The State Quaternion
How a single algebraic object encodes everything an electron or proton carries
How a single algebraic object encodes everything an electron or proton carries
2026 · It Is All One — Working Paper Series
Introduction
A particle in quantum mechanics is described by a wave function — a complex-valued function of position and time that encodes the probability of finding the particle in any given state. Wave functions live in an abstract infinite-dimensional space called Hilbert space. They are powerful but opaque. You cannot look at a wave function and immediately read off the particle’s charge, its spin, its angular momentum.
The structure of the state quaternion is not arbitrary. It follows directly from the octonion assignment established in Paper 5 of this series (Neutron Decay as Octonion Algebra), where each quark was assigned an octonion with eight components split into two orthogonal quaternions. The first quaternion Q₁ carries the electroweak quantum numbers: mass-energy, electric charge, spin, and angular momentum. The second quaternion Q₂ carries the colour quantum numbers: confinement, red, green, blue. For the electron, which has no colour, Q₂ is empty. The electron lives entirely in Q₁.
1The Four Axes and What They Carry
The state quaternion has one real axis and three imaginary axes. We label them e₀ (real), e₁, e₂, e₃ (imaginary). Each axis carries one conserved quantity.
(1)Q = (E/Λ)·e₀ + q·e₁ + mₛ·e₂ + (ℓ + i·mℓ)·e₃
Figure 1. The state quaternion Q shown as four axes. The real axis e₀ (blue, pointing up) carries energy E/Λ. The three imaginary axes carry charge on e₁ (green), spin on e₂ (amber), and the complex angular momentum ℓ + imℓ on e₃ (purple). The full state quaternion is written at the bottom.
11 e₀ — mass-energy (real axis)
The real axis carries E/Λ, where E is the particle’s rest energy in MeV and Λ = 310 MeV is the confinement scale established in Paper 5. For the electron, E = mᵉ
12 e₁ — electric charge
The first imaginary axis carries the electric charge in units of the elementary charge e. For the electron q = −1. For the proton q = +1. For a neutron q = 0. For an up quark q = +2/3. For a down quark q = −1/3. Charge is a discrete quantum number — an integer or simple fraction — fixed for each particle species. It does not take different values in different quantum states of the same particle.
13 e₂ — spin projection
The second imaginary axis carries mₛ, the spin projection along the quantisation axis. For a spin-½ particle (electron, proton, quark), mₛ = +½ or mₛ = −½. Spin is the intrinsic angular momentum of the particle — not associated with orbital motion, but an internal degree of freedom. The two values +½ and −½ are the only options for all the particles we consider here.
14 e₃ — orbital angular momentum (complex)
The third imaginary axis carries a complex number. This is the key structural insight of this paper.
(2)L = ℓ + i·mℓ ℓ = 0, 1, 2, ... −ℓ ≤ mℓ ≤ +ℓ
The real part ℓ is the orbital angular momentum quantum number — the total magnitude of the orbital angular momentum, independent of which direction you choose as your quantisation axis. It takes non-negative integer values: 0, 1, 2, 3, ... In atomic physics these are labelled s, p, d, f for ℓ = 0, 1, 2, 3. The imaginary part mℓ is the projection of the orbital angular momentum along the quantisation axis. It is coordinate-dependent: if you rotate your frame, mℓ changes. It takes integer values from −ℓ to +ℓ, giving 2ℓ + 1 possible values for each ℓ. This is exactly the structure of a complex number: the real part is the invariant magnitude, the imaginary part is the projection. The modulus |ℓ + imℓ| = √(ℓ² + mℓ²) combines both. This is also the same structure as the spacetime biquaternion established in Paper 1, where the time component W = iτ carries a complex number on the real axis: an invariant on the real part, a coordinate-dependent projection on the imaginary part.
15 Why charge and spin are not complex
Charge (e₁) does not have a magnitude and a projection. It is a topological quantum number that does not change when you rotate the coordinate system. There is no direction to project charge along. It is the same from every angle. So e₁ carries a real integer or fraction. Spin (e₂) does have a projection, but for a spin-½ particle the magnitude is fixed at ½ always — it never changes between states. Only the sign of the projection varies: +½ or −½. So e₂ carries a real number with two possible values, not a complex number. Angular momentum (e₃) has both a variable magnitude ℓ and a variable projection mℓ, both changing between quantum states. Only e₃ needs two real degrees of freedom, and a complex number is the natural way to carry them on a single axis. The assignment — why each axis carries what it carries e₀ (real): energy E/Λ — a real positive magnitude. The observable. e₁ (imaginary): charge q — an integer. No direction to project. Always the same. e₂ (imaginary): spin mₛ — fixed magnitude ½, only sign varies. Two values only. e₃ (imaginary): angular momentum ℓ + imℓ — both magnitude and projection vary. Complex number on one axis: two degrees of freedom, one slot.
