Moment map

In plain terms

You can accept that probabilities are volumes and still find the link between a quantum state and its outcome weights opaque, and wonder why these particular numbers. Classical mechanics has an answer waiting. The state space of a classical system, its phase space, has a geometry in which every continuous symmetry comes paired with a conserved quantity, so turning the system about an axis pairs with angular momentum and shifting it in time pairs with energy. When a system has several such symmetries at once, their conserved quantities can be collected into a single map, the moment map, which sends each state to the list of values those quantities take there.

The space of quantum states has exactly the right kind of symmetry. Each possible outcome contributes to the state with a size and an angle, the angle is called the phase, and turning the phase of one outcome's contribution changes nothing that can be measured. Turning each outcome's phase independently gives a family of symmetries, one circle of rotations per outcome, and together those circles form a torus. The moment map of that torus symmetry sends a state to a list of numbers, one per outcome, that add up to one, and if you compute the list it's the Born weights, the probabilities the textbook obtains by squaring the size of each contribution.

So the probabilities were sitting in the geometry all along, as the conserved quantities of phase rotation.

In CSD

The moment map is the corpus's route to the Born numbers that draws no regions by hand. The weight of an outcome is the corresponding coordinate of the moment map, a quantity fixed by the geometry of the state space and forced by the requirement that it generate the torus symmetry, with no partition drawn and no operational input consulted. It also supplies the canonical measurement context, whose rate field is the moment map itself, so even the rates of the record layer are geometric.

Downstream, pushing the symmetric measure on the state space forward through the moment map gives the uniform law on the space of probability lists. This is the Duistermaat-Heckman fact that converts statements about volumes into integrals over that space, and it closes the programme of obtaining the Born weights from volume in every finite dimension.

Mathematically

For the torus action on CP^(N-1) the moment map sends a ray to its vector of squared amplitudes, the identity is proved as momentMap_mk_eq_inner_sq, and the moment map is forced by torus generation (torusGenerated_eq_momentMap). The joint pushforward of the Fubini-Study measure is (N-1)-factorial times Lebesgue measure on the open simplex, the flat Dirichlet law, proved via the Gaussian representation and a product-index bridge, and consumed by the unconditional volume-ratio and frequency theorems at every N.

The defining equation is proved at manifold level, as the velocity field of the torus action is the Hamiltonian vector field of 2 sum theta_k mu_k for the Fubini- Study symplectic form (torusField_isHamiltonianVectorField), a moment map is unique up to a constant and the normalisation pins it, its image is exactly the standard simplex (range_momentMap), and the torus orbits are its integral curves. For any Hermitian H the Schrodinger flow is the Hamiltonian flow of minus twice the expectation of H (schrodingerField_isHamiltonianVectorField), and the torus is the diagonal case.

The moment map is also a map between geometries. Its differential kills the torus directions and, on their Fubini-Study orthogonal complement, carries the Fubini-Study metric to the Fisher-Rao metric of the weights with constant one (fsMetric_eq_fisherRaoInner), so the Born weights of a state sit in the simplex at the statistical distances the quantum distances dictate.

Related entries

• 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.
• 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.
• 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.
• 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.
• 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.