation relations remain of the type "field (anti)commutator=a distribution". We illustrate the results by choosing as examples interacting non-relativistic and free relativistic QFT on Moyal space(time)s. Show Chinese abstract

Abstract: 从极简的角度来看，使用非交换坐标仅仅可以看作是一种更好地表达某种特殊非局域相互作用的方式： 在外场存在的情况下，运动方程的1粒子解（波函数）作为非交换坐标函数时可能看起来更简单。 事实证明，如果我们将n个这样的玻色子/费米子系统的波动力学描述及其二次量子化用它们的变形对应物来表示，则会简化。 后者是通过一种通用的由扭变换诱导的* - 变形过程获得的，该过程不仅协调地扭曲了时空代数，还扭曲了由任意数量n个时空坐标副本和粒子产生与湮灭算符生成的大代数。 在变形代数上，原始时空变换的作用看起来被扭曲了。 因此，在非保守观点下，我们在相应的非交换时空上得到了一个与量子力学公理和玻色-费米统计一致的扭曲协变QFT框架。 一个显著特征是，场对易关系仍然保持“场（反）对易子=分布”的形式。 我们通过选择Moyal时空上的相互作用非相对论和自由相对论QFT为例来说明这些结果。