2The Electron State Quaternion — Explicit States
The hydrogen atom has energy levels labelled by the principal quantum number n = 1, 2, 3, ... The energy of level n is:
(3)Eₙ = −13.606 eV / n² n = 1, 2, 3, ...
The e₀ component of the state quaternion uses the electron’s total energy — rest energy plus orbital energy — divided by Λ. The orbital correction −13.606/n² eV is only parts per million of the rest energy 0.511 MeV, and at display precision every hydrogen state prints as e₀ = 0.511/310 = 0.001648. But the correction must not be truncated from the state itself: e₀(n) = (mₑ
21 The ground state
The ground state of hydrogen has n=1, ℓ=0, mℓ=0, mₛ=+½:
(4)Q(1s,↑) = 0.001648·e₀ − e₁ + ½·e₂ + 0·e₃
The e₃ component is zero because both ℓ = 0 and mℓ = 0. The electron has no orbital angular momentum. Its orbital is spherically symmetric — the 1s orbital. A second electron can occupy the same spatial orbital with opposite spin mₛ = −½. Its quaternion differs only on e₂. These two electrons, one spin-up and one spin-down, fill the n=1 shell. A third electron cannot enter: it would have to duplicate one of these two on all four axes, and the encoding has no third address to give — e₂ takes two values and no more.
22 The n=2 shell — eight distinct states
The n=2 shell contains four distinct orbital states before spin is included: the 2s orbital (ℓ=0, mℓ=0) and three 2p orbitals (ℓ=1, mℓ=−1,0,+1). Each combines with two spin states, giving eight states total. Their state quaternions:
(5)Q(2s) = 0.001648·e₀ − e₁ + mₛ·e₂ + (0 + 0i)·e₃
(8)Q(2p, mℓ=−1) = 0.001648·e₀ − e₁ + mₛ·e₂ + (1 − i)·e₃
(6)Q(2p, mℓ=0) = 0.001648·e₀ − e₁ + mₛ·e₂ + (1 + 0i)·e₃
(7)Q(2p, mℓ=+1) = 0.001648·e₀ − e₁ + mₛ·e₂ + (1 + i)·e₃
Each of these four orbital states combines with mₛ = +½ and mₛ = −½, giving eight quaternions in total. Every one is distinct — no two are identical on all four axes. The shell's capacity is legible by inspection: a ninth electron would have to duplicate one of the eight, and the antisymmetrised two-particle state then has norm 1 − |⟨α|β⟩|² = 0. Note what the complex e₃ component achieves. The 2s state has e₃ = 0. The three 2p states all have real part ℓ = 1. They are distinguished by their imaginary parts mℓ = −1, 0, +1. Without the complex structure, the three 2p states (all with ℓ = 1) would be indistinguishable on e₃, and two of the eight states would be identical. Pauli would break. The complex angular momentum on e₃ is not a formal convenience — it is necessary for Pauli exclusion to work.
[ Figure 2 — Eight n=2 hydrogen states as state quaternions ]
Figure 2. The eight state quaternions of the n=2 hydrogen shell. Top row (spin up): 2s, 2p mℓ=−1, 2p mℓ=0, 2p mℓ=+1. Bottom row (spin down): same four orbitals with e₂ flipped to −½. Each box shows all four components. No two boxes are identical. Pauli exclusion is satisfied for all 28 pairs.
3The Label, the Sign, and the Rule
Exclusion has exactly two ingredients, and they belong to different owners. There is a sign — a two-fermion state changes sign when the particles are exchanged — and there is a rule — that no two fermions may occupy the same state, which is the consequence of that sign. This series can claim the sign as genuinely quaternionic. It cannot claim the rule, and it does not need to: what Sections 1–2 establish is a third thing, prior to both, and a real result in its own right.
