<aside> 🧸 本文简单说明了如何用轮椅跑路。 QFT
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Mark Srednick: Quantum Field theory https://web.physics.ucsb.edu/~mark/ms-qft-DRAFT.pdf
Peskin Schroeder: Introduction to QFT
Steven Weinberg: Quantum theory of Fields Vol I II
position space(configuration space)
$$ \int[dg(t)] $$
phase space
$$ \int [dg(t)][dp(t)] $$
→ Integration in functional space, such as
$$ C=\{x(t)\mid x(t_a)=a,x(t_b)=b\} $$
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变分法基本引理
借助另一个东西描述此函数：
弱导数
取$[a,b]$上的任意阶可微函数$\eta(x)$，若$\eta(a)=\eta(b)=0$
$$ \forall \eta(x)\in\Omega_0^\infty[a,b],\int_a^b f(x)\eta(x)dx=0 $$
则零函数。
$\Omega_0^\infty[a,b]$ 是一个线性空间。
证明：保号性
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From Young’s Double split experiment
将全空间$(x,t)$划分称dV直至路径选择仅有一个（Linear）⇒
phase space $(q,p),t$
transition amplitude $\bra {\psi_f,t_f}\ket{\psi_i,t_i}$
$$ \bra{q_{final},t_f}\ket{q_i,0}=\bra{q_{f},t_f}\exp -i\hat H t\ket{q_i,0}\\=\Pi {n=}^N=\bra{q{n+}}\exp -i\hat H \epsilon\ket {q_n} \mid {q=q_i}^{q_{N+}=q_f},N\cdot\epsilon =t $$
insert Identity $\int_q dq\ket q\bra q=\hat I$
$$ =\Pi\int_{-\infty}^{+\infty}dp_n\bra{q_{n+}}\ket {p_n}\bra{p_n} $$
solve that $\bra{q}\ket p=\exp ipq/\hbar$
$$ =\Pi\int\exp ip_n q_{n+}/\hbar \bra {p_n} $$
Using BCH formula and $\hat H=\frac {p^2}{2m}+V(q)$ [$\partial _t\hat H\mid _{t=t_i}^{t=t_i+\epsilon}=0$ so just extand to $\hat H(\hat p,\hat q)$]
<aside> 🥰
quantum version H should place p before q, then below could be right[Weyl ordering]
$$ \bra{p}\exp \hat p\hat q\epsilon\ket q=\bra{p}\sum\frac{(\hat p\hat q \epsilon)^n}{n！}\ket q $$
$$ \hat p\hat q\hat p\hat q=\hat p(\hat p\hat q+i\hbar)\hat q $$
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$$ \exp(\hat A)\exp \hat B\exp(-\hat A)=\exp\hat B\exp \left[[\hat A,\hat B]+\frac 1 {2!}[\hat A,[\hat A,\hat B]]+\frac 1 {3!}[\hat A.[\hat A,[\hat A,\hat B]]]+\cdots\right] $$
$$ \exp i\hat H \epsilon =\exp i\frac{\hat p^2}{2m}\epsilon\cdot\exp i\frac{V}{\hbar}\epsilon\cdot \exp O(\epsilon^2) $$
$$ =^{N\to \infty}\Pi_{n=2}^N \int dq_n\Pi_{n=}^N\int dp_n \exp \frac i\hbar \{p_n(q_{n+}-q_n)-H(p_n,q_n)\epsilon\} \\=\boxed{\int[dq(t)]\int [dp(t)]\exp \left[\frac{i}{\hbar}\int_{t_i}^{t_f} dt\left (p(t)\dot q(t)-H(p(t),q(t),t)\right)\right]}. $$
which $p_i,p_f$ is bounded and $p(t),q(t)$ inside integral is real function.
=⇒ more general
$$ \bra{\psi_f,t_f}\ket{\psi_i,t_i}=\int dq_f\int dq_i\psi^*_f(t_f)\psi_i(t_i)\bra{q_f,t_f}\ket{q_i,t_i} $$
→ path integral