nLab wave polarization

for other concepts of a similar name see at polarization

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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 $k$, but also by that component vector $e$. Broadly speaking this $e$ is the polarization of the wave.

Specifically a plane wave with coefficients in the cotangent bundle of a Minkowski spacetime $\Sigma$, such as an electromagnetic field history “vector potential” $A \in \Gamma_\Sigma(T^\ast \Sigma)$ is given by

$$A_\mu(x) = e_\mu e^{i k_\mu x^\mu} \,.$$

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

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

This is also called the space of transversal polarizations.

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

References

• Wikipedia, Polarization (waves)

• Radovan Dermisek, Quantum Electrodynamics (QED) (pdf, pdf)