Quantum Foundations II

Relational Transport

How continuous quantum evolution can emerge from simple relational transport.

Phase
Begin Lesson

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.