$\renewcommand{\rm}{\mathrm} \newcommand{\bm}[0]{\boldsymbol} \newcommand{\brac}[1]{{\left(#1\right)}} \newcommand{\bracc}[1]{{\left[#1\right]}} \newcommand{\braccc}[1]{{\left\{#1\right\}}} \newcommand{\ii}[0]{\mathrm{i}} \newcommand{\dag}[0]{\dagger} \renewcommand{\cal}{\mathcal} \renewcommand{\scr}{\mathscr} \newcommand{\frak}{\mathfrak} \newcommand{\bb}{\mathbb} \renewcommand{\sf}{\mathsf} \newcommand{\pprime}[0]{{\prime\prime}} \newcommand{\ppprime}[0]{{\prime\prime\prime}} \newcommand{\eqref}[1]{{\mbox{式}\,\left(\ref{#1}\right)}} \newcommand{\|}[1]{{\left.#1\right|}}$

$S$ 矩阵

$d=D+1$ $D$ 为空间维数, 并且以实标量场为例进行分析.

自由场系统的生成泛函

对于自由场算子, 设其满足自由场方程

\begin{matrix} (1) & D (x) ϕ (x) = 0, \end{matrix}

$\scr{D}\brac{x}=-\brac{\partial^2+m^2}$ $\phi$ 可以展开为

\begin{matrix} (4) & ϕ (x) = \int d^{D} k [a_{k} φ_{k} (x) + a_{k}^{†} φ_{k}^{*} (x)], \end{matrix}

$\varphi_\bm{k}\brac{x}=\varphi_\bm{k}\brac{t,\bm{x}}=\frac{1}{\sqrt{\brac{2\pi}^32\omega_\bm{k}}}e^{-\ii\brac{\omega_\bm{k}t-\bm{k}\cdot\bm{x}}}$ $\omega_\bm{k}=\sqrt{\left|\bm{k}\right|^2+m^2}$ $\bracc{a_{\bm{k}^\prime},a_{\bm{k}^\pprime}^\dag}=\delta^\brac{D}\brac{\bm{k}^\prime-\bm{k}^\pprime}$ $a_\bm{k}\ket{0}=0$ .

$Z_0\brac{J}$ 定义为

\begin{matrix} (5) & Z_{0} (J) = ⟨ 0 | T \exp [i \int d^{d} x J (x) ϕ (x)] | 0 ⟩, \end{matrix}

可以证明

\begin{matrix} (6) & Z_{0} (J) = \exp {\frac{1}{2} \int d^{d} x d^{d} y i J (x) i Δ_{F} (x, y) i J (y)} . \end{matrix}

相互作用场系统的生成泛函

$\cal{L}$ $\cal{L}_0$ $\cal{L}_I$ $\cal{L}=\cal{L}_0+\cal{L}_I$ $S=S_0+S_I$ . 其场算子的运动方程为

\begin{matrix} (7) & D (x) ϕ (x) = - (\frac{\partial L_{I}}{\partial ϕ} - \partial_{μ} \frac{\partial L_{I}}{\partial \partial_{μ} ϕ}) . \end{matrix}

$Z\brac{J}$ 定义为

\begin{matrix} (8) & Z (J) = ⟨ 0 | T \exp [i \int d^{d} x J (x) ϕ (x)] | 0 ⟩, \end{matrix}

可以证明

\begin{matrix} (9) & Z (J) = \exp [i S_{I} (\frac{δ}{i δ J}, \partial \frac{δ}{i δ J})] Z_{0} (J) . \end{matrix}

渐近场与渐近态

设系统的哈密顿量可以分为自由部分和相互作用部分,

\begin{matrix} (10) & H (t) = H_{0} (t) + H^{'} (t) . \end{matrix}

由力学量的运动方程,

\begin{matrix} (11) & i \dot{O} (t) = [O (t), H (t)], \end{matrix}

$\dot{H}\brac{t}=0$ $H\brac{t}$ $H\brac{t}=H$ $H_0\brac{t}$ $H^\prime\brac{t}$ $H$ $\dot{H}_0\brac{t}$ $\dot{H}^\prime\brac{t}$ 并不为零, 总结起来即有

\begin{matrix} (12) & H = H_{0} (t) + H^{'} (t), \end{matrix}

即两个含时的部分相加得到一个不含时的总哈密顿量.

