particles

Established

T1 Transport — Spin on S³

The T1 transport programme establishes how spin-½ wavefunctions are parallel transported on S³. The spin connection on S³ gives SU(2) holonomy, recovering the half-angle rotation behaviour of spinors.

What is T1 Transport?

The T1 transport programme describes the parallel transport of a T(1,1) torus knot seat around a closed loop on S³. The resulting holonomy phase isthe particle's spin. This provides a purely geometric origin for quantum spin.

The Spin Connection on S³

The 3-sphere S³ is isomorphic to the SU(2) group manifold. The spin connection is the Maurer-Cartan form on SU(2). Parallel transport around a closed loop gives a holonomy in SU(2).

SU(2) Maurer-Cartan Form

Holonomy

Half-Angle Behaviour

Transporting a spinor around a loop on S³ returns it to minus itself (). Transporting it around a loop returns it to . This is the defining characteristic of spin-½ behaviour, which here is derived geometrically rather than postulated.

Spinor Rotation

The 5 Slices of the T1 Programme

  • Slice 0: Setup — S³ as SU(2), spin connection formalism.
  • Slice 1: Parallel transport around the Hopf fibre (the basic exchange).
  • Slice 2: Holonomy group computation yielding SU(2).
  • Slice 3: Spinor representation extracted from holonomy corresponding to spin-½.
  • Slice 4: Clebsch-Gordan decomposition detailing spin addition rules.
  • Slice 5: Explicit construction of the T(1,1) wavefunction.

Connection to the Dirac Equation

The T1 transport programme shows that the Dirac equation on S³ naturally emerges from the spin connection. The matrices are exactly the SU(2) generators in the torus knot basis.

Key Result: Minimal Non-Trivial Holonomy

Spin-½ is the minimum non-trivial holonomy on S³. Any particle with a Hopf fibre seat has exactly spin-½. This is a strict mathematical theorem within the framework — particles cannot possess arbitrary spin without higher-order knot structure.

Source Documents

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