Lesson 2 of 10
Quantum Foundations IV · Relational Coupling and Dynamical Sector Structure
Sector-Preserving Transport
Having several sectors does not mean they interact.
Transport can change each sector internally while still preserving the separation between them.
Total Relational State Space
IllustrationEach sector evolves continuously inside its own boundary.
U(t) Vi ⊆ Vi
Given the QM3 exact-invariance definition, transport preserves each sector.
If a state begins in sector V₁, exact sector preservation prevents transport from carrying it into V₂.
Caution: QM4 does not independently establish that such sectors exist.
U(t) = e−iHt uses natural units with ħ = 1 throughout.
What this means
Continuous evolution is possible.
Internal phase can change.
But no transport occurs between different sectors.
Internal Evolution
✓ Allowed
Movement Between Sectors
✗ Not reconstructed
Mathematical Status
- ✓ Transport preserves individual sectors.
- ✓ Internal phase evolution occurs.
- ✓ Sectors remain dynamically separate.
- ✗ Inter-sector transport
- ✗ Coupling operators
- ✗ Off-diagonal generators
- ✗ Hamiltonian coupling
These remain open reconstruction questions.
Quick check
If two sectors evolve independently, does that mean they exchange amplitude?
Lesson summary
Sector-preserving transport allows smooth evolution inside each relational sector.
Nothing crosses between sectors.
Understanding this limitation is the starting point for QM4.
Status key
- Establishedsupported by the preceding reconstruction when its assumptions hold
- Illustrationused to explain the mathematical idea visually
- Opennot yet reconstructed from PDT