The Algebra of Light - Ettore Majorana’s Hidden Tower

Quantum Quill
Jun 18
2 min read

Updated: Sep 9

CHAPTER 2

For decades, paraparticle statistics lived only in equations: today they finally have a lab address.

Majorana’s Real Breakthrough

Ettore Majorana is famous for the self-conjugate “Majorana fermion,” yet his bolder insight was an infinite-spin tower of equations that extends well beyond bosons and fermions. Until now, nobody had turned that tower into a working, verifiable platform.

Hype vs Demonstration

In the last ten years 150 + papers and thousands of news articles have trumpeted zero-bias peaks and topological braids in exotic superconductors—Microsoft’s “Project Majorana” included. Not one of those efforts has yielded a clear, reproducible demonstration of true Majorana statistics.

What We Deliver Instead

Four statistics—bosonic, fermionic, plus two paraparticle types—formalized by a Z₂ × Z₂-graded algebra.
All four sectors encoded in a single photon’s SAM + OAM modes.
Deterministic CNOT and Toffoli on that four-level system—no ancilla photons, no cryogenics.
Direct verification: we measure each commutation/anticommutation rule on the optics bench.

(Preprint under peer review, arXiv: 2505.23232.)

This is the first physics-based demonstration of paraparticle behavior—no “tantalizing hints,” but data you can repeat.

How We Built It

Layer Key Idea

Grading Z₂ × Z₂ creates four sectors: bosonic (0,0), fermionic (1,1), and two genuine paraparticle sectors (1,0) and (0,1).

Algebra The graded commutator sign (−1)a⋅b(-1)^{a·b} dictates whether two operations commute, anticommute, or land in between.

Photonics We map those sectors onto structured photons (SAM + OAM) and use integrated waveguides to enact and read each rule: no superconductors, no dilution fridges.