7 Experiment 6: The 𝜃-Scaling Theorem

Python: theta_scaling.py

We verify the sin(2𝜃)∕ formula analytically and confirm it holds across all chain lengths and site separations.

Proof

For ψ(𝜃) = cos𝜃e_i + isin𝜃e_j (normalised, i≠j), the density matrix ρ = |ψ⟩⟨ψ| has the block structure (showing only the i,j subspace, all other entries zero):

Removing the diagonal gives the boson matrix restricted to the same subspace:

In full, the components are:

ρii	= cos2𝜃, ρ jj = sin2𝜃,
ρij	= ψiψj∗ = cos𝜃 ⋅ (−isin𝜃) = − sin2𝜃,
ρji	= + sin2𝜃.

All other entries of ρ are zero since ψ_k = 0 for k≠i,j. Therefore:

giving

This proof uses only the definition B = ρ − diag(ρ) and the normalisation of ψ. It is independent of L, the separation |i −j|, and any overall phase factor on e_j. Numerical verification confirms agreement to within floating-point precision (< 4 × 10⁻¹⁶) for L ∈{4,8,16,32,64,128} and all hop distances.

Connection to Born rule

Let P = sin²𝜃 be the Born-rule probability that the fermion occupies site j. Then:

∥B∥ is times the geometric mean of the hop and no-hop probabilities. This is the interference term of the Born rule, made visible as a matrix norm. The boson amplitude is not an approximation or an emergent average: it is the exact quantum interference encoded in the density matrix.

Lifecycle of a virtual boson

The formula ∥B(𝜃)∥ = sin(2𝜃)∕ traces the complete lifecycle:

• 𝜃 = 0: fermion stationary, ∥B∥ = 0. No boson.
• 𝜃 = π∕4: fermion in equal superposition, ∥B∥ = 1∕. Boson at maximum amplitude.
• 𝜃 = π∕2: fermion arrived, ∥B∥ = 0. Boson has vanished.

The boson is born when the fermion begins to move, peaks at mid-hop, and vanishes when the fermion arrives. This is the lifecycle of a virtual particle — a propagator in the language of quantum field theory, derived here without postulating field theory.