Wigner's classification

Massless case, ${\displaystyle \mathrm {SE} (2)}$

As aforewritten, for massless particles with lightlike momentum (${\displaystyle k^{2}=0}$), the Wigner little group is a double cover of ${\displaystyle \mathrm {SE} (2)}$—the group of isometries of 2-dimensional Euclidean space. The geometric reasoning behind this fact was originally a mystery to Wigner, but has since been resolved.^[1][2][3][4] The answer to this mystery is best presented in the language of the spacetime algebra (STA), ${\displaystyle \mathbb {G} _{1,3}=\mathbb {R} [\gamma _{0},\gamma _{1},\gamma _{2},\gamma _{3}]}$, which is the Clifford or geometric algebra of (1+3)-dimensional spacetime. It is generated by the orthonormal basis vectors ${\displaystyle {\gamma _{0},\gamma _{1},\gamma _{2},\gamma _{3}}}$ satisfying the Minkowski metric, $${\displaystyle \gamma _{\mu }\cdot \gamma _{\nu }={\frac {1}{2}}(\gamma _{\mu }\gamma _{\nu }+\gamma _{\nu }\gamma _{\mu })=\eta _{\mu \nu }.}$$ The vectors are written with gammas as their matrix representation is given by the Dirac matrices. Without loss of generality: For a lightlike momentum vector ${\displaystyle k=k_{0}(\gamma _{0}+\gamma _{3})\in \mathbb {G} _{1,3}^{1}}$ there exists some spacelike polarization vector ${\displaystyle s=s_{1}\gamma _{1}+s_{2}\gamma _{2}}$ satisfying ${\displaystyle s\cdot k=0}$. And for this ${\displaystyle k}$, there is a subset of the (spin) Lorentz group ${\displaystyle \mathrm {Spin} (1,3)\subset \mathbb {G} _{1,3}}$ that, for ${\displaystyle \Lambda _{k}\in \mathrm {Spin} (1,3)}$, satisfies $${\displaystyle \Lambda _{k}k\Lambda _{k}^{-1}=k.}$$ This is the defining relation of the massless little group that is a double cover of ${\displaystyle \mathrm {SE} (2)}$: $${\displaystyle \mathrm {Spin} (0,2,1)/{\pm 1}\approx \mathrm {SE} (2).}$$ Elements of this group are exponentials of the Lie algebra generators, ${\displaystyle J_{3}=\gamma _{12},\quad N_{1}=\gamma _{1}(\gamma _{0}+\gamma _{3}),\quad N_{2}=\gamma _{2}(\gamma _{0}+\gamma _{3}).}$ It follows that, when transforming under ${\displaystyle \Lambda _{J}=\exp[{-\omega J_{3}/2}]}$, $${\displaystyle s'=\Lambda _{J}s\Lambda _{J}^{-1}=e^{-{\frac {\omega }{2}}\gamma _{12}}se^{{\frac {\omega }{2}}\gamma _{12}}.}$$ Then, when transforming under ${\displaystyle \Lambda _{N}=\exp {[-(\nu _{1}N_{1}+\nu _{2}N_{2})/2]}}$, and for appropriate real scalars ${\displaystyle \alpha }$ and ${\displaystyle \beta }$,$${\displaystyle s''=\Lambda _{N}s\Lambda _{N}^{-1}=s-(\alpha s_{2}+\beta s_{1})(\gamma _{0}+\gamma _{3}).}$$ Thus the bivector ${\displaystyle s\wedge k=sk}$ is invariant under the little group transformation$${\displaystyle \Lambda _{N}sk\Lambda _{N}^{-1}=s''k=sk,}$$ but not invariant under$${\displaystyle \Lambda _{J}sk\Lambda _{J}^{-1}=s'k\neq sk.}$$ The transformations ${\displaystyle \Lambda _{N}}$ leave the bivector ${\displaystyle sk}$ invariant and are therefore not physically-realizable. It so happens that this bivector is algebraically equivalent to the electromagnetic field.^[4][5] This serves as a derivation of the electromagnetic field from the invariance of lightlike momentum, and proves that the only physically-realizable transformations within the massless little group are the rotations of $${\displaystyle \Lambda _{J}\in \mathrm {Spin} (2)\approx \mathrm {U} (1)\subset \mathrm {Spin} (0,2,1).}$$ Indeed, these are the helicity degrees of freedom for the photon.

Using the mirror view of geometry (also plane-based geometry)—wherein vectors represent hyperplanes—results in a direct geometric interpretation for the massless little group. The resulting pictures, when projectively modeled in relative 3-dimensional space show that the lightlike momentum vector ${\displaystyle k}$ is a plane tangent to the lightsphere while the spacelike polarization vector ${\displaystyle s}$ and its transformation ${\displaystyle s'}$ are planes orthogonal to ${\displaystyle k}$ and passing through the origin of the lightsphere. The bivector ${\displaystyle sk}$ is the intersection of the planes ${\displaystyle s}$ and ${\displaystyle k}$, which is a line tangent to the lightsphere. The physical-realizability of the subgroup ${\displaystyle \mathrm {Spin} (2)}$ is geometrically seen as a rotation of ${\displaystyle sk}$ into ${\displaystyle s'k}$ about the axis orthogonal to the plane ${\displaystyle k}$ and passing through the origin of the lightsphere. Said otherly, the physical-realizability is seen as the set of all lines tangent to the lightsphere at a specific point. The transformations of ${\displaystyle s}$ to ${\displaystyle s''}$ preserve the intersection line, and are therefore not physically-realizable; they are but gauge transformations.