Kahler form

In plain terms

The space of quantum states carries three structures that could, in principle, have nothing to do with one another. There's a way of measuring distances between states, called a metric. There's a way of measuring areas, called a form. And there's a quarter-turn operation, which comes from multiplying by the imaginary unit, the number whose square is minus one, so applying it twice reverses a direction, just as two quarter-turns make a half-turn.

On this space the three aren't independent, as each one can be recovered from the other two. Turn the metric through a quarter-turn and you get the area form, and so on around the triangle. A space in which the three lock together in this way is called Kahler, and the lock has a payoff, as such a space owns a volume of its own. Take the area form, raise it to the highest power the dimension allows, and the result measures volume, so no external ruler is required.

That's why the measure used throughout this programme isn't an extra ingredient someone had to supply. The geometry already contains it.

In CSD

This underwrites the claim that CSD's typicality measure, the measure used to say how large a region of states is, is geometric rather than chosen, which is the difference between deriving the Born rule and restating it with more steps.

The corpus proves the locking of the three structures at two levels. At each point, in the space of directions through that point, it establishes the compatibility relations, so the quarter-turn squares to minus one, the area form is the metric composed with the turn and vice versa, the form survives turning both of its inputs, and pairing a direction with its own quarter-turn returns its squared length. For the whole space of states, it establishes that the area form has the two properties that make a space symplectic, the setting in which classical mechanics is done, that the quarter-turn is the complex structure the space naturally carries, so the space is Kahler in both of the standard senses, and that the top power of the form is the Fubini-Study measure up to a fixed constant.

What this doesn't change is what carries the Born derivation, which still runs through the invariance characterisation of the measure, the fact that it's the unique measure left unchanged by the unitary symmetries. The geometry is proved to agree with that measure and not to replace it, and the mathematical section gives the exact claims.

Mathematically

With g the real part of the inner product, omega the imaginary part, and J multiplication by i, J squared is minus one, omega equals g after J, g equals omega after J, omega is of type (1,1), and omega(u, Ju) equals the squared norm of u. All of this is proved axiom-free, pointwise, on the tangent model.

The manifold statements are all proved in the corpus's own manifold layer, as Mathlib has no differential forms on manifolds at the pin. They are fsForm_isSymplectic (the Fubini-Study form is closed and non-degenerate on CP^n), fsForm_isKahler (with J the multiplication by i, read through the holomorphic atlas), IsKahler.nijenhuis_eq_zero (the Nijenhuis tensor vanishes), fsVolume_eq_smul_fsMeasure (the top power of the form is (4 pi)^n times the Fubini-Study measure), and riemannianVolume_fsMetric (the Riemannian volume of the compatible metric is that top power over n factorial). The sector's posited Liouville measure is proved to be this volume (LF4/SectorManifold.lean).

One restriction remains as stated. The corpus's permitted-claims list forbids calling Kahler geometry load-bearing for the Born rule, because the uniqueness argument that carries the programme runs through Haar measure, so the geometry is proved to agree with it and not to replace it.

The name

Erich Kahler (1906-2000) was born in Leipzig and took his doctorate there at twenty- two. After the war, which included a spell as a French prisoner of war, he held chairs at Berlin and Hamburg.

The manifolds come from one short 1933 paper. He had noticed that when metric, complex structure and form are compatible, a single potential function generates everything, a simplification whose consequences he didn't live to see fully spent in Hodge theory, the classification of complex surfaces, Calabi-Yau spaces and string theory. Andre Weil attached the name in the 1950s, two decades late. Kahler's own later work drifted into a private philosophical system that attracted essentially no following, which is a reminder that mathematicians get one name on the map and rarely two.

Related entries

• 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.
• Liouville measureA deterministic substrate needs a notion of volume that its own motion can't distort, and on CSD's sector that volume is forced and it's the Born measure.
• Moment mapThe Born probabilities are coordinates, and the moment map reads them straight off the geometry of the space of quantum states, with no measurement postulate consulted.
• Poisson bracketThe Poisson bracket is the one rule that tells classical mechanics how fast any quantity changes, and the quantum commutator obeys exactly the same algebra, so that shared grammar is what lets a deterministic flow present itself in quantum form.
• Duistermaat-Heckman formulaProject the curved space of quantum states down to its outcome probabilities and the curvature disappears exactly, which is what makes the Born rule computable in every dimension.
• Fisher-Rao metricThe natural way to measure distance between quantum states agrees with the statistician's way to measure distance between the probability distributions they produce, exactly and with no constant left to fix.
• geometric phaseA quantum state taken around a closed loop and brought back to where it started keeps a leftover phase that depends only on the shape of the loop, so the phase is a record of the journey rather than of the time it took.
• Darboux theoremEvery symplectic space looks the same near any point, as there's always a chart in which the structure takes the textbook form, so symplectic geometry has no local invariants beyond the dimension.