Epistemic measure

In plain terms

You prepared the system carefully, so you know its quantum state exactly, and on the arena picture that knowledge is radically incomplete. One quantum state corresponds to a whole crowd of points on the arena, all projecting to the same state, and that crowd is called the fibre over the state, so you can ask what you should believe about the part you can't see.

The natural answer is to condition on what you know. You know the visible coordinate exactly, so you put all your credence, your degree of belief, there, and you know nothing about the hidden coordinate, so you spread your credence evenly over the fibre, using the fibre's own symmetry to say what "evenly" means. That combined object is the epistemic measure.

The satisfying part is that "evenly" isn't a choice. Take the arena's own volume and disintegrate it along the projection, which means slicing it into the volumes it induces on each fibre, and those slices are exactly this measure, for almost every state. What the programme first adopted as a stated definition turned out to be the unique answer the geometry had waiting.

In CSD

The epistemic measure is what "isolation is conditioning" means in practice. The isolation-conditioned state of knowledge at a given preparation places all its weight on that one state in the base, the projected state space, and spreads uniformly over the fibre above it, uniformly in the sense fixed by the fibre's symmetry, and every record-layer probability, including the Born weights of the context-fixed basins, is computed against it.

Conditioning on a single state means conditioning on a set the base measure calls null, one of size zero, which is why the corpus first took this form as a stated definition, and the derivation then followed. The Liouville measure, the arena's own volume, disintegrates along the base projection, its disintegration kernel, the family of fibre volumes it induces, is almost everywhere the uniform one, and the epistemic measure is the fibre of that disintegration planted at its base point. The modelling choice turned out to be the theorem.

Mathematically

epistemicMeasure p is the product of the Dirac measure at p with normalised Haar on the torus fibre, a probability measure on the arena for every p.

The derivation runs as follows. kMuL_condKernel_ae identifies the disintegration kernel of the Liouville measure with the constant Haar kernel, almost everywhere in the Fubini-Study base measure, via the standard-Borel conditional-kernel machinery and almost-everywhere uniqueness of disintegrations, and epistemicMeasure_eq_disintegration then equates the epistemic measure with the Dirac-paired disintegration fibre. The almost-everywhere qualifier is intrinsic, as disintegration kernels are only determined up to a null set of base points.

Related entries

• Ontic vs epistemicWhat exists is one question and what a measurement context lets you say is another, and most quantum paradoxes are category errors between the two.
• Psi-epistemicThe quantum state is what you know and not what exists, and the hard part was always saying what it's knowledge of, so CSD's whole architecture is an answer to that.
• Born ruleQuantum probabilities aren't a law of nature in CSD, as they're the sizes of regions in the space of states, measured with a ruler nobody was free to choose.
• 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.
• Preparation regionRepeating the same experiment is a physical claim, and this is what it refers to, a region of the arena and not a single point, that the preparation procedure lands the world in.