论文标题
物理中的双对信函：振荡器实现和表示
Dual Pair Correspondence in Physics: Oscillator Realizations and Representations
论文作者
论文摘要
我们研究还原双对对应关系的一般方面，也称为Howe二元性。我们进行了明确和系统的处理，首先我们得出所有不可还原双对的振荡器实现：$（gl（m，\ mathbb r），gl（n，\ mathbb r））$，$（gl（m，\ mathbb c），gl（gl（n，\ mathbb c）） $（u（m _+，m_-），u（n _+，n _-））$，$（o（n _+，n _-），sp（2m，\ mathbb r））$，$（o（n，\ mathbb c），sp（2m，\ mathbb c c）然后，对于双对中的一个成员紧凑以及第一个非平整的非平凡情况，我们将FOCK空间分解为双对中每组不可约的表示。在整个分析过程中，我们在几种物理应用中讨论了这些表示的相关性。特别是，我们讨论其分支特性的特殊性。最后，建立了与一对两组的所有Casimir操作员有关的封闭式表达式。
We study general aspects of the reductive dual pair correspondence, also known as Howe duality. We make an explicit and systematic treatment, where we first derive the oscillator realizations of all irreducible dual pairs: $(GL(M,\mathbb R), GL(N,\mathbb R))$, $(GL(M,\mathbb C), GL(N,\mathbb C))$, $(U^*(2M), U^*(2N))$, $(U(M_+,M_-), U(N_+,N_-))$, $(O(N_+,N_-),Sp(2M,\mathbb R))$, $(O(N,\mathbb C), Sp(2M,\mathbb C))$ and $(O^*(2N), Sp(M_+,M_-))$. Then, we decompose the Fock space into irreducible representations of each group in the dual pairs for the cases where one member of the pair is compact as well as the first non-trivial cases of where it is non-compact. We discuss the relevance of these representations in several physical applications throughout this analysis. In particular, we discuss peculiarities of their branching properties. Finally, closed-form expressions relating all Casimir operators of two groups in a pair are established.
