「時間が同時(
)の場合は、次の(2.20)式により、交換関係がゼロになることは既に見ました」
![{ \displaystyle
\begin{eqnarray}
[\phi({\bf{x}}), \pi({\bf{y}})]&=&i\delta^{(3)}({\bf{x}}-{\bf{y}});\\
[\phi({\bf{x}}), \phi({\bf{y}})]&=&[\pi({\bf{x}}), \pi({\bf{y}})]=0.
\end{eqnarray}
}](http://chart.apis.google.com/chart?cht=tx&chl=%7B%20%5Cdisplaystyle%0A%5Cbegin%7Beqnarray%7D%0A%5B%5Cphi%28%7B%5Cbf%7Bx%7D%7D%29%2C%20%5Cpi%28%7B%5Cbf%7By%7D%7D%29%5D%26%3D%26i%5Cdelta%5E%7B%283%29%7D%28%7B%5Cbf%7Bx%7D%7D-%7B%5Cbf%7By%7D%7D%29%3B%5C%5C%0A%5B%5Cphi%28%7B%5Cbf%7Bx%7D%7D%29%2C%20%5Cphi%28%7B%5Cbf%7By%7D%7D%29%5D%26%3D%26%5B%5Cpi%28%7B%5Cbf%7Bx%7D%7D%29%2C%20%5Cpi%28%7B%5Cbf%7By%7D%7D%29%5D%3D0.%0A%5Cend%7Beqnarray%7D%0A%7D)
(2.20)
「次に、任意の時刻を想定した、より一般的な計算をしてみましょう。(2.47)式から、一般的なクライン−ゴルドン場
の交換関係を計算することができます」

(2.47)
![{ \displaystyle
\begin{eqnarray}
\big[\phi(x), \phi(y)\big]=\int\frac{d^3p}{(2\pi)^3}&\frac{1}{\sqrt{2E_{\bf{p}}}}&
\int\frac{d^3q}{(2\pi)^3}\frac{1}{\sqrt{2E_{\bf{q}}}}\\
&\times&\big[( a_{\bf{p}}e^{-ip\cdot x}+ a_{\bf{p}}^\dagger e^{ip\cdot x}), (a_{\bf{q}}e^{-iq\cdot y}+ a_{\bf{q}}^\dagger e^{iq\cdot y})\big]
\end{eqnarray}
}](http://chart.apis.google.com/chart?cht=tx&chl=%7B%20%5Cdisplaystyle%0A%5Cbegin%7Beqnarray%7D%0A%5Cbig%5B%5Cphi%28x%29%2C%20%5Cphi%28y%29%5Cbig%5D%3D%5Cint%5Cfrac%7Bd%5E3p%7D%7B%282%5Cpi%29%5E3%7D%26%5Cfrac%7B1%7D%7B%5Csqrt%7B2E_%7B%5Cbf%7Bp%7D%7D%7D%7D%26%0A%5Cint%5Cfrac%7Bd%5E3q%7D%7B%282%5Cpi%29%5E3%7D%5Cfrac%7B1%7D%7B%5Csqrt%7B2E_%7B%5Cbf%7Bq%7D%7D%7D%7D%5C%5C%0A%26%5Ctimes%26%5Cbig%5B%28%20a_%7B%5Cbf%7Bp%7D%7De%5E%7B-ip%5Ccdot%20x%7D%2B%20a_%7B%5Cbf%7Bp%7D%7D%5E%5Cdagger%20e%5E%7Bip%5Ccdot%20x%7D%29%2C%20%28a_%7B%5Cbf%7Bq%7D%7De%5E%7B-iq%5Ccdot%20y%7D%2B%20a_%7B%5Cbf%7Bq%7D%7D%5E%5Cdagger%20e%5E%7Biq%5Ccdot%20y%7D%29%5Cbig%5D%0A%5Cend%7Beqnarray%7D%0A%7D)
「上式では、2×2の4つの生成・消滅演算子の交換関係の組み合わせが生じますが、このうち、生成演算子同士の交換関係は、
となり、また、消滅演算子同士の交換関係も、
となって消えるため、結局、生成演算子と消滅演算子を掛け合わせた2つの交換関係の組み合わせのみが残ります」
![{ \displaystyle
\begin{eqnarray}
\big[\phi(x), \phi(y)\big]=\int\frac{d^3p}{(2\pi)^3}&\frac{1}{\sqrt{2E_{\bf{p}}}}&
\int\frac{d^3q}{(2\pi)^3}\frac{1}{\sqrt{2E_{\bf{q}}}}\\
&\times&\big[( a_{\bf{p}}e^{-ip\cdot x}+ a_{\bf{p}}^\dagger e^{ip\cdot x}), (a_{\bf{q}}e^{-iq\cdot y}+ a_{\bf{q}}^\dagger e^{iq\cdot y})\big]\\
=\int\frac{d^3p}{(2\pi)^3}&\frac{1}{\sqrt{2E_{\bf{p}}}}&
\int\frac{d^3q}{(2\pi)^3}\frac{1}{\sqrt{2E_{\bf{q}}}}\\
&\times&\bigg\{\big[ a_{\bf{p}}e^{-ip\cdot x}, a_{\bf{q}}^\dagger e^{iq\cdot y}\big]+\big[ a_{\bf{p}}^\dagger e^{ip\cdot x}, a_{\bf{q}}e^{-iq\cdot y}\big]\bigg\}\\
=\int\frac{d^3p}{(2\pi)^3}&\frac{1}{\sqrt{2E_{\bf{p}}}}&
\int\frac{d^3q}{(2\pi)^3}\frac{1}{\sqrt{2E_{\bf{q}}}}\\
&\times&\bigg\{\big[ a_{\bf{p}}, a_{\bf{q}}^\dagger \big] e^{-ip\cdot x} e^{iq\cdot y}+\big[ a_{\bf{p}}^\dagger, a_{\bf{q}}\big] e^{ip\cdot x}e^{-iq\cdot y}\bigg\}\\
\end{eqnarray}
}](http://chart.apis.google.com/chart?cht=tx&chl=%7B%20%5Cdisplaystyle%0A%5Cbegin%7Beqnarray%7D%0A%5Cbig%5B%5Cphi%28x%29%2C%20%5Cphi%28y%29%5Cbig%5D%3D%5Cint%5Cfrac%7Bd%5E3p%7D%7B%282%5Cpi%29%5E3%7D%26%5Cfrac%7B1%7D%7B%5Csqrt%7B2E_%7B%5Cbf%7Bp%7D%7D%7D%7D%26%0A%5Cint%5Cfrac%7Bd%5E3q%7D%7B%282%5Cpi%29%5E3%7D%5Cfrac%7B1%7D%7B%5Csqrt%7B2E_%7B%5Cbf%7Bq%7D%7D%7D%7D%5C%5C%0A%26%5Ctimes%26%5Cbig%5B%28%20a_%7B%5Cbf%7Bp%7D%7De%5E%7B-ip%5Ccdot%20x%7D%2B%20a_%7B%5Cbf%7Bp%7D%7D%5E%5Cdagger%20e%5E%7Bip%5Ccdot%20x%7D%29%2C%20%28a_%7B%5Cbf%7Bq%7D%7De%5E%7B-iq%5Ccdot%20y%7D%2B%20a_%7B%5Cbf%7Bq%7D%7D%5E%5Cdagger%20e%5E%7Biq%5Ccdot%20y%7D%29%5Cbig%5D%5C%5C%0A%3D%5Cint%5Cfrac%7Bd%5E3p%7D%7B%282%5Cpi%29%5E3%7D%26%5Cfrac%7B1%7D%7B%5Csqrt%7B2E_%7B%5Cbf%7Bp%7D%7D%7D%7D%26%0A%5Cint%5Cfrac%7Bd%5E3q%7D%7B%282%5Cpi%29%5E3%7D%5Cfrac%7B1%7D%7B%5Csqrt%7B2E_%7B%5Cbf%7Bq%7D%7D%7D%7D%5C%5C%0A%26%5Ctimes%26%5Cbigg%5C%7B%5Cbig%5B%20a_%7B%5Cbf%7Bp%7D%7De%5E%7B-ip%5Ccdot%20x%7D%2C%20a_%7B%5Cbf%7Bq%7D%7D%5E%5Cdagger%20e%5E%7Biq%5Ccdot%20y%7D%5Cbig%5D%2B%5Cbig%5B%20a_%7B%5Cbf%7Bp%7D%7D%5E%5Cdagger%20e%5E%7Bip%5Ccdot%20x%7D%2C%20a_%7B%5Cbf%7Bq%7D%7De%5E%7B-iq%5Ccdot%20y%7D%5Cbig%5D%5Cbigg%5C%7D%5C%5C%0A%3D%5Cint%5Cfrac%7Bd%5E3p%7D%7B%282%5Cpi%29%5E3%7D%26%5Cfrac%7B1%7D%7B%5Csqrt%7B2E_%7B%5Cbf%7Bp%7D%7D%7D%7D%26%0A%5Cint%5Cfrac%7Bd%5E3q%7D%7B%282%5Cpi%29%5E3%7D%5Cfrac%7B1%7D%7B%5Csqrt%7B2E_%7B%5Cbf%7Bq%7D%7D%7D%7D%5C%5C%0A%26%5Ctimes%26%5Cbigg%5C%7B%5Cbig%5B%20a_%7B%5Cbf%7Bp%7D%7D%2C%20a_%7B%5Cbf%7Bq%7D%7D%5E%5Cdagger%20%5Cbig%5D%20e%5E%7B-ip%5Ccdot%20x%7D%20e%5E%7Biq%5Ccdot%20y%7D%2B%5Cbig%5B%20a_%7B%5Cbf%7Bp%7D%7D%5E%5Cdagger%2C%20a_%7B%5Cbf%7Bq%7D%7D%5Cbig%5D%20e%5E%7Bip%5Ccdot%20x%7De%5E%7B-iq%5Ccdot%20y%7D%5Cbigg%5C%7D%5C%5C%0A%5Cend%7Beqnarray%7D%0A%7D)