31 What the label achieves: faithfulness
The encoding of Sections 1–2 gives every physical state of the electron a four-component address [e₀, e₁, e₂, e₃], and the property it must have — the only property a labelling scheme is obliged to have — is that distinct states receive distinct addresses. Call that faithfulness. It is not the exclusion principle; it is the precondition for exclusion to be legible in this notation at all, and Section 3.4 shows it is not automatic: one component has to be complex before it holds. Helium, in the ground state. Its two electrons carry Q↑ = [0.001648, −1, +½, 0] and Q↓ = [0.001648, −1, −½, 0]: identical on three axes, opposed on e₂. Two distinct addresses, so two distinct states, so a shell that holds two. A third electron would need a third address at n = 1, and the encoding has none to give — e₂ takes two values and no more. It goes to n = 2. The shell structure of the periodic table is the arithmetic of how many distinct addresses each shell admits, and that arithmetic is what the encoding performs.
32 The sign, earned from the double cover
Exchanging two identical particles is, followed through in configuration space, a 2π rotation of the pair about their midpoint. The unit quaternions are
33 The rule, and why it does not live in the algebra of labels
From the sign the rule follows by a construction that is standard, and that construction lives in the tensor product of the two one-particle Hilbert spaces rather than in any product of their labels. For two particles in states |α⟩ and |β⟩ the antisymmetrised state is (|α⟩|β⟩ − |β⟩|α⟩)/√2, whose squared norm is
(9)‖α ∧ β‖² = 1 − |⟨α|β⟩|²
34 Why the third axis must be complex
4Selection Rules from Conservation on Each Axis
Every atomic transition — every photon emitted or absorbed — must preserve the state quaternion’s components subject to what the photon carries away. The photon’s state quaternion is:
(10)Qγ = (hν/Λ)·e₀ + 0·e₁ + 0·e₂ + (1 + i·m)·e₃ m = −1, 0, +1
The photon carries energy (e₀), no charge (e₁ = 0), and one unit of angular momentum — its spin — with projection m = −1, 0, +1, entered once, on the angular-momentum axis e₃. Entering the photon’s spin a second time as e₂ = ±1 would double-count it; worse, an additively conserved e₂ = ±1 would force Δmₛ = ∓1, which is impossible for a spin-½ electron. Conservation at each axis gives the selection rules:
(11)Δe₀: E_initial = E_final + hν (energy conservation)
(12)Δe₁: q_initial = q_final (charge conservation)
(13)Δe₂: Δmₛ = 0 (the dipole operator does not act on the spin axis)
(14)Δe₃: Δℓ = ±1 and Δmℓ = 0, ±1 (angular momentum)
The selection rule Δℓ = ±1 — the most important rule in atomic spectroscopy, the one that determines which transitions are electric dipole allowed — falls directly from equation (
The rule Δmℓ = 0, ±1 comes from the imaginary part of e₃: the photon’s imaginary part is m = −1, 0, or +1, so the electron’s mℓ changes by at most ±1. Spectroscopic selection rules from axis conservation Δℓ = ±1: from conservation of the real part of e₃ (photon carries ℓγ = 1) Δmℓ = 0,±1: from conservation of the imaginary part of e₃ (photon carries m = −1,0,+1) Δmₛ = 0: the electric-dipole operator does not act on the spin axis e₂ Δq = 0: from conservation of e₁ (photon has no charge) Each rule lives on one axis: the four-axis structure organizes the bookkeeping. The addition law and the parity input behind the rules come from the underlying operator algebra.
5The Quaternion Norm
The squared norm of the state quaternion is:
(15)|Q|² = (E/Λ)² + q² + mₛ² + ℓ² + mℓ²
For the hydrogen ground state (n=1, ℓ=0, mℓ=0, mₛ=+½):
|Q(1s,↑)|² = (0.001648)² + 1 + ¼ + 0 ≈ 5/4
For the n=2, ℓ=1, mℓ=+1, spin-up state:
|Q(2p,mℓ=+1,↑)|² = (0.001648)² + 1 + ¼ + (1²+1²) ≈ 13/4
The norm grows with excitation. When a photon is emitted, the components of the electron’s state quaternion change by what the photon carries away, axis by axis — the bookkeeping of Chapter 4. The norms themselves do not subtract: |Q|² is quadratic, and norms are not additive. For 2p → 1s the electron’s squared norm drops by 13/4 − 5/4 = 2, while the photon’s squared norm from equations (10) and (15) is ≈ 3. The balance is componentwise, not norm-wise. Energy, angular momentum, and spin are transferred from particle to particle, and each axis adjusts accordingly. The norm is the measure of how much quantum state the particle is carrying.
