附录一：Lehmann–Symanzik–Zimmermann(LSZ)约化公式#
实标量场#
对于实标量场，在相对论性归一化下，其产生湮灭算符在无穷原的过去和未来可以写为
(1028)#\[\begin{align}
a_{\vec{k}}^\dagger(+ \infty) - a_{\vec{k}}^\dagger(-\infty) =&\
-i \int d^4x \, e^{- i k \cdot x} (\partial^2 + m^2) \phi(x) \\
a_{\vec{k}}(+ \infty) - a_{\vec{k}}(-\infty) =&\
i \int d^4x \, e^{i k \cdot x} (\partial^2 + m^2) \phi(x)
\end{align}\]
狄拉克场#
对于狄拉克场，我们以\(a^\dagger, a\)标记粒子态产生湮灭算符，以\(b^\dagger, b\)标记反粒子的产生湮灭算符，则
(1029)#\[\begin{align}
a_{\vec{k}, s}^\dagger(+ \infty) - a_{\vec{k}, s}^\dagger(-\infty) =&\
- i \int d^4x \, \bar\psi(x) (i \overset{\leftarrow}{\partial_\mu} \gamma^\mu + m ) u_s(k) e^{- i k \cdot x} \\
a_{\vec{k}, s}(+ \infty) - a_{\vec{k}, s}(-\infty) =&\ i \int d^4x\, e^{ikx} \bar u_s(k) (- i \partial_\mu \gamma^\mu + m ) \psi(x)
\\
b_{\vec{k}, s}^\dagger(+ \infty) - b_{\vec{k}, s}^\dagger(-\infty) =&\
i \int d^4x \, e^{ - i k \cdot x} \bar v_s(k ) ( - i \partial_\mu \gamma^\mu + m ) \psi(x) \\
b_{\vec{k}, s}(+ \infty) - b_{\vec{k}, s}(-\infty) =&\
- i \int d^4x \, \bar\psi(x) (i \overset{\leftarrow}{\partial_\mu} \gamma^\mu + m ) v_s(k) e^{i k \cdot x}
\end{align}\]
矢量场#
在费曼规范下，
(1030)#\[\begin{align}
a^{\lambda,\dagger}(k) (+ \infty) - a^{\lambda,\dagger}(k) (- \infty) =&\
-i \int d^4x \, \epsilon^\mu(k, \lambda) \eta^{\lambda \lambda} e^{- ikx} \partial^2 A_\mu(x)\\
a^\lambda(k) (+ \infty) - a^\lambda(k) (- \infty) =&\
i \int d^4x \, \epsilon^\mu(k, \lambda) \eta^{\lambda \lambda} e^{ikx} \partial^2 A_\mu(x)
\end{align}\]