Woit · §33.2 Supersymmetric quantum mechanics with a superpotential
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Introductory supersymmetric discussion of non-invariant lowest-energy states; Chapter39 gives further treatment.
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For any quantum system, an important question is that of whether it has a unique lowest energy state. If the lowest energy state is not unique, and a symmetry group acts non-trivially on the space of lowest energy states, the symmetry is said to be “spontaneously broken”, a situation that will be discussed in section 39.4. In supersymmetric quantum mechanics systems, thinking in terms of Lie superalgebras, one calls the generator of the action of a supersymmetry, with invariant under the supersymmetry in the sense that the commutator of and is zero. The question of how the supersymmetry acts on the lowest energy state depends on whether or not solutions can be found to the equation
which will be a lowest energy state with zero energy. If such a solution does exist, one describes the ground state |0⟩ as “invariant under the supersymmet . If no such solution exists, will take a lowest energy state to another, diferent, lowest energy stat in which case one says that one has “spontaneously broken supersymmetry”. The question of whether a given supersymmetric theory has its supersymmetry spontaneously broken or not is one that has become of great interest in the case of much more sophisticated supersymmetric quantum field theories. There, hopes (so far unrealized) of making contact with the real world rely on finding theories where the supersymmetry is spontaneously broken.
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正文：英文 OCR 草稿
来源版本：2025-10-20
原文位置：§33.2；PDF 页 372；印刷页 355
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中文机译草稿
Introductory supersymmetric discussion of non-invariant lowest-energy states; Chapter39 gives further treatment.
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对于任何量子系统，一个重要的问题是它是否具有唯一的能量最低态。如果能量最低态不唯一，并且一个对称群在能量最低态空间上的作用是非平凡的，则称该对称是“自发破缺的”，这一情形将在 39.4 节中讨论。在超对称量子力学系统中，从李超代数的角度考虑，人们称 为某个超对称作用的生成元，而 在该超对称下不变，其含义是 与 的对易子为零。超对称如何作用于能量最低态的问题取决于能否找到方程的解
它将是一个能量为零的最低能量态。如果这样的解确实存在，人们就把基态 |0⟩ 描述为“在超对称 下不变”。如果不存在这样的解， 将把一个最低能量态变为另一个不同的最低能量态 ，在这种情况下，人们说超对称被“自发破缺”。对于一个给定的超对称理论，其超对称是否自发破缺这一问题，在更为复杂的超对称量子场论情形中已引起极大兴趣。在那里，与真实世界建立联系（迄今尚未实现）的希望依赖于找到超对称自发破缺的理论。
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正文：中文机译草稿
来源版本：2025-10-20
原文位置：§33.2；PDF 页 372；印刷页 355
处理记录：MinerU OCR；machine-translation/deepseek-v4.1-flash
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Woit · §39.4 Spontaneous symmetry breaking
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and the vacuum state not an eigenstate of , so it does not have a well-defined particle number. If , the states |⟩ will all have the same energy as and there will be a multiplicity of diferent vacuum states, labeled by . In such a case the symmetry is said to be “spontaneously broken”. This phenomenon occurs when non-relativistic quantum field theory is used to describe a superconductor. There the lowest energy state will be a state without a definite particle number, with electrons pairing up in a way that allows them to lower their energy, “condensing” in the lowest energy state.
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正文：英文 OCR 草稿
来源版本：2025-10-20
原文位置：§39.4；PDF 页 440；印刷页 423
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中文机译草稿
Author-named discussion; machine draft source identity and human-review status are preserved.
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而真空态不是的本征态，因此它没有确定的粒子数。如果，那么态|⟩都将具有与相同的能量，并且将存在多个不同的真空态，由标记。在这种情况下，称对称性是“自发破缺的”。当非相对论量子场论被用于描述超导体时，就会出现这种现象。在那里，最低能量态将是一个没有确定粒子数的态，电子以某种方式配对，使它们能够降低能量，在最低能量态中“凝聚”。
来源与版本
正文：中文机译草稿
来源版本：2025-10-20
原文位置：§39.4；PDF 页 440；印刷页 423
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