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R04

Quantum states probability and dynamics

Quantum probability and dynamics under explicitly stated operational premises.

Current foundationsPaper overview and reading guideManuscript PDF availableReview draft 1.0
Document version
Review draft 1.0
Date printed on the document
24 September 2026
Public release date
Not confirmed

This page is a reading guide written for the website. It is not the manuscript and does not reproduce its proofs. The manuscript PDF is linked below. It is a review draft and carries its own review notice. Authorship is not stated in the document and has not been confirmed. The public release date on this website has not been confirmed; the date shown is the date printed on the document. “Current foundations” does not by itself mean peer reviewed or experimentally confirmed.

Purpose

The question addressed

R04 examines how quantum probability and quantum dynamics follow once explicitly stated operational premises are adopted.

Mathematical starting point

Objects and premises used

  • A finite-dimensional complex Hilbert space H\mathcal{H}, density operators ρ\rho and effects 0≤Er≤I0 \le E_r \le I.
  • Assumption Q1 on effects and normalisation, and operational composition assumptions discussed in R06.

Main results

What the paper establishes

  • Finite effect probability p(r)=Tr⁡(ρEr)p(r) = \operatorname{Tr}(\rho E_r) (Theorem R04.1).
  • Reversible channels on one sector are unitary (Theorem R04.2).
  • With an identified physical clock and action scale ℏ\hbar, continuous reversible evolution takes the form U(t)=e−iH^t/ℏU(t) = e^{-i\hat{H}t/\hbar}.

Theorem R04.1: Finite effect probability

Result statement (summary)
Under assumption Q1 (the full effect space has a physical interpretation, the probability of an effect is independent of which complete measurement contains it, and probabilities normalise across each such measurement), outcome probabilities on a finite-dimensional Hilbert space take the form p(r)=Tr⁡(ρEr)p(r) = \operatorname{Tr}(\rho E_r).
Assumptions and scope: Assumption Q1 and a finite-dimensional complex Hilbert space.

Rests on established effect-theorem mathematics; the original references are given in the foundational paper.

Link to this statement

The full proof is given in the R04 manuscript (PDF).

Theorem R04.2: Reversible channels on one sector

Result statement (summary)
A channel with a completely positive, trace-preserving inverse on the same finite sector is unitary.
Assumptions and scope: A finite sector and a completely positive, trace-preserving inverse on that same sector.
Link to this statement

The full proof is given in the R04 manuscript (PDF).

Explanation

Worked example: a qubit

For ∣ψ⟩=32∣0⟩+12∣1⟩|\psi\rangle = \tfrac{\sqrt{3}}{2}|0\rangle + \tfrac12|1\rangle, the projective measurement gives p(0)=34p(0) = \tfrac34 and p(1)=14p(1) = \tfrac14, which ∑\sum to 1.
The effects fix the probabilities; the instrument also fixes the state after each outcome.

More on the mathematics page: Quantum probabilities and instruments →

Physical interpretation and scope

What connects this to physics

The physical Hilbert space, Hamiltonian and instrument are supplied through the model’s premises. The continuous-time form requires the clock identification examined in R08.

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Using this paper

View the R04 manuscript (PDF)Download PDF

R04. “Quantum states probability and dynamics.” Phase Differential Theory current foundations. Review draft 1.0, document dated 24 September 2026. https://www.phasedifferentialtheory.com/foundations/r04

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Reading order is expository and is not a claim that each premise is derived in the preceding paper.