Why Not Five?
Why Not Three?
The universe has four dimensions. This paper derives that number from scratch — it is not assumed.
the Lorentz algebra's representation theory imposes
no constraint the postulate did not generate."
The Chirality Argument
The representations of a Lie algebra — like its Killing form — are determined entirely by the structure constants. They are not an additional input. In each spacetime dimension, the question is: does the algebra impose a constraint on its spinor representations that the postulate did not demand? Any such constraint is external structure — forbidden.
The Dynkin Classification of so(N, 1)
The complexified Lorentz algebra determines spinor representations. Its type changes with N. Only at N = 3 does it factorise — giving two fully independent sectors.
D ≤ 2: The Lorentz algebra is abelian or trivial. Already eliminated by Cartan's criterion (degenerate Killing form).
Odd D ≥ 3: Type A1 (D = 3) or type Bn (D ≥ 5). No chirality operator exists. No Weyl spinors. The algebra forecloses distinguishing left from right — a constraint the postulate did not generate. Eliminated.
Even D ≥ 6: Type Dn with n ≥ 3: simple. Weyl spinors exist, but they are conjugate representations related by an outer automorphism. The algebra forces left and right to be mirror images — a relationship the postulate did not generate. Eliminated.
D = 4: Type D2 = A1 × A1 — the unique non-simple case. It factorises into two independent copies of sl(2, C). Representations (jL, jR) live in independent algebras. No relationship between left and right is imposed. No constraint is generated. Survives.
Every Other Dimension Is Eliminated
The Dynkin classification is exhaustive. The complexified Lorentz algebra so(N+1, ℂ) has a definite Cartan type in every dimension. In each case except D = 4, the algebra imposes a constraint on spinor representations that the postulate never demanded.
D ≤ 2: Abelian or Trivial
Already eliminated by Cartan's criterion: the Killing form is degenerate, requiring external structure. The algebra cannot be self-contained. Ruled out before reaching the representation theory argument.
Odd D ≥ 3: No Chirality Operator
The complexified Lorentz algebra is type A1 (D = 3) or type Bn (D ≥ 5). In odd spacetime dimensions, no chirality operator exists. There are no Weyl spinors. The algebra forecloses distinguishing left from right — a constraint the postulate did not generate. Eliminated.
Even D ≥ 6: Forced Mirror Symmetry
The complexified Lorentz algebra is type Dn with n ≥ 3: simple. Weyl spinors exist, but they are conjugate representations of a single simple algebra, related by an outer automorphism. The algebra forces left and right to be mirror images of each other — a relationship the postulate did not generate. Eliminated.
D = 4: The Unique Case
The complexified Lorentz algebra is type D2 = A1 × A1: the unique case that isn't simple. It factorises into two independent copies of sl(2, ℂ). Weyl spinors (jL, 0) and (0, jR) are representations of independent algebras. No relationship between left and right is imposed. No constraint is generated. Survives.
| D (spacetime) | N (spatial) | Algebra type | Constraint imposed | Result |
|---|---|---|---|---|
| 2 | 1 | Abelian | Degenerate Killing form | ✗ Excluded |
| 3 | 2 | A₁ (odd) | No Weyl spinors | ✗ Excluded |
| 4 | 3 | D₂ = A₁×A₁ | None — two independent sectors | ✓ Selected |
| 5 | 4 | B₂ (odd) | No Weyl spinors | ✗ Excluded |
| 6 | 5 | D₃ (even, simple) | Forced mirror symmetry | ✗ Excluded |
| 7+ | 6+ | Bₙ / Dₙ | No Weyl / forced mirror | ✗ Excluded |
the Lorentz algebra's representation theory imposes
no constraint the postulate did not generate.
Mathematics is the art of giving the same name to different things.
Poincaré intuited deep unification in physics and mathematics. This paper shows that the 'different things' — special relativity, general relativity, the cosmological constant, gravitational waves — are all the same thing: consequences of one postulate.