The classical expression of a plane electromagnetic wave (electric part)(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\bar{E}(t,\bar{x})=\bar{E}_{0}e^{i(\bar{k}\cdot \bar{x}-\omega t)}

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looks a lot like the basis function of the Fourier decomposition in Minkowski space-time

[tex]

\bar{E}(\bar{w})=\int_{-\infty}^{\infty}\hat{\bar{E}}(\bar{\nu})e^{2\pi i\ \eta(\bar{\nu},\bar{w})}d\bar{\nu}

[/tex]

where [itex]\bar{E}\colon\mathbb{R}^{4}\to\mathbb{R}^{4}[/itex] and [itex]\bar{w}=(ct,\bar{x})[/itex]. If I write [itex]\bar{\nu}=\frac{\nu}{c}(1,\bar{n})[/itex] with [itex]\left\|\bar{n}\right\|=1[/itex] then we get

[itex]\eta(\bar{\nu},\bar{w})=\frac{\nu}{c}\bar{n}\cdot \bar{x}-\nu t=\frac{1}{2\pi}(\bar{k}\cdot \bar{x}-\omega t)[/itex] and

[tex]

\bar{E}(\bar{w})=\int_{-\infty}^{\infty}\hat{\bar{E}}(\bar{\nu})e^{ i\ (\bar{k}\cdot \bar{x}-\omega t)}d\bar{\nu}

[/tex]

and for a monochromatic wave

[tex]

\bar{E}(\bar{w})=\bar{E}(ct,\bar{x})=\hat{\bar{E}}(\bar{\nu})e^{ i\ (\bar{k}\cdot \bar{x}-\omega t)}

[/tex]

which is close to the classical expression, but not exactly. So the point is, I feel Fourier decomposition in Minkowski space and the classical plane wave are related, but I'm not sure how. Can someone clarify?

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# Plane wave in Minkowski space-time

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