Single trajectory

In plain terms

A deterministic account of quantum mechanics faces a fork, as either the world splits, or there's a vast space of possible world-states and one actual path through it. CSD takes the second branch without hedging, with a single trajectory on a single arena evolving deterministically, and this is what the programme means by a single-trajectory theory.

You can then ask where statistics come from, and the answer is repetition. Each run of an experiment re-prepares the system, which lands the world at a fresh point of the preparation region, the set of world-states compatible with the preparation procedure, so a thousand runs sample the region a thousand times. The frequencies those samples produce are what the volume theorems compute, so an outcome's share of the runs matches its share of the region's volume.

What statistics don't come from is one run watched forever. Whether a single evolving path eventually visits every part of its space in fair proportion is a property called ergodicity, and it's a different and much stronger property than repeated preparation, so the programme doesn't rely on it.

In CSD

CSD is a single-trajectory theory, so physical reality is one deterministic path on the arena. Its typicality is the typicality of repeated preparation, ignorance spread across runs and priced by the arena's natural measure restricted to the preparation region, and not the averaging of one path over time.

The corpus even proves the cautionary half. The unitary flow projected onto the space of quantum states isn't ergodic, so a time-average route to the Born weights is closed at that level. That's a narrow fact about one flow on the projective space and not an objection to the mechanism, because the frequency theorems never invoke ergodicity, and that they don't need it is exactly the point.

Mathematically

The LF1 chain formalises the repetition, with i.i.d. draws from the conditioned Liouville measure, indicator processes per outcome, and the strong law of large numbers delivering almost-sure convergence of empirical frequencies to volume weights.

The non-ergodicity of the epistemic unitary flow on the projective sector is a proved theorem kept as a boundary marker, as the volume route and the time-average route aren't the same route, and only the first is claimed.

Related entries

• DeterminismThe substrate has no dice, so the spread in the data is variation in what your preparation couldn't pin down.
• Constraint surfaceThere's one arena, it's deterministic, and it doesn't grow, and everything else in the programme is bookkeeping about how much of it a measurement context lets you see.
• If CSD is deterministic, where do probabilities come from?They come from ignorance of the exact starting point, as the dynamics never rolls dice, and the probabilities are the sizes of the sets of microstates a preparation can't tell apart, measured with a ruler nobody was free to choose.
• First-passage raceWhen a measurement has several possible outcomes, the one that happens is the one whose clock fires first, and the law those clocks obey isn't a modelling choice, as it's the only law that makes the race come out right, and that's proved in both directions.
• TypicalityTypicality is probability without randomness, as the world is deterministic, you don't know which state you have, and the odds are the sizes of the possibilities.