Schrodinger from records

In plain terms

Reconstructions of quantum mechanics usually spend their effort on probability and quietly assume the dynamics, so states evolve by the Schrodinger equation because that's the theory being reconstructed. If the arena picture is doing real work, the dynamics should be earned too.

The corpus's route earns it from records. Suppose the arena's dynamics preserves record statistics, the pairwise outcome frequencies experiments actually accumulate. A rigidity theorem going back to Wigner says a map preserving those statistics must be either unitary, one of the reversible transformations quantum mechanics ordinarily uses, or antiunitary, a mirror-image variant that also reverses the sign of the imaginary unit. A selection argument rules out the antiunitary branch along a continuous flow. A phase-lifting step then passes from the transformation of states to a transformation of the vectors that represent them, choosing consistently the phase of each, the undetectable angle by which such a vector may be rotated without changing the state, and a continuity theorem of Stone's type delivers a self-adjoint generator, which is the Hamiltonian.

The endpoint is the exponential form of the Schrodinger equation, derived, with a premise about the statistics of records rather than about the structure of Hilbert space, the abstract vector space in which quantum states live. The chain is assembled from separate theorems rather than proved in one stroke, and the final link carries real hypotheses, a smoothness condition, and a consistency condition on the phase, called a coboundary condition, which the corpus shows is unavoidable rather than convenient.

In CSD

This series of results is the dynamics spine of the programme. The flow on the projected sector, the visible quantum state space obtained by projecting the arena, is conjugation by a one-parameter group of unitary maps, proved from the descent structure, the way the arena's flow descends to the projected sector, rather than assumed. A record-level premise conversion makes the hypothesis operational, as transition probabilities are themselves record observables, so preserving record statistics and preserving transition probabilities are the same condition, by theorem.

The scope is tracked mechanically rather than by prose. An automated guard enforces that the descent equation is genuinely consumed by the proof, and a connectivity manifest, not this page, is the single source of truth for which end-to-end connective claims are backed.

Mathematically

The capstone sigmaFlow_schrodinger_form states that for sector data with a projectable flow, the projected dynamics is exp(-itH)-conjugation on rays. Read the hypotheses, because they're where the content sits. The theorem takes the projected flow as already given by a unitary family, so Wigner rigidity and the Bargmann branch discriminator stand upstream supplying that premise rather than inside this statement, and it then takes a cocycle for the family, a unit-modulus phase satisfying a coboundary condition, an anti-Hermitian generator, and a C^1 derivative hypothesis. The coboundary condition isn't a convenience, as the corpus proves it necessary, so the phase lift exists exactly when it holds.

The same module carries the form that needs neither datum, as projectedFlow_schrodinger_form_of_continuous_flow reaches the same conclusion from continuity of the lifted family alone, with the coboundary supplied by the one- parameter Bargmann lift and the generator by the finite-dimensional continuity-only Stone theorem.

What the capstone adds over the ray-level form is that it genuinely consumes the sector fields, the flow, the projection and projectability, rather than restating a result about an already-projected map. The record-level conversion equates the operational premise with the geometric one and re-derives the flow theorem from record statistics, with the invariance of the Fubini-Study measure obtained with U(N) in the proof and never in the statement.

Related entries

• Schrodinger equationIn CSD the Schrodinger equation isn't postulated, as it's what any continuous motion that preserves the measurable relations between quantum states is forced to look like.
• Wigner rigidityPreserve the overlaps between quantum states and you have no freedom left, as the rearrangement must be a unitary or an anti-unitary transformation whether or not you intended it to be.
• Stone's theoremAsk only that time evolution never jumps, and differentiability, a generator and the Hamiltonian all follow for free.
• Bargmann invariantOne phase survives every change of convention, the phase of the product of three overlaps taken around a cycle, and it's also the fingerprint that tells a unitary symmetry from an anti-unitary one.
• ProjectabilityProjectability is the one genuine postulate at the interface, that the ontic arena projects onto the space quantum mechanics is written on, and downstream of that symmetry does the choosing.