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        "quote": "\n> [!definition] Definition · Dirac operator\n>\n> [<u class=\"concept-term\" data-concept-reference=\"concepts/quantum-mechanics/dirac-operator\" data-concept-identity=\"urn:uuid:555c6cfc-d07b-463a-9c7d-5fc1fecfd3e4\" data-concept-route=\"/concept-identities#concept-555c6cfc-d07b-463a-9c7d-5fc1fecfd3e4\">Dirac operator</u>](../../../../../concepts/quantum-mechanics/dirac-operator) is the operator\n>\n> $$\n> \\not\\partial=\\sum_{j=1}^d\\gamma_j\\frac\\partial{\\partial q_j}.\n> $$\n",
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            "text": "在维数为 $d$、签名为 $(r,s)$ 的空间中，$r+s=d$ 且前 $r$ 个方向取正号、后 $s$ 个取负号；由 Clifford 生成元构成的一阶狄拉克算符为 $\\partial\\!/ = \\sum_{j=1}^{d}\\gamma_j\\frac{\\partial}{\\partial q_j}$。它作用于取值在旋量向量空间 $S$ 中的函数，而非标量函数。",
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            "text": "生成元的 Clifford 关系使该一阶算符平方为带签名的二阶拉普拉斯：正号方向的二阶导数与负号方向的二阶导数相减。",
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            "text": "选定 $\\gamma_j$ 的矩阵表示后，算符成为常系数的一阶微分算符，作用于具有 $\\dim S$ 个分量的波函数；$r=3,s=1$ 的情形将在后文专门讨论。",
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