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5-5 導関数結合とラグランジアン法
5-5 DERIVATIVE COUPLINGS AND LAGRANGIAN METHOD
これまで本章では、次のように仮定できる単純な相互作用のみを議論してきた。
$$\mathscr{H}_{\text{int}}(x)=-\mathscr{L}_{\text{int}} \tag{5-160}$$So far in this chapter we have disucussed only simple interactions for which we may assume
しかしながら、\(\mathscr{L}_{\text{int}}\)が場の演算子の導関数を含む場合、もはやその限りではない。全ラグランジアン密度を次のように定義する。
$$\mathscr{L}(x)=\mathscr{L}_\text{f}(x)+\mathscr{L}_{\text{int}}(x) \tag{5-161}$$This is, however, no longer the case when \(\mathscr{L}_{\text{int}}\) involves the derivatives of field operators.The total Lagrangian density as
与えられたラグランジアン密度から、相互作用表示におけるハミルトニアン密度の相互作用項\(\mathscr{H}_{\text{int}}(x)\)を求めよう。
Let us find the interaction Hamiltonian density \(\mathscr{H}_{\text{int}}(x)\) in the interaction representation from the given Lagrangian density.
\(\mathscr{L}_{\text{int}}(x)\) がスピンを持たない実スカラー場\(\varphi_\alpha\)の一次導関数を含むと仮定する。複素場に対しては、常に2-4節で導入された実スカラー場の表現を用いることができる。
We assume that \(\mathscr{L}_{\text{int}}(x)\) involves the first order derivatives of spinless real fileds \(\phi_\alpha\).For complex fileds we can always employ the real filed representations introduced in Section 2-4.
\(\varphi_\alpha(x)\)に対して正準共役な演算子は、自由場において次のように定義される。
$$\pi_\alpha(x)=\frac{\partial \mathscr{L}_\text{f}}{\partial\dot{\varphi}_\alpha(x)}=\dot{\varphi}_\alpha(x)\tag{5-162}$$The operator that is canonlcally conjugate to \(\varphi_\alpha(x)\) is defind, for a free field, as
そして、相互作用する場においては次のように定義される。
$$\pi’_\alpha(x)=\frac{\partial\mathscr{L}}{\partial\dot{\varphi}_\alpha(x)}=\dot{\varphi}_\alpha(x)+\frac{\partial\mathscr{L}_\text{int}}{\partial\dot{\varphi}_\alpha(x)}\tag{5-163}$$and for an interacting field, as
これから、相互作用描像とハイゼンベルグ描像の関係を勉強する。ヤン＝フェルドマン形式と関連して示されたように、その関係は次のように与えられる。
$$\varphi_\alpha(x)=\text{U}(x_0,-\infty)^{-1}\varphi_\alpha^\text{in}(x)\text{U}(x_0,-\infty)\tag{5-164}$$We now shall study the relationship between the interaction and Heisenberg representations. As has been shown in connection with the Yang-Feldman formalism, the relationship is given by
\(\varphi_\alpha\)に対して正準共役な演算子については、次の関係が成り立つ。
$$\pi_\alpha'(x)=\text{U}(x_0,-\infty)^{-1}\pi_\alpha^\text{in}(x)\text{U}(x_0,-\infty)\tag{5-165}$$For the operator that is canonlcally conjugate to \(\varphi_\alpha\) we have
相互作用描像における演算子は、上付き添字inで表される。ハイゼンベルク描像におけるハミルトニアン密度は、次のように与えられる。
$$\mathscr{H}(x)=\sum_\alpha\pi’_\alpha(x)\dot{\varphi}_\alpha(x)-\mathscr{L}(\varphi_\alpha(x),\dot{\varphi}_\alpha(x))\tag{5-166}$$The operators in the interaction representation are designated by a superscript, in. The Hamiltonian density in the Heisenberg representation is given by
逆変換公式を使用して
$$\text{U}(x_0,-\infty)\varphi_\alpha(x)\text{U}(x_0,-\infty)^{-1}=\varphi_\alpha^\text{in}(x)\tag{5-168}$$Using the inverse transformation formulas
そして
$$\text{U}(x_0,-\infty)\pi’_\alpha(x)\text{U}(x_0,-\infty)^{-1}=\pi_\alpha^\text{in}(x)\tag{5-169}$$and
私たちは\(\dot{\varphi}_\alpha\)が次のように変換されることを見出だす。
\begin{eqnarray} \text{U}(x_0,-\infty)\dot{\varphi}_\alpha(x)\text{U}(x_0,-\infty)^{-1}&=&\text{U}(x_0,-\infty)\left(\pi’_\alpha(x)-\frac{\partial\mathscr{L}_\text{int}}{\partial\dot{\varphi}_\alpha(x)}\right)\text{U}(x_0,-\infty)^{-1}\\ &=&\pi^\text{in}_\alpha(x)-\frac{\partial\mathscr{L}_\text{int}}{\partial\dot{\varphi}^\text{in}_\alpha(x)}\\ &=&\dot{\varphi}^\text{in}_\alpha(x)-\frac{\partial\mathscr{L}_\text{int}}{\partial\dot{\varphi}^\text{in}(x)}\tag{5-170} \end{eqnarray}we find that \(\dot{\varphi}_\alpha\) is transformed as
