From Dyadic Complex Marks to 1+3 Minkowski Spacetime

A reconstruction — one result, assumptions exposed. One narrow claim, fully worked, with every assumption stated and the central objection — "have you smuggled the physics in by choosing spinors?" — met directly in §4. No gravity, no quantum dynamics, no constants, no consciousness. Just this: from three explicit assumptions about an interaction's mark, the 1+3-dimensional Lorentzian structure and the Lorentz group follow. The word in the title is reconstruction, deliberately.

0. What is and is not claimed

Claimed. Three assumptions — (A1) an interaction has exactly two poles; (A2) its mark is a complex amplitude; (A3) physical quantities read from a mark are real, quadratic, and phase-invariant — suffice to force: the space of such quantities is 4-dimensional, carries a Lorentzian quadratic form of signature (1,3), its symmetry group is the Lorentz group SO⁺(1,3) (doubly covered by SL(2,ℂ)), and the spatial rotations form SO(3) (doubly covered by SU(2)). The "1" and the "3" are identified explicitly.

Not claimed. That spacetime arises "from interaction alone" with no further input. The three assumptions are real, and one of them (A3) is the classical gateway to spinor–vector structure. This note shows the Lorentzian structure is the unique consequence of A1–A3 — an economical reconstruction — not that it appears from nothing. §4 is devoted to exactly this distinction, honestly.

Provenance of the mathematics. The correspondence between Hermitian 2×2 matrices and Minkowski space, and between SL(2,ℂ) and the Lorentz group, is classical (Pauli; Weyl; Penrose–Rindler). Nothing in the mathematics of §2–§3 is new. The only candidate for novelty is in §1 and §4: that the inputs to that classical correspondence can be motivated from an interaction ontology rather than postulated. The reader should weigh that claim, and only that claim.

1. The three assumptions

(A1) The mark is a 2-dimensional state. The mark is a vector in a 2-dimensional space V. Corrected: the "2" is not the two poles — that was a non-sequitur (arity ≠ amplitude dimension). The 2 comes from the mark being a binary distinction (two sides); that those sides form a superposable state space at all is a separate, quantum premise. Both are treated in the companion note Why the Mark is ℂ²; here A1 is taken as given.

(A2) Complex amplitude. The field is ℂ, so V = ℂ². Justification: the amplitude field is taken to be an associative normed division algebra (so that marks compose without zero-divisors — a vanishing composite would be a completed interaction conveying nothing). By Hurwitz's theorem the options are ℝ, ℂ, ℍ, 𝕆. ℝ carries no continuous phase (no interference between alternative completion-orderings); ℍ, 𝕆 have no associative tensor product, so composite systems are ill-defined. ℂ is the unique survivor. Honest flag: the premise "we want a continuous phase and well-defined composites" is partly physics-motivated — it is the desideratum of a theory that supports superposition. A2 is therefore selected, not ontologically forced. (Empirically, real-amplitude quantum theory was falsified by Renou et al., Nature 2021, which supports the selection but is external evidence, not part of the derivation.)

(A3) Observables are phase-invariant real quadratic forms. A real physical quantity q read from a mark ψ is (i) real, (ii) quadratic in ψ (bilinear in ψ, ψ̄ — the minimal nontrivial degree, since linear-in-ψ quantities are complex and not phase-invariant), and (iii) invariant under the global phase ψ ↦ e^iθψ (the phase distinguishes no pole and carries no observable content — it is a redundancy). Honest flag: A3 is the load-bearing, physics-adjacent assumption. "Real quadratic phase-invariant function of a spinor" is precisely the classical recipe that yields a 4-vector; this is where the spinor–vector correspondence is entered. It is motivated (those three adjectives are natural), but a skeptic should press here, and §4 does.

2. The construction

By (A3), a phase-invariant real quadratic form in ψ ∈ ℂ² is

q_M(ψ) = ψ^† M ψ, M = M^† (Hermitian),

because ψ^†Mψ is real iff M is Hermitian, is quadratic by construction, and is phase-invariant. The observables are thus indexed by the Hermitian 2×2 matrices, a real 4-dimensional vector space Herm₂(ℂ).

The mark's own content (its "momentum") is the operator that generates all its observables — the Hermitian density

P_ψ = ψψ^† ∈ Herm₂(ℂ),

the unique (up to scale) Hermitian operator with q_M(ψ) = tr(M P_ψ) for all M. So a mark is a point of the 4-dimensional real space Herm₂(ℂ). Choose the basis σ_μ = (I, σ_x, σ_y, σ_z) (identity + Pauli matrices); they span Herm₂. Write any P ∈ Herm₂ as P = p^μσ_μ with real coordinates p^μ = (p⁰, p¹, p², p³):

P = [ p⁰+p³ p¹−ip² ; p¹+ip² p⁰−p³ ]

3. The theorem

Proposition 1 (the metric). The determinant is a quadratic form on Herm₂ of signature (1,3):

det P = (p⁰)² − (p¹)² − (p²)² − (p³)² = η_μν p^μ p^ν, η = diag(+,−,−,−).