$H_{0\rm{as}}\brac{t}$ 演化, 即

\begin{matrix} (13) & i {\dot{O}}_{as} (t) = [O_{as} (t), H_{0 as} (t)], \end{matrix}

由此可知,

\begin{matrix} (14) & {\dot{H}}_{0 as} (t) = 0, \end{matrix}

$H_{0\rm{as}}\brac{t}$ $H_{0\rm{as}}\brac{t}=H_{0\rm{as}}$ $H_\rm{as}\brac{t}$ $H^\prime_\rm{as}\brac{t}$ $H_{0\rm{as}}$ $\dot{H}_\rm{as}\brac{t}$ $\dot{H}^\prime_\rm{as}\brac{t}$ 并不为零, 总结起来即有

\begin{matrix} (15) & H_{0 as} = H_{as} (t) - H_{as}^{'} (t), \end{matrix}

即两个含时的部分相减得到一个不含时的自由哈密顿量.

$\ket{\rm{in}}$ $\ket{\rm{out}}$ , 即

\begin{matrix} (16) & \begin{aligned} H_{0 in} | α, in ⟩ = E_{α} | α, in ⟩; \\ H_{0 out} | α, out ⟩ = E_{α} | α, out ⟩ . \end{aligned} \end{matrix}

$H_{0\rm{as}}$ $H$ $\cal{O}$ 有如下渐近条件,

\begin{matrix} (17) & \begin{aligned} O_{in} = O (- \infty); \\ O_{out} = O (+ \infty) . \end{aligned} \end{matrix}

$\cal{O}_\rm{as}\brac{t}$ 的演化方程

\begin{matrix} (18) & i {\dot{O}}_{as} (t) = [O_{as} (t), H_{0 as}] \end{matrix}

可以解得

\begin{matrix} (19) & O_{as} (t) = e^{i H_{0 as} (t - t_{0})} O_{as} (t_{0}) e^{- i H_{0 as} (t - t_{0})} . \end{matrix}

$\cal{O}\brac{t}$ 的演化方程

\begin{matrix} (20) & i \dot{O} (t) = [O (t), H] \end{matrix}

可以解得

\begin{matrix} (21) & O (t) = e^{i H (t - t_{0})} O (t_{0}) e^{- i H (t - t_{0})} . \end{matrix}

由渐近条件可知,

\begin{matrix} (22) & O_{in} (t) = e^{i H_{0 in} [t - (- \infty)]} e^{- i H [t - (- \infty)]} O (t) e^{i H [t - (- \infty)]} e^{- i H_{0 in} [t - (- \infty)]} . \end{matrix}

从而

\begin{matrix} (23) & \begin{matrix} H_{0 in} = e^{i H_{0 in} [(+ \infty) - (- \infty)]} e^{- i H [(+ \infty) - (- \infty)]} H_{0} (+ \infty) \\ e^{i H [(+ \infty) - (- \infty)]} e^{- i H_{0 in} [(+ \infty) - (- \infty)]}, \end{matrix} \end{matrix}

$U\brac{t,t_0}=e^{\ii H_{0\rm{in}}\bracc{t-t_0}}e^{-\ii H\bracc{t-t_0}}$ $H_{0\rm{in}}=U\brac{+\infty,-\infty}H_0U^\dag\brac{+\infty,-\infty}$ . 而由于

\begin{matrix} (24) & H_{0 out} = H_{0} (+ \infty), \end{matrix}

因此

\begin{matrix} (25) & \begin{matrix} H_{0 in} = e^{i H_{0 in} [(+ \infty) - (- \infty)]} e^{- i H [(+ \infty) - (- \infty)]} H_{0 out} \\ e^{i H [(+ \infty) - (- \infty)]} e^{- i H_{0 in} [(+ \infty) - (- \infty)]}, \end{matrix} \end{matrix}

由此可得

\begin{matrix} (26) & H_{0 in} e^{- i H [(+ \infty) - (- \infty)]} | α, out ⟩ = E_{α} e^{- i H [(+ \infty) - (- \infty)]} | α, out ⟩, \end{matrix}

从而

\begin{matrix} (27) & e^{- i H [(+ \infty) - (- \infty)]} | α, out ⟩ \propto | α, in ⟩, \end{matrix}

$\ket{\alpha,\rm{in}}$ $\ket{\alpha,\rm{out}}$ $1$ $e^{\ii\varphi_\alpha}$ , 即

\begin{matrix} (28) & e^{- i H [(+ \infty) - (- \infty)]} | α, out ⟩ = e^{i φ_{α}} | α, in ⟩ . \end{matrix}