したがって
\begin{eqnarray} \mathscr{H}^\text{in}(x)&=&\sum_\alpha\pi^\text{in}_\alpha(x)\left(\pi^\text{in}_\alpha(x)-\frac{\partial\mathscr{L}_\text{int}}{\partial\dot{\varphi}^\text{in}_\alpha(x)}\right)\\ &&-\mathscr{L}\left(\varphi^\text{in}_\alpha(x),\dot{\varphi}^\text{in}_\alpha(x)-\frac{\partial\mathscr{L}_\text{int}}{\partial\dot{\varphi}^\text{in}_\alpha(x)}\right)\tag{5-171} \end{eqnarray}Hence
第2項を適切なテイラー級数に展開します。
\begin{eqnarray} \mathscr{L}\left(\varphi^\text{in}_\alpha(x),\dot{\varphi}^\text{in}_\alpha(x)-\frac{\partial\mathscr{L}_\text{int}}{\partial\dot{\varphi}^\text{in}_\alpha(x)}\right)&=&\mathscr{L}(\varphi^\text{in}_\alpha(x),\dot{\varphi}^\text{in}_\alpha(x))\\ &&-\sum_\alpha\frac{\partial\mathscr{L}_\text{int}}{\partial\dot{\varphi}^\text{in}_\alpha(x)}\\ &&\times\frac{\partial(\varphi^\text{in}_\alpha(x),\dot{\varphi}^\text{in})}{\partial\dot{\varphi}^\text{in}}\\ &&+\frac{1}{2}\sum_{\alpha,\beta}\frac{\partial\mathscr{L}_\text{int}}{\partial\dot{\varphi}^\text{in}_\alpha(x)}\frac{\partial\mathscr{L}_\text{int}}{\partial\dot{\varphi}^\text{in}_\beta(x)}\\ &&\times\frac{\partial^2\mathscr{L}(\varphi^\text{in}_\alpha,\dot{\varphi}^\text{in}_\alpha)}{\partial\dot{\varphi}^\text{in}_\alpha(x)\partial\dot{\varphi}^\text{in}_\beta(x)} \end{eqnarray}We expand the second term into the appropriate Taylor series:
そして、次のことを使用します。
\begin{eqnarray} \frac{\partial^2\mathscr{L}}{\partial\dot{\varphi}^\text{in}_\alpha\partial\dot{\varphi}^\text{in}_\beta}&=&\delta_{\alpha\beta}\\ \frac{\partial\mathscr{L}}{\partial\dot{\varphi}^\text{in}_\alpha}&=&\dot{\varphi}^\text{in}+\frac{\partial\mathscr{L}_\text{int}}{\partial\dot{\varphi}^\text{in}_\alpha}\\ &=&\pi^\text{in}_\alpha+\frac{\partial\mathscr{L}_\text{int}}{\partial\dot{\varphi}^\text{in}_\alpha}\tag{5-172} \end{eqnarray}When then use
これらの結果を\(\mathscr{L}\)に代入すると
\begin{eqnarray} \mathscr{L}\left(\varphi^\text{in}_\alpha(x),\dot{\varphi}^\text{in}_\alpha(x)-\frac{\partial\mathscr{L}_\text{int}}{\partial\dot{\varphi}^\text{in}_\alpha(x)}\right)\\ &=&\mathscr{L}(\varphi^\text{in}_\alpha(x),\dot{\varphi}^\text{in}_\alpha(x))-\sum_\alpha\pi^\text{in}_\alpha(x)\frac{\partial{L}_\text{int}}{\partial\dot{\varphi}^\text{in}_\alpha(x)}\\ &&-\frac{1}{2}\sum_\alpha\frac{\partial{L}_\text{int}}{\partial\dot{\varphi}^\text{in}_\alpha(x)}\cdot\frac{\partial{L}_\text{int}}{\partial\dot{\varphi}^\text{in}_\alpha(x)}\tag{5-173} \end{eqnarray}Substitution of these results into \(\mathscr{L}\) yields
そして、したがって
\begin{eqnarray} \mathscr{H}^\text{in}(x)&=&\sum_\alpha\pi^\text{in}_\alpha\pi^\text{in}_\alpha-\mathscr{L}(\varphi^\text{in}_\alpha(x),\varphi^\text{in}_\alpha(x))\\ &&+\frac{1}{2}\sum_\alpha\frac{\partial\mathscr{L}_\text{int}}{\dot{\partial}^\text{in}_\alpha(x)}\cdot\frac{\partial\mathscr{L}_\text{int}}{\dot{\partial}^\text{in}_\alpha(x)}\\ &=&\mathscr{H}^\text{in}_\text{f}(x)+\mathscr{H}^\text{in}_\text{int}(x)\tag{5-174} \end{eqnarray}and hence
ここで
$$\mathscr{H}^\text{in}_\text{int}(x)=-\mathscr{L}^\text{in}_\text{int}(x)+\frac{1}{2}\sum_\alpha\left(\frac{\partial\mathscr{L}_\text{int}}{\partial\dot{\varphi}^\text{in}_\alpha(x)}\right)^2\tag{5-175}$$where
例(5-1) \(\mathscr{L}_\text{int}=-iF\bar{\psi}\gamma\psi(\partial\varphi/\partial X_\mu)\)
Example(5-1) \(\mathscr{L}_\text{int}=-iF\bar{\psi}\gamma\psi(\partial\varphi/\partial X_\mu)\)
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