I'm preparing a little sketch on the irreps of the Poincaré group according to Wigner's classification. Unfortunately, this subject seems to be not too popular in the textbooks and in review articles. No author whom I found describes clearly and correctly the structure of the little group of the photon. The most part of them resort just to rescaling the angle (i.e. \phi\to\phi/2), some give wrong statements on the semidirect structure.
I spent some time to puzzle everything together. Now, it's quite easy to explain. I try to make some comments. I am much indebted to the comments of George Jones.
According to Simms, I will denote the little group of the photon by Δ.
George Jones said:
The universal cover of the restricted Lorentz group is SL\left( 2,\mathbb{C}\right).
That indicates, that Δ is also a sort of covering group. At least, its center will be \left\{ I_2,-I_2\right\}, I_2 being the identity element of SL\left( 2,\mathbb{C}\right).
The little group of the lightlike 4-vector X=1+\sigma_3 is the subgroup of SL\left( 2,\mathbb{C}\right) that consists of matrices of the form
<br />
\begin{pmatrix}<br />
e^{i\theta} & b\\<br />
0 & e^{-i\theta}<br />
\end{pmatrix},<br />
where \theta is an arbitrary real number and b is an arbitrary complex number.
This is the group Δ.
To get rid of using a chart of U(1) (and thereby to avoid any problems with rescaling), I prefer the equivalent definition of the group Δ as follows:
The little group Δ of the lightlike 4-vector X=1+\sigma_3 is the subgroup of SL\left( 2,\mathbb{C}\right) that consists of matrices of the form
<br />
\begin{pmatrix}<br />
u & u^{-1}b\\ 0 & u^{-1}<br />
\end{pmatrix},<br />
where u,b are arbitrary complex numbers but with \left|u\right|=1.
The introduction of an additional factor u^{-1} in the entry (1,2) simplifies the exhibition of the semidirect structure.
The key point is that the product in Δ resembles much to that in the Euclidean group E(2), but not fully. Namely we have
<br />
\begin{pmatrix}<br />
u_1 & u_1^{-1}b_1\\ 0 & u_1^{-1}<br />
\end{pmatrix}\begin{pmatrix}<br />
u_2 & u_2^{-1}b_2\\ 0 & u_2^{-1}<br />
\end{pmatrix}<br />
=<br />
\begin{pmatrix}<br />
u_1u_2 & u_2^{-1}u_1b_2+u_1^{-1}b_1\\ 0 & u_1^{-1}u_2^{-1}<br />
\end{pmatrix}<br />
=\begin{pmatrix}<br />
u_1u_2 & (u_1u_2)^{-1}(u_1^2b_2+b_1)\\ 0 & (u_1u_2)^{-1}<br />
\end{pmatrix}.<br />
The difference lies in the u_1^2.
Let's describe the special Euclidean group SE(2) by real 3\times3-matrices: SE(2) consists of the elements
<br />
\begin{pmatrix} R & b \\ 0 & 1 \end{pmatrix}<br />
with R\in SO(2), b\inℝ^2. Note: Reflections should be excluded since they cannot be represented by any e^{i\theta}.
Ordinary matrix multiplication reproduces the structure as semidirect product of rotations and translations in two dimensions correctly:
<br />
\begin{pmatrix} R_1 & b_1 \\ 0 & 1 \end{pmatrix}<br />
\begin{pmatrix} R_2 & b_2 \\ 0 & 1 \end{pmatrix}<br />
=<br />
\begin{pmatrix} R_1R_2 & R_1b_2+b_1 \\ 0 & 1 \end{pmatrix}.<br />
Here, the rotation R_1 does not appear as R_1^2.
To have then some sort of isomorphism to Δ, in the entry (1,2) should be a u_1 instead of u_1^2. It seems that the elements u and -u have to be identified.
This identification is achieved by the following homomorphism
<br />
\begin{pmatrix}<br />
e^{i\theta} & b\\<br />
0 & e^{-i\theta}<br />
\end{pmatrix}<br />
\rightarrow<br />
\begin{pmatrix}<br />
e^{i2\theta} & b\\<br />
0 & 1<br />
\end{pmatrix},<br />
that means that I have got the homomorphism \phi:
<br />
\phi:\quad<br />
\begin{pmatrix}<br />
u & b \\ 0 & u^{-1}<br />
\end{pmatrix}<br />
\mapsto<br />
\begin{pmatrix}<br />
u^2 & b \\ 0 & 1<br />
\end{pmatrix}.<br />
(That I now suppress the additional factor u^{-1} at entry (1,2) is of course of no significance.)
Note: I don't make use of any angle \theta nor of some rescaling. Defining the group multiplication with elements written in the form g(\theta,b) suffers from tackling the addition of two angles to a value exceeding 2\pi or even 4\pi. Somewhere in the literature, the mere extension of the interval \left[0,2\pi\right] to \left[0,4\pi\right] together with the replacement \theta by \theta/2 seems to imitate the descent to a group covered by Δ.
The kernel of \phi is obviously \left\{I_2,-I_2\right\}.
So, in taking the cosets of this kernel we get an isomorphism \iota
<br />
\iota:\quad<br />
\left\{\begin{pmatrix}<br />
u & b \\ 0 & u^{-1}<br />
\end{pmatrix},<br />
\begin{pmatrix}<br />
-u & -b \\ 0 & -u^{-1}<br />
\end{pmatrix}<br />
\right\}<br />
\mapsto<br />
\begin{pmatrix}<br />
u^2 & b \\ 0 & 1<br />
\end{pmatrix}.<br />
The group formed by the elements \begin{pmatrix}u^2 & b \\ 0 & 1\end{pmatrix} with \left|u\right|=1 is of course the same as the group formed by the elements \begin{pmatrix}v & b \\ 0 & 1\end{pmatrix} with \left|v\right|=1.
To prove that this group is isomorphic to the proper Euclidean group SE(2), one needs now the identification v=e^{i\theta} with \theta\in[0,2\pi[. Then one can proceed in the manner as demonstrated by George Jones. No factor 2 is needed.
I conclude that the group Δ is a twofold covering group of the Euclidean group SE(2). The covering homomorphism (Δ onto an isomorphic copy of SE(2)) is given by \phi defined above.
I do not state that the little group Δ is the universal covering group of SE(2) since I don't know whether Δ is simply connected. So, I refrain from writing \widetilde{SE(2)} instead of Δ.