$S$ $S$ 矩阵

$S$ $\ket{\alpha,\rm{out}}$ $\ket{\alpha,\rm{in}}$ 之间的幺正变换算子,

\begin{matrix} (29) & | α, in ⟩ = S | α, out ⟩, \end{matrix}

$S$ 算子在 in 态的矩阵元为

\begin{matrix} (30) & S_{β α} = ⟨ β, in | S | α, in ⟩ = ⟨ β, out | α, in ⟩, \end{matrix}

$t\to-\infty$ $\ket{\alpha,\rm{in}}$ $t\to+\infty$ $\ket{\beta,\rm{out}}$ $\eqref{带相位的S算子}$ $S$ 矩阵元可以有一个任意的相位因子, 由此可知

\begin{matrix} (31) & S_{β α} = e^{- i φ_{β}} ⟨ β, in | e^{- i H [(+ \infty) - (- \infty)]} | α, in ⟩, \end{matrix}

$\varphi_\beta$ $-E_\beta\bracc{\brac{+\infty}-\brac{-\infty}}$ $S$ 矩阵元写为

\begin{matrix} (32) & \begin{aligned} S_{β α} & = ⟨ β, in | e^{i E_{β} [(+ \infty) - (- \infty)]} e^{- i H [(+ \infty) - (- \infty)]} | α, in ⟩ \\ = ⟨ β, in | e^{i H_{0 in} [(+ \infty) - (- \infty)]} e^{- i H [(+ \infty) - (- \infty)]} | α, in ⟩ \\ = ⟨ β, in | U (+ \infty, - \infty) | α, in ⟩, \end{aligned} \end{matrix}

$S$ 算子定义为

\begin{matrix} (33) & S = U (+ \infty, - \infty) . \end{matrix}

$U\brac{t,t_0}$ $S$ $t>t_0$ $t_0$ $-\infty$ , 则

\begin{matrix} (34) & U (t, - \infty) = e^{i H_{0 in} [t - (- \infty)]} e^{- i H [t - (- \infty)]} . \end{matrix}

$U\brac{t,-\infty}$ $-\ii t$ 求导,得

\begin{matrix} (35) & \begin{aligned} i \frac{\partial}{\partial t} U (t, - \infty) & = e^{i H_{0 in} [t - (- \infty)]} (H - H_{0 in}) e^{- i H [t - (- \infty)]} \\ = e^{i H_{0 in} [t - (- \infty)]} H e^{- i H [t - (- \infty)]} - H_{0 in} U (t, - \infty) \\ = e^{i H_{0 in} [t - (- \infty)]} H e^{- i H_{0 in} [t - (- \infty)]} U (t, - \infty) - H_{0 in} \\ = (H_{in} (t) - H_{0 in}) U (t, - \infty) \\ = H_{in}^{'} (t) U (t, - \infty), \end{aligned} \end{matrix}

$U\brac{-\infty,-\infty}=1$ , 可以解得

\begin{matrix} (36) & U (t, - \infty) = T \exp [- i \int_{- \infty}^{t} d t^{'} H_{in}^{'} (t^{'})], \end{matrix}

由此可知

\begin{matrix} (37) & S = U (+ \infty, - \infty) = T \exp [- i \int_{- \infty}^{+ \infty} d t H_{in}^{'} (t)] . \end{matrix}

$H^\prime_\rm{in}\brac{t}=\int{\dd}^Dx\cal{H}^\prime\brac{\phi_\rm{in}\brac{t,\bm{x}},\pi_\rm{in}\brac{t,\bm{x}}}=-\int{\dd}^Dx\cal{L}_I\brac{\phi_\rm{in}\brac{t,\bm{x}},\partial\phi_\rm{in}\brac{t,\bm{x}}}$ $\cal{H}^\prime$ $\cal{L}_I$ 分别为系统哈氏密度和拉氏密度的相互作用部分, 由此可知,

\begin{matrix} (38) & \begin{aligned} S & = T \exp [- i \int_{- \infty}^{+ \infty} d t H_{in}^{'} (t)] \\ = T \exp [- i \int_{- \infty}^{+ \infty} d t H^{'} (ϕ_{in} (t), π_{in} (t))] \\ = T \exp [i \int d^{d} x L_{I} (ϕ_{in} (x), \partial ϕ_{in} (x))] \\ = {T e^{i \int d^{d} x L_{I} (\frac{δ}{i δ J (x)}, \partial \frac{δ}{i δ J (x)})} e^{i \int d^{d} x J (x) ϕ_{in} (x)} |}_{J = 0} \\ = {e^{i \int d^{d} x L_{I} (\frac{δ}{i δ J (x)}, \partial \frac{δ}{i δ J (x)})} T e^{i \int d^{d} x J (x) ϕ_{in} (x)} |}_{J = 0} \\ = {e^{i S_{I} (\frac{δ}{i δ J}, \partial \frac{δ}{i δ J})} T e^{i \int d^{d} x J (x) ϕ_{in} (x)} |}_{J = 0}, \end{aligned} \end{matrix}

