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"quote": "- 第一个群在 $\\mathbf{C}^2$ 上的表示，最容易看作 $\\mathrm{SU}(2)$ 的[定义表示(defining representation)](../../../../../concepts/representation-theory/defining-representation)。它不是 $\\mathrm{SO}(3)$ 的表示：从单位元出发，沿不可收缩回路绕行一周后，该回路在 $\\mathrm{SU}(2)$ 中的提升会到达负单位元，而非单位元，因此这种作用不能下降为 $\\mathrm{SO}(3)$ 的表示。这称为“[自旋 1/2 表示(spin-1/2 representation)](../../../../../concepts/representation-theory/spinor-representation)”或“[旋量表示(spinor representation)](../../../../../concepts/representation-theory/spinor-representation)”，并将在[第 7 章](../chapter-07/index#section-7)中进一步讨论。",
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"quote": "\n> [!definition] 定义（[旋量表示(spinor representation)](../../../../../concepts/representation-theory/spinor-representation)）\n> $\\mathrm{Spin}(3)=\\mathrm{SU}(2)$ 的旋量表示是在 $\\mathbf{C}^2$ 上由下式给出的表示：\n>\n> $$\n> g \\in \\mathrm{SU}(2) \\longmapsto \\pi_{\\mathrm{spinor}}(g)=g\n> $$\n",
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"quote": "\n> [!definition] Definition ([Spinor representation](../../../../../concepts/representation-theory/spinor-representation))\n> The spinor representation of $\\mathrm{Spin}(3)=\\mathrm{SU}(2)$ is the representation on $\\mathbf{C}^2$ given by\n>\n> $$\n> g \\in \\mathrm{SU}(2) \\longmapsto \\pi_{\\mathrm{spinor}}(g)=g\n> $$\n",
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"quote": "For a spin-$1/2$ particle, the [Spin group (Spin(3))](../../../../../concepts/group-theory/spin-group) group, which is isomorphic to $SU(2)$, acts on states through the [spinor representation](../../../../../concepts/representation-theory/spinor-representation) with the element $\\Omega ( \\theta , \\mathbf { w } ) \\in S U ( 2 )$ acting as",
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"quote": "对具有 $d$ 个自由度的费米子振子，算符代数是 $\\operatorname{Cliff}(2d,\\mathbb C)$，由 $a_{Fj},a_{Fj}^\\dagger$ 生成。这些算符作用于 $2^d$ 维复向量空间 $\\mathcal H_F=\\mathcal F_d^+$，对应于玻色子情形中作用于 $\\mathcal F_d$ 的 $\\Gamma_{BF}^\\prime$。自旋群由Clifford代数中的可逆元素组成，因此在 $\\mathcal F_d^+$ 上有一个表示，称为“[旋量表示(spinor representation)](../../../../../concepts/representation-theory/spinor-representation)”。它可仿照玻色子情形的元辛表示来构造。本章还研究Schrödinger表示的费米子类似物；它在酉性上有问题，却可用于描述物理学中的“鬼”自由度。",
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"quote": "$\\gamma_j$ 的二次组合给出 $\\mathfrak{so}(2d)=\\mathfrak{spin}(2d)$ 的表示。指数化后，得到 $SO(2d)$ 的至多相差一个符号的表示，以及其二重覆盖 $\\operatorname{Spin}(2d)$ 的真正表示。这个在 $\\mathcal F_d^+$ 上构造的表示称为[旋量表示(spinor representation)](../../../../../concepts/representation-theory/spinor-representation)；有时把带此群作用的空间记为 $S$。",
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