nLab dispersion relation

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Contents

Idea

In harmonic analysisa dispersion relation is a relation between the frequency and the wavelength of plane waves. Typically this relation expresses the frequency $\nu(\lambda)$ as a function of the wavelength.

In special relativity the frequency of a plane wave is proportional to its energy, and its wave vector is proportional to its momentum, so that now a dispersion relation becomes an energy-momentum relation.

on

$\array{ \mathbb{R}^{p,1} &\overset{\psi_k}{\longrightarrow}& \mathbb{C} \\ x &\mapsto& \exp\left( \, i k_\mu x^\mu \, \right) \\ (\vec x, x^0) &\mapsto& \exp\left( \, i \vec k \cdot \vec x + i k_0 x^0 \, \right) \\ (\vec x, c t) &\mapsto& \exp\left( \, i \vec k \cdot \vec x - i \omega t \, \right) }$
symbolname
$c$
$\hbar$
$\,$$\,$
$m$
$\frac{\hbar}{m c}$
$\,$$\,$
$k$, $\vec k$
$\lambda = 2\pi/{\vert \vec k \vert}$
${\vert \vec k \vert} = 2\pi/\lambda$
$\omega \coloneqq k^0 c = -k_0 c = 2\pi \nu$
$\nu = \omega / 2 \pi$
$p = \hbar k$, $\vec p = \hbar \vec k$
$E = \hbar \omega$
$\omega(\vec k) = c \sqrt{ \vec k^2 + \left(\frac{m c}{\hbar}\right)^2 }$
$E(\vec p) = \sqrt{ c^2 \vec p^2 + (m c^2)^2 }$