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"quote": " > [!definition] 定义（[正规序乘积(normal ordered product)](../../../../../concepts/quantum-mechanics/normal-ordering)）\n >\n > 给定由 $a$、$a^\\dagger$ 构成的乘积 $P$，其正规序形式记作 $:P:$，定义为把所有 $a^\\dagger$ 因子移到所有 $a$ 因子的左侧。例如，\n > $$\n > :a^2a^\\dagger a(a^\\dagger)^3:=(a^\\dagger)^4a^3.\n > $$\n >\n > 对哈密顿算符 $H$，可以选取正规序形式\n > $$\n > :H:=:\\frac12(aa^\\dagger+a^\\dagger a):=a^\\dagger a.\n > $$\n > 它在 $|0\\rangle$ 上作用为零，并且在 $\\mathcal F$ 上的特征值为整数而非半整数。将它指数化后得到 $U(1)$ 的表示，不再有符号歧义，因此无须引入双覆盖。缺点是，$:H:$ 给出 $\\mathfrak u(1)$ 的表示，却不能延拓为 $\\mathfrak{sl}(2,\\mathbb R)$ 的表示。\n",
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"quote": " > [!definition] Definition ([Normal ordered product](../../../../../concepts/quantum-mechanics/normal-ordering))\n >\n > Given any product $P$ of the operators $a$ and $a^\\dagger$, its normal-ordered form, written $:P:$, is obtained by moving every factor $a^\\dagger$ to the left of every factor $a$. For example,\n > $$\n > :a^2a^\\dagger a(a^\\dagger)^3:=(a^\\dagger)^4a^3.\n > $$\n >\n > For the Hamiltonian $H$, the normal-ordered version\n > $$\n > :H:=:\\frac12(aa^\\dagger+a^\\dagger a):=a^\\dagger a\n > $$\n > may be chosen. It acts trivially on $|0\\rangle$ and has integer rather than half-integer eigenvalues on $\\mathcal F$. Exponentiating it gives a representation of $U(1)$ with no sign ambiguity, so no double cover is needed. The disadvantage is that $:H:$ represents $\\mathfrak u(1)$ but does not extend to a representation of $\\mathfrak{sl}(2,\\mathbb R)$.\n",
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