$S\brac{J}=e^{\ii S_I\brac{\frac{\delta}{\ii\delta J},\partial\frac{\delta}{\ii\delta J}}}\sf{T}e^{\ii\int{\dd}^dxJ\brac{x}\phi_\rm{in}\brac{x}}$ $S=S\brac{J=0}$ .

$A$ $B$ $\bracc{\bracc{A,B},A}=\bracc{\bracc{A,B},B}=0$ $e^Ae^B=e^{A+B}e^{\frac{1}{2}\bracc{A,B}}$ .

$A_\rm{in}\brac{t}$ $\bracc{A_\rm{in}\brac{t^\prime},A_\rm{in}\brac{t^\pprime}}$ $c$ 数, 有

\begin{matrix} (39) & T e^{\int_{t_{0}}^{t} d t^{'} A_{in} (t^{'})} = N e^{\int_{t_{0}}^{t} d t^{'} A_{in} (t^{'})} e^{\frac{1}{2} \int_{t_{0}}^{t} d t^{'} d t^{''} ⟨ 0, in | T A_{in} (t^{'}) A_{in} (t^{''}) | 0, in ⟩}, \end{matrix}

$\ket{0,\rm{in}}$ 为 in 场的基态. 又由于

\begin{matrix} (40) & \begin{aligned} ⟨ 0, in | T ϕ_{in} (x) ϕ_{in} (y) | 0, in ⟩ = i Δ_{F} (x, y); \\ D (x) ϕ_{in} (x) = 0; \\ D (x) Δ_{F} (x, y) = δ^{(d)} (x, y), \end{aligned} \end{matrix}

因此,

\begin{matrix} (41) & \begin{aligned} S (J) & = e^{i S_{I} (\frac{δ}{i δ J}, \partial \frac{δ}{i δ J})} T e^{i \int d^{d} x J (x) ϕ_{in} (x)} \\ = e^{i S_{I} (\frac{δ}{i δ J}, \partial \frac{δ}{i δ J})} N e^{i \int d^{d} x J (x) ϕ_{in} (x)} e^{\frac{1}{2} \int d^{d} y d^{d} z i J (y) i Δ_{F} (y, z) i J (z)} \\ = e^{i S_{I} (\frac{δ}{i δ J}, \partial \frac{δ}{i δ J})} N e^{- \int d^{d} x ϕ_{in} (x) D (x) \frac{δ}{δ J (x)}} e^{\frac{1}{2} \int d^{d} y d^{d} z i J (y) i Δ_{F} (y, z) i J (z)} \\ = N e^{- \int d^{d} x ϕ_{in} (x) D (x) \frac{δ}{δ J (x)}} e^{i S_{I} (\frac{δ}{i δ J}, \partial \frac{δ}{i δ J})} e^{\frac{1}{2} \int d^{d} y d^{d} z i J (y) i Δ_{F} (y, z) i J (z)} \\ = N e^{- i \int d^{d} x ϕ_{in} (x) D (x) \frac{δ}{i δ J (x)}} e^{i S_{I} (\frac{δ}{i δ J}, \partial \frac{δ}{i δ J})} Z_{0} (J) \\ = N e^{- i \int d^{d} x ϕ_{in} (x) D (x) \frac{δ}{i δ J (x)}} Z (J), \end{aligned} \end{matrix}

从而

\begin{matrix} (42) & S = {{N [e^{- i \int d^{d} x ϕ_{in} (x) D (x) \frac{δ}{i δ J (x)}}] Z (J)} |}_{J = 0}, \end{matrix}

$S$ 算子的 LSZ 约化公式.

散射问题的渐近理论与 SS 矩阵

自由场系统的生成泛函

相互作用场系统的生成泛函

渐近场与渐近态

SS 算子与 SS 矩阵

$S$ 矩阵

$S$ $S$ 矩阵