Quantum Foundations III · Lesson 5
The Missing Connections
The last lesson identified something surprising.
Phase evolution inside each sector is available.
Coupling between sectors is not.
Assuming n one-dimensional complex sectors have been embedded into a combined state space…
- Total complex matrix dimensions: n²
- Diagonal sector-preserving dimensions: n
- Off-diagonal coupling dimensions: n² − n
For n one-dimensional complex sectors. The count n² − n is not the universal coupling count for sectors of arbitrary higher dimension.
- Full available Hermitian dimensions: n²
- Sector-preserving dimensions: n
- Unavailable coupling dimensions: n² − n
1 sector
- Full available Hermitian dimensions = 1
- Sector-preserving dimensions = 1
- Unavailable coupling dimensions = 0
2 sectors
- Full available Hermitian dimensions = 4
- Sector-preserving dimensions = 2
- Unavailable coupling dimensions = 2
3 sectors
- Full available Hermitian dimensions = 9
- Sector-preserving dimensions = 3
- Unavailable coupling dimensions = 6
5 sectors
- Full available Hermitian dimensions = 25
- Sector-preserving dimensions = 5
- Unavailable coupling dimensions = 20
Diagonal cells are within-sector phase terms. Off-diagonal cells are coupling terms unavailable under exact invariance.
Currently showing 3 sectors: 9 full, 3 sector-preserving, 6 unavailable.
Where the missing connections come from
- Exact invarianceDEFINED HERE
- Block diagonalityDERIVED CONDITIONALLY
- No off-diagonal couplingDERIVED CONDITIONALLY
- Coupling obstructionIDENTIFIED
Interactions of this off-diagonal form are unavailable while exact sector invariance is maintained. Coupling would require additional structure, or a modification of the exact sector-invariance definition.
Assuming multiple invariant sectors exist and compatible embeddings are available:
9
Full Hermitian dimensions
3
Sector-preserving dimensions
6
Unavailable coupling dimensions
What to notice
- ✓More sectors would allow more possible relationships.
- ✓Within-sector phase terms remain available.
- ✓The unavailable coupling dimensions grow faster still.
Assuming several one-dimensional complex sectors have been embedded into a combined state space, sector-preserving transport is block-diagonal: relative phase can evolve between fixed sector components, while off-diagonal coupling terms are unavailable. Whether such sectors and compatible embeddings exist remains open, as does the adequacy of exact invariance itself.
What mathematical structure could unlock these connections?
QM3 leaves this as an open question.