SAPPHON SPINOR · THE TWO-LAP RETURN

real SU(2): U(θ) = exp(−iθ σ·n / 2) acting on a spin-½ state · his −++− → +−−+ rule, measured against the physics LITcomputed live, fail-loud AMBERmodel-local / assigned mapping WALLlimit of the claim

Bloch sphere · SO(3) picture

Gold arrow = spin direction ⟨σ⟩ (a vector: back at 360°). Purple = rotation axis n. The flag at the arrow tip shows the spinor's sign relative to the start: + / −. At 360° the arrow is home but the flag reads −. Only at 720° does it read + again.

Möbius band · the same topology ANALOGY

Carry the normal arrow once around the band (one lap) and it points the other way. Twice around, it is back. SO(3) has the same double cover structure (π₁(SO(3)) = ℤ₂). The picture is an analogy, not a derivation.

Amplitudes over two laps (0° → 720°)

speed θ axis n

His four signs MAPPING

The mapping used here (chosen for this page, so AMBER): his four signs are the signs of the four real numbers in a spin-½ state, (Re ψ↑, Im ψ↑, Re ψ↓, Im ψ↓). The start state ψ₀ = ((−1+i)/2, (1−i)/2) reads −++−. That state is spin along −x.

Row 1, the raw spinor ψ(θ) = U(θ)ψ₀. Each component turns at half the rotation angle, so at 360° every one of the four numbers has changed sign: +−−+, because U(2π) = −1 for every axis. At 720° they are back to −++−, because U(4π) = +1. The four numbers never vanish together, since the state stays normalized, and each crosses zero at its own angle.

Row 2, the part of ψ(θ) along its start, A(θ)·ψ₀ with A(θ) = ⟨ψ₀|ψ(θ)⟩. For an axis ⊥ to the spin (z or y), A(θ) = cos(θ/2) exactly. That makes row 2 his rule to the letter: −++− fades gradually, all four cross zero at 180°, it reads +−−+ at 360°, crosses zero again at 540°, and returns at 720°. A(θ) is the quantity a neutron interferometer measures: the rotated beam interferes with the unrotated one, I ∝ 1 + cos(θ/2) (Rauch 1975, Werner 1975).

Why the photon is different WALL

A photon is spin-1, a boson. Rotating a helicity state by θ multiplies it by e^(−iθ), so the gold curve cos θ is back to +1 at 360°, not 720°. The physical polarization of light returns after one turn. The two-lap sign return is a spin-½ (fermion) property. For light, the same SU(2) mathematics appears only in the description of polarization: Jones vectors are spinors on the Poincaré sphere (Pancharatnam 1956), and light picks up geometric phase (see the Berry panel). This page therefore does not claim that a photon is a spinor. It shows where his two-lap rule is exact physics, which is spin-½, and where it is only an analogy, which is the photon.

Two clocks, side by side

AMBERHis lap frame: one lap = 8φ = 12.944 (0.035956 per degree), t = 0 at photon birth (+electrum), from A:\CONSTANTS.txt. AMBERBridge: this page assumes a lap is 8φ Planck lengths travelled at c, so one lap = 8φ·tP. LITtP = 5.391247(60)e-44 s (CODATA 2018). His decided lP = 1e-35 m with tP = 5.39e-44 s gives lP/tP = 1.855e8 m/s ≠ c, so the bridge only closes with the CODATA pair (see CONSTANTS_CODATA.txt).

REALA lab clock for the same rotation: a neutron precessing in a field B turns at ω = γnB, with γn = 1.83247174e8 s⁻¹T⁻¹ (CODATA 2022, NIST). Werner et al. used this effect to see the 4π return.

B (mT)

Self-checks LIT

Berry / geometric phase for light

Guide light through a helically wound fibre. Its direction k traces a cone on the sphere of directions, sweeping a solid angle Ω = 2π(1 − cos ϑ) per turn. The plane of linear polarization rotates by Ω. That is a geometric phase, independent of wavelength (Tomita & Chiao 1986; Berry 1984). A spin-½ particle taken around the same loop picks up phase Ω/2. The ½ is the spinor again.

pitch ϑturns

Sources

  • H. Rauch et al., “Verification of coherent spinor rotation of fermions”, Phys. Lett. A 54, 425 (1975).
  • S. A. Werner et al., “Observation of the phase shift of a neutron due to precession in a magnetic field”, Phys. Rev. Lett. 35, 1053 (1975).
  • A. Tomita & R. Y. Chiao, “Observation of Berry's topological phase by use of an optical fiber”, Phys. Rev. Lett. 57, 937 (1986).
  • M. V. Berry, Proc. R. Soc. Lond. A 392, 45 (1984). S. Pancharatnam, Proc. Indian Acad. Sci. A 44, 247 (1956).
  • CODATA 2018: E. Tiesinga et al., Rev. Mod. Phys. 93, 025010 (2021). NIST CUU physics.nist.gov/cuu (tP, lP, γn).
  • J. J. Sakurai & J. Napolitano, Modern Quantum Mechanics, §3.2 (2π rotation of spin-½ gives −1).
  • Model: A:\CONSTANTS.txt (8φ lap, −++− → +−−+, t = 0 at +electrum), D. L. Wise.