Duistermaat-Heckman formula

In plain terms

The space of quantum states is curved, and computing the volume of a region in a curved space is usually hard work of the kind that gets harder quickly as the number of dimensions grows. CSD says probabilities are volumes of regions in this space, so you might expect every probability calculation to be a small project in differential geometry.

It isn't, and the reason is the Duistermaat-Heckman formula. Fix a measurement, and each state then has two kinds of content, a list of probabilities with one for each outcome of the measurement, and a further angle-like piece of information called phase, which affects how states interfere with one another but not the probability of any single outcome. There's a natural way to compress a state down to its list of probabilities and discard the phases, and under that compression the curved space flattens completely. The volume of a region of states becomes an ordinary flat volume inside a simplex, the shape formed by all lists of probabilities that add up to one, so for three outcomes the simplex is a triangle and the calculation is the kind you could do with school geometry.

Mathematicians know the general phenomenon as exact stationary phase. The method of stationary phase is normally an approximation, a first guess at an integral, and in the presence of enough symmetry it turns out to be exactly right.

In CSD

This is the engine that makes the Born result computable rather than merely true. The compression that sends each state to its list of outcome probabilities is called the moment map, and pushing the Fubini-Study measure through the moment map gives a measure on the simplex of probability lists that is flat, so every part of the simplex counts equally. A curved volume ratio in a high-dimensional space becomes a flat volume inside a simplex.

Without this, the Born theorem at general dimension would have been assembled case by painful case, or not at all, and with it the central claim of the programme is a finite computation at every dimension. It's also a pleasing historical accident, as a piece of symplectic geometry, the geometry that underlies classical mechanics, developed for entirely unrelated reasons, turned out to be exactly the tool a question in the foundations of quantum mechanics needed.

Mathematically

For a measurable region R of the open free moment simplex, the Fubini-Study measure of its preimage under the normalised moment map equals M factorial times the Lebesgue volume of R, so the pushforward of the Fubini-Study measure along the moment map is the flat Dirichlet distribution on the simplex.

This is the Duistermaat-Heckman theorem in measure-theoretic clothing. For a Hamiltonian torus action the pushforward of Liouville measure is piecewise polynomial, and here the polynomial is constant. It's proved unconditionally at general N.

The name

Hans Duistermaat (1942-2010) was born in Den Helder and took his doctorate at Utrecht in 1968. After a chair at Nijmegen he returned to Utrecht in 1975 and became the leading Dutch geometer of his generation. Gerrit Heckman was his student at Leiden and Utrecht, and later a professor at Nijmegen.

The formula appeared in Inventiones in 1982, with an addendum the following year that fixed the degenerate cases, which is a small lesson in how even exact results can be published slightly wrong. Atiyah and Bott recast it almost immediately in the language of equivariant cohomology, which is how most geometers now know it. Duistermaat died suddenly of lymphoma at sixty-seven, still supervising students.

Related entries

• 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.
• 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.
• 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.
• Dirichlet distributionBeing fairly ignorant about proportions that must add up to one has a precise mathematical form, and by a small miracle it's exactly the form that quantum state space flattens into.