Beyond the ordinary

1. Geometry and measure on S³

Definition of S³

• S³ = { x ∈ ℝ⁴ | ‖x‖ = 1 }

Volume element

• dΩ₃ = invariant volume element on S³ (the natural “area” measure of the 3‑sphere).

 

2. Scalar Hilbert space and spherical harmonics

Scalar Hilbert space

• H_scalar = L²(S³)

• Elements: complex functions ψ(x) on S³ with finite norm.

• Inner Product ⟨ψ, φ⟩ = ∫ ψ*(x) φ(x) dΩ₃
  

Scalar spherical harmonics

• Yₙₗₘ(x) are eigenfunctions of the Laplace–Beltrami operator Δ on S³:

−Δ Yₙₗₘ(x) = λₙ Yₙₗₘ(x)

• Orthonormality:

∫ Yₙₗₘ*(x) Yₙ'ₗ'ₘ'(x) dΩ₃ = δₙₙ' δₗₗ' δₘₘ'

Expansion of a scalar state

Any ψ(x) ∈ L²(S³) can be written as: ψ(x) = Σₙₗₘ cₙₗₘ Yₙₗₘ(x)

with coefficients: cₙₗₘ = ∫ Yₙₗₘ*(x) ψ(x) dΩ₃

3. Spinor and vector Hilbert spaces

Spinor fields (fermions)

Spinor Hilbert space

• H_spinor = L²(S³, S) (square‑integrable spinor fields on S³)

Inner product

⟨Ψ, Φ⟩ = ∫ Ψ†(x) Φ(x) dΩ₃

(† = Hermitian conjugate)

Dirac operator and spinor harmonics

• Dirac operator D on S³:  D Ξₖₐ(x) = μₖ Ξₖₐ(x)

• Ξₖₐ(x) = spinor spherical harmonics (orthonormal, complete).

Expansion of a spinor state

Ψ(x) = Σₖₐ aₖₐ Ξₖₐ(x)

Vector fields (gauge bosons)

Vector Hilbert space

• H_vector = L²(S³, T S³) (square‑integrable vector fields on S³)

Inner product

⟨A, B⟩ = ∫ gᵢⱼ(x) Aᵢ*(x) Bⱼ(x) dΩ₃

(gᵢⱼ = metric on S³)

Vector spherical harmonics

• Vₙₗₘᵢ₍ₐ₎(x) = vector harmonics (index i for components, a for polarisation).

Expansion of a vector state

Aᵢ(x) = Σₙₗₘₐ bₙₗₘₐ Vₙₗₘᵢ₍ₐ₎(x)

4. Single‑particle quantum states

For each particle type:

• Scalar (Higgs‑like):

• Vector bosons (photon, W/Z, gluons):H_A = L²(S³, T S³)

State as superposition of harmonics

Scalar example: |ψ⟩ ↔ ψ(x) = Σₙₗₘ cₙₗₘ Yₙₗₘ(x)

Spinor example: |Ψ⟩ ↔ Ψ(x) = Σₖₐ aₖₐ Ξₖₐ(x)

Vector example: |A⟩ ↔ Aᵢ(x) = Σₙₗₘₐ bₙₗₘₐ Vₙₗₘᵢ₍ₐ₎(x)

The coefficients (cₙₗₘ, aₖₐ, bₙₗₘₐ) are the quantum amplitudes in the harmonic basis.

5. Time evolution and unitarity

Free scalar Hamiltonian

H = √( −Δ + m² ) (acting on L²(S³))

Time evolution

|ψ(t)⟩ = e^(−i H t) |ψ(0)⟩ 

Because H is self‑adjoint, evolution is unitary: ⟨ψ(t), φ(t)⟩ = ⟨ψ(0), φ(0)⟩

⟨ψ(t), φ(t)⟩ = ⟨ψ(0), φ(0)⟩

Analogous constructions hold for spinors (using Dirac operator) and vectors (using gauge‑covariant operators).

6. Quantum fields and Fock space

Mode expansion of a scalar field

Field operator Φ(x): Φ(x) = Σₙₗₘ ( aₙₗₘ Yₙₗₘ(x) + aₙₗₘ† Yₙₗₘ*(x) )

with commutation relations: [ aₙₗₘ, aₙ'ₗ'ₘ'† ] = δₙₙ' δₗₗ' δₘₘ'

Fock space

For bosons: F_boson = ⊕_{N=0}^∞ H^{⊗_s N}

(symmetrised tensor products)

For fermions: F_fermion = ⊕_{N=0}^∞ H^{⊗_a N}

(antisymmetrised tensor products)

Interactions are implemented by adding coupling terms that mix different harmonic modes in the Hamiltonian or action.

7. One‑sentence bridge

The allowed vibrations of the hyperspherical boundary (harmonics on S³) form orthonormal bases of L²‑type spaces; the space of all complex superpositions of these modes, with the natural inner product, is the quantum Hilbert space, and unitary evolution is generated by self‑adjoint geometric operators (Laplace–Beltrami, Dirac, gauge‑covariant).