nLab
wave polarization

for other concepts of a similar name see at polarization


Contents

Idea

A plane wave with more than one component, i.e. being a section of a trivial vector bundle, is characterized not just by its wave vector kk, but also by that component vector ee. Broadly speaking this ee is the polarization of the wave.

Specifically a plane wave with coefficients in the cotangent bundle of a Minkowski spacetime Σ\Sigma, such as an electromagnetic field historyvector potentialAΓ Σ(T *Σ)A \in \Gamma_\Sigma(T^\ast \Sigma) is given by

A μ(x)=e μe ik μx μ. A_\mu(x) = e_\mu e^{i k_\mu x^\mu} \,.

In the case of free electromagnetic waves the wave vector kk is light-like, k μk μ=0k_\mu k^\mu = 0, and in Gaussian-averaged Lorenz gauge the polarization vector ee has to satisfy e μk μ=0e^\mu k_\mu = 0 and polarization vectors proportional to the wave vector e μk μe_\mu \propto k_\mu are gauge equivalent to zero. Therefore in this case the space of physically distinguishable polarizations for given wave vector kk is the quotient space

{e|e μk μ=0}{e|e μk μ}. \frac{ \left\{ e \,\vert\, e^\mu k_\mu = 0 \right\} }{ \left\{ e \,\vert\, e_\mu \propto k_\mu \right\} } \,.

This is also called the space of transversal polarizations.

If one chooses coordinates such that k=(κ,0,,0,κ)k = (\kappa, 0, \cdots, 0, \kappa) then this may be identified with the space of vectors of the form e=(0,*,,*,0)e = (0, *, \cdots, *, 0).

References

Last revised on December 18, 2017 at 09:12:31. See the history of this page for a list of all contributions to it.