Proof. Direct expansion of the 2×2 determinant. ∎ So Herm₂(ℂ) ≅ ℝ^1,3 as a quadratic space: 4 dimensions, Lorentzian signature. The split is canonical: p⁰ = ½ tr P (the trace part) is the unique rotation-invariant coordinate — the time direction (1); the traceless part pⁱ spans the 3 space directions.

Proposition 2 (mark momenta are null). For a single mark, P_ψ = ψψ^† has rank one, so det P_ψ = 0, i.e. η_μνp^μp^ν = 0 with p⁰ = ½‖ψ‖² > 0: a future-pointing null (lightlike) vector. A bare mark is massless.

Proposition 3 (the symmetry group). The transformations ψ ↦ Sψ with S ∈ SL(2,ℂ), acting on Herm₂ by P ↦ SPS^†, preserve Hermiticity and (since det S = 1) the determinant, hence the Minkowski form. The map SL(2,ℂ) → SO⁺(1,3) is a 2-to-1 surjective homomorphism (±S give the same Lorentz transformation): the Lorentz group is recovered, doubly covered. Its maximal compact subgroup SU(2) fixes p⁰ and acts on the traceless part as SO(3), again 2-to-1: spatial rotations, with the spinor's double-valuedness (2π ↦ −1) built in.

Summary. From A1–A3: a mark lives in a 4-dimensional real space with a (1,3) Lorentzian metric; bare marks are null; the symmetry group is the Lorentz group with SU(2) → SO(3) rotations. The dimension count is 1+3, with the 1 = trace and the 3 = the three Pauli directions.

4. Is this circular? Assumed vs. derived, stated plainly

This is the section the result lives or dies by. The objection: spinors already encode Lorentz structure, so you assumed your conclusion. Here is the honest accounting.

What was NOT assumed (genuine outputs). No metric was put in; no signature; no dimension count; no Lorentz group; no notion of spacetime. We did not write down η_μν, the number 4, the split 1+3, or SL(2,ℂ). Every one of these is an output of Propositions 1–3. In particular the signature (that time and space enter with opposite sign) and the dimension split (1+3, not 4+0 or 2+2) are not inputs — they are forced by det on Herm₂.

What WAS assumed (the inputs), ranked by how much physics they carry:

A1 (the mark is a 2-dim state): partly ontological, partly quantum. The number 2 is ontological (a binary distinction has two sides). That the sides form a superposable state space is the quantum premise (see Why the Mark is ℂ², P0). The earlier "two poles ⟹ dim 2" was a non-sequitur and is withdrawn.
A2 (field = ℂ): selected, premise mildly physics-flavored. The division-algebra argument is real, but its premise ("we want a continuous phase and good composites") is the wish-list of a theory that will support superposition. Not ontologically forced; honestly, selected.
A3 (observables = phase-invariant real quadratic forms): the gateway, physics-adjacent. "Real, quadratic, phase-invariant function of a 2-spinor" is, to a physicist, exactly the construction that produces a 4-vector. This is where the classical spinor–vector correspondence is entered. Each adjective is independently motivated — but together they are the door through which Lorentz structure walks in.

The honest verdict. This is a reconstruction, not a from-nothing derivation. Its content is: the 1+3 Lorentzian structure and the Lorentz group are the unique consequence of {two poles, complex amplitude, phase-invariant Hermitian observables}. That is economical and non-obvious — three modest assumptions, only one of which is overtly physical, force the entire kinematic skeleton of special relativity, including the signature and the exact dimension split, which are usually postulated. But it is not the stronger claim "spacetime from interaction alone," and this note does not make it. A skeptic who grants A1–A3 must grant the conclusion (the math is airtight); a skeptic who suspects circularity should aim at A3 specifically — that is the load-bearing wall, and it is named, not hidden.

What would strengthen it past reconstruction. A derivation of A3 from something deeper — e.g., showing that the awareness relation (the aware pole registering the mark) must read the mark through a phase-invariant Hermitian pairing because awareness cannot access the global phase — would move A3 from "motivated assumption" toward "consequence." That is open, and worth stating as the next target rather than papering over it.

5. Scope

This note proves one thing: A1–A3 ⟹ (1+3 Minkowski + Lorentz group + SU(2) → SO(3)). It says nothing here about quantum dynamics, mass generation, gravity, gauge structure, the constants, or anything above the kinematic skeleton — those are separate claims, in separate documents, with their own (often weaker) status. Keeping this result isolated is the point: it is the one rung clean enough to hand to a skeptic intact.

One result, three assumptions, one of them pure ontology and two of them honestly flagged. Given them, the kinematics of special relativity follow — signature, dimension, group. Called by its right name: a reconstruction, and a tight one.

Standalone note. The mathematics of §2–§3 is the classical spinor–Lorentz correspondence (Pauli, Weyl, Penrose–Rindler); the only thing offered as new is the ontological sourcing of its inputs (§1) and the explicit assumption-accounting (§4). Part of the Interaction Theory project; deliberately separable from it. Source: papers/minkowski_from_dyadic_marks.md in the project repository. Back to The Formal Theory.