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      "label": "路径积分(path integral)",
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        "quote": "然而，还有一种截然不同的方法将经典理论与量子理论联系起来，它完全绕过了哈密顿形式，仅使用拉格朗日量。这就是[<u class=\"concept-term\" data-concept-reference=\"concepts/quantum-mechanics/path-integral\" data-concept-identity=\"urn:uuid:d8842a51-427e-436c-8476-372fb5a37bfd\" data-concept-route=\"/concept-identities#concept-d8842a51-427e-436c-8476-372fb5a37bfd\" id=\"first-ee1f874ac6780dc7b095-zh\">路径积分(path integral)</u>](../../../../../concepts/quantum-mechanics/path-integral)形式，它基于一种计算时间演化算符的矩阵元的方法",
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        "quote": "There is however a very different approach to relating classical and quantum theories, which completely bypasses the Hamiltonian formalism, just using the Lagrangian. This is the [<u class=\"concept-term\" data-concept-reference=\"concepts/quantum-mechanics/path-integral\" data-concept-identity=\"urn:uuid:d8842a51-427e-436c-8476-372fb5a37bfd\" data-concept-route=\"/concept-identities#concept-d8842a51-427e-436c-8476-372fb5a37bfd\" id=\"first-ee1f874ac6780dc7b095-en\">path integral</u>](../../../../../concepts/quantum-mechanics/path-integral) formalism, which is based upon a method for calculating matrix elements of the time evolution operator",
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            "text": "Woit 把路径积分介绍为一种绕开哈密顿形式、从拉格朗日量处理量子理论的方式：在一维情形，将时间演化算符的固定位置端点矩阵元表示为路径空间上的积分。",
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            "text": "Schwartz 将演化振幅分成许多短时间片，并用高斯积分 \n\n$$\n\\int_ {- \\infty} ^ {\\infty} d p e ^ {- \\frac {1}{2} a p ^ {2} + J p} = \\sqrt {\\frac {2 \\pi}{a}} e ^ {\\frac {J ^ {2}}{2 a}}. \\tag {14.6}\n$$\n\n 求得短时矩阵元 \n\n$$\n\\langle x _ {j + 1} | e ^ {- i H \\delta t} | x _ {j} \\rangle = N e ^ {- i V (x _ {j}, t _ {j}) \\delta t} e ^ {i \\frac {m}{2} \\delta t \\frac {(x _ {j + 1} - x _ {j}) ^ {2}}{(\\delta t) ^ {2}}} = N e ^ {i L (x, \\dot {x}) \\delta t}, \\tag {14.12}\n$$\n\n；代回时间片乘积后再取连续极限。",
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            "text": "连续极限给出固定端点的路径积分表达式 \n\n$$\n\\langle f | i \\rangle = N \\int_ {x \\left(t _ {i}\\right) = x _ {i}} ^ {x \\left(t _ {f}\\right) = x _ {f}} \\mathcal {D} x (t) e ^ {i S [ x ]}, \\tag {14.15}\n$$\n\n，其中作用量为 $S[x] = \\int dt\\mathcal{L}[x(t),\\dot{x} (t)]$。Schwartz 在式 (14.15) 中将被积因子写作 $e^{iS[x]}$，该式未显式写出 $\\hbar$；归一化 $N$ 经重定义后形式上发散、但会从物理量中消去；该段给出物理推导表达，并未展开测度存在性的论证。",
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            "conditions": "本段的路径积分说明针对一维情形；矩阵元带有显式 $\\hbar$，并使用位置端点 $q_0,q_T$。",
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            "notation": "$\\mathcal{D}x(t)$ 表示满足边界条件的路径求和；$S[x]=\\int dt\\,\\mathcal{L}[x(t),\\dot{x}(t)]$。",
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            "text": "随后对照 Woit §35.3 的一维固定位置矩阵元说明与 Schwartz 的固定端点路径表达，留意 Woit 的矩阵元显式写出 $\\hbar$、而 Schwartz 式 (14.15) 未显式写出 $\\hbar$，并比较两处的叙述范围。",
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