Contents

Idea

Let $n \in \mathbb{N}$ and write $\mathbb{R}^n$ for the Cartesian space of dimension $n$. Then a plane wave on $\mathbb{R}^n$ is a function

$\mathbb{R}^n \longrightarrow \mathbb{C}$

given by the exponential function with complex argument of the form

$\vec x \;\mapsto\; A e^{2 \pi i \vec k \cdot \vec x} \,.$

Here $\vec k \in \mathbb{R}^n$ is the wave vector of this plane wave (and $A \in \mathbb{C}$ is its amplitude).

In Fourier analysis over Cartesian space, the Fourier transform expresses every function with rapidly decreasing partial derivatives as a superposition of plane waves.

If here $\mathbb{R}^n \simeq \mathbb{R}^{p,1}$ is identified with Minkowski spacetime with canonical coordinates labeled $(x^0, x^1, \cdots x^p)$, then the 0-component of the wave vector

$\nu \coloneqq k_0$

is called the frequency of the wave (in this chosen coordinate system). If in this situation the wave vector satisfies $k_\mu k^\mu = 0$, then the plane wave is a solution to the wave equation on Minkowski spacetime. If more generally it satisfies $k_\mu k^\mu + m^2 = 0$ for some $m^2 \in \mathbb{R}$, the it is a solution to the Klein-Gordon equation on Minkowski spacetime.

plane waves on Minkowski spacetime

$\array{ \mathbb{R}^{p,1} &\overset{\psi_k}{\longrightarrow}& \mathbb{C} \\ x &\mapsto& \exp\left( \, i k_\mu x^\mu \, \right) \\ (\vec x, x^0) &\mapsto& \exp\left( \, i \vec k \cdot \vec x + i k_0 x^0 \, \right) \\ (\vec x, c t) &\mapsto& \exp\left( \, i \vec k \cdot \vec x - i \omega t \, \right) }$
symbolname
$c$speed of light
$\hbar$Planck's constant
$\,$$\,$
$m$mass
$\frac{\hbar}{m c}$Compton wavelength
$\,$$\,$
$k$, $\vec k$wave vector
$\lambda = 2\pi/{\vert \vec k \vert}$wave length
${\vert \vec k \vert} = 2\pi/\lambda$wave number
$\omega \coloneqq k^0 c = -k_0 c = 2\pi \nu$angular frequency
$\nu = \omega / 2 \pi$frequency
$p = \hbar k$, $\vec p = \hbar \vec k$momentum
$E = \hbar \omega$energy
$\omega(\vec k) = c \sqrt{ \vec k^2 + \left(\frac{m c}{\hbar}\right)^2 }$Klein-Gordon dispersion relation
$E(\vec p) = \sqrt{ c^2 \vec p^2 + (m c^2)^2 }$energy-momentum relation