Woit · §39.1 Inequivalent irreducible representations
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英文 OCR 草稿
Recalls non-number-preserving canonical transformations outside U(d); the original finite-dimensional discussion is section25.5.
机器草稿中的相关讨论摘录
In both the standard oscillator case with acting, and the fermionic oscillator case with acting, we found that there were “Bogoliubov transformations”: elements of the group not in the subgroup distinguished by the choice of which acted non-trivially on , taking it to a diferent state. in the case of the Heisenberg group action on coherent states above, such action by Bogoliubov transformations can, in the limit of , take |0⟩ to an orthogonal state. This introduces the possibility of inequivalent representations of the commutation relations, built by applying operators to orthogonal ground states. The physical interpretation again is that such states correspond to condensates of quanta. For the usual bosonic oscillator case, this phenomenon occurs in the theory of superfluidity, for fermionic oscillators it occurs in the theory of superconductivity. It was in the study of such systems that Bogoliubov discovered the transformations that now bear his name.
来源与版本
正文：英文 OCR 草稿
来源版本：2025-10-20
原文位置：§39.1；PDF 页 436、437；印刷页 419、420
处理记录：MinerU OCR
核对状态：英文 OCR 草稿 · 待校对
中文机译草稿
Recalls non-number-preserving canonical transformations outside U(d); the original finite-dimensional discussion is section25.5.
机器草稿中的相关讨论摘录
在 作用的标准振子情形与 作用的费米振子情形中，我们都发现存在“Bogoliubov 变换”：群中不属于 子群的元素，由 的选择所区分，这些元素非平凡地作用于 ，将其变为不同的态。 在上述 Heisenberg 群作用于相干态的情形中，这种由 Bogoliubov 变换给出的作用在 的极限下可以把 |0⟩ 变为正交态。这引入了对易关系的非等价表示的可能性，这些表示通过将算符作用于正交基态而构造。物理解释同样是，这样的态对应于量子的凝聚体。对于通常的玻色振子情形，这一现象出现在超流理论中；对于费米振子，它出现在超导理论中。正是在对此类系统的研究中，Bogoliubov 发现了如今以他名字命名的变换。
来源与版本
正文：中文机译草稿
来源版本：2025-10-20
原文位置：§39.1；PDF 页 436、437；印刷页 419、420
处理记录：MinerU OCR；machine-translation/deepseek-v4.1-flash
核对状态：中文机译草稿 · 待校对