论文标题
$ u（3）$的三个面孔
The Three Faces of $U(3)$
论文作者
论文摘要
$ u（n）$是一个半独立产品组，其特征是$ su（n）$的自动形态组中$ u（1）$映射$ u（1）$的特征。对于$ u（3）$，有三种非平凡的同态诱导三个独立的定义表示形式。在$ u（3）$ yang-mills（赋予合适的内部产品）的玩具型号中，这与大量费米子相连，这使得作用于单个物质领域的三个不同的协变量。通过采用$ \ mathrm {mod} \，3 $载体空间的排列，携带由``大''量规变换引起的定义表示形式，可以将三个协变量衍生物和一个物质领域作为单个共同衍生物表达为单个共衍生的衍生品，该衍生物在具有同一$ u（3）$ u（3）$ U（3）$ umum的三个不同的物质领域上。人们可以将其解释为定义表示中的三种物质领域。
$U(n)$ is a semi-direct product group that is characterized by non-trivial homomorphisms mapping $U(1)$ into the automorphism group of $SU(n)$. For $U(3)$, there are three non-trivial homomorphisms that induce three separate defining representations. In a toy model of $U(3)$ Yang-Mills (endowed with a suitable inner product) coupled to massive fermions, this renders three distinct covariant derivatives acting on a single matter field. By employing a $\mathrm{mod}\,3$ permutation of the vector space carrying the defining representation induced by a ``large'' gauge transformation, the three covariant derivatives and one matter field can alternatively be expressed as a single covariant derivative acting on three distinct species of matter fields possessing the same $U(3)$ quantum numbers. One can interpret this as three species of matter fields in the defining representation.
