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        "quote": "对玻色振子的 $Sp(2d,\\mathbb R)$ 和费米振子的 $SO(2d,\\mathbb R)$，此前都遇到过“[<u class=\"concept-term\" data-concept-reference=\"concepts/quantum-mechanics/bogoliubov-transformation\" data-concept-identity=\"urn:uuid:78e418b2-6084-4135-979b-115fe4364239\" data-concept-route=\"/concept-identities#concept-78e418b2-6084-4135-979b-115fe4364239\" id=\"first-69f33431470f559279d3-zh\">Bogoliubov变换(Bogoliubov transformations)</u>](../../../../../concepts/quantum-mechanics/bogoliubov-transformation)”：它们是不属于由 $J$ 选出的 $U(d)$ 子群的群元素，非平凡地作用于 $|0\\rangle_J$。与相干态位移类似，这种作用在 $d\\to\\infty$ 时可能把真空变为与之正交的态，因而可能得到建立在不同基态上的不等价表示。物理上，这些态对应量子的凝聚体；玻色振子对应超流，费米振子对应超导。Bogoliubov正是在研究这些系统时发现了这些变换。",
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        "quote": "For both the bosonic $Sp(2d,\\mathbb R)$ and fermionic $SO(2d,\\mathbb R)$ oscillators, “[<u class=\"concept-term\" data-concept-reference=\"concepts/quantum-mechanics/bogoliubov-transformation\" data-concept-identity=\"urn:uuid:78e418b2-6084-4135-979b-115fe4364239\" data-concept-route=\"/concept-identities#concept-78e418b2-6084-4135-979b-115fe4364239\" id=\"first-69f33431470f559279d3-en\">Bogoliubov transformations</u>](../../../../../concepts/quantum-mechanics/bogoliubov-transformation)” lie outside the $U(d)$ subgroup selected by $J$, acting nontrivially on $|0\\rangle_J$. As with coherent-state displacements, in the $d\\to\\infty$ limit they may take the vacuum to an orthogonal state, allowing inequivalent representations built on different ground states. Physically these describe condensates: superfluidity for bosons, superconductivity for fermions. These systems led Bogoliubov to the transformations bearing his name.",
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