6The Proton State Quaternion — at the Surface
Before an electron probe penetrates the proton’s interior, the proton appears as a point particle with definite quantum numbers. At the surface, it has a state quaternion with the same structure as the electron’s:
(16)Qₚ = 3.0267·e₀ + (+1)·e₁ + mₛ·e₂ + (0 + 0i)·e₃ [surface]
The proton is 1836.1 times heavier than the electron. This ratio appears directly as the ratio of their e₀ components: 3.0267/0.001648 = 1836.1. The mass ratio is transparent in the state quaternion.
61 The electron-proton interaction at the surface
The Coulomb potential energy between electron and proton is given by the product of their e₁ components:
(17)U(r) = (qᵉ·qₚ·
The product of e₁ components is qᵉ × qₚ = (−1)(+1) = −1. A negative product means attraction. Two protons give (+1)(+1) = +1: repulsion. The sign of the Coulomb interaction is the product of the e₁ components. The magnitude is αℋ
7The Boundary: Where the Quaternion Hands Over to the Octonion
The state quaternion Q₁ gives a complete description of the electron and of the proton’s surface as long as the electron’s de Broglie wavelength is large compared to the proton radius rₚ = 0.8414 fm. The transition occurs at:
(18)p_transition = 2πℋ
Below this momentum, Q₁ is all that matters. The proton is a point with a charge, a spin, and a mass — a state quaternion exactly like equation ( (16). Above this momentum, the electron’s wavelength shrinks below the proton radius. It begins to see inside. The proton’s Q₂ — the colour quaternion, the second orthogonal quaternion of the full octonion — becomes visible. The four components of Q₁ are no longer sufficient. You need all eight components of the octonion:
(19)Oₚ = Q₁ + Q₂·e₄ Q₂ = g·e₄ + r·e₅ + g·e₆ + b·e₇
where Q₂ carries the confinement strength and the colour charges of the three constituent quarks. The full component assignment for the proton’s octonion is given in Paper 5 (Section 2). The six quark presets, the closure angle, the spring constant, and the neutron-proton mass difference all emerge from the structure of Q₂.
[ Figure 3 — Quaternion to octonion boundary at p = 1474 MeV/
Figure 3. The boundary between quaternion and octonion physics on a logarithmic momentum axis. Left of the red dashed line (p < 1474 MeV/c, λ > rₚ): the proton looks like a point and Q₁ carries all information. Right of the boundary (p > 1474 MeV/
Equation Index
(1) Q = (E/Λ)·e₀ + q·e₁ + mₛ·e₂ + (ℓ + i·mℓ)·e₃
(2) L = ℓ + i·mℓ ℓ = 0, 1, 2, ... −ℓ ≤ mℓ ≤ +ℓ
(3) Eₙ = −13.606 eV / n² n = 1, 2, 3, ...
(4) Q(1s,↑) = 0.001648·e₀ − e₁ + ½·e₂ + 0·e₃
(5) Q(2s) = 0.001648·e₀ − e₁ + mₛ·e₂ + (0 + 0i)·e₃
(6) Q(2p, mℓ=0) = 0.001648·e₀ − e₁ + mₛ·e₂ + (1 + 0i)·e₃
(7) Q(2p, mℓ=+1) = 0.001648·e₀ − e₁ + mₛ·e₂ + (1 + i)·e₃
(8) Q(2p, mℓ=−1) = 0.001648·e₀ − e₁ + mₛ·e₂ + (1 − i)·e₃
(9) ‖α ∧ β‖² = 1 − |⟨α|β⟩|² — the antisymmetrised two-particle norm; zero iff the states coincide
(10) Qγ = (hν/Λ)·e₀ + 0·e₁ + 0·e₂ + (1 + i·m)·e₃ m = −1, 0, +1
(11) Δe₀: E_initial = E_final + hν (energy conservation)
(12) Δe₁: q_initial = q_final (charge conservation)
(13) Δe₂: Δmₛ = 0 (the dipole operator does not act on the spin axis)
(14) Δe₃: Δℓ = ±1 and Δmℓ = 0, ±1 (angular momentum)
(15) |Q|² = (E/Λ)² + q² + mₛ² + ℓ² + mℓ²
(16) Qₚ = 3.0267·e₀ + (+1)·e₁ + mₛ·e₂ + (0 + 0i)·e₃ [surface]
(17) U(r) = (qᵉ·qₚ·
(18) p_transition = 2πℋ
(19) Oₚ = Q₁ + Q₂·e₄ Q₂ = g·e₄ + r·e₅ + g·e₆ + b·e₇