89 lines
2.5 KiB
TeX
89 lines
2.5 KiB
TeX
\nopagenumbers
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\magnification=1200
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\beginsection{3.1.1.tex}
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$$\phi({\bf x},t)=\sum_{\bf k}
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\left[
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\overbrace{
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a({\bf k})\exp(-i\omega({\bf k})t+{\bf k}\cdot{\bf x})
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\over\sqrt{2V\omega({\bf k})}
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}^A
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+
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\overbrace{
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a^\dagger({\bf k})\exp(i\omega({\bf k})t-{\bf k}\cdot{\bf x})
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\over\sqrt{2V\omega({\bf k})}
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}^B
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\right]
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$$
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$$\dot\phi({\bf x^\prime},t)=\sum_{\bf k^\prime}
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\left[
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\overbrace{
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-i\omega({\bf k^\prime})a({\bf k^\prime})
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\exp(-i\omega({\bf k^\prime})t+{\bf k^\prime}\cdot{\bf x^\prime})
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\over\sqrt{2V\omega({\bf k^\prime})}
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}^C
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+
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\overbrace{
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i\omega({\bf k^\prime})a^\dagger({\bf k^\prime})
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\exp(i\omega({\bf k^\prime})t-{\bf k^\prime}\cdot{\bf x^\prime})
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\over\sqrt{2V\omega({\bf k^\prime})}
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}^D
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\right]
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$$
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$$\eqalignno{
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\Delta=
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-&
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%{i\omega({\bf k^\prime})\over2V\sqrt{\omega({\bf k})\omega({\bf k^\prime})}}
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a({\bf k})a({\bf k^\prime})
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\exp(-i\omega({\bf k})t-i\omega({\bf k^\prime})
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+{\bf k}\cdot{\bf x}+{\bf k^\prime}\cdot{\bf x^\prime})
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&AC\cr
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+&
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%{i\omega({\bf k^\prime})\over2V\sqrt{\omega({\bf k})\omega({\bf k^\prime})}}
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a({\bf k^\prime})a({\bf k})
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\exp(-i\omega({\bf k})t-i\omega({\bf k^\prime})
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+{\bf k}\cdot{\bf x}+{\bf k^\prime}\cdot{\bf x^\prime})
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&-CA\cr
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+&
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%{i\omega({\bf k^\prime})\over2V\sqrt{\omega({\bf k})\omega({\bf k^\prime})}}
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a({\bf k})a^\dagger({\bf k^\prime})
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\exp(-i\omega({\bf k})t+i\omega({\bf k^\prime})
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+{\bf k}\cdot{\bf x}-{\bf k^\prime}\cdot{\bf x^\prime})
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&AD\cr
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-&
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%{i\omega({\bf k^\prime})\over2V\sqrt{\omega({\bf k})\omega({\bf k^\prime})}}
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a^\dagger({\bf k^\prime})a({\bf k})
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\exp(-i\omega({\bf k})t+i\omega({\bf k^\prime})
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+{\bf k}\cdot{\bf x}-{\bf k^\prime}\cdot{\bf x^\prime})
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&-DA\cr
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-&
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%{i\omega({\bf k^\prime})\over2V\sqrt{\omega({\bf k})\omega({\bf k^\prime})}}
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a^\dagger({\bf k})a({\bf k^\prime})
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\exp(i\omega({\bf k})t-i\omega({\bf k^\prime})
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-{\bf k}\cdot{\bf x}+{\bf k^\prime}\cdot{\bf x^\prime})
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&BC\cr
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+&
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%{i\omega({\bf k^\prime})\over2V\sqrt{\omega({\bf k})\omega({\bf k^\prime})}}
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a({\bf k^\prime})a^\dagger({\bf k})
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\exp(i\omega({\bf k})t-i\omega({\bf k^\prime})
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-{\bf k}\cdot{\bf x}+{\bf k^\prime}\cdot{\bf x^\prime})
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&-CB\cr
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+&
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%{i\omega({\bf k^\prime})\over2V\sqrt{\omega({\bf k})\omega({\bf k^\prime})}}
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a^\dagger({\bf k})a^\dagger({\bf k^\prime})
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\exp(i\omega({\bf k})t+i\omega({\bf k^\prime})
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-{\bf k}\cdot{\bf x}-{\bf k^\prime}\cdot{\bf x^\prime})
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&BD\cr
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-&
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%{i\omega({\bf k^\prime})\over2V\sqrt{\omega({\bf k})\omega({\bf k^\prime})}}
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a^\dagger({\bf k^\prime})a^\dagger({\bf k})
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\exp(i\omega({\bf k})t+i\omega({\bf k^\prime})
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-{\bf k}\cdot{\bf x}-{\bf k^\prime}\cdot{\bf x^\prime})
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&-DB\cr
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}$$
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\end
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