nLab plane wave

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

$$\array{ \mathbb{R}^{p,1} &\overset{\psi_k}{\longrightarrow}& \mathbb{C} \\ x &\mapsto& \exp\left( \, i k_\mu x^\mu \, \right) \\ (\vec x, x^0) &\mapsto& \exp\left( \, i \vec k \cdot \vec x + i k_0 x^0 \, \right) \\ (\vec x, c t) &\mapsto& \exp\left( \, i \vec k \cdot \vec x - i \omega t \, \right) }$$

symbol	name
$c$	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

Related concepts

• wave

• wave polarization

• Fourier transform

• wave function

• pp-wave spacetime

References

See also

• Wikipedia, Plane wave