Projective sector

In plain terms

The space of quantum states isn't a vector space, whatever the opening chapters of the textbooks suggest. A quantum state is written as a vector, but two vectors that differ only by an overall phase, a rotation of the vector's mathematical description that changes no prediction at all, are the same physical state. Once you treat all such vectors as a single point, quotienting the redundancy away, you're standing on complex projective space, a compact arena, meaning one of finite extent, that's curved and highly symmetric.

Most formulations treat that space as the fundamental stage, and CSD demotes it. The projective space is a sector, the image of the real arena under a many-to-one projection, a map under which many points of the arena share one quantum state, so states are shadows and the surface casting them is bigger.

The demotion has a payoff. Questions that are awkward even to pose on the state space, such as which outcome actually happened and what carries the record of it, have room to be answered one level down.

In CSD

The sector is where the epistemic half of the reconstruction lives, the half that concerns descriptions rather than what exists. Measurement contexts partition the sector into outcome regions, and the Fubini-Study measure prices them, as it's the natural volume on complex projective space and the unique one left unchanged by unitary symmetry, the transformations that preserve every quantum probability. The ontic half stays below, as the weight of a sector region is the Liouville volume, the arena's own volume, of its preimage, the set of arena points that project into it.

The sector is posited, through projectability, and then heavily constrained, as it's compact, it's Kahler, carrying the curved geometry complex projective space is known for, and it has a unique invariant measure. The corpus's working instance takes the sector to be complex projective space exactly, with the projection's fibre, the set of arena points lying above a single state, carrying what the sector can't.

Mathematically

ProjectiveSector bundles the measurable projection to CP^(N-1) with the pushforward law connecting the ontic measure to the sector measure.

The Kahler instance instantiates it on CP^(N-1) x T^2 with projection onto the first factor, and the pushforward of the Liouville measure is the Fubini-Study measure, which is a theorem and not a stipulation.

Related entries

• 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.
• 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.
• Fubini-Study measureThe space of quantum states allows exactly one way of measuring size that respects its symmetry, and that one measure is what turns regions of the space into Born-rule probabilities.
• Hilbert spaceHilbert space is the stage on which quantum mechanics is usually performed, and CSD's claim is that it's scenery, the shadow cast by the physical object and not the object itself.
• Kahler formOn the space of quantum states, distance, area and a quarter-turn coming from the complex numbers lock together, and that lock is what makes the volume of the space nobody's choice.