The Wigner Little Group for Photons Is a Projective Subalgebra

Abstract

This paper presents the Geometric Algebra approach to the Wigner little group for photons using the Spacetime Algebra, incorporating a mirror-based view for physical interpretation. The shift from a point-based view to a mirror-based view is a modern movement that allows for a more intuitive representation of geometric and physical entities, with vectors and their higher-grade counterparts viewed as hyperplanes. This reinterpretation simplifies the implementation of homogeneous representations of geometric objects within the Spacetime Algebra and enables a relative view via projective geometry. Then, after utilizing the intrinsic properties of Geometric Algebra, the Wigner little group is seen to induce a projective geometric algebra as a subalgebra of the Spacetime Algebra. However, the dimension-agnostic nature of Geometric Algebra enables the generalization of induced subalgebras to (1+n)-dimensional Minkowski geometric algebras, termed little photon algebras. The lightlike transformations (translations) in these little photon algebras are seen to leave invariant the (pseudo)canonical electromagetic field bivector. Geometrically, this corresponds to Lorentz transformations that do not change the intersection of the spacelike polarization hyperplane with the lightlike wavevector hyperplane while simultaneously not affecting the lightlike wavevector hyperplane. This provides for a framework that unifies the analysis of symmetries and substructures of point-based Geometric Algebra with mirror-based Geometric Algebra.

Figures (4)

• Figure 1: List of geometric interpretation assigned to different grades in both the point-based view and mirror-based view.
• Figure 2: An example of the mirror-based view in the $3$-dimensional relative space $\mathbb{G}_3\approx\mathbb{G}_{1,3}^+$. The red plane is $\gamma_{10}$, the green plane is $\gamma_{20}$, and the blue plane is $\gamma_{30}$. The arrows at the ends of the planes denote the extrinsic orientation of the (hyper)planes.
• Figure 3: Visualization in $\mathbb{G}_{1,2}$. Left: The lightlike worldplane (gray), $k$, lies tangent to the lightcone (red), $|v|=c$, with the spacelike worldplane (blue), $s$. Right: An observer's relative frame at time $ct$, where the lightlike worldplane (gray), $k$, looks like a line tangent to the light-circle (red), $|v|=c$, with the spacelike worldplane (blue), $s$, which looks like a line orthogonal to $k$ intersecting on the light-circle.
• Figure 4: Left: Visualization in the relative view of the Spacetime Algebra. Spacelike $s$ (light blue) is rotated about $\gamma_{12}=e_{12}$ into $s'$ (dark blue), and separately Lorentz transformed into $s"$ (pink) by the rotor in EQ. \ref{['EQ:sec2.1: Lorentz Rotor']}. The canonical bivector $sk$ (black line) is unchanged by $s\mapsto s"$. The red sphere represents the light-sphere at moment $ct$. Top right: In the cylindrical group, $s\mapsto s'$ is equivalent to moving on a circle orthogonal to the height, and does not leave $sk$ invariant. Bottom right: In the cylindrical group, $s\mapsto s"$ is equivalent to moving vertically, and leaves $sk$ invariant.