Haar measure

In plain terms

Many questions ask for an average over all the symmetries of something, the average effect of a random rotation say, or whether one family of transformations is larger than another. To answer them you need a notion of volume on the set of transformations itself and not on the objects they act on, and that volume had better not depend on the coordinates you happen to use to label the transformations, or your average is an artefact of your labelling rather than a fact about the symmetries.

Haar's theorem concerns groups, families of transformations that can be combined and undone, such as all the rotations of a sphere. The demand he placed on a volume is invariance, meaning that the size of any collection of transformations doesn't change when every member of the collection is followed by one fixed transformation, so for rotations this says that turning a whole set of rotations by a further fixed rotation leaves its size alone. The demand sounds mild. For compact groups, those that are closed and bounded in the way the rotations of a sphere are and the shifts of an infinite line aren't, it's remarkably restrictive, as once the total volume is set to one exactly one invariant volume exists, and not one natural choice among several but one, full stop.

So a question such as what a random symmetry does, on average, has a definite answer that doesn't depend on anyone's choice of coordinates. That's Haar's gift, and a large part of modern mathematics quietly relies on it.

In CSD

Trace the programme's central claim back far enough and it rests on this theorem. The transformations quantum mechanics allows on a system, the unitary transformations that preserve lengths and angles between states, form a compact group, so there's exactly one invariant way to assign volume to them. The space of quantum states can be built from that group, as you fix any one state and every other state is what some unitary transformation turns it into. Sending each transformation to the state it produces carries the unique volume on transformations down to a volume on states, and its uniqueness comes with it, so that's the whole proof that the Fubini-Study measure, the notion of volume on state space the programme uses, is forced rather than chosen.

Remove this link and everything downstream changes character. The measure would become a preference, the derivation of the Born rule would rest on that preference, and the reply to the objection that the programme smuggles in its measure would evaporate. The same fact is what makes the choice of starting state immaterial, as two different starting states give two ways of labelling states by transformations, but the two labellings differ by a fixed transformation and an invariant volume can't tell the difference.

Mathematically

A compact topological group carries a unique translation-invariant Borel probability measure. On a homogeneous space of such a group, invariant measures are unique up to scale, and normalisation fixes the scale.

The corpus uses the unitary group's Haar measure directly and pushes it forward along the orbit map, so uniqueness downstairs is inherited from uniqueness upstairs. No geometric argument enters the uniqueness proof, as it's group theory and measure theory all the way down.

The name

Alfred Haar (1885-1933) was born in Budapest and took his doctorate at Gottingen under Hilbert in 1909, and the Haar wavelet, now the first example in every wavelet course, comes from that thesis. With Frigyes Riesz he built the mathematics institute at Szeged into a serious research centre, and its journal, Acta Scientiarum Mathematicarum, into a serious journal.

The measure theorem dates from 1933, the last year of his life, as he died of cancer at forty-seven. Von Neumann, who had been sceptical that such a measure existed in general, used it almost immediately to solve Hilbert's fifth problem for compact groups, and Weil built the general theory soon after, so the result outlived its author before the year was out.

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.
• 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.
• 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.