Final Stage

QM2 Complete

Relational Transport and Continuous Quantum Evolution

Review everything you’ve discovered.

The Full Journey

  1. State-Space Structure(shown)

    QM1 provides the complex, Hermitian state space.

  2. Transport Changes Smoothly(shown)

    Small changes do not contain sudden jumps.

  3. Geometry Is Preserved(shown)

    Lengths and relationships remain unchanged.

  4. Transport Steps Combine(shown)

    Separate steps agree with one combined step.

  5. Motion Has a Generator(shown)

    The initial direction is well defined.

  6. The Generator Defines H(shown)

    H = iA (H equals i times A)

  7. Continuous Quantum Evolution(shown)

    U(t) = exp(−iHt) (U of t equals the exponential of minus i H t)

Interactive Roadmap

Open any lesson summary — no need to replay the lesson.

Established in QM2

  • Continuous transport
  • Geometry-preserving transport is unitary
  • One-parameter transport group
  • Generator exists
  • Hamiltonian — H = iA
  • Continuous evolution — U(t) = exp(−iHt) (natural units, ħ = 1)

These results hold within the assumptions introduced in QM2.

Starting Assumptions

  • D1

    Continuous transport.

  • D2

    Hermitian geometry is preserved.

  • D3

    Transport combines consistently.

  • D4

    The motion begins with a well-defined direction.

These assumptions are explicit starting points for QM2.

Still To Be Solved

  • How much of the QM2 transport structure can be reconstructed from the preceding PDT relational foundations?
  • Higher-dimensional state spaces.
  • Composition of multiple sectors.
  • Measurement.
  • Born rule.
  • Collapse.
  • Decoherence.
  • Experimental predictions.

These questions are intentionally left open for later papers.

Where Does QM2 Fit?

  1. PDT0Primitive relational phase framework
  2. R-SeriesRelational and transport foundations
  3. QM1Complex and Hermitian state-space structure
  4. QM2Continuous transport of quantum states
  5. Later QM PapersFurther reconstruction

QM1 established the mathematical structure in which quantum states can be represented. QM2 asks the next question: How can those states be transported continuously while preserving the structure already established?

INHERITEDASSUMED HERE

QM2 therefore begins with inherited mathematical structure and introduces explicit conditions on continuous transport.

The Transport Conditions

ASSUMED HERE

  • D1 — Continuity

    Transport varies continuously with its parameter.

  • D2 — Hermitian-structure preservation

    Transport preserves the Hermitian inner product.

  • D3 — One-parameter composition

    U(t+s) = U(t)U(s), with U(0) = I.

  • D4 — Differentiability

    The family has a well-defined infinitesimal rate of change.

QM2 asks what follows if quantum transport satisfies these four conditions. The power of the result lies in the fact that once these conditions are imposed, the mathematical form of the transport becomes highly constrained.

These are conditions imposed here in QM2. They are not presented as results already proved from the earlier foundations.

The Logic of QM2

Inherited structure, stated conditions, and what follows from them.

  1. INHERITED

    Complex / Hermitian State-Space Structure

    from QM1

  2. ASSUMED HERE

    The Transport Conditions

    D1 Continuity

    D2 Hermitian Preservation

    D3 One-Parameter Composition

    D4 Differentiability

  3. DERIVED

    Unitary Transport

    U(t)†U(t) = I

  4. DERIVED

    Infinitesimal Generator

    dU/dt at t = 0

  5. DERIVED

    Hermitian Generator H

    H† = H

  6. DERIVED

    Exponential Transport

    U(t) = e^(−iHt)

  7. DERIVED

    Schrödinger-Type Evolution

    i dψ/dt = Hψ

  8. NEXT

    Specific Physical Dynamics

    and later reconstruction

Natural units may be used in these lessons. Restoring ħ gives U(t) = e^(−iHt/ħ) and iħ dψ/dt = Hψ.

What QM2 Actually Establishes

DERIVED

Given the inherited QM1 state-space structure and the stated QM2 transport conditions, continuous structure-preserving transport is represented by a one-parameter unitary family generated by a Hermitian operator.

  1. D1–D4
  2. Unitary One-Parameter Transport
  3. Hermitian Generator
  4. Quantum Evolution Form

This is a conditional mathematical reconstruction: the consequences are derived once the stated transport conditions are supplied.

Scope of This Result

SCOPE CONDITION

The derivation shown here establishes the result within the mathematical setting specified by QM2. It must not automatically be extrapolated to arbitrary higher-dimensional or interacting quantum systems without the additional reconstruction required by later work.

QM2 reconstructs the form of continuous quantum transport under its assumptions. The deeper question of uniquely selecting underlying relational dynamics belongs to the broader reconstruction programme.

Full Dependency Chain

Select a level to see what it contributes.

QM2 provides

  • • Continuous unitary transport
  • • Hermitian generator H
  • • U(t) = e^(−iHt)

Each stage builds on the one before it. Some results in the earlier series remain conditional on stated assumptions.

From Transport to Further Reconstruction

NEXT
  1. QM1State-Space Structure
  2. QM2Continuous Unitary Transport
  3. Later QM FoundationsAdditional Physical Structure

QM2 establishes how continuous quantum transport behaves once its structural conditions are supplied. The later foundation papers ask what further physical and geometric structure can be reconstructed from this framework.

Theorem Viewer

Theorem 1

DERIVED

Under D2, structure-preserving transport is unitary.

Theorem 2

DERIVED

Under D1 and D3, transport forms a one-parameter unitary family.

Theorem 3

DERIVED

Under D1–D4, the generator is Hermitian and plays the role of the Hamiltonian.

Final Knowledge Quiz

Ten questions covering every lesson. Retry as often as you like.

  1. 1. In QM2, what does transport describe?

  2. 2. Continuous transport means the state can jump suddenly between values.

  3. 3. Which transport preserves the relationship between two states?

  4. 4. Geometry-preserving transport is called unitary.

  5. 5. Two steps of transport, one after the other, should equal…

  6. 6. A transport step of zero duration leaves the state unchanged.

  7. 7. Zooming into the very start of the motion reveals…

  8. 8. The Hamiltonian in QM2 is…

  9. 9. QM2 already explains measurement, the Born rule and collapse.

  10. 10. Put the QM2 chain in order.

    Explore Next

    • Return to QM1

      The complex state space and Hermitian geometry.

      Coming soon

    • Continue to QM3

      Higher dimensions, sectors and measurement.

      Coming soon

    • Explore PDT Research

      The primitive ontology behind the programme.

      Coming soon

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