The matrix elements of operators transforming as irreducible representations of an unbroken symmetry group $G$ are governed by the well-known Wigner-Eckart relations. In the case of infinitely-extended systems, with $G$ spontaneously broken, we prove that the corrections to such relations are provided by symmetry breaking Ward identities, and simply reduce to a tadpole term involving Goldstone bosons. The analysis extends to the case in which an explicit symmetry breaking term is present in the Hamiltonian, with the tadpole term now involving pseudo Goldstone bosons. An explicit example is discussed, illustrating the two cases.

中文翻译：

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通过自发对称破坏修正 Wigner-Eckart 关系

变换为完整对称群 $G$ 的不可约表示的算子的矩阵元素由众所周知的 Wigner-Eckart 关系控制。在无限扩展系统的情况下，$G$ 自发破缺，我们证明对这种关系的修正是通过对称破缺沃德恒等式提供的，并且简单地简化为涉及戈德斯通玻色子的蝌蚪项。分析扩展到哈密顿量中存在显式对称破缺项的情况，蝌蚪项现在涉及伪戈德斯通玻色子。讨论了一个明确的例子，说明了这两种情况。