(1) In the canonical quantization of the free electromagnetic field, the Coulomb gauge condition $$A^0=0,~~ \nabla\cdot\textbf{A}=0\tag{1}$$ implies that the polarization vector $\epsilon^\mu$ satisfies $$\epsilon^0=0,~~\boldsymbol{\epsilon}\cdot\hat{\textbf{p}}=0\tag{2}$$ which says that the electromagnetic field has two independent transverse states of polarization.
(2) From the representation theory of Poincare group, it is known that for photons $\textbf{S}\cdot\hat{\textbf{p}}$ has eigenvalues $h=\pm 1$ where $\textbf{S}$ denotes the spin operator.
Both the descriptions above make it clear that the electromagnetic field has two independent degrees of freedom. The description (1) says the electromagnetic field has two independent states of polarization and the description (2) says that it has two independent helicity states.
Question
$\bullet$ Does it mean states of polarization are identical to states of helicity?
$\bullet$ Is there is a unique one-to-one correspondence between the states of helicity and the independent states of polarization? In that case, $h=+1$ corresponds to which polarization and $h=-1$ corresponds to which? How can such a correspondence, if exists, be understood?
A similar question was asked here.
No comments:
Post a Comment