Photon vs Electron

The photon lives in the simplest non-trivial sector: \(\mathbb{CP}^1\) (the complex projective line, topologically \(S^2\)). Its two polarization states are the complete internal geometry of this 2D space — there are no additional sector coordinates beyond the two transverse directions. The \(n=0\) mode with \(S(0,2) = 0\) makes it exactly massless.

Because it is a pure gauge mode on the U(1) Hopf fiber, an observer at the \(d=3\) coordinate level resolves the photon as a clean transverse wave. This is why light behaves so classically and propagates freely over cosmological distances. The two polarizations of light are not an arbitrary choice — they are the full set of degrees of freedom of the \(d=2\) sector, giving a geometric origin for helicity and why photons have exactly two polarization states and no others.

The photon's two sector coordinates are both transverse — they specify the polarization directions perpendicular to travel. There is no third sector coordinate pointing along the propagation axis. This has a direct consequence for interaction geometry: in IDWT, a particle couples to a partner only where all of its sector coordinates are "in contact" with the partner's sector. For the photon, both coordinates are transverse and the contact condition is satisfied along the entire line of travel — there is no longitudinal handle that needs to align. The photon appears extended along its path not because the wave is physically spreading, but because it has nothing to localise itself there. When it encounters an electron (\(d=6\)), whose sector contains all of the photon's coordinates, the coupling is available at every transverse cross-section. The electron's \(d=6\) sector includes full \(d=3\) coordinate structure, which pins it to spatial regions — producing the familiar orbital shapes instead.

The electron lives in a much richer \(d=6\) sector (\(\mathbb{CP}^3\) geometry). It is a small, compact resonance executing a genuine orbit in all 6 sector dimensions — the electron itself is not spread across the atom. What is spread across the atom is the 6D orbit: the 6D path the electron traces, projected down to the three directions we can observe.

The familiar s, p, d, f orbital shapes are the 3D projections of the electron's 6D orbit paths at each angular momentum level. Each angular momentum eigenstate of the 6D orbit — classified by the SU(4) ⊃ SU(3) ⊃ SO(3) representation chain — has a specific 6D path whose 3D shadow gives the observed shape. The richness of 6D orbit structure, the intricacy of heavy-element chemistry, and the nodal geometry of high-L orbitals all come from the higher multiplicity and complex topology of 6D mode structure, not from a wave smeared across the orbit volume.

Dimensional hierarchy. Photons are "simple" because they live in the lowest-dimensional sector. Electrons are "complicated" because they live in a higher-dimensional one. The electromagnetic interaction couples them via the shared U(1) Hopf fiber — the \(d=2\) photon coordinates are inside the \(d=6\) electron sector, which is why charge coupling exists at all.

Chemistry as 6D structure. What 3D chemistry calls atomic orbitals, chemical bonding patterns, and molecular shapes are the 3D shadows of 6D orbit paths. Nodes and lobes arise because the 6D path visits some 3D regions but not others; hybridization reflects different angular momentum combinations of the 6D orbit; heavier atoms have more electrons filling higher angular momentum levels, so their chemistry involves the more intricate 3D shadows that high-L 6D orbits project.