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        "quote": "\n\n> [!definition] 定义（协变导数）\n> 给定一个联络 $A$，沿 $\\mu$ 方向的相伴协变导数是算符\n>\n> $$\n> D _ {\\mu} ^ {A} = \\frac {\\partial}{\\partial x _ {\\mu}} - i e A _ {\\mu} (x)\n> $$\n>\n> 有了这个定义，规范变换的效应是\n>",
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        "quote": "\n\n> [!theorem] Definition (Covariant derivative)\n> Given a connection $A$, the associated covariant derivative in the $\\mu$ direction is the operator\n>\n> $$\n> D _ {\\mu} ^ {A} = \\frac {\\partial}{\\partial x _ {\\mu}} - i e A _ {\\mu} (x)\n> $$\n>\n> With this definition, the efect of a gauge transformation is\n>\n> $$\n> \\begin{aligned}\n> D _ {\\mu} ^ {A} \\psi &\\to \\left(D _ {\\mu} ^ {A} - i e \\frac {\\partial \\varphi}{\\partial x _ {\\mu}}\\right) e ^ {i e \\varphi (x)} \\psi = e ^ {i e \\varphi (x)} \\left(D _ {\\mu} ^ {A} + i e \\frac {\\partial}{\\partial x _ {\\mu}} - i e \\frac {\\partial}{\\partial x _ {\\mu}}\\right) \\psi \\\\[0.45em]\n> &= e ^ {i e \\varphi (x)} D _ {\\mu} ^ {A} \\psi\n> \\end{aligned}\n> $$\n>\n> If one replaces derivatives by covariant derivatives, terms in a Hamiltonian such as\n>\n> $$\n> | \\nabla \\psi | ^ {2} = \\sum_ {j = 1} ^ {3} \\frac {\\overline {{\\partial \\psi}}}{\\partial x _ {j}} \\frac {\\partial \\psi}{\\partial x _ {j}}\n> $$